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Set-valued stochastic differential equations with unbounded coefficients and applications

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Set-valued stochastic differential equations with unbounded coefficients and applications

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Content Set-Valued Stochastic Differential Equations with Unbounded Coefficients and Applications by Atiqah Hamoud S. Almuzaini Dissertation Advisor: Professor Jin Ma A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) May 2024 Copyright 2024 Atiqah Hamoud S. Almuzaini Abstract In this dissertation, we focus on the set-valued (stochastic) analysis on the space of convex, closed, but possibly unbounded sets. By establishing a new theoretical framework on such sets, which is beyond the existing theory of set-valued analysis, we shall study the set-valued SDEs (SV-SDEs) with unbounded coefficients, and their applications in the super-hedging problem of a continuous-time model with transaction costs in finance. The space that we will be focusing on is convex, closed sets that are “generated” by a given cone with certain constraints. We shall argue that, for such a special class of unbounded sets, the cancellation law could still be valid, and many algebraic and topological properties of the existing theory of set-valued analysis on compact sets and standard techniques for studying SV-SDEs can be extended to the case with unbounded (drift) coefficients. In the super-hedging problem of discrete-time models with transaction costs, the set of self-financing portfolios are often described by the (unbounded) “solvency cone”. Our study of unbounded sets is therefore crucial in extending the theory to the continuous-time model. In the model with transaction costs and vector-valued contingent claims, the set of super-hedging positions is inherently a closed convex unbounded set. We shall argue that the (dynamic) super-hedging set can be expressed as set-valued integrals of the solvency cones, and define a set-valued dynamic risk measure. Finally, after some refinement, we show that the dynamic super-hedging sets satisfy a recursive relation which can be considered as a geometric dynamic programming principle (DPP). ii Acknowledgments I would like to express my deepest gratitude to my advisor, professor Jin Ma, for their unwavering support, patience and mentorship throughout my Ph.D. journey. Their expertise, encouragement and constructive feedback have been invaluable in shaping this dissertation. I am grateful for the countless hours of discussions and the encouragement to explore new ideas. Their passion for the subject matter and care for their students have been a constant source of inspiration. I would like to extend my gratitude to professor C¸ a˘gın Ararat for their significant contribution to my academic journey. Their course on Risk Measures and Related Stochastic Analysis not only enriched my understanding of the subject matter, but also provided a solid foundation for my research. I am particularly grateful for the insightful comments and engaging discussions we had, which greatly influenced the development of my research. Furthermore, I would like to express my sincere appreciation to the University of Jeddah for their generous sponsorship throughout my academic journey. Their investment in my education has provided invaluable opportunities that have played a crucial role in the realization of my academic goals. Finally, my sincere appreciation goes to my family and friends for their constant love, encouragement, and understanding throughout this journey. Their support has been a source of strength and motivation, and I am profoundly grateful for their presence in my life. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii 1 Chapter One: Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Chapter Two: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Spaces of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Set-Valued Measurable Mappings and Decomposable Sets . . . 7 2.3 Set-Valued Integrals . . . . . . . . . . . . . . . . . . . . . . . 10 3 Chapter Three: Set-Valued Stochastic Analysis for Unbounded Sets . 14 3.1 Examples: Set-Valued Risk Measures and Solvency Cones . . 14 3.2 LC-Space and Basic Properties . . . . . . . . . . . . . . . . . 17 3.3 Set-Valued Lebesgue Integrals on LC-Space . . . . . . . . . . . 23 4 Chapter Four: Set-Valued SDEs with Unbounded Coefficients . . . . 34 4.1 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . 34 4.2 Connections to SDIs . . . . . . . . . . . . . . . . . . . . . . . 44 5 Chapter Five: Continuous-time Super-hedging Problem with Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Consistent Price Processes . . . . . . . . . . . . . . . . . . . . 55 5.3 Super-hedging Problem . . . . . . . . . . . . . . . . . . . . . . 60 iv Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 v 1 Chapter One: Introduction Set-valued analysis, both deterministic and stochastic, has been widely applied in optimizations, optimal control, as well as economics and finance. However, the existing literature in set-valued analysis predominantly focuses on compact sets. Although such a limitation is often technically necessary and does not pose fundamental challenge in many applications, it becomes more pronounced when the underlying objects under investigation are unbounded in nature. There are many practical applications that involve set-valued subjects that are necessarily unbounded. A well-known example is the set-valued risk measures, the extension of the univariate risk measure when the risk appears in the form of random vectors, which is often referred to as systemic risk in the context of default contagion (see, e.g., [28]) or multi-asset markets with transaction costs (see, e.g, [24],[21]). Mathematically, the set-valued risk measure takes the form of functions with values being the “upper sets”, that is, the convex sets that are “additively invariant” with respect to a fixed convex cone, and hence unbounded in nature. It is worth noting that a univariate dynamic risk measure (the family of risk measures indexed by time) satisfying the so-called time-consistency is often representable by the backward stochastic differential equation (BSDE) (cf. [10, 12, 64]), but the extension of such representation seems to be quite remote without appropriate technical tools in setvalued analysis that can deal with unbounded sets. Another example that involves upper sets is the so-called “solvency cone” proposed in, e.g., Kabanov [37, 39] and Scharchmayer [65], which are frequently used in the study of super-hedging problem that involves the transection costs. It is quite conceivable that any dynamics whose coefficients involve the solvency cone would naturally require a framework for unbounded sets. 1 In this dissertation, we attempt to establish a workable theoretical basis of setvalued analysis that is suitable for the study of set-valued SDEs that have unbounded coefficients. There are, however, several main technical obstacles that need to be recognized, before we try to build the reasonable framework. The first obstacle regarding the space of unbounded sets is the lack of the so-called cancellation law in its algebraic structure under the Minkowski addition. More precisely, for nonempty closed convex sets A, B, and D ⊆ R d with A + B = D + B, it is not true in general that A = D, unless B is compact. The second main obstacle in dealing with unbounded sets is in its topological structure with respect to the Hausdorff metric. More precisely, there is no single set (bounded or unbounded) that has a finite distance from all unbounded sets. Therefore, our first task is to try to identify an appropriate space of unbounded sets that possesses a “reference point”, that is, an unbounded set that has finite distance to every other elements in the space. It turns out that the space of unbounded sets that have the “upper-set” nature with respect to a certain fixed cone is one that fits our purpose well. Not only does such a space; which we call the LC-space; contain many well-known application, we can actually argue that such a space of unbounded sets that can be equipped with appropriate topological and algebraic structures on which our set-valued analysis will be based, and that many desired properties of set-valued analysis can be established with some efforts. Moreover, we shall extend some necessary concepts, such as setvalued Lebesgue integrals and their path regularities under the new framework of LC-valued unbounded sets. Another main objective of this dissertation is to establish the existence and uniqueness of the solution to a class of set-valued stochastic differential equations with unbounded drift coefficients. We recall that the first papers dealing with deterministic set-valued differential equations with compact set-valued coefficients can be traced 2 back to the early works of [5, 14, 15]. The stochastic set-valued differential equations with compact drift and absolutely summable diffusion terms and their path-regularity can be found in, e.g., [50]. However, to the best of our knowledge, the set-valued SDEs or even stochastic differential inclusions (SDIs) with unbounded coefficients, even only on the drift part, is novel in the literature. We should remark that we shall only content ourselves with the case when the “diffusion” coefficient of the set-valued SDE to be absolutely summable (see §2 for details), mainly due to the path-regularity issue. The Last part of this dissertation concerns a super-hedging problem on markets with transaction costs in continuous-time. By a super-hedging theorem we mean a result giving a description of the set of initial endowments allowing to super-hedge a contingent claim by the terminal value of a self-financing portfolio. For the classical models of frictionless markets, theorems of this kind are well-known (for the continuous-time setting see, e.g., [17], [18],[52]). In the model with transaction costs, a contingent claim is a vector-valued random variable. Similarly, the initial endowment may contain various assets and their conversion implies the loss of the value due to the market friction. The “dual” description of the super-hedging set was first proved in Cvitani´c and Karatzas [13] for a continuous-time model with transaction costs with only one risky asset. The dual description for a continuous-time currency market with transaction costs in the d-dimensional case was then proved by Kabanov and Last in [38]. In [39], Kabanov and Stricker proved a dual description for an abstract continuous-time model with transaction costs in the d-dimensional case. The “recursive” description of the super-hedging set in the discrete-time case was proved in Feinstein and Rudloff [20]. This is done by making use of the multiportfolio time-consistency of set-valued dynamic risk measures in discrete time. In this dissertation, we consider a d-dimensional continuous-time super-hedging 3 problem with transaction costs. More precisely, we consider a d-dimensional stock market model with constant transaction costs in continuous time (see e.g. [39]). In particular, this model generalizes the one risky asset model in [13] to the d-dimensional case. Similar to Feinstein and Rudloff [20] in the discrete time setting, we shall make use of continuous-time set-valued dynamic risk measure. In particular, we shall show that the dynamic super-hedging sets define a normalized set-valued dynamic risk measure. More importantly, we show that the dynamic super-hedging sets, after some refinement, satisfy recursive relations similar to those of dynamic programming principle (DPP). 4 2 Chapter Two: Preliminaries In this section, we give a brief introduction to set-valued analysis and all the necessary notations associated to it. We refer to [51],[47], and [33] for more details, but we shall present all the results in a self-contained way. 2.1 Spaces of Sets Let d ≥ 1, and consider the Euclidean space (R d , |.|). Denote by P(R d ) the family of all nonempty subsets of R d and by C (R d ) the family of all nonempty closed subsets of R d . For a set A ∈ P(R d ), we shall denote the closure of A by A or clA, interchangeably. For A, B ∈ P(R d ) and α ∈ R, we define the Minkowski addition and scalar multiplication; respectively; by A + B := {a + b : a ∈ A, b ∈ B}, αA := {αa : a ∈ A}. (2.1) It is important to note that in general, A + (−1)A ̸= {0} unless A is a singleton. Consequently, theses operations do not establish a vector space structure on P(R d ). Moreover, if A, B ∈ C (R d ), then A+B may not be a closed set, unless one of A or B is compact. We shall introduce the extended notion of Minkowski addition, defined by A ⊕ B := A + B, A, B ∈ C (R d ). (2.2) Clearly, A ⊕ B = A + B if either A or B is compact. A set A ∈ P(R d ) is called convex if λx + (1 − λ)y ∈ A for any λ ∈ (0, 1) and x, y ∈ A. Given a set A ∈ P(R d ), we define the convex hull of A; denoted by coA; as the smallest convex set containing A, and the closed convex hull of A; denoted by 5 coA; as the smallest closed convex set containing A. We denote by G (R d ) the family of all nonempty closed convex subsets of R d . Proposition 2.1. Let A, B ∈ P(R d ) and let α ∈ R. Then 1. co(A + B) = co(A) + co(B); 2. co(αA) = αco(A); 3. coA = cl(coA). The following definition and result will be essential in our future discussions. Definition 2.2. A set C ∈ P(R d ) is called a cone if αx ∈ C for any α > 0 and x ∈ C. A cone C ∈ P(R d ) is called a convex cone if it is convex. Proposition 2.3. Let C ∈ P(R d ). Then C is a convex cone if and only if C +C = C and αC = C, for any α > 0. Let A, B ∈ P(X ). The Hausdorff distance between A and B is defined by h(A, B) := max{h(A, B), h(B, A)} = inf{ε > 0 : A ⊆ B + B¯ ε(0), B ⊆ A + B¯ ε(0)}, (2.3) where B¯ r(0) ⊂ R d is the closed ball centered at 0 with radius r > 0, and h(A, B) := sup{d(a, B) : a ∈ A} := sup{inf{d(a, b) : b ∈ B} : a ∈ A}. It is clear by definition that h(A, B) = 0 if and only if A = B. Moreover, we have the following theorem. Theorem 2.4. (C (R d ), h) is a complete metric space. 6 The following result will be used frequently in our future discussions. Proposition 2.5. Let A, B, D, E ∈ P(R d ). Then 1. h(A, B) ≤ h(A, B); 2. h(coA, coB) ≤ h(A, B). 3. h(A + B, D + E) ≤ h(A, D) + h(B, E). 2.2 Set-Valued Measurable Mappings and Decomposable Sets Given a nonempty set X, by a set-valued mapping defined on X with values in P(R d ) we mean a mapping F : X → P(R d ). A function f : X → R d such that f(x) ∈ F(x) for all x ∈ X is called a selector for F. Let (X,M, µ) be a finite measure space. We shall make use of the following definition of set-valued “measurable” mappings. Definition 2.6. A set-valued mapping F : X → P(R d ) is said to be measurable if for each E ∈ C (R d ), it holds that {x ∈ X : F(x) ∩ E ̸= ∅} ∈ M. Equivalently, a set-valued mapping F : X → P(R d ) is measurable if for each ∅ ̸= U ∈ P(R d ), it holds that {x ∈ X : F(x) ∩ U ̸= ∅} ∈ M. Proposition 2.7 ([51]). (i) If F : X → C (R d ) is measurable, then F admits a measurable selector, i.e. there exists an M/B(R d )-measurable selector f : X → R d of F; (ii) (Castaing Representation) F : X → C (R d ) is measurable if and only if there exists a sequence {fn} ∞ n=1 of measurable selectors for F such that F(x) = cl{fn(x) : n ∈ N}, x ∈ X. Let us denote by L 0 (X, R d ) = L 0 M (X, R d ) the set of all M/B(R d )-measurable functions; and by L 0 (X, C (R d )) = L 0 M (X, C (R d )) the set of all measurable set-valued 7 mappings. For F ∈ L0 (X, C (R d )), we consider the set of measurable selectors S(F) := {f ∈ L 0 (X, R d ) : f(x) ∈ F(x) µ-a.e. x ∈ X}. Clearly; by Proposition 2.7; if F, G ∈ L0 (X, C (R d )), then F = G, µ-a.e. if and only if S(F) = S(G). Recall that a function f : X × R d → R d is said to be Carath`eodory if f(x, ·) is continuous for fixed x ∈ X, and f(·, a) is measurable for fixed a ∈ R d . It can be shown that a Carath`eodory function f : X×R d → R d must be M ⊗B(R d )-measurable (see, e.g., [31, Theorem 6.1]). Accordingly, we have the following set-valued version: Definition 2.8. A set-valued mapping F : X×R d → C (R d ) is said to be a Carath`eodory set-valued mapping if F(x, ·) is continuous for fixed x ∈ X, and F(·, a) is measurable for every fixed a ∈ R d . Theorem 2.9. If F : X × R d → G (R d ) is a Carath`eodory set-valued mapping, then it is M ⊗ B(R d )-measurable. Let p ∈ [1,∞) and denote L p (X, R d ) := L p M (X, R d ) := {f ∈ L 0 (X, R d ) : ∥f∥ p p := Z X |f(x)| pµ(dx) < ∞}. For any F ∈ L0 (X, C (R d )), we define S p (F) := S(F) ∩ L p (X, R d ), and consider the set A p (X, C (R d )) := A p M (X, C (R d )) := {F ∈ L0 (X, C (R d )) : S p (F) ̸= ∅}. We say F is p-integrable if F ∈ A p (X, C (R d )). Theorem 2.10. Let F ∈ A p (X, C (R d )). Then there exits a sequence {fn} ∞ n=1 ⊆ S p (F) such that F(x) = cl{fn(x) : n ∈ N} for a.e. x ∈ X. Moreover, S p (F) = 8 dec{fn : n ≥ 1}. Consequently, by Theorem 2.10, if F, G ∈ A p (X, C (R d )), then F = G µ-a.e. if and only if S p (F) = S p (G). Moreover, we have the following result: Theorem 2.11. Let F, G ∈ A p (X, C (R d )). Then 1. coS p (F) = S p coF ; 2. S p F ⊕ G = S p (F) ⊕ S p (G). Given a collection K ⊆ L p (X, R d ), we want to see under which conditions we can guarantee the existence some F ∈ A p (X, C (R d )) such that K = S p (F). The following notion is essential for this purpose. Definition 2.12. A set K ⊆ L p (X, R d ) is said to be decomposable if IAf + IAc g ∈ K for every f, g ∈ K and A ∈ M. Given a set K ⊆ L p (X, R d ), we define the decomposable hull of K, denoted by dec(K), to be the smallest decomposable set in L p (X, R d ) containing V. We shall often consider the smallest closed decomposable set in L p (X, R d ) containing K, which we call the closed decomposable hull of K, and denote it by dec(K). Proposition 2.13. Let K, H ⊆ L p (X, R d ) and α ∈ R. Then 1. dec(K + H) = dec(K) + dec(H); 2. dec(αK) = α dec(K); 3. dec[co(K)] = co[dec(K)]. Theorem 2.14. Let K be a nonempty closed subset of L p (X, R d ). Then there exists a measurable set-valued mapping F ∈ A p (X, C (R d ))such that K = S p (F) if and only if K is decomposable. From now on, we shall consider a given complete, filtered probability space (Ω, F, P, F = {Ft}t∈[0,T]), on which is defined a standard m-dimensional Brownian motion B = {Bt}t∈[0,T] , T > 0. We shall sometimes write W to denote the Brownian Motion B. The notions of set-valued random variables and stochastic processes, etc., can be defined in an analogues way as the usual “single-valued” concepts in probability. For example, a set-valued random variable X : Ω → C (R d ) is an F-measurable set-valued mapping; and a set-valued stochastic process Φ = {Φt}t∈[0,T] is a family of set-valued random variables. A set-valued process is called measurable if it is B([0, T]) ⊗ Fmeasurable; it is called F-adapted if Φt is Ft-measurable for each t ∈ [0, T]; it is called F-nonanticipative if it is both measurable and F-adapted. Given a set-valued F-non-anticipative process Φ, we denote SF(Φ) to be the set of all F-non-anticipative selectors of Φ. For p ≥ 1, we denote S p F (Φ) := SF(Φ) ∩ L p ([0, T] × Ω, R d ), and A p F ([0, T] × Ω; C (R d )) = {F ∈ L0 ([0, T] × Ω; C (R d )) : S p F (F) ̸= ∅}. (2.4) 2.3 Set-Valued Integrals In this section, we shall introduce the notions of set-valued Lebesgue and Itˆo integrals. Let us assume that F = F B be the natural filtration generated by B, augmented by all P-null sets of F so that it satisfies the usual hypothesis. Lebesgue Set-Valued Integral: Let Φ ∈ A 2 F ([0, T] × Ω; C (R d )), and define J0,T (Φ) := Z T 0 ϕtdt : ϕ ∈ S 2 F (Φ) ⊆ L 2 FT (Ω, R d ). (2.5) Since decFT (J0,T (Φ)) is closed and FT -decomposable, by Theorem 2.14, there exists a unique FT -measurable set-valued random variable I0,T (Φ) ∈ A 2 FT (Ω, C (R d )) such that S 2 FT (I0,T (Φ)) = decFT (J0,T (Φ)). We call I0,T (Φ) the Lebesgue set-valued integral 10 of Φ on [0, T], and denote it by R T 0 Φtdt := I0,T (Φ). Similarly, for t ∈ (0, T], we define R t 0 Φsds by S 2 Ft ( R t 0 Φsds) = decFt (J0,t(Φ)). It is important to note that the set-valued integral R t 0 Φsds is defined, almost surely, for each t ∈ (0, T]. Moreover, we have the following Proposition: Proposition 2.15. Let Φ ∈ A 2 F ([0, T] × Ω; C (R d )). Then 1. The set-valued integral R t 0 Φsds is F-adapted; 2. If Φ is convex-valued, so is R t 0 Φsds; 3. [50] If Φ is compact convex-valued, then the process { R t 0 Φsds}t∈[0,T] is continuous with respect to the Hausdorff metric h. Remark 2.16. For Φ ∈ A 2 F ([0, T] × Ω; C (R d )) that is not compact-valued, the continuity of the process { R t 0 Φsds}t∈[0,T] is yet to be discussed in the literature, and will be one of the main focusses of this dissertation. Aumann-Itˆo Set-Valued Stochastic Integral: Let Ψ ∈ A 2 F ([0, T]×Ω; C (R d×m)). Define J0,T (Ψ) := n Z T 0 ψtdBt : ψ ∈ S 2 F (Ψ)o ⊆ L 2 FT (Ω, R d ). (2.6) Again, by Theorem 2.14, there exists a unique set-valued random variable I0,T (Ψ) ∈ A 2 FT (Ω, C (R d )) such that S 2 FT (I0,T (Ψ)) = decFT (J0,T (Ψ)). We call IT (Ψ) the AumannItˆo set-valued stochastic integral of Ψ on [0, T], and denote it by R T 0 ΨtdBt := I0,T (Ψ). Similarly, for t ∈ [0, T], we define R t 0 ΨsdBs by S 2 Ft ( R t 0 ΨtdBt) = decFt (J0,t(Ψ)). Again, the set-valued Integral R t 0 ΨsdBs is defined, almost surely, for each t ∈ (0, T]. Proposition 2.17. Let Ψ ∈ A 2 F ([0, T] × Ω; C (R d×m)). Then 11 1. The set-valued integral R t 0 ΨsdBs is F-adapted; 2. If Ψ is convex-valued, so is R t 0 ΨsdBs. Let us denote by L 2 F ([0, T] × Ω, R d×m) the set of all non-anticipative processes of L 2 ([0, T] × Ω, R d×m), and let G := {g n}n≥1 ⊆ L 2 F ([0, T] × Ω, R d×m). Since for any E ∈ C (R d×m), it holds that {(t, ω) : G(t, ω) ∩ E ̸= ∅} = ∪ ∞ n=1{(t, ω) : g n (t, ω) ∈ E}, and each g n is F-non-anticipative, then G is an F-non-anticipative set-valued process. The following notions are essential for establishing the continuity of Aumann-Itˆo setvalued integrals for such set-valued mappings. Definition 2.18. A set G := {g n}n≥1 ⊆ L 2 F ([0, T] × Ω, R d×m) is called absolutely summable if P∞ n=1 |g n (t, ω)| < ∞ for a.e. (t, ω) ∈ [0, T] × Ω. If furthermore, E R T 0 P∞ n=1 |g n (t, .)| 2dt < ∞, then it is called square integrable. Theorem 2.19. For every absolutely summable square integrable set G := {g n}n≥1 ⊆ L 2 F ([0, T] × Ω, R d×m), the set-valued process { R t 0 GsdBs}t∈[0,T] is continuous with respect to the Hausdorff metric h. We end this section by some technical results that will be useful in our later sections. Proposition 2.20 ([71]). Let (T, A) be a measurable space, X be a Polish space and Y be a topological space. If F : T ×X → P(Y ) is A⊗B(X)-measurable and g : T → X is A-measurable, then for every open set U ⊆ Y , the set {t ∈ T : F(t, g(t)) ∩ U ̸= ∅} ∈ A. If F is also a Carath`eodory set-valued mapping, then for every measurable set-valued mapping G : T → P(X), the set-valued mapping FG : T → P(Y ) defined as FG(t) := ∪x∈G(t)F(t, x), t ∈ T is A-measurable. 12 Proposition 2.21. Let (X,M, µ) be a finite measure space, and let F ∈ A p M (X, C (R d )) and p ≥ 1. Let ϕ : X × R d → [−∞, ∞] be a jointly measurable function. Suppose that the integral Tϕ(f) := R X ϕ(x, f(x))µ(dx) is well-defined for each f ∈ S p (F), and Tϕ(f) < +∞ (resp. > −∞) for at least one f ∈ S p (F). Suppose further that either (i) ϕ(x, ·) is upper semicontinuous for every fixed x ∈ X or (ii) (X,M, µ) is a complete measure space and ϕ(x, .) is lower semicontinuous for every fixed x ∈ X. Then, inf f∈Sp(F) Tϕ(f) = Z X inf y∈F(x) ϕ(x, y)µ(dx) (resp. sup f∈Sp(F) Tϕ(f) = Z X sup y∈F(x) ϕ(x, y)µ(dx)). 13 3 Chapter Three: Set-Valued Stochastic Analysis for Unbounded Sets 3.1 Examples: Set-Valued Risk Measures and Solvency Cones In this section, we provide examples in finance of sets that are naturally unbounded. We refer to [20],[37], [39] and [65] for more details, but we shall present all the results in a self-contained way. For p, d ≥ 1 and t ∈ [0, T], let L p d (Ft) := L p Ft (Ω; R d ). For Dt ∈ L p d (Ft), let L p d (Ft ; Dt) := {X ∈ L p d (Ft) : X ∈ Dt P-a.s.} and (Dt)+ := {X ∈ Dt : X ∈ R d + P-a.s.}. Let Mt ⊆ L p d (Ft) and denote by P Mt ; (Mt)+ := {D ⊆ Mt : D = D + (Mt)+}. Definition 3.1. A function Rt : L p d (FT ) → P Mt ; (Mt)+ is a conditional set-valued risk measure at time t if it is 1. Mt-translative: ∀mt ∈ Mt : Rt(X + mt) = Rt(X) − mt ; 2. L p d (FT )+-monotone: Y − X ∈ L p d (FT )+ =⇒ Rt(Y ) ⊇ Rt(X); 3. Finite at zero: ∅ ̸= Rt(0) ̸= Mt . If Rt is a conditional set-valued risk measure for every t ∈ [0, T], then (Rt)t∈[0,T] is a called a dynamic set-valued risk measure. A conditional risk measure Rt : L p d (FT ) → P Mt ; (Mt)+ is called normalized if Rt(X) = Rt(X) +Rt(0) for every X ∈ L p d (FT ). A dynamic risk measure (Rt)0≤t≤T is said to be normalized if Rt is normalized for every t ∈ [0, T]. A dynamic risk measure (Rt)t∈[0,T] is called multi-portfolio time consistent if, for all 0 ≤ t < r ≤ T and A, B ⊆ L p d (FT ), the implication [ X∈A Rr(X) ⊆ [ Y ∈B Rr(Y ) =⇒ [ X∈A Rt(X) ⊆ [ Y ∈B Rt(Y ) (3.1) is satisfied. Given a conditional set-valued risk measure Rt , we shall denote by At := {X ∈ L p d (FT ) : 0 ∈ Rt(X)} and At,r := At ∩ L p d (Fr), for 0 ≤ t < r ≤ T. Theorem 3.2. [Theorem 3.4, [20]] For a normalized dynamic risk measure (Rt)t∈[0,T] , the following are equivalent: 1. (Rt)t∈[0,T] is multi-portfolio time consistent; 2. For 0 ≤ t < r ≤ T and X ∈ L p d (FT ), the recursive relation Rt(X) = ∪Z∈Rr(X)Rt(−Z) holds. If, additionally, Mt ⊆ Mr for 0 ≤ t < r ≤ T, then all of the above is equivalent to 3. At = Ar + At ∩ Mr for 0 ≤ t < r ≤ T. Another situation where the space of unbounded sets can be of interest is in the so called solvency cones used by Schachermayer [65] and Kabanov [37], [39], which we now describe. By a bid-ask matrix we mean a matrix Π = (π ij )1≤i,j≤d that satisfies the following conditions: 1. π ij > 0; 2. π ij = 1; 3. π ij ≤ π ikπ kj . 15 An adapted process (Π)t∈[0,T] taking values in the set of bid-ask matrices will be called a bid-ask process. The typical interpretation of the element π ij is that it is the number of units of asset i which can be exchanged for 1 unit of asset j. The interpretation of the conditions above goes as follows: 1. Any trade has a positive exchange rate; 2. Can always trade 1 unit with itself; 3. A direct exchange is always at most as expensive as a chain of exchanges. Definition 3.3. Given a bid-ask matrix Π = (π ij )1≤i,j≤d. The solvency cone K(Π) is defined to be the convex cone in R d spanned by {π ijei − ej , ei : 1 ≤ i, j ≤ d}. The solvency cone of a given bid-ask process is defined in an obvious way. 16 3.2 LC-Space and Basic Properties In this section we introduce the main subject of our framework, which we refer to as the LC-Space. As we observed in the previous section, most of the concepts of the standard set-valued analysis and stochastic analysis are defined on C (R d ) (or G (R d )), the space of all closed (convex) sets. However, many results in the set-valued analysis literature are valid only in K (R d ), the space of all compact, convex sets. Our main task is to remove the boundedness constraint in the analysis and establish a workable framework of set-valued analysis on unbounded sets. To begin with, we note that the first main issue for allowing unbounded sets is the choice of the reference element, since there is no single element in the space C (R d ), to which the Hausdorff distance from all elements is finite. Consequently, the commonly used reference point {0}, as well as the “norm” ∥A∥ = h(A, {0}), is no longer valid. To overcome this difficulty and proceed with our analysis, a reasonable remedy would be to consider a subspace of C (R d ) for which there exists a fixed “reference point” that has finite Hausdorff distance to all other elements. In light of the applications mentioned in the Introduction, in this paper we shall consider the space LC, which we now describe. Let C ∈ G (R d ) be a fixed cone in R d , such that {0} ̸= C ̸= R d . We consider the set: L ∞ C = L ∞ C (R d ) := {A ∈ C (R d ) : A = A ⊕ C}. (3.2) In what follows we do not distinguish L ∞ C and L ∞ C (R d ) when the context is clear. We endow the space L ∞ C with the usual Hausdorff metric h, then it is not hard to check that L ∞ C is “closed” under countable union and intersections. Indeed, if {An} ∞ n=1 ⊆ L ∞ C , then ∪∞ n=1An ∈ L ∞ C and ∩ ∞ n=1An ∈ L ∞ C . We note that spaces similar 17 to L ∞ C have appeared in the literature (e.g., [29]), as a result of a preorder relation defined on linear spaces. In fact, we have the following result. Theorem 3.4. (L ∞ C , h) is a complete metric space. Proof. Clearly, (L ∞ C , h) is a subspace of (C (R d ), h). It suffices to show the completeness. To see this, let {An} ∞ n=1 be a Cauchy sequence in (L ∞ C , h), whence a Cauchy sequence in (C (R d ), h). It then follows from [51, Theorem 1.3.1] that {An} ∞ n=1 converges to A := ∩ ∞ n=1∪∞ m=nAm in (C (R d ), h). Since each Am ∈ L ∞ C , so is A. We note that, however, unlike the bounded case there is a main deficiency in the definition of L ∞ C , that is, it is very possible for a set A ∈ L ∞ C to have h(A, C) = ∞. Therefore, to facilitate our discussion, say, on SDE below, it is more desirable to consider the following subset of L ∞ C : LC = LC(R d ) := {A ∈ L ∞ C (R d ) : h(A, C) < ∞}. (3.3) Clearly, for all A, B ∈ LC we have h(A, B) < ∞, thanks to the triangle inequality. However, we note that the space LC is no longer “closed” under countable unions. The following simple example is thus worth noting: Example 3.5. Let A ∈ L ∞ C (R d ) such that h(A, C) = ∞. For each n ∈ N, define An := (A ∩ Bn(0)) + C, where Bn(0) is the closed ball in R d centered at 0 with radius n. Then, An ∈ LC, as h(An, C) ≤ n < ∞, n ∈ N, but A = ∪ ∞ n=1An ∈/ LC. We should note that, Example 3.5 notwithstanding, LC is still a complete metric space. Proposition 3.6. The space LC is a closed subspace of (L ∞ C , h), hence a complete metric space. 18 Proof. We need only check that any limit point of LC belongs to LC. To see this, let (An)n∈N ⊂ LC and limn→∞ h(An, A) = 0. Then A ∈ L ∞ C , and for N ∈ N large enough, we have h(AN , A) ≤ 1. Thus we have h(A, C) ≤ h(A, AN ) + h(AN , C) < ∞, since AN ∈ LC. To wit, A ∈ LC. We shall mainly focus on the space (LC, h). Since the cone C plays a special role in the space LC, in what follows we refer to it as the generating cone. Next, let us pay attention to a special type of elements in LC. We say that a set B ∈ LC has a compact component B˜ ∈ K (R d ) if B = B˜ + C. Clearly, if A ∈ L ∞ C ∩ G (R d ) in Example 3.5 , then the sequence (An) all have compact components. An important feature of such sets can be seen from the following extension of the so-called Cancellation Law1 to the case involving unbounded sets. Proposition 3.7 (Cancellation law). Let A, B, D ∈ L ∞ C ∩ G (R d ) such that B has a compact component. Then A ⊕ B = D ⊕ B implies that A = D. Proof. First assume that B = B˜ + C, where B˜ ∈ K (R d ). Since the Minkowski sum of a closed set and a compact set is closed, we have A ⊕ B = A ⊕ (B˜ + C) = A + (B˜ + C) = (A⊕ C) + B˜ = A + B˜. Similarly, we have D ⊕ B = D + B˜. Therefore A ⊕ B = D ⊕ B implies that A + B˜ = D + B˜. Since B˜ is compact, we conclude that A = D, thanks to the usual cancellation law. We should note from Proposition 3.7 and Example 3.5 that the sets in LC that have compact components provides a useful subspace of LC for our discussion. In fact, such sets resemble the so-called Motzkin decomposable sets in the literature (see e.g. [25]). To see the relation between an element in LC that has compact component and a Motzkin decomposable set, we first recall the so-called recession cone of a set 1The “Cancellation Law” states that for A, B, C ∈ G (R d ), A + B = C + B implies A = C, provided B is compact (cf. [61]). 19 A ∈ P(R d ), defined by 0 +A := {y ∈ R d : x + λy ∈ A, x ∈ A, λ ≥ 0}. (3.4) It is easy to see that 0 ∈ 0 +A, A = A + 0+A, and 0+A ̸= {0} implies that A is unbounded. Moreover, if A ∈ G (R d ) and 0+A ̸= {0}, then 0+A is a non-trivial closed convex cone. A set A ∈ G (R d ) with 0+A ̸= {0} is called Motzkin decomposable if A = A˜ + 0+A for some A˜ ∈ K (R d ). We have the following result regarding the generating cone C and the recession cone of A ∈ L ∞ C . Proposition 3.8. For any A ∈ L ∞ C , the following hold: (i) 0 +A ⊇ C; (ii) If A ∈ LC and A has a compact component, then A is Motzkin-decomposable and 0 +A = C; (iii) If A ∈ LC ∩ G (R d ), then 0 +A = C. Proof. (i) For any c ∈ C, x ∈ A, and λ ≥ 0, it is readily seen that x + λc ∈ A + C ⊆ A. Thus C ⊆ 0 +A. To see (ii), let A ∈ LC, and let the compact component be A˜. Then we have A˜ + C = A = A ⊕ 0 +A = A˜ + (C ⊕ 0 +A). Since A˜ is compact, by Proposition 3.7 we obtain C = C ⊕ 0 +A. Now since 0 ∈ C, we have 0 +A = 0+A + {0} ⊂ 0 +A + C ⊆ C, whence C = 0+A. It remains to check (iii). First, by (i), C ⊆ 0 +A. On the other hand, since h(A, C) < ∞, by definition of Hausdorff distance (2.3), we have A ⊆ C+B¯ ε(0) =: Dε, for some ε > 0. Since B¯ ε(0) is compact, by (ii) Dε is Motzkin-decomposable and 0 +Dε = C. Now note that A ⊆ Dε, by definition (3.4) we see that C ⊆ 0 +A ⊆ 0 +Dε = C, proving (iii). We should note that in general, C ⊊ 0 +A for A ∈ L ∞ C . For example, let C := 20 {(x, y) ∈ R 2 : x = 0, y ≥ 0} and A := {(x, y) ∈ R 2 : x ≥ 1, y ≥ 1}. Then A = A ⊕ C but 0+A = R 2 +. We shall consider set-valued random variables taking values in the space LC. Let us define    LC(Ω, F; R d ) := {F ∈ L0 (Ω, C (R d )) : F ∈ LC, P-a.s.}; L 2 C (Ω, F; R d ) := {F ∈ LC(Ω, F) : E[h 2 (F, C)] < ∞}. (3.5) We shall also often drop R d from the notations of the spaces above when the context is clear. Next, for any F1, F2 ∈ L 2 C (Ω, F), define d(F1, F2) := (E[h 2 (F1, F2)]) 1 2 . Theorem 3.9. (L 2 C (Ω, F), d) is a complete metric space. Proof. We first check that d(·, ·) defines a metric. Let F1, F2, F3 ∈ L 2 C (Ω, F). Then clearly d(F1, F1) = 0 since h(F1, F1) = 0. Moreover, if d(F1, F2) = 0, then h(F1, F2) = 0, P-a.s., and therefore F1 = F2, P-a.s. Clearly, we have d(F1, F2) = d(F2, F1). It remains to check the triangle inequality. Note that d 2 (F1, F3) = E[h 2 (F1, F2)] ≤ E hh(F1, F2) + h(F2, F3) 2i = E[h 2 (F1, F2)] + E[h 2 (F2, F3)] + 2E[h(F1, F2)h(F2, F3)] ≤ E[h 2 (F1, F2)] + E[h 2 (F2, F3)] + 2 E[h 2 (F1, F2)] 1 2 E[h 2 (F2, F3)] 1 2 = d 2 (F1, F2) + d 2 (F2, F3) + 2d(F1, F2)d(F2, F3) = d(F1, F2) + d(F2, F3) 2 , hence d(F1, F3) ≤ d(F1, F2) + d(F2, F3), and (L 2 C (Ω, F), d) is a metric space. To show the completeness, we assume that (Fn) ∞ n=1 is a Cauchy sequence in (L 2 C (Ω, F), d). For any ε > 0, by Chebyshev’s inequality we have P{h(Fn, Fm) > 21 ϵ} ≤ d 2 (Fn,Fm) ϵ 2 . Thus (Fn) ∞ n=1 is Cauchy under Hausdorff distance h, in probability. We first claim that there is a subsequence (Fnk ) ∞ k=1 ⊆ (Fn) ∞ n=1 such that it converges P-a.s. in Hausdorff metric h to some F ∈ L 2 C (Ω, F). Since (Fn) ∞ n=1 is Cauchy under Hausdorff distance h, in probability, then for k ≥ 1, there exists N ∈ N such that P{h(Fn, Fm) > 2 −k} < 2 −k , for n, m ≥ N. Let n1 := 1 and for k > 1, nk := inf{N > nk−1 : P{h(Fn, Fm) > 2 −k} < 2 −k , n, m ≥ N}. Let Ek := {ω ∈ Ω : h(Fnk+1 , Fnk ) > 2 −k}. Then P∞ k=1 P(Ek) < P∞ k=1 2 −k = 1. Denoting Nl := ∪ ∞ k=lEk and N := ∩ ∞ l=1Nl , we have P(N ) = 0, thanks to Borel-Cantelli lemma. Now for anyl ≥ 1, let ω ∈ N c l . Then for any j > i > l, it holds that h Fni (ω), Fnj (ω) ≤ X j−1 k=i h Fnk+1 (ω), Fnk (ω) ≤ X j−1 k=i 2 −k ≤ X∞ k=i 2 −k = 2−(i−1) . Thus (Fnk (ω))∞ k=1 is a Cauchy sequence in (LC, h) for ω ∈ N c = ∪ ∞ l=1N c l . Since (LC, h) is complete, by the proof of Theorem 3.9, (Fnk (ω))∞ k=1 converges to ∩ ∞ l=1∪∞ k=lFnk (ω) in (LC, h) for ω ∈ N c . Now let us define F(ω) := C1N (ω) + ∩ ∞ l=1∪∞ k=lFnk (ω)1N c (ω), ω ∈ Ω, then F(ω) ∈ LC for any ω ∈ Ω and (Fnk ) ∞ k=1 converges to F-P-a.s. We claim that F is measurable and (Fn) ∞ n=1 converges to F in (L 2 C (Ω, F), d). Indeed, for ∅ ̸= U ⊆ R d open, we have {ω ∈ Ω : F(ω) ∩ U ̸= ∅} = {ω ∈ N : C ∩ U ̸= ∅} ∪ {ω ∈ N c : ∩ ∞ l=1∪∞ k=lFnk (ω) ∩ U ̸= ∅}. Note that {ω ∈ N : C ∩ U ̸= ∅} is a null set, hence measurable. On the other hand, ω ∈ N c : ∩ ∞ l=1∪∞ k=lFnk (ω) ∩ U ̸= ∅ = ∩ ∞ l=1 ω ∈ N c : ∪∞ k=lFnk (ω) ∩ U ̸= ∅ = ∩ ∞ l=1 ω ∈ N c : ∪ ∞ k=lFnk (ω) ∩ U ̸= ∅ = ∩ ∞ l=1 ∪ ∞ k=l ω ∈ N c : Fnk (ω) ∩ U ̸= ∅ is also measurable. Thus {ω ∈ Ω : F(ω) ∩ U ̸= ∅} is measurable, to wit, ω 7→ F(ω) is measurable. It remains to show that E[h 2 (F, C)] < ∞ and that (Fn) ∞ n=1 converges to F in (L 2 C (Ω, F), d). First, for n ≥ 1, we have E[h 2 (F, C)] ≤ 2E[h 2 (Fn, F)] + 2E[h 2 (Fn, C)], where E[h 2 (Fn, C)] < ∞ as (Fn) ∞ n=1 ⊆ L 2 C (Ω, F). On the other hand, since (Fn) ∞ n=1 is a Cauchy sequence in (L 2 C (Ω, F), d), then for any ε > 0 there is N := N(ε) ∈ N such that E[h 2 (Fn, Fm)] < ε for any n, m ≥ N. Thus, for any n > N, we can apply Fatou’s lemma to get E[h 2 (Fn, F)] = E[h 2 (Fn, lim nk→∞ Fnk )] = E lim nk→∞ h 2 (Fn, Fnk ) ≤ lim nk→∞ E[h 2 (Fn, Fnk )] < ε. Thus, E[h 2 (F, C)] < ∞ and (Fn) ∞ n=1 converges to F in (L 2 C (Ω, F), d). This completes the proof. 3.3 Set-Valued Lebesgue Integrals on LC-Space In this section, we show some results regarding set-valued Lebesgue integrals that will be useful for our discussion but not covered by the existing literature. Theorem 3.10. Let F : [0, T] × Ω → C (R d ) be a “constant” set-valued mapping with F ≡ A, for some A ∈ C (R d ). Then F ∈ A 2 F ([0, T] × Ω; C (R d )), and for 0 ≤ t0 < t ≤ T, it holds that R t t0 Fsds = (t − t0)A, P-a.s. In particular, if A = C is a convex cone, then we have R t t0 Fsds = C, P-a.s. Proof. Let F(t, ω) ≡ A, (t, ω) ∈ [0, T] × Ω for some A ∈ C (R d ). Then obviously F ∈ A 2 F ([0, T] × Ω; C (R d )). Define F˜(ω) := (t − t0)A, ω ∈ Ω. Then F ∈ A 2 Ft ([0, T] × 23 Ω; C (R d )). Thus, to show that R t t0 Fsds = (t − t0)A, P-a.s., it is enough to show that S 2 Ft ( R t t0 Fsds) = S 2 Ft (F˜). Assume that 0 ∈ A and let {an : n ≥ 1} := A ∩ Qd . Define f n (t, ω) ≡ an,(t, ω) ∈ [0, T] × Ω, and ˜f n (ω) ≡ (t − t0)an, ω ∈ Ω. Then (f n ) ∞ n=1 ⊆ S 2 F (F), ( ˜f n ) ∞ n=1 ⊆ S 2 Ft (F˜), and R t t0 f n s ds = ˜f n. Moreover, F(t, ω) = cl{f n (t, ω) : n ≥ 1}, (t, ω) ∈ [0, T] × Ω, and F˜(ω) = cl{ ˜f n (ω) : n ≥ 1}, ω ∈ Ω. By Theorem 2.10, we have S 2 B([0,T])⊗F (F) = decB([0,T])⊗F {f n : n ≥ 1} and S 2 Ft (F˜) = decFt{ ˜f n : n ≥ 1}. Thus S 2 Ft (F˜) = decFt Z t t0 f n s ds : n ≥ 1 ⊆ decFt Z t t0 fsds : f ∈ S 2 F (F) = S 2 Ft Z t t0 Fsds . We shall show that decFt R t t0 fsds : f ∈ S 2 F (F) ⊆ decFt R t t0 f n s ds : n ≥ 1 , which in turn implies that S 2 Ft R t t0 Fsds ⊆ S 2 Ft (F˜). Let f ∈ S 2 F (F) ⊆ decB([0,T])⊗F {f n : n ≥ 1}. If f ∈ decB([0,T])⊗F {f n : n ≥ 1}, then f = Pm i=1 IAi×Bi f i = Pm i=1 IAi×Biai , where {Ai × Bi} m i=1 is a measurable partion of [0, T] × Ω. Since f is F-progressively measurable, and {ai} m i=1 ⊆ B(R d ), then for 0 ≤ t ≤ T and 1 ≤ i ≤ m, we have {(s, ω) ∈ [0, t] × Ω : f(s, ω) = ai} ∈ B([0, t]) ⊗ Ft . Note that {(s, ω) ∈ [0, t] × Ω : f(s, ω) = ai} =    ([0, t] ∩ Ai) × Bi ; [0, t] ∩ Ai ̸= ∅ ∅ × Ω ; otherwise. Thus, Bi ∈ Ft whenever [0, t] ∩ Ai ̸= ∅. Let B˜ i :=    Bi , [0, t] ∩ Ai ̸= ∅ ∅, otherwise. Then {B˜ i} m i=1 ⊆ Ft and for (s, ω) ∈ [0, T] × Ω, we have I[0,t]×Ω(s, ω)f(s, ω) = Xm i=1 I[0,t]∩Ai (s)aiIB˜ i (ω) = Xm j=1 Xm i=1 I[0,t]∩Ai (s)aj IB˜ j (ω). Thus, Z t t0 fsds = Xm j=1 Xm i=1 Z t t0 I[0,t]∩Ai dsaj IB˜ j = Xm j=1 (t − t0)aj IB˜ j = Xm j=1 Z t t0 ajdsIB˜ j . Hence R t t0 fsds = Pm j=1 R t t0 ajdsIB˜ j + 0.I(∪m i=1B˜) c ∈ decFt R t t0 f n s ds : n ≥ 1 . If f ∈ decB([0,T])⊗F {f n : n ≥ 1}, then there is a sequence {g m} ∞ m=1 ⊆ decB([0,T])⊗F {f n : n ≥ 1} such that it converges to f in L 2 ([0, T] × Ω, B([0, T] ⊗ F). By Holder’s inequality, we obtain Z t t0 g m s ds − Z t t0 fsds 2 = Z t t0 (g m s − fs)ds 2 ≤ (t − t0) Z t t0 |g m s − fs| 2 ds, and therefore E h R t t0 g m s ds − R t t0 fsds 2i ≤ (t − t0)E[ R t t0 |g m s − fs| 2ds]. Let ϵ > 0. Since {g m} ∞ m=1 converges to f in L 2 ([0, T] × Ω, B([0, T] ⊗ F), then there is N = N(ϵ) ∈ N such that E[ R t t0 |g m s −fs| 2ds] < ϵ t−t0 whenever m > N. Hence E h R t t0 g m s ds− R t t0 fsds 2i < ϵ for any m > N. That is, { R t t0 g m s ds} ∞ m=1 converges to R t t0 fsds in L 2 (Ω, F). Since {g m} ∞ m=1 ⊆ decB([0,T])⊗F {f n : n ≥ 1}, then by the previous step { R t t0 g m s ds} ∞ m=1 ⊆ decFt n R t t0 f n s ds : n ≥ 1 o , and therefore R t t0 fsds ∈ decFt n R t t0 f n 0 ds : 25 n ≥ 1 o . Thus, n Z t t0 fsds : f ∈ S 2 F (F) o ⊆ decFt n Z t t0 f n s ds : n ≥ 1 o , and hence decFt n R t t0 fsds : f ∈ S 2 F (F) o ⊆ decFt n R t t0 f n 0 ds : n ≥ 1 o . That is, S 2 Ft R t t0 Fsds ⊆ S 2 Ft (F˜), and therefore S 2 Ft R t t0 Fsds = S 2 Ft (F˜) for F ≡ A ∋ 0. If 0 ̸∈ A, consider B := A − a0 := A + (−a0), for some a0 ∈ A. Then 0 ∈ B and therefore S 2 Ft Z t t0 Bds = S 2 Ft ((t − t0)B) = S 2 Ft ((t − t0)A) − (t − t0)a0. On the other hand, S 2 Ft Z t t0 Bds = S 2 Ft Z t t0 Ads + S 2 Ft Z t t0 −a0ds = S 2 Ft Z t t0 Ads − (t − t0)a0. Thus, S 2 Ft R t t0 Ads = S 2 Ft (t − t0)A . Finally, , if A = C is a convex cone, then we have (t − t0)C = C, proving the theorem. Proposition 3.11. Let F ∈ A 2 F ([0, T] × Ω; C (R d )). Then for 0 ≤ t0 < t ≤ T, it holds that Z t 0 Fsds = Z t0 0 Fsds ⊕ Z t t0 Fsds, P-a.s. Proof. It is enough to show that S 2 Ft Z t 0 Fsds = S 2 Ft Z t0 0 Fsds ⊕ Z t t0 Fsds . By Proposition 2.11, we have S 2 Ft Z t0 0 Fsds ⊕ Z t t0 Fsds = S 2 Ft Z t0 0 Fsds ⊕ S 2 Ft Z t t0 Fsds . By definition of Lebesgue integrals and Proposition 2.13, we have S 2 Ft Z t0 0 Fsds ⊕ S 2 Ft Z t t0 Fsds =decFt n Z t 0 f 1 s ds : f 1 ∈ S 2 F (I[0,t0]F) o ⊕ decFt n Z t 0 f 2 s ds : f 2 ∈ S 2 F (I[t0,t]F) o = decFt n Z t 0 f 1 s ds : f 1 ∈ S 2 F (I[0,t0]F) o ⊕ decFt n Z t 0 f 2 s ds : f 2 ∈ S 2 F (I[t0,t]F) o =decFt n Z t 0 (f 1 s + f 2 s )ds : f 1 ∈ S 2 F (I[0,t0]F), f 2 ∈ S 2 F (I[t0,t]F) o . Let f ∈ S 2 F (F), then f 1 := I[0,t0]f ∈ S 2 F (I[0,t0]F) and f 2 := I[t0,t]f ∈ S 2 F (I[t0,t]F). Thus decFt n Z t 0 fsds : f ∈ S 2 F (F) o ⊆ decFt n Z t 0 (f 1 s +f 2 s )ds : f 1 ∈ S 2 F (I[0,t0]F), f 2 ∈ S 2 F (I[t0,t]F) o , and therefore S 2 Ft Z t 0 Fsds ⊆ S 2 Ft Z t0 0 Fsds ⊕ S 2 Ft Z t t0 Fsds . On the other hand, for any f 1 ∈ S 2 F (I[0,t0]F) and f 2 ∈ S 2 F (I[t0,t]F), we have f := f 1 + f 2 ∈ S 2 F (F). Thus, decFt n Z t 0 (f 1 s +f 2 s )ds : f 1 ∈ S 2 F (I[0,t0]F), f 2 ∈ S 2 F (I[t0,t]F) o ⊆ decFt n Z t 0 fsds : f ∈ S 2 F (F) o , 27 and therefore S 2 Ft Z t0 0 Fsds ⊕ S 2 Ft Z t t0 Fsds ⊆ S 2 Ft Z t 0 Fsds , proving the result. Next, we present an important inequality regarding set-valued Lebesgue integrals. The compact counterpart of this ineqaulity can be found in [50]. Theorem 3.12. Suppose that F, F˜ ∈ A 2 F ([0, T]×Ω; G (R d )) such that E R T 0 h 2 (Fs, F˜ s)ds < ∞. Then, for any 0 ≤ t0 < t ≤ T, it holds that h 2 Z t t0 Fsds, Z t t0 F˜ sds ≤ (t − t0) Z t t0 h 2 (Fs, F˜ s)ds, P-a.s. (3.6) Proof. Fix 0 ≤ t0 < t ≤ T and let U be an arbitrary Ft measurable set. It is enough to show that Z U h 2 Z t t0 Fsds, Z t t0 F˜ sds dP ≤ (t − t0) Z U Z t t0 h 2 (Fs, F˜ s)dsdP, P-a.s. (3.7) By definition of Hausdorff distance, we have Z U h 2 Z t t0 Fsds, Z t t0 F˜ sds dP = Z U sup n inf n |x−y| 2 : y ∈ Z t t0 F˜ sdso : x ∈ Z t t0 Fsdso dP. Let θ : U × R d → [−∞,∞] be such that θ(ω, x) := inf n |x − y| 2 : y ∈ R t t0 F˜ sdso . Since the infimum of an arbitrary collection of continuous functions is upper-semi continuous, then θ is upper semi-continuous in x. On the other hand, By Proposition 2.7, there exists {g n}n∈N ⊆ SFt ( R t t0 F˜ sds) such that R t t0 F˜ sds = cl{g n : n ∈ N}. Thus θ(ω, x) = inf n |x − g n (ω)| 2 : n ∈ N o is measurable in ω, and therefore jointly measurable. Moreover, for every u ∈ S 2 Ft ( R t t0 Fsds), we have R U θ(ω, u(ω))dP(ω) is well 28 defined and is nonnegative. Thus by Proposition 2.21, Z U sup n θ(ω, x) : x ∈ Z t t0 Fsdso dP(ω) = sup n Z U θ(ω, u(ω))dP(ω) : u ∈ S 2 Ft Z t t0 Fsdso. Hence, Z U h 2 Z t t0 Fsds, Z t t0 F˜ sds dP = sup n Z U inf n |u−y| 2 : y ∈ Z t0 t F˜ sdso dP : u ∈ S 2 Ft Z t t0 Fsdso. Let u ∈ S 2 Ft R t t0 Fsds be fixed and let θ : U × R d → [−∞,∞] be such that θ(ω, y) := |u(ω)−y| 2 . Then for every fixed y ∈ R d , θ(ω, y) is measurable and for every fixed ω ∈ U, θ(ω, y) is continuous. Thus by Proposition 2.9, θ is jointly measurable. Moreover, for every v ∈ S 2 Ft ( R t t0 F ds ˜ ), we have R U θ(ω, v(ω))dP is well defined and is finite. Thus by Proposition 2.21, we have Z U inf n θ(ω, y) : y ∈ Z t t0 F˜ sdso dP = inf n Z U |u − v| 2 dP : v ∈ S 2 Ft Z t t0 F˜ sdso, and therefore Z U inf n |u − y| 2 : y ∈ Z t t0 F˜ sdso dP = inf n Z U |u − v| 2 dP : v ∈ S 2 Ft Z t t0 F˜ sdso. Hence Z U h 2 Z t t0 Fsds, Z t t0 F˜ sds dP = sup n inf n Z U |u − v| 2 dP : v ∈ S 2 Ft Z t t0 F˜ sdso : u ∈ S 2 Ft Z t t0 Fsdso = sup n inf n Z U |u − v| 2 dP : v ∈ decFtJt0,t(F˜) o : u ∈ decFtJt0,t(F) o . (3.8) 2 By H¨older’s inequality and Proposition 2.21, we also have sup n inf n Z U |u − v| 2 dP : v ∈ Jt0,t(F˜) o : u ∈ Jt0,t(F) o = sup n inf n Z U Z t t0 fsds − Z t t0 ˜fsds 2 dP : ˜f ∈ S 2 F (F˜) o : f ∈ S 2 F (F) o ≤(t − t0) sup n inf n Z U Z t t0 |fs − ˜fs| 2 dsdP : ˜f ∈ S 2 F (F˜) o : f ∈ S 2 F (F) o =(t − t0) sup n inf n Z U×[t0,t] |fs − ˜fs| 2 dP × ds : ˜f ∈ S 2 F (F˜) o : f ∈ S 2 F (F) o =(t − t0) Z U Z t t0 h 2 (Fs, F˜ s)dsdP. (3.9) Next, we show that sup n inf n Z U |u − v| 2 dP : v ∈ decFtJt0,t(F˜) o : u ∈ decFt Jt0,t(F) o ≤ (t − t0) Z U Z t t0 h 2 (Fs, F˜ s)dsdP. Let u ∈ decFt Jt0,t(F). Then u = Pm i=1 uiIUi , for some {ui} m i=1 ⊆ Jt0,t(F) and an Ft-measurable partition {Ui} m i=1 of Ω. Then by (3.9), we see that inf n Z U | Xm i=1 ui1Ui − v| 2 dP : v ∈ decFtJt0,t(F˜) o ≤ inf n Z U | Xm i=1 ui1Ui − v| 2 dP : v ∈ Jt0,t(F˜) o ≤ Xm i=1 sup n inf n Z U∩Ui |ui − v| 2 dP : v ∈ Jt0,t(F˜) o : ui ∈ Jt0,t(F) o (3.10) ≤ Xm i=1 (t − t0) Z U∩Ui h Z t t0 h 2 (Fs, F˜ s)dsi dP = (t − t0) Z U h Z t t0 h 2 (Fs, F˜ s)dsi dP. Since this is true for all u ∈ decFt Jt0,t(F), we see that (3.10) implies that sup n inf n Z U |u − v| 2 dP : v ∈ decFtJt0,t(F˜) o : u ∈ decFt Jt0,t(F) o ≤ (t − t0) Z U Z t t0 h 2 (Fs, F˜ s)dsdP. 30 Now let u ∈ decFtJt0,t(F) and let {un} ∞ n=1 ⊆ decFt Jt0,t(F) be such that limn→∞ E|u− un| 2 = 0. For any n ≥ 1, we have inf n Z U |un − v| 2 dP : v ∈ decFtJt0,t(F˜) o ≤ (t − t0) Z U Z t t0 h 2 (Fs, F˜ s)dsdP. (3.11) In particular, (3.11) implies that for ε > 0 and n ≥ 1, we can find vn ∈ decFtJt0,t(F˜) such that Z U |un − vn| 2 dP ≤ (t − t0) Z U Z t t0 h 2 (Fs, F˜ s)dsdP + ε ≤ T Z T 0 E[h 2 (Fs, F˜ s)]ds + ε < ∞, thanks to the assumption of the theorem. Since {un} ∞ n=1 is convergent in L 2 (Ω, Ft , P) by definition, whence bounded, this in turn implies that {vn1U } ∞ n=1 is bounded in L 2 (Ω, Ft , P) as well. We can then apply Banach-Saks-Mazur theorem to obtain a subsequence {vnk } ∞ k=1 such that {vnk } ∞ k=1 converges to, say, v ∈ L 2 (Ω, Ft , P), where vnk := 1 k Pk j=1 vnj1U . Since F˜ takes values in G (R d ), we can easily see that Jt0,t(F˜) is convex. Thus, by Proposition 2.13, it follows that decFtJt0,t(F˜) is convex, and therefore vnk ∈ decFtJt0,t(F˜), k ≥ 1. Hence, v ∈ decFtJt0,t(F˜). Moreover, since {un} ∞ n=1 ⊆ decFt Jt0,t(F) converges to u, and decFt Jt0,t(F) is convex, then unk := 1 k Pk j=1 unj converges to u in L 2 (Ω, Ft , P) as well. Thus by Jensen’s inequality and (3.11) we have Z U |unk − vnk | 2 dP = Z U X k j=1 1 k (unj − vnj ) 2 dP ≤ X k j=1 1 k Z U |unj − vnj | 2 dP ≤ (t − t0) Z U Z t t0 h 2 (Fs, F˜ s)dsdP + ε. 31 Sending k → ∞ we obtain that inf n Z U |u−v| 2 dP : v ∈ decFtJt0,t(F˜) o ≤ Z U |u−v| 2 dP ≤ (t−t0) Z t t0 Z U h 2 (F, F˜)dPds+ε. Since ε > 0 and u ∈ decFtJt0,t(F) are arbitrary, and noting (3.8), we can first take a “sup” in the inequality above and then let ε → to conclude that (3.7) holds, proving the theorem. Remark 3.13. A main technical difficulty in this proof compared to the compact case is in the application of Banach-Saks-Mazur theorem. In the compact case, the sequence {vn1U } ∞ n=1 is bounded since the whole set decFtJt0,t(F˜) is bounded. However, we do not require decFtJt0,t(F˜) to be bounded. Instead, we show that the sequence {vn1U } ∞ n=1 is bounded by making use of the condition E R T 0 h 2 (Fs, F˜ s)ds < ∞. Setting F˜ ≡ C in Theorem 3.12 and noting that R t t0 C = C, we have the following corollary. Corollary 3.14. Suppose that F ∈ A 2 F ([0, T]×Ω; G (R d )) such that E R T 0 h 2 (Ft , C)dt < ∞. Then for any 0 ≤ t0 < t ≤ T, it holds that h 2 Z t t0 Fsds, C ≤ (t − t0) Z t t0 h 2 (Fs, C)ds, P-a.s. To conclude this section, let us consider, similar to the spaces in (3.5), the following spaces of F-nonanticipative set-valued mappings:    LC,F([0, T] × Ω; R d ) := {F ∈ A 2 F ([0, T] × Ω; C (R d )) : Ft ∈ LC, P-a.s., t ∈ [0, T]}; L 2 C,F ([0, T] × Ω; R d ) := {F ∈ LC,F([0, T] × Ω; R d ) : E[ R T 0 h 2 (Fs, C)ds] < ∞}. (3.12) 32 Proposition 3.15. Let F ∈ L 2 C,F ([0, T] × Ω; R d ). Then R t 0 Fsds ∈ L 2 C (Ω, Ft), t ∈ [0, T]. Furthermore, the process { R t 0 Fsds}0≤t≤T has Hausdorff continuous paths, Pa.s. Proof. Let F ∈ L 2 C,F ([0, T] × Ω; R d ). We first argue the adaptedness of the indefinite integral. Fix t ∈ [0, T], by definition of Lebesgue set-valued integral, it is clear that R t 0 Fsds is an Ft-measurable set-valued random variable. Furthermore, since C is a convex cone containing 0, we have R t 0 Cds = Ct = C. This, together with the fact that Ft ∈ LC, P-a.s. t ∈ [0, T], yields Z t 0 Fsds ⊕ C = Z t 0 Fsds ⊕ Z t 0 Cds = Z t 0 (Fs ⊕ C)ds = Z t 0 Fsds. Moreover, by defintion 3.12 and Corollary 3.14 we have E[h 2 ( R t 0 Fsds, C)] ≤ tE[ R t 0 h 2 (Fs, C)ds] < ∞. Thus R t 0 Fsds ∈ L 2 C (Ω, Ft), t ∈ [0, T]. To see the continuity, let 0 ≤ t0 < t ≤ T. By Theorem 3.10, Propositions 3.11, 3.15, Corollary 3.14, and properties of Hausdorff metric, we have, P-almost surely, h 2 Z t 0 Fsds, Z t0 0 Fsds = h 2 Z t0 0 Fsds ⊕ Z t t0 Fsds, Z t0 0 Fsds ⊕ C ≤ h 2 Z t0 0 Fsds + Z t t0 Fsds, Z t0 0 Fsds + C ≤ h 2 Z t t0 Fsds, C ≤ (t − t0) Z T 0 h 2 (Fs, C)ds. Thus h 2 ( R t 0 Fsds, R t0 0 Fsds) → 0, as |t − t0| → 0, proving the result. 33 4 Chapter Four: Set-Valued SDEs with Unbounded Coefficients 4.1 Existence and Uniqueness In this section, we consider stochastic differential equations of the form: Xt = ξ ⊕ Z t 0 F(s, Xs)ds ⊕ Z t 0 G ◦ XdBs, t ∈ [0, T], P-a.s. , (4.1) where ξ ∈ L 2 C (Ω, F0) and F, G are set-valued mappings taking values in C (R d ) (hence possibly unbounded), which we now describe. Definition 4.1. A mapping F : [0, T] × Ω × R d → LC is called a non-anticipating Carath´eodory set-valued random field if it enjoys the following properties: (i) F ∈ L0 ([0, T] × Ω × R d ; LC); (ii) For fixed a ∈ R d , F(·, ·, a) ∈ LC,F([0, T] × Ω; R d ); and (iii) For P-a.s. ω ∈ Ω, F(·, ω, ·) is a Carath`eodory set-valued function. Furthermore, for a given non-anticipating Carath´eodory set-valued random field F, we define, with a slight abuse of notations, its set-to-set version F : [0, T]×Ω×LC → LC by F(t, ω, A) := co(∪a∈AF(t, ω, a)), A ∈ LC, (t, ω) ∈ [0, T] × Ω. (4.2) Clearly, if the coefficient F in SDE (4.1) is a non-anticipating Carath´eodory setvalued random field of the form (4.2), then it follows from Proposition 2.20 that for any X ∈ L0 C,F ([0, T] × Ω; R d ), F(·, X·) ∈ L0 F ([0, T] × Ω; R d ). Furthermore, we make the following assumption. 34 Assumption 4.2. F : [0, T] × Ω × LC → LC is a non-anticipative Carath´eodory random field such that for some constants β > 0, the following conditions hold: (1) h 2 (F(t, ·, A), C) ≤ β(1 + h 2 (A, C)), A ∈ LC, P-a.s.; (2) h 2 (F(t, ·, A), F(t, ·, A˜)) ≤ βh2 (A, A˜), A, A˜ ∈ LC, P-a.s. Next, we specify the coefficient G ◦ X in (4.1). We note that in SDE (4.1) the stochastic integral is defined in Aumann-Itˆo sense, and we shall try to define the process G ◦ X so that Theorem 2.19 can be applied. We therefore begin with the following definition. Let (X, ρ) be a given metric space. Definition 4.3. A mapping g : [0, T] × Ω × X → R d×m is called a non-anticipating Carath´eodory random field if it enjoys the following properties: (i) g ∈ L 0 ([0, T] × Ω × X; R d×m); (ii) for fixed x ∈ X, g(·, ·, A) ∈ L 0 F ([0, T] × Ω; R d×m); and (iii) for fixed (t, ω), the mapping x 7→ g(t, ω, x) is continuous. Now let us consider the class of sequences {g n} of non-anticipating Carath´eodory random fields g n : [0, T] × Ω × LC → R d×m that further satisfy the following uniform Lipschitz condition. Assumption 4.4. There exist αn > 0, n = 1, 2, · · · , such that P∞ n=1 α 2 n < ∞, and (1) |g n (t, ·, C)| ≤ αn, P-a.s.; (2) |g n (t, ·, A) − g n (t, ω, A˜)| ≤ αnh(A, A˜), A, A˜ ∈ LC, P-a.s. We remark that Assumption 4.4 implies the following growth condition. |g n (t, ·, A)| ≤ αn(1 + h(A, C)), A ∈ LC. (4.3) Furthermore, if G := {g n} is a sequence of non-anitcipating Carath´eodory random fields satisfying Assumption 4.4, and X ∈ L 2 C,F ([0, T] × Ω; R d×m), then in light of 35 Proposition 2.20, for each n the mapping g n ◦ X ∈ L 0 F ([0, T] × Ω; R d×m), where (g n ◦ X)(t, ω) := g n (t, ω, X(t, ω)), (t, ω) ∈ [0, T]×Ω. Furthermore, since X ∈ L 2 C,F ([0, T]× Ω; R d×m), we have E[ R T 0 h 2 (Xt , C)dt] < ∞, thus E h Z T 0 X∞ n=1 |(g n ◦ X)(t, ·)| 2 dti ≤ E h Z T 0 X∞ n=1 α 2 n [1 + h 2 (X(t, ·), C)]dti (4.4) = X∞ n=1 α 2 n E h Z T 0 [1 + h 2 (X(t, ·), C)]dti < ∞. In what follows, for G = {g n} and X ∈ L 2 C,F ([0, T] × Ω; R d ) as above we denote G ◦X := co{g n ◦X : n ≥ 1} = co(G◦X), then the Aumann-Itˆo integral R t 0 G ◦XdBs = co(R t 0 G ◦ XdBs) is well-defined for each t ∈ [0, T], and it follows from Theorem 2.19 that the process of indefinite integrals { R t 0 G ◦ XdBs}t∈[0,T] has Hausdorff continuous paths. Our main result of the well-posedness for set-valued SDE (4.1) is the following theorem. Theorem 4.5. Assume that the Assumptions 4.2 and 4.4 hold for the coefficients F and G, respectively. Then, for each G (R d )-valued random variable ξ ∈ L 2 C (Ω, F0), there exists a unique convex, (Hausdorff ) continuous process X = (Xt)0≤t≤T ∈ L 2 C,F ([0, T]× Ω; R d ), such that (4.1) holds P-a.s. Proof. We follow the standard Picard iteration: let Y (0) ≡ ξ, and for k ≥ 0, we define Y (k+1) t := ξ ⊕ Z t 0 F(s, Y (k) s )ds ⊕ Z t 0 G ◦ Y (k) dBs, t ∈ [0, T]. (4.5) We first claim that for each k ≥ 0, Y (k) ∈ L 2 C,F ([0, T] × Ω; R d ) and has Hausdorff continuous paths. Indeed, since ξ ∈ L 2 C (Ω, F0; R d ), the claim is trivial for k = 0. Now 36 assume that the claim is true for Y (k) . Then, by using Assumptions 4.2, 4.4, along with the estimates similar to (4.4) and an induction argument, if follows that F(·, Y (k) · ) ∈ L 2 C,F ([0, T] × Ω; R d ) and G ◦ Y (k) ∈ L 2 ([0, T] × Ω; R d×m). Then by Proposition 3.15 and Theorem 2.19, we further deduce that all the indefinite integrals R t 0 F(s, Y (k) )ds and R t 0 G ◦ Y (k)dBs, t ∈ [0, T], are well-defined F-adapted set-valued processes with Hausdorff continuous paths. Thus so is Y (k+1) . To show that Y (k+1) ∈ L 2 C,F ([0, T] × Ω; R d ), it remains to check E[h 2 (Y (k+1 t , C)] < ∞. To see this, let t ∈ [0, T] be fixed, then we have E[h 2 (Y (k+1) t , C)] = E h h 2 ξ ⊕ Z t 0 F(s, Y (k) s )ds ⊕ Z t 0 G ◦ Y (k) dBs}, Ci ≤ E h h 2 ξ + Z t 0 F(s, Y (k) s )ds + Z t 0 G ◦ ξdBs, C + C + {0} i (4.6) ≤ 2E h h 2 ξ + Z t 0 F(s, Y (k) s )ds, C + C i + 2E h h 2 Z t 0 G ◦ Y (k) dBs, {0} i ≤ 4E[h 2 (ξ, C)] + 4E h h 2 Z t 0 F(s, Y (k) s )ds, Ci + 2E h h 2 Z t 0 G ◦ Y (k) dBs, {0} i. Note that, by Corollary 3.14 and the inductional assumption, we have E h h 2 Z t 0 F(s, Y (k) s )ds, Ci ≤ tE h Z t 0 h 2 (F(s, Y (k) s ), C)dsi < ∞. (4.7) Moreover, by [51, Theorem 5.4.2] and Assumption 4.4, we have E h h 2 Z t 0 G ◦ Y (k) dBs, {0} i ≤ E h Z t 0 X∞ n=1 |g n ◦ Y (k) s | 2 dsi < ∞. (4.8) Combining (4.6)-(4.8) we obtain E[h 2 (Y (k+1) t , C)] < ∞, t ∈ [0, T], whence the claim. 37 Next, let us estimate E[h 2 (Y (k+1) t , Y (k) t )], for k ≥ 1, we have E[h 2 (Y (k+1) t , Y (k) t )] (4.9) ≤ E h h 2 ξ + Z t 0 F(s, Y (k) s )ds + Z t 0 G ◦ Y (k) dBs, ξ + Z t 0 F(s, Y (k−1) s )ds + Z t 0 G ◦ Y (k−1)dBs i ≤ 2E h h 2 Z t 0 F(s, Y (k) s )ds, Z t 0 F(s, Y (k−1) s )dsi + 2E h h 2 Z t 0 G ◦ Y (k) dBs, Z t 0 G ◦ Y (k−1)dBs i. Now, applying Theorem 3.12 and using the Assumption 4.2, we have E h h 2 Z t 0 F(s, Y (k) s )ds, Z t 0 F(s, Y (k−1) s )dsi ≤ TE h Z t 0 h 2 (F(s, Y (k) s ), F(s, Y (k−1) s )ds dsi ≤ T βE h Z t 0 h 2 (Y (k) s , Y (k−1) s )dsi = T β Z t 0 E[h 2 (Y (k) s , Y (k−1) s )]ds. (4.10) Furthermore, by [51, Theorem 5.4.2]) and Assumption 4.4 we obtain E h h 2 Z t 0 G ◦ Y (k) dBs, Z t 0 G ◦ Y (k−1)dBs i ≤ E hX∞ n=1 Z t 0 [g n ◦ Y (k) − g n ◦ Y (k−1)]dBs 2i ≤ E hX∞ n=1 Z t 0 αnh 2 (Y (k) , Y (k−1))dsi = X∞ n=1 αn Z t 0 E[h 2 (Y (k) , Y (k−1))]ds. (4.11) Combining (4.9)-(4.11) we obtain E h 2 Y (k+1) t , Y (k) t ≤ 2 h T β + X∞ n=1 αn i Z t 0 E h 2 Y k s , Y (k−1) s ds. (4.12) Since Y (0) ≡ ξ, by Assumptions 4.2 and 4.4 we can easily check that E h 2 Y (1) t , Y (0) t ≤ 2 h βT + X∞ n=1 αn i (1 + E[h 2 (ξ, C)])t, t ∈ [0, T]. (4.13) Repeatedly applying (4.12) and noting (4.13), an induction argument leads to that E h 2 Y (k+1) t , Y (k) t ≤ Mk+1(1 + E[h 2 (ξ, C)]) t k+1 (k + 1)!, (4.14) where M := 2 T β+ P∞ n=1 αn . Now consider the complete metric space (LC(Ω, Ft), d) in Theorem 3.9. In terms of the metric d we see that (4.14) implies that, for m > n, d(Y (m) t , Y (n) t ) ≤ mX−1 k=n d(Y (k+1) t , Y (k) t ) = mX−1 k=n E h 2 (Y (k+1) t , Y (k) t ) 1 2 ≤ (1 + E[h 2 (ξ, C)]) 1 2 mX−1 k=n Mk+1t k+1 (k + 1)! 1 2 → 0, as m, n → ∞. Thus {Y (k) t }k∈N is a Cauchy sequence in (LC(Ω, Ft), d). Thus there exists Yt ∈ LC(Ω, Ft) such that limk→∞ d(Yt , Y k t ) = 0. We shall argue that the limit Y has Hausdorff continuous paths. More precisely, we claim that {Y (k)}k∈N convergences to Y uniformly on [0, T]. Indeed, denoting ∆(k) g n (t, ·) := g n (t, Y (k) t ) − g n (t, Y (k−1) t ), ∆(k) (h ◦ F)(t, ·) := h(F(t, Y (k) t ), F(t, Y (k−1) t )), k ∈ N, and applying Theorem 3.12 we can easily show that, for any k ≥ 0, P-almost surely, sup 0≤t≤T h 2 (Y (k+1) t , Y (k) t ) ≤ 2 sup 0≤t≤T t Z t 0 [∆(k) (h ◦ F)]2 (s, ·)ds + 2 sup 0≤t≤T X∞ n=1 Z t 0 |∆ (k) g n (s, ·)]dBs 2 ≤ 2T β Z T 0 h 2 (Y (k) s , Y (k−1) s )ds + 2 sup 0≤t≤T X∞ n=1 Z t 0 |∆ (k) g n (s, ·)|dBs 2 . (4.15) Here the second inequality above is due to Assumption 4.2. Thus, for each k ∈ N we have P sup 0≤t≤T h 2 (Y (k+1 t , Y k t ) > 1 2 k ≤ P 2T β Z T 0 h 2 (Y (k) s , Y (k−1) s )ds > 1 2 k +P 2 sup 0≤t≤T X∞ n=1 Z t 0 [∆(k) g n (s, ·)]dBs 2 > 1 2 k . (4.16) Now, by Markov’s inequality we have P 2T β Z T 0 h 2 (Y (k) s , Y (k−1) s )ds > 1 2 k ≤ 2 k+1T βE h Z T 0 h 2 (Y (k) s , Y (k−1) s )dsi . (4.17) Furthermore, by Burkholder-Davis-Gundy’s inequality, for some K > 0, it holds that P 2 sup 0≤t≤T X∞ n=1 Z t 0 [∆(k) g n (s, ·)]dBs 2 > 1 2 k ≤ 2 k+1E hX∞ n=1 sup 0≤t≤T Z t 0 [∆(k) g n (s, ·)]dBs 2i ≤ 2 k+1K X∞ n=1 E h Z T 0 |∆ (k) g n (s, ·)| 2 dsi ≤ 2 k+1K X∞ n=1 αn E h Z T 0 h 2 (Y (k) s , Y (k−1) s )dsi .(4.18) Combining (4.16)-(4.18) and noting (4.14) we deduce that P sup 0≤t≤T h 2 (Y k+1 t , Y k t ) > 1 2 k ≤ 2 k+1T βE h Z T 0 h 2 Y k s , Y k−1 s dsi + 2k+1K X∞ n=1 αn E h Z T 0 h 2 (Y (k) s , Y (k−1) s )dsi = 2k+1h T β + K X∞ n=1 αn i Z T 0 E[h 2 (Y (k) s , Y (k−1) s )]ds (4.19) ≤ 2 k+1h T β + K X∞ n=1 αn i (1 + E[h 2 (ξ, C)])MkT k+1 (k + 1)! . Therefore, we conclude that P∞ k=1 P(sup0≤t≤T h 2 (Y (k+1) t , Y (k) t ) > 1 2 k ) < ∞. BorelCantelli lemma then yields that the sequence {Y (k) t }k∈N converges uniformly on [0, T] to the Y ∈ L 2 C,F ([0, T] × Ω), P-a.s., and thus Y has (Hausdorff) continuous paths. 40 Finally, the similar argument shows that, for fixed t ∈ [0, T], lim k→∞ E h h 2 Z t 0 F(s, Y (k) s )ds, Z t 0 F(s, Ys)dsi + lim k→∞ E h h 2 Z t 0 G ◦ Y (k) dBs, Z t 0 G ◦ Y dBs i ≤ h T β + X∞ n=1 αn i lim k→∞ E h Z t 0 h 2 (Y (k) s , Ys)dsi = 0. That is, Y satisfies the SDE (4.1). For the uniqueness, suppose there exists another convex, (Hausdorff) continuous process Y˜ = (Y˜ t)0≤t≤T ∈ L 2 C,F ([0, T] × Ω; R d ), such that (4.1) holds P-a.s. Then by similar arguments we get E h 2 (Yt , Y˜ t) ≤ 2E h h 2 Z t 0 F(s, Ys)ds, Z t 0 F(s, Y˜ s)dsi + 2E h h 2 Z t 0 G ◦ Y dBs, Z t 0 G ◦ Y dB ˜ s i ≤ 2TE h Z t 0 h 2 F(s, Ys), F(s, Y˜ s) dsi + 2E hX∞ n=1 Z t 0 h g n (s, Ys) − g n (s, Y˜ s) i dBs 2i ≤ 2T βE h Z t 0 h 2 (Ys, Y˜ s)dsi + 2X∞ n=1 E h Z t 0 g n (s, Ys) − g n (s, Y˜ s) 2 dsi ≤ 2T βE h Z t 0 h 2 (Ys, Y˜ s)dsi + 2X∞ n=1 βnE h Z t 0 h 2 (Ys, Y˜ s)dsi = 2h T β + 4X∞ n=1 βn i Z t 0 E h h 2 (Ys, Y˜ s) i ds. Let ε > 0, then by Gronwall’s inequality we have E h h 2 (Yt , Y˜ t) i ≤ εeAt, where A = 2 h T β + 4P∞ n=1 βn i . Hence E h 2 (Yt , Y˜ t) = 0, and therefore d(Yt , Y˜ t) = 0. 41 To end this section, we give the following stability results of the solution to the SDE (4.1). Proposition 4.6. Assume that Assumptions 4.2 and 4.4 are in force, and let X be the solution to SDE (4.1). Then for any 0 ≤ s < t ≤ T, it holds that    E[sup0≤t≤T h 2 (Xt , C)] ≤ KT 1 + E[h 2 (ξ, C)] ; E[h 2 (Xt , Xs)] ≤ KT (1 + E[h 2 (ξ, C)])(t − s). (4.20) where KT > 0 is a generic constant depending only on T, β in Assumption 4.2, and {αn} in Assumption 4.4. Proof. Let us denote KT to be a generic constant depending only on T, β, and {αn}, which is allowed to vary from line to line. Then, similar to (4.6) we have, for t ∈ [0, T], E h sup s≤t h 2 (Xs, C) i = E h sup s≤t h 2 nξ ⊕ Z s 0 F(τ, Xτ )dτ ⊕ Z s 0 G ◦ XdBτ o , Ci (4.21) ≤ 4 n E h h 2 (ξ, C) i + E h sup s≤t h 2 Z s 0 F(τ, Xτ )dτ, Ci + E h sup s≤t h 2 Z s 0 (G ◦ X)τdBτ , {0} io. By Theorem 3.12 and Assumption 4.2 we have E h sup s≤t h 2 Z s 0 F(τ, Xτ )dτ, C ≤ E h sup s≤t s Z s 0 h 2 F(τ, Xτ ), C dτi (4.22) ≤ KTE h Z t 0 (1 + h 2 (Xτ , C))dτi ≤ KT 1 + E h Z t 0 sup s≤τ h 2 (Xs, C)dτi. 42 Similarly, by Assumption 4.4 and Burkholder-Davis-Gundy inequality we have E h sup s≤t h 2 Z s 0 (G ◦ X)dBτ , {0} i ≤ E h sup s≤t X∞ n=1 Z s 0 (g n ◦ X)τdBτ 2i (4.23) ≤ X∞ n=1 E h Z t 0 |(g n ◦ X)τ | 2 dτi ≤ KT 1 + E h Z t 0 sup s≤τ h 2 (Xs, C)dτi. Combining (4.21)-(4.23) we obtain E h sup s≤t h 2 (Xs, C) i ≤ KT 1 + Z t 0 E h sup s≤τ h 2 (Xs, C) i dτ , t ∈ [0, T]. The first inequality in (4.20) then follows from the Gronwall inequality. To see the second inequality of (4.20) we note that, for 0 ≤ s < t ≤ T, E[h 2 (Xt , Xs)] = E h h 2 Z t 0 F(τ, Xτ )dτ ⊕ Z t 0 (G ◦ X)dBτ , Z s 0 F(τ, Xτ )dτ ⊕ Z s 0 (G ◦ X)dBτ i (4.24) ≤ 2E h h 2 Z t 0 F(τ, Xτ )dτ, Z s 0 F(τ, Xτ )dτi + 2E h h 2 Z t 0 (G ◦ X)dBτ , Z s 0 (G ◦ X)dBτ i. Now applying Proposition 3.15 and noting the first inequality of (4.20) we have E h h 2 Z t 0 F(τ, Xτ )dτ, Z s 0 F(τ, Xτ )dτi ≤ E h h 2 Z s 0 F(τ, Xτ )dτ + Z t s F(τ, Xτ )dτ, Z s 0 F(τ, Xτ )dτ + C i (4.25) ≤ E h h 2 Z t s F(τ, Xτ )dτ, Ci ≤ TE h Z t s h 2 (F(τ, Xτ ), C)dτi ≤ (t − s)KT 1 + E[h 2 (ξ, C)] . 4 Similarly, we have E h h 2 Z t 0 (G ◦ X)dBτ , Z s 0 (G ◦ X)dBτ i ≤ X∞ n=1 E h Z t s |g n (τ, Xτ )| 2 dτi ≤ KTE h Z t s 1 + h 2 (Xτ , C))dτi ≤ (t − s)KT 1 + E h sup 0≤τ≤T h 2 (Xτ , C) i(4.26) ≤ (t − s)KT 1 + E[h 2 (ξ, C)] . Combining (4.24)-(4.26) we derive the second inequality of (4.20). The proof is now complete. 4.2 Connections to SDIs Let us begin with the description of an SDI associated with the SVSDE (4.1). Let F : [0, T] × Ω × R d → LC and {g n : n ≥ 1} be a family of functions g n : [0, T] × Ω × R d → R d×m satisfying the following Standing Assumptions: Assumption 4.7. The functions F and {g n} are non-anticipative Carath´eodory random fields such that: (i) there exists β > 0, such that for fixed t ∈ [0, T], x, y ∈ R d , and A ∈ LC, it holds that    supx∈A h 2 (F(t, ω, x), C) ≤ β(1 + h 2 (A, C)), h 2 (F(t, ω, x), F(t, ω, y)) ≤ β|x − y|, P-a.e. ω ∈ Ω. (4.27) (ii) there exist constants αn > 0 such that P∞ n=1 α 2 n < ∞, and for fixed t ∈ [0, T] and x, y ∈ R d , it holds that |g n (t, ω, x)| ≤ αn, |g n (t, ω, x) − g n (t, ·, y)| ≤ αn|x − y|, P-a.e. ω ∈ Ω. (4.28) Assuming that F and {g n} satisfy Assumption 4.7, and for a given process x ∈ L 2 F ([0, T] × Ω; R d×m), we define (G ◦ x)t := co{(g n ◦ x)t : n ≥ 1}. Then, by Theorem 2.19, the Aumann-Itˆo indefinite integral { R t 0 (G ◦ x)dB} is well-defined and has (Hausdorff) continuous paths. We can now consider the the SDI in the following sense. Definition 4.8. A process x = (xt)0≤t≤T ∈ L 2 F ([0, T] × Ω; R d ) is said to be a solution to a Stochastic Differential Inclusion if (i) x has continuous paths; and (ii) the following relations holds: xt − x0 ∈ Z t 0 F(s, xs)ds ⊕ Z t 0 (G ◦ x)dBs, 0 < t ≤ T, P-a.s. (4.29) The SDI (4.29) is closely related to the SDE studied in the previous section. For example, let F and {g n} satisfy Assupmtion 4.7, and let us define F(t, ω, A) := co[∪x∈AF(t, ω, x)], g n (t, ω, A) := sup x∈A g n (t, ω, x), (t, ω) ∈ [0, T]×Ω, A ∈ LC. Here supx∈A g n (·, ·, x) := supx∈A g n ij (·, ·, x) d,m i,j=1. We claim that the mappings F and G := {g n} are Carath´eodory set-valued random fields satisfying Assumptions 4.2 and 4.4, respectively. Indeed, note that by Assumptions 4.7-(2) we have h 2 (F(t, ω, A), C) ≤ h 2 (∪x∈AF(t, ω, x), C) ≤ sup x∈A h 2 (F(t, ω, x), C) ≤ β(1 + h 2 (A, C)) < ∞. (4.30) Furthermore, since F(t, ω, x) ∈ LC, we have F(t, ω, x) = F(t, ω, x) ⊕ C. Thus, F(t, ω, A) = co[∪x∈A(F(t, ω, x) ⊕ C)] = co(∪x∈AF(t, ω, x)) ⊕ C = F(t, ω, A) ⊕ C. (4.31) 4 Here in the above we used the fact that co(C) = C. Clearly, (4.30) and (4.31) imply that F is a non-anticipative Carath´eodory LC-valued random field defined on [0, T] × Ω × LC, and that Assumption 4.2-(1) holds. Moreover, Assumption 4.2-(2) follows from (4.27) and [50, Lemma (4.1,1)]. To see Assumption 4.4, we note that (4.28) implies that |g n (t, ·, A)| ≤ supx∈A |g n (t, ·, x)| ≤ αn, P-a.s., whence Assumption 4.4-(1). To verify Assumption 4.4-(2), we first note that all Euclidean norms on R d×m are equivalent, so we shall use the “maximum” norm on R d×m, that is, |M| := maxi,j |Mi,j |, M ∈ R d×m. Now let A, A˜ ∈ LC, then, for any x ∈ A, y ∈ A˜, by (4.28) we have g n ij (t, ω, x) ≤ g n ij (t, ω, y) + αn|x − y| ≤ g¯ n ij (t, ω, A˜) + αn|x − y|, 1 ≤ i ≤ d, 1 ≤ j ≤ m. Since x ∈ A, y ∈ A˜, and i, j are arbitrary, it is readily seen that the inequality above yields g¯ n ij (t, ω, A) = sup x∈A g n ij (t, ω, x) ≤ g¯ n ij (t, ω, A˜) + αnh¯(A, A˜). (4.32) Switching the position of A and A˜ in (4.32) and recalling the definition of the maximum norm we see that {g¯ n} satisfies Assumption 4.4-(2). We can then consider the following set-valued SDE: Xt = ξ ⊕ Z t 0 F(s, Xs)ds ⊕ Z t 0 (G ◦ X)dBs, t ∈ [0, T], (4.33) which is well-posed, thanks to Theorem 4.5. 46 5 Chapter Five: Continuous-time Super-hedging Problem with Transaction Costs 5.1 The Model Consider a financial market consisting of one riskless asset, called bank account (or bond) with price S 1 := B given by dBt = Btrtdt, B0 = 1, (5.1) and d−1 risky assets, called stocks, with price per share S i governed by the stochastic equation dSi t = S i t [b i tdt + σ i tdWt ], Si (0) = s i ∈ (0,∞), i = 2, 3, ..., d, (5.2) for t ∈ [0, T], where r is bounded F-progressively measurable process and for i ∈ {1, 2, ..., d}, the coefficients b i and σ i are bounded F-progressively measurable processes and σ i > 0 . Suppose that the market allows transferring funds form one asset to another. For i, j ∈ {1, 2, ..., d}, let L ij be an F-adapted process with left continuous nondecreasing paths such that L ij t represents the total amount of funds transferred from the i th to the j th asset by time t ∈ [0, T]. Thus, it is natural to assume that L ij 0 = 0 and L ii t = 0 for i, j ∈ {1, 2, ..., d} and t ∈ [0, T]. Suppose further that for a transfer from the i th to the j th asset, a fee has to be deducted from the bank account according to a given proportional transaction cost µ ij ∈ [0, 1] for i, j ∈ {1, 2, ..., d} with µ ii = 0. Given initial holdings v := (v 1 , ..., vd ), for 0 ≤ t ≤ T, the portfolio holdings V i t evolve 47 according to the equations: V 1 t = v 1 − X d j=1 (1 + µ 1j )L 1j t + X d j=1 (1 − µ j1 )L j1 t − X d i,j=2 µ ijL ij t + Z t 0 V 1 s rsds (5.3) V i t = v i + X d j=1 (L ji t − L ij t ) + Z t 0 V i s [b i sds + σ i sdWs], 2 ≤ i ≤ d. (5.4) Before introducing the Solvency Cone of this model, we first discuss its construction in the the special two-dimensional case. Example 1. Consider the two-dimensional case of this model originally proposed in [13]. Namely, a financial market consisting of one riskless asset, called bank account (or bond) with price B and of one risky assets, called stocks, with price per share S whose dynamics are given by    dSt = St [btdt + σtdWt ], S(0) = p ∈ (0,∞); dBt = rtBtdt, B0 = 1. (5.5) with L := L 12, M := L 21, λ := µ 12, µ := µ 21, X := V 1 , Y := V 2 , x := v 1 and y = v 2 , where dynamics of (X, Y ) for 0 ≤ t ≤ T is given by    dXt = rtXtdt − (1 + λ)dLt + (1 − µ)dMt , X0 = x; dYt = Yt [btdt + σtdWt ] + dLt − dMt , Y0 = y. (5.6) Let us now describe the Solvency Cone of this special case as proposed in [37, 39] and [65]. Let us denote π 12 t to be the number of units of B that can be exchanged for one share of S at time t (including its transaction cost). That is, we have, P-a.s., π 12 t Bt = (1 + λ)St or π 12 t = (1 + λ) St Bt , t ∈ [0, T]. (5.7) 48 Similarly, we let π 21 t denote the number of shares of S that can be exchanged for one unit of B at time t (including its transaction cost), so that the following identities hold: π 21 t St = 1 1 − µ Bt or π 21 t = 1 1 − µ Bt St , t ∈ [0, T], P-a.s. (5.8) Now let us define π 11 t = π 22 t :≡ 1 and consider the matrix-valued process Πt := (π ij t ) 2 i,j=1, t ∈ [0, T]. Clearly, the components π ij t satisfy the following properties: π ij t > 0, i, j = 1, 2; π 11 t = π 22 t = 1; π 12 t π 21 t > 1, t ∈ [0, T], P-a.s. (5.9) We shall follow [37, 65] and call any matrix Π satisfying (5.9) the bid-ask matrix, and the process Π = (Πt)0≤t≤T taking values in bid-ask matrices the bid-ask process. Next we defined the so-called Solvency Cone associated with the bid-ask process Π = (Πt)0≤t≤T . Recall that for a given set of vectors {ξ 1 , · · · , ξn} ⊂ R d , a cone generated by {ξ i} n i=1 is define by K := cone {ξ 1 , · · · , ξn} = { Pn i=1 αiξ i : αi ≥ 0}. Now for a given bid-ask process Πt = {π ij t } 2 i,j=1, t ∈ [0, T], we define Kˆ t := K(Πt) := cone{e 1 , e2 , π12 t e 1−e 2 , π21 t e 2−e 1 } = cone{(1, 0),(0, 1),(π 12 t , −1),(−1, π21 t )}. Note that by (5.7) and (5.8), π 12 t = (1 + λ) St Bt > 0, and π 21 = 1 1−µ Bt St > 0, we see that (0, 1),(1, 0) ∈ cone{(π 12 t , −1),(−1, π21 t )}, therefore, using the conic properties we can easily deduce that Kˆ t = = conen(1 + λ) St Bt ,−1 , −1, 1 1 − µ Bt St o=conen(1 + λ) 1 Bt , − 1 St , −(1 − µ) 1 Bt , 1 St o = n(1 + λ) α Bt − (1 − µ) β Bt , β − α St : α, β ≥ 0 o . (5.10) 49 Define K := {x ∈ R 2 : ˆx := ( x 1 B , x 2 S ) ∈ Kˆ }. Then K = {x ∈ R 2 : x 1 = (1 + λ)α − (1 − µ)β, x2 = β − α, α, β ≥ 0}. We shall now consider the “constant” solvency cone of the d-dimensional model; initially presented in Kabanove and Stricker [39]; which we now describe. Let M be be the convex cone that consists of all x ∈ R d for which there exists a matrix (a ij )1≤i,j≤d with nonnegative entries such that x 1 − X d j=1 a 1j (1 + µ 1j ) +X d j=1 a j1 (1 − µ 1j ) − X d i,j=2 a ijµ ij = 0, (5.11) x i + X d j=1 (a ji − a ij ), 2 ≤ i ≤ d. (5.12) The solvency cone is then defined as K := M + R d +. Next, we give the following proposition which will be useful for our discussion. Proposition 5.1. K = M. Proof. Let x ∈ K. Then x 1 − X d j=1 a 1j (1 + µ 1j ) +X d j=1 a j1 (1 − µ j1 ) − X d i,j=2 a ijµ ij = α 1 and x i + X d j=1 (a ji − a ij ) = α i , i = 2, 3, ..., d, for some (a ij )1≤i,j≤d with nonnegative entries and α i ≥ 0 for 1 ≤ i ≤ d. If α i = 0 for all 1 ≤ i ≤ d, then we are done. Thus, we shall assume that α i > 0 for some i. Note 50 that if we can show that there exists matrix (b ij )1≤i,j≤d with nonnegative entries such that α 1 = X d j=1 b 1j (1 + µ 1j ) − X d j=1 b j1 (1 − µ j1 ) + X d i,j=2 b ijµ ij and α i = − X d j=1 (b ji − b ij ), i = 2, 3, ..., d Then we will have x 1 − X d j=1 (1 + µ 1j )(a 1j + b 1j ) +X d j=1 (1 − µ j1 )(a j1 + b j1 ) − X d i,j=2 µ ij (a ij + b ij ) = 0 and x i + X d j=1 (a ji + b ji) − (a ij + b ij ) = 0, i = 2, 3, ..., d where the matrix (a ij + b ij )1≤i,j≤d is of nonnegative entries, and hence x ∈ M. Let X := b 12, b13, ..., b1d , b21, b23, ..., b2d , ..., bd1 , bd2 , ..., bd(d−1)T ∈ R d(d−1) and let A be the matrix whose rows a i are such that a 1X = X d j=1 (1 + µ 1j )b 1j − X d j=1 (1 − µ j1 )b j1 + X d i,j=2 µ ij b ij and a iX = − X d j=1 (b ji − b ij ); i = 2, 3, ..., d. Then A has the form A :=   1 + µ 12 1 + µ 13 · · · 1 + µ 1d −(1 − µ 21) · · · −(1 − µ 31) · · · − (1 − µ d1 ) · · · −1 0 · · · 0 1 0 −1 · · · 0 0 · · · 1 · · · . . . . . . . . . . . . 0 0 · · · −1 0 · · · 0 1   . Let α := (α 1 , α2 , ..., αd ) T and let Y ∈ R d . If AT Y ≥ 0, then (1 + µ 12)y 1 − y 2 ≥ 0 and −(1 − µ 21)y 1 + y 2 ≥ 0. Thus (µ 12 + µ 21)y 1 ≥ 0 and therefore y 1 ≥ 0. Since −(1 − µ j1 )y 1 + y j ≥ 0, then y j ≥ (1 − µ j1 )y 1 ≥ 0 for all j = 2, .., d. Thus α T Y ≥ 0. Thus, by Farkas’ lemma, X ≥ 0. That is, (b ij )1≤i,j≤d is of nonnegative entries, which completes the proof. Moreover, we have the following result. Proposition 5.2. Assume that the portfolio process V := (V 1 , ..., V d ) satisfies (5.3) and (5.4), and the price process S := (S 1 , ..., Sd ) satisfies (5.1) and (5.2). Assume further that for i, j ∈ {1, 2, ..., d}, there exists a process θ ij ∈ L 2 F ([0, T] × Ω; R) such that θ ij t ≥ 0 P-a.s., 0 ≤ t ≤ T and that L ij t = R t 0 θ ij s ds, 0 ≤ t ≤ T. Then V satisfies the following relation P-almost surely: Vt ∈ v + R t 0 hV 1 s rs, V 2 s b 2 s , ..., V d s b d s − K i ds + R t 0 0, V 2 s σ 2 s , ..., V d s σ d s dWs, t ∈ [0, T]. Proof. Since for i, j ∈ {1, 2, ..., d}, θ ij ∈ L 2 F ([0, T] × Ω; R) with θ ij t ≥ 0 P-a.s. 0 ≤ t ≤ T and by (5.11) and (5.12), we have k := − X d j=1 (1+µ 1j )θ 1j+ X d j=1 (1−µ j1 )θ j1− X d i,j=2 µ ijθ ij , X d j=1 (θ j2−θ 2j ), ...,X d j=1 (θ jd−θ dj ) ∈ S 2 F (−K). 52 Thus R t 0 ksds ∈ S 2 Ft ( R t 0 −Kds) = S 2 Ft (K). Since L ij t = R t 0 θ ij s ds, 0 ≤ t ≤ T, we have Z t 0 ksds = − X d j=1 (1+µ 1j )L 1j+ X d j=1 (1−µ j1 )L j1− X d i,j=2 µ ijL ij , X d j=1 (L j2−L 2j ), ...,X d j=1 (L jd−L dj ) . Thus, V 1 t , V 2 t , ..., V d t = v + Z t 0 ksds + Z t 0 V 1 s rsds, Z t 0 V 2 s [b 2 sds + σ 2 sdWs], ..., Z t 0 V d s [b d sds + σ d sdWs] ∈ v + Z t 0 −Kds + Z t 0 V 1 s rsds, Z t 0 V 2 s [b 2 sds + σ 2 sdWs], ..., Z t 0 V d s [b d sds + σ d sdWs] = v + Z t 0 hV 1 s rs, V 2 s b 2 s , ..., V d s b d s − K i ds + Z t 0 0, V 2 s σ 2 s , ..., V d s σ d s dWs. We shall also consider the random solvency cone Kˆ := {xˆ : x ∈ K}, where xˆ := x 1 B , x 2 S2 , ..., x d Sd for x ∈ R d . For portfolio process V := (V 1 , ..., V d ) and the initial holdings v := (v 1 , v2 , ..., vd ), let Vˆ := (V 1 S1 , ..., V d Sd ) and ˆv := v 1 , v 2 s 2 , ..., v d s d . Proposition 5.3. Assume that the portfolio process V := (V 1 , ..., V d ) satisfies (5.3) and (5.4), and the price process S := (S 1 , ..., Sd ) satisfies (5.1) and (5.2). Assume further that for i, j ∈ {1, 2, ..., d}, there exists a process θ ij ∈ L 2 F ([0, T] × Ω; R) such that θ ij t ≥ 0 P-a.s., 0 ≤ t ≤ T and that L ij t = R t 0 θ ij s ds, 0 ≤ t ≤ T. Then the corresponding unit portfolio process Vˆ := (V 1 S1 , ..., V d Sd ) satisfies the following relation P-almost surely: Vˆ t ∈ vˆ + Z t 0 −Kˆ sds, t ∈ [0, T]. (5.13) 53 Proof. By Itˆo’s formula, d V 1 t Bt = 1 Bt dV 1 t + V 1 t d 1 Bt = 1 Bt h − X d j=1 (1 + µ 1j )dL1j t + X d j=1 (1 − µ j1 )dLj1 t − X d i,j=2 µ ijdLij t + V 1 t rtdti − V 1 t rt Bt dt = 1 Bt h − X d j=1 (1 + µ 1j )dL1j t + X d j=1 (1 − µ j1 )dLj1 t − X d i,j=2 µ ijdLij t i = − X d j=1 (1 + µ 1j ) dL1j t Bt + X d j=1 (1 − µ j1 ) L j1 t Bt − X d i,j=2 µ ij dLij t Bt , and d V i t S i t = 1 S i t dV i t + V i t d 1 S i t − σ i t 2 V i t S i t dt = 1 S i t hX d j=1 (dLji t − dLij t ) + V i t [b i tdt + σ i tdWt ] i + V i t S i t [(σ i t 2 − b i t )dt − σ i tdWt ] − σ i t 2 V i t S i t dt = 1 S i t X d j=1 (dLji t − dLij t ) = X d j=1 dLji t S i t − dLij t S i t , for i = 2, 3, ..., d. By assumptions on θ ij for i, j ∈ {1, 2, ..., d}, we obtain V 1 t Bt = v 1 + Z t 0 h − X d j=1 (1 + µ 1j ) θ 1j s Bs + X d j=1 (1 − µ j1 ) θ j1 s Bs − X d i,j=2 µ ij θ ij s Bs i ds, and V i t S i t = v 1 s 1 + Z t 0 X d j=1 θ ji s S i t − θ ij s Si s ds, 54 for i = 2, 3, ..., d. Since θ ij ∈ L 2 F ([0, T] × Ω; R), i, j ∈ {1, 2, ..., d}, then − X d j=1 (1 + µ 1j ) θ 1j B + X d j=1 (1 − µ j1 ) θ j1 B − X d i,j=2 µ ij θ ij B ∈ L 2 F ([0, T] × Ω; R) and X d j=1 θ ji Si − θ ij Si ∈ L 2 F ([0, T] × Ω; R), i = 2, 3, ..., d. Moreover, since θ ij t ≥ 0 P-a.s., 0 ≤ t ≤ T for i, j ∈ {1, 2, ..., d}, and by (5.11) and (5.12), we have ˆk := − X d j=1 (1+µ 1j ) θ 1j B + X d j=1 (1−µ j1 ) θ j1 B − X d i,j=2 µ ij θ ij B , X d j=1 θ j2 S2 − θ 2j S2 , ...,X d j=1 θ jd Sd − θ dj Sd ∈ S 2 F (−Kˆ ). Thus R t 0 ˆksds ∈ S 2 Ft ( R t 0 −Kˆ sds), and Vˆ t = ˆv + Z t 0 ˆksds ∈ vˆ + Z t 0 −Kˆ sds. 5.2 Consistent Price Processes By Proposition 5.1 and the definition of Kˆ , it follows that Kˆ consists of all x ∈ R d for which there exists (a ij )1≤i,j≤d with nonnegative entries such that x 1 = 1 B hX d j=1 (1 + µ 1j )a 1j − X d j=1 (1 − µ j1 )a j1 + X d i,j=2 µ ija iji , (5.14) x i = 1 Si X d j=1 (a ij − a ji), 2 ≤ i ≤ d. (5.15) 55 Consider the dual cone Kˆ ∗ of Kˆ given by Kˆ ∗ := {x ∈ R d : x ⊺ y ≥ 0, y ∈ Kˆ }. (5.16) Next we introduce the notion of consistent price processes in an analogous way to its discrete-time counterpart given by Schachermayer [65]. Definition 5.4. An adapted R d +-valued process Z = (Zt)0≤t≤T is called consistent price process for the solvency cone Kˆ if Z is a martingale under P, and Zt ∈ Kˆ ∗ t \ {0} P-a.s. for each t ∈ [0, T]. The following proposition shows that these consistent price processes are in line with the Auxiliary martingales introduced in the two dimensional continuous-times case by Cvitani´c and Karatzas [13]. Proposition 5.5. x ∈ Kˆ ∗ if and only if (1 − µ i1 ) x 1 B ≤ x i Si ≤ (1 + µ 1i ) x 1 B , (5.17) x j Sj − x i Si ≤ µ ij x 1 B , (5.18) for 2 ≤ i, j ≤ d. Proof. Since µ ii = 0, then we may assume that a ii = 0 in (5.14) and (5.15). Thus, there are d(d−1) generating vectors of the convex cone Kˆ each corresponding to some a ij with i ̸= j. Note that a 1i appears only in the first and the i-th components of x ∈ Kˆ . Thus, the generating vector corresponding to a 1i is 1+µ 1i B e 1 − 1 Si e i . Similarly, since a i1 appears only in the first and the i-th components of x, then the generating vector corresponding to a i1 is −(1−µ i1 ) B e 1 + 1 Si e i . For i, j ≥ 2, we have a ij in the first, 56 the i-th and the j-th components of x. The generating vector corresponding to a ij is then given by µ ij B e 1 + 1 Si e i − 1 Sj e j . Now let x ∈ R d . Then x ⊺ y ≥ 0 for all y ∈ Kˆ if and only if x ⊺ y ≥ 0 for every generating vector of Kˆ . Thus for 2 ≤ i, j ≤ d, we have 1+µ 1i B x 1 − 1 Si x i ≥ 0, −(1−µ i1 ) B x 1 + 1 Si x i ≥ 0 and µ ij B x 1 + 1 Si x i − 1 Sj x j ≥ 0. Hence, the result. For a strictly positive consistent price process Z = (Z 1 , ..., Zd ), let Rj := Zˆj Zˆ1 and Rij := Rj − Ri , for 1 ≤ i, j ≤ d. Theorem 5.6. Let Z = (Z 1 , ..., Zd ) be a strictly positive consistent price process and let V = (V 1 , ..., V 2 ) be given by (5.3) and (5.4). Then V 1+ Pd j=2 RjV j B is a P 1 - supermartingale, where P 1 (A) := E[Z 1 T IA] for A ∈ F provided that V 1 t + Pd j=2 R j tV j t ≥ 0, for t ∈ [0, T]. Proof. Let Z = (Z 1 , ..., Zd ) be a strictly positive consistent price process. Then for each 1 ≤ i ≤ d, there exists F-progressively measurable processes θ i with R T 0 θ i tdt < ∞ P-a.s. and Z i t = Z i o exp n Z t 0 θ i sdWs − 1 2 Z t 0 (θ i s ) 2 dso . (5.19) By Itˆo’s formula, we obtain Z 1 t Vˆ 1 t = Z 1 0 vˆ 1 + Z t 0 h Z 1 s dVˆ 1 s + Vˆ 1 s dZ1 s i =Z 1 0 vˆ 1 + Z t 0 Z 1 s hX d i=1 (1 − µ i1 ) dLi1 s Bs − X d i=1 (1 + µ 1i ) dL1i s Bs − X i,j≥2 µ ij dLij s Bs + Vˆ 1 s θ 1 sdWs i , Z j t Vˆ j t = Z j 0 vˆ j + Z t 0 h Z j sdVˆ j s + Vˆ j s dZj s i =Z j 0 vˆ j + Z t 0 Z j s hX d i=1 1 S j s (dLij s − dLji s ) + Vˆ j s θ j sdWs i , 2 ≤ j ≤ d. 57 Let Rj := Zˆj Zˆ1 and Rij := Rj − Ri , for 1 ≤ i, j ≤ d. Then for 2 ≤ j ≤ d, we have Z j t Vˆ j t = Z j 0 vˆ j + Z t 0 h Zˆ1 sR j s X d i=1 (dLij s − dLji s ) + Zˆ1 sR j sV j s θ j sdWs i . Then X d j=1 Z j t Vˆ j t = X d j=1 Z j 0 vˆ j + Z t 0 h Zˆ1 s X d i=1 (1 − µ i1 )dLi1 s − X d i=1 (1 + µ 1i )dL1i s − X i,j≥2 µ ijdLij s + Vˆ 1 s Z 1 s θ 1 sdWs i + X d j=2 Z t 0 h Zˆ1 sR j s (dL1j s − dLj1 s ) + Zˆ1 sR j s X d i=2 (dLij s − dLji s ) + Vˆ j s Z j s θ j sdWs i = X d j=1 Z j 0 vˆ j + X d j=2 Z t 0 Zˆ1 s [(1 − µ j1 ) − R j s ]dLj1 s + X d j=2 Z t 0 Zˆ1 s [R j s − (1 + µ 1j )]dL1j s + X i,j≥2 Z t 0 Zˆ1 s [R ij s − µ ij ]dLij s + X d j=1 Z t 0 Zˆ1 sR j sV j s θ j sdWs. By Itˆo’s formula, we have 1 Z 1 t X d j=1 Z j t Vˆ j t = 1 Z 1 0 X d j=1 Z j 0 vˆ j + Z t 0 1 Z1 s d X d j=1 Z j t Vˆ j t + Z t 0 X d j=1 Z j sVˆ j s d 1 Z1 s + Z t 0 d X d j=1 Z j t Vˆ j s d 1 Z1 s . By the previous step, we obtain Z t 0 1 Z1 s d X d j=1 Z j t Vˆ j t = X d j=2 Z t 0 (1 − µ j1 ) − Rj s Bs dLj1 s + X d j=2 Z t 0 Rj s − (1 + µ 1j ) Bs dL1j s + X i,j≥2 Z t 0 Rij s − µ ij Bs dLij s + X d j=1 Z t 0 Rj sV j s θ j s Bs dWs. By Itˆo’s formula, it follows that d 1 Z1 t = θ 1 t Z1 t [θ 1 t dt − dWt ]. Thus, Z t 0 X d j=1 Z j sVˆ j s d 1 Z1 s = Z t 0 X d j=1 Z j sVˆ j s θ 1 s Z1 s θ 1 sds − dWs = Z t 0 X d j=1 Rj sV j s θ 1 s Bs θ 1 sds − dWs . 58 Moreover, Z t 0 d X d j=1 Z j t Vˆ j t d 1 Z1 s = − X d j=1 Z t 0 Zˆ1 sR j sV j s θ j s θ 1 s Z1 s ds = − X d j=1 Z t 0 Rj sV j s θ j s θ 1 s Bs ds. Note that X d j=1 Z t 0 Rj sV j s θ j s Bs dWs + Z t 0 X d j=1 Rj sV j s θ 1 s Bs θ 1 sds − dWs − X d j=1 Z t 0 Rj sV j s θ j s θ 1 s Bs ds = X d j=1 Z t 0 Rj sV j s Bs (θ j s − θ 1 s )(dWs − θ 1 sds). Thus 1 Z 1 t X d j=1 Z j t Vˆ j t = 1 Z 1 0 X d j=1 Z j 0 vˆ j + X d j=2 Z t 0 (1 − µ j1 ) − Rj s Bs dLj1 s + X d j=2 Z t 0 Rj s − (1 + µ 1j ) Bs dL1j s + X i,j≥2 Z t 0 Rij s − µ ij Bs dLij s + X d j=1 Z t 0 Rj sV j s Bs (θ j s − θ 1 s )(dWs − θ 1 sds), and therefore 1 Z 1 t X d j=1 Z j t Vˆ j t + X d j=2 Z t 0 Rj s − (1 − µ j1 ) Bs dLj1 s + X d j=2 Z t 0 (1 + µ 1j ) − Rj s Bs dL1j s + X i,j≥2 Z t 0 µ ij − Rij s Bs dLij s = 1 Z 1 0 X d j=1 Z j 0 vˆ j + X d j=1 Z t 0 Rj sV j s Bs (θ j s − θ 1 s )(dWs − θ 1 sds) = 1 Z 1 0 X d j=1 Z j 0 vˆ j + X d j=1 Z t 0 Rj sV j s Bs (θ j s − θ 1 s )dW1 s is a P 1 -local martingale where W1 t := Wt − R t 0 θ 1 sds is a Brownian motion under the probability measure P 1 (A) := E[Z 1 T IA] for A ∈ F. Since Z takes values in Kˆ ∗ , then 1 − µ i1 ≤ Ri ≤ 1 + µ 1i and Rij ≤ µ ij for 1 ≤ i, j ≤ d. Thus the second, third, and forth terms in the L.H.S. of the previous equation are nonnegative and are increasing. 59 Assuming V 1 t + Pd j=2 R j tV j t ≥ 0 for t ∈ [0, T], we have 1 Z 1 t X d j=1 Z j t Vˆ j t = Vˆ 1 t + X d j=2 Z j t Z 1 t Vˆ j t = V 1 t + Pd j=2 R j tV j t Bt is a nonnegative P 1 -local martingale, and therefore a P 1 -supermartingale. 5.3 Super-hedging Problem Before formulating our super-hedging problem, we shall first shed some lights on its discrete-time counterpart discussed in [20]. Given a discrete-time market model given by an (Ft) T t=0 adapted process (Kˆ t) T t=0 of solvency cones. An R d -valued adapted process (Vt) T t=0 is called a self-financing portfolio process for the market given by (Kˆ t) T t=0 if for all t ∈ {0, ..., T}, it holds that Vt−Vt−1 ∈ −Kˆ t P-a.s. with the convention V−1 = 0. Let CT be the set of random variables VT ∈ L p d (FT ) each being the value of a self-financing portfolio at time T. By definition of self-financing, it follows that CT = − PT s=0 L p d (Fs; Kˆ s). By a contingent claim we mean any X ∈ L p d (FT ). An initial portfolio u ∈ L p d (F0) is said to super-hedge the contingent claim X ∈ L p d (FT ) at time t = 0 if there exists VT ∈ CT such that u + VT = X P-a.s., i.e., u − X ∈ PT s=0 L p d (Fs; Kˆ s). The super-hedging set of the contingent claim X ∈ L p d (FT ) at time t = 0 is given by SHP0(X) := {u ∈ L p d (F0) : u − X ∈ PT s=0 L p d (Fs; Kˆ s)}. The dynamic super-hedging set at any time t is then given by SHPt(X) := n u ∈ L p d (Ft) : u − X ∈ X T s=t L p d (Fs; Kˆ s) o . Now we shall describe our super-hedging problem for the financial model described in 5.1 with its continuous-time solvency cones (Kˆ t)t∈[0,T] . An R d -valued adapted process V = (Vt)t∈[0,T] is called a self-financing portfolio process for the market given 60 by (Kˆ t)t∈[0,T] if for all t ∈ [0, T], it holds that V ∈ − R t 0 Kˆ sds. An initial portfolio u ∈ L p d (F0) is said to super-hedge the contingent claim X ∈ L p d (FT ) at time t = 0 if there exists VT ∈ −S p FT ( R T 0 Kˆ sds) such that u+VT = X P-a.s., i.e., u−X ∈ S p FT ( R T 0 Kˆ sds). The super-hedging set of the contingent claim X ∈ L p d (FT ) at time t = 0 is given by Γ0(X) := {u ∈ L p d (F0) : u − X ∈ S p FT ( R T 0 Kˆ sds)}. The dynamic super-hedging set at any time t ∈ [0, T) is then given by Γt(X) := n u ∈ L p d (Ft) : −X + u ∈ S p FT Z T t Kˆ sdso. (5.20) Remark 5.7. 1. Since 0 ∈ Kˆ , then 0 ∈ S p FT ( R T t Kˆ sds) for t < T; 2. Since Kˆ is a convex cone, then S p FT ( R T t Kˆ sds) is a convex cone; 3. For Y, X ∈ L p d (FT ) and t ∈ [0, T), define the relation X ⪰t Y if and only if X − Y ∈ S p FT ( R T t Kˆ sds). Since 0 ∈ S p FT ( R T t Kˆ sds), then X ⪰ X. Moreover, if X ⪰t Y and Y ⪰t Z for some Z ∈ L p d (FT ), then X−Z = X−Y +Y −Z ∈ S p FT Z T t Kˆ sds +S p FT Z T t Kˆ sds ⊆ S p FT Z T t Kˆ sds , where the last inclusion follows since S p FT ( R T t Kˆ sds) is a convex cone. Therefore, X ⪰t Z and ⪰t defines a preorder relation. Thus, a portfolio u ∈ L p d (Ft) super-hedges the contingent claim X ∈ L p d (FT ) at time t ∈ [0, T) simply means u ⪰t X. We shall denote At := S p FT ( R T t Kˆ sds) and At,r := At ∩ L p d (Fr), for 0 ≤ t < r ≤ T. 61 Proposition 5.8. For any 0 ≤ t < r ≤ T. 1. Γt(X) is closed Ft-decomposable; 2. Γt(X) = L p d (Ft) ∩ (Γt(X) + At); 3. At,r = L p d (Fr) ∩ S p FT R r t Kˆ sds ⊕ Ar . Proof. 1. Let {u i} m i=1 ⊆ Γt(X) and let {Ai} m i=1 be an Ft-measurable partition of Ω. Then u := Pm i=1 IAiu i ∈ L p d (Ft) and −X + u ∈ S p FT R T t Kˆ sds , and hence Γt(X) is decomposable. Moreover, for any sequence {u i} ∞ i=1 ⊆ Γt(X) converging to some u ∈ L p d (Ft), we must have u ∈ Γt(X) since X +S p FT R T t Kˆ sds is a closed subset of L p d (FT ) and each u i is Ft-measurable. Thus, Γt(X) is closed. 2. Since 0 ∈ At , then clearly Γt(X) ⊆ L p d (Ft)∩(Γt(X) + At). To see that L p d (Ft)∩ (Γt(X) + At) ⊆ Γt(X), let u ∈ L p d (Ft) ∩ (Γt(X) + At). Then u ∈ L p d (Ft) and u = u 1+u 2 , where u 1 ∈ Γt(X) and u 2 ∈ At . Therefore, −X+u 1 ∈ S p FT ( R T t Kˆ sds) and u 2 ∈ S p FT ( R T t Kˆ sds). Thus, −X + u = −X + u 1 + u 2 ∈ S p FT ( R T t Kˆ sds) + S p FT ( R T t Kˆ sds) ⊆ S p FT ( R T t Kˆ sds). That is, u ∈ Γt(X). 3. Note that At := S p FT Z T t Kˆ sds = S p FT Z r t Kˆ sds ⊕S p FT Z T r Kˆ sds = S p FT Z r t Kˆ sds ⊕Ar. Since At,r := L p d (Fr) ∩ At , then At,r = L p d (Fr) ∩ S p FT Z r t Kˆ sds ⊕ Ar . Lemma 5.9. The set-valued functions Rt : L p d (FT ) → P L p d (Ft); (L p d (Ft))+ defined as Rt(X) := Γt(−X) is a normalized set-valued dynamic risk measure. Proof. Clearly, Rt(X) ⊆ Rt(X) + (L p d (Ft))+. To see that Rt(X) + (L p d (Ft))+ ⊆ Rt(X), let u ∈ Rt(X) and v ∈ (L p d (Ft))+. Then X +u ∈ S p FT ( R T t Kˆ sds), and therefore X + u + v ∈ S p FT Z T t Kˆ sds + v. Thus X + u + v ∈ S p FT Z T t Kˆ sds + S p FT R d + ⊆ S p FT Z T t Kˆ sds , and therefore u + v ∈ Rt(X). Next, we check the properties of risk measures: 1. L p d (Ft)-translative: Let mt ∈ L p d (Ft). Then Rt(X + mt) = n u ∈ L p d (Ft) : X + mt + u ∈ S p FT Z T t Kˆ sdso = n u − mt : u ∈ L p d (Ft) : X + u ∈ S p FT Z T t Kˆ sdso = n u ∈ L p d (Ft) : X + u ∈ S p FT Z T t Kˆ sdso − mt = Rt(X) − mt . 2. L p d (FT )+-monotone: Let X, Y ∈ L p d (FT ) and Z := Y − X ∈ (L p d (FT ))+. Then Rt(X) = n u ∈ L p d (Ft) : X + u ∈ S p FT Z T t Kˆ sdso = n u ∈ L p d (Ft) : Y − Z + u ∈ S p FT Z T t Kˆ sdso = n u ∈ L p d (Ft) : Y + u ∈ S p FT Z T t Kˆ sds + Z o ⊆ n u ∈ L p d (Ft) : Y + u ∈ S p FT Z T t Kˆ sds + S p FT (R d +) o ⊆ n u ∈ L p d (Ft) : Y + u ∈ S p FT Z T t Kˆ sdso = Rt(Y ). 3. Finite at zero: Follows by the fact the Kˆ a is a cone and Rt(0) = n u ∈ L p d (Ft) : u ∈ S p FT Z T t Kˆ sdso. 4. Normalized: Since 0 ∈ Rt(0), then Rt(X) ⊆ Rt(X) + Rt(0). To see that Rt(X) + Rt(0) ⊆ Rt(X), let u ∈ Rt(X) and v ∈ Rt(0). Then u, v ∈ L p d (Ft) and X +u, v ∈ S p FT R T t Kˆ sds . Thus X +u+v ∈ S p FT ( R T t Kˆ sds) +S p FT ( R T t Kˆ sds) ⊆ S p FT ( R T t Kˆ sds). Thus, u + v ∈ Rt(X). Theorem 5.10. For any r > t, we have 1. Γt(X) ⊇ L p d (Ft) ∩ (Γr(X) + At); 2. L p d (Ft) ∩ (Γr(X) + Ar) = L p d (Ft) ∩ Γr(X); 3. L p d (Ft) ∩ Γr(X) + Ar + S p Ft ( R r t Kˆ sds) = L p d (Ft) ∩ Γr(X) + S p Ft ( R r t Kˆ sds) . 64 Proof. 1. Let r > t and let u ∈ L p d (Ft)∩ Γr(X) + At . Then u ∈ L p d (Ft) and u ∈ Z + At for some Z ∈ Γr(X). Since Z ∈ Γr(X), then −X +Z ∈ Ar. Thus, u ∈ Z +At ∈ X + Ar + At = X + At . Hence, u ∈ Γt(X). 2. Since 0 ∈ Ar, then clearly L p d (Ft)∩Γr(X) ⊆ L p d (Ft)∩(Γr(X)+Ar). To show that L p d (Ft) ∩ (Γr(X) + Ar) ⊆ L p d (Ft) ∩ Γr(X), let u ∈ L p d (Ft) ∩ (Γr(X) + Ar). Then u ∈ L p d (Ft) and u = u 1+u 2 , for some u 1 ∈ Γr(X) and u 2 ∈ Ar := S p FT ( R T r Kˆ sds). Thus, −X + u 1 ∈ S p FT ( R T r Kˆ sds), and therefore −X+u = −X+u 1+u 2 ∈ S p FT Z T r Kˆ sds +S p FT Z T r Kˆ sds ⊆ S p FT Z T r Kˆ sds . Since u ∈ L p d (Ft) ⊆ L p d (Fr), then u ∈ L p d (Ft) ∩ Γr(X). 3. Sinec 0 ∈ Ar, then clearly L p d (Ft)∩ Γr(X) + Ar + S p Ft Z r t Kˆ sds ⊇ L p d (Ft)∩ Γr(X) + S p Ft ( Z r t Kˆ sds) . Conversely, note that u ∈ L p d (Ft) ∩ Γr(X) + Ar + S p Ft ( R r t Kˆ sds) if and only if u ∈ L p d (Ft) and u = u 1 + u 2 , for some u 1 ∈ Γr(X) + Ar and u 2 ∈ S p Ft ( R r t Kˆ sds). Since u, u2 ∈ L p d (Ft), then u 1 ∈ L p d (Ft)∩(Γr(X)+Ar) and by the previous step, we have u 1 ∈ L p d (Ft) ∩ Γr(X). Thus, u = u 1+u 2 ∈ L p d (Ft)∩Γr(X)+S p Ft Z r t Kˆ sds = L p d (Ft)∩ Γr(X)+S p Ft Z r t Kˆ sds. Next, we shall refine the definition of Γt(X) to achieve more desirable properties. 65 Let us first consider the subset Γ1 t (X) ⊆ Γt(X) defined as Γ 1 t (X) := n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), −X + u = Z T t ˆksdso . Here for simplicity, we use the special case of p = 2. We start with the following Proposition. Proposition 5.11. Let Z ∈ (L 2 d (FT ))+. Then there exists Z˜ ∈ S 2 F (R d +) such that Z = R T t Z˜ sds. Proof. Let T : S 2 F (R d +) → (L 2 d (FT ))+ such that T (Z˜) := R T t Z˜ sds for Z˜ ∈ S 2 F (R d +). Then T is a continuous linear mapping and V := T (S 2 F (R d +)) is a closed subspace of (L 2 d (FT ))+. Thus (L 2 d (FT ))+ = V ⊕ V ⊥. Let Y ∈ V ⊥. Then E[Y R T t Z˜ sds] = 0 for every Z˜ ∈ S 2 F (R d +). Thus R T t E[Y Z˜ s]ds = 0 for every Z˜ ∈ S 2 F (R d +). Choose Z˜ t := E[Y |Ft ] ∀t. Then Z˜ ∈ S 2 F (R d +) and E[Y Z˜ t ] = E E[Y Z˜ t |Ft ] = E E[Y |Ft ]Z˜ t = E[Z˜2 t ] ∀t. Thus R T t E[Z˜2 s ]ds = 0 and therefore Z˜2 t dt×dP. Since Z˜2 is continuous, it follows that Z˜ t = 0 P-a.s.∀t. By taking t = T, we obtain Y = E[Y |FT ] = Z˜ T = 0. Thus V ⊥ = {0} and the proof is complete. Lemma 5.12. The set-valued functions Rt : L 2 d (FT ) → P L 2 d (Ft); (L 2 d (Ft))+ defined as Rt(X) := Γ1 t (−X) is a normalized multi-portfolio time-consistent set-valued dynamic risk measure. Proof. Clearly, Rt(X) ⊆ Rt(X) + (L 2 d (Ft))+. To see that Rt(X) + (L 2 d (Ft))+ ⊆ Rt(X), let u ∈ Rt(X) and v ∈ (L 2 d (Ft))+. Then u = −X + R T t ˆksds, ˆk ∈ S 2 F (Kˆ ) and 6 by Proposition 5.11, v = R T t vˆsds, vˆ ∈ S 2 F (R d +). Thus u + v = −X + Z T t ( ˆk + ˆv)sds, where ˆk + ˆv ∈ S 2 F (Kˆ ) + S 2 F (R d +) ⊆ S 2 F (Kˆ ), and therefore u + v ∈ Rt(X). Next, we check the properties of risk measure: 1. L 2 d (Ft)-translative: Let mt ∈ L 2 d (Ft). Then Rt(X + mt) = n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), X + mt + u = Z T t ˆksdso = n u − mt : u ∈ L 2 d (Ft), ∃ˆk ∈ S 2 F (Kˆ ), X + u = Z T t ˆksdso = n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), X + u = Z T t ˆksdso − mt = Rt(X) − mt . 2. L 2 d (FT )+-monotone: Let X, Y ∈ L 2 d (FT ) and Z := Y − X ∈ (L 2 d (FT ))+. Then Rt(X) = n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), X + u = Z T t ˆksdso = n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), Y − Z + u = Z T t ˆksdso = n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), Y + u = Z T t ˆksds + Z o ⊆ n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), Y + u = Z T t ˆksdso = Rt(Y ), where the last inclusion follows by Proposition 5.11. 67 3. Finite at zero: Follows by the fact that Kˆ is a solvency cone and Rt(0) = n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), u = Z T t ˆksdso . 4. Normalized: Since 0 ∈ Rt(0), then Rt(X) ⊆ Rt(X) + Rt(0). To see that Rt(X) +Rt(0) ⊆ Rt(X), let u 1 ∈ Rt(X) and u 2 ∈ Rt(0). Then u 1 +u 2 ∈ L 2 d (Ft) and u 1 = −X + R T t ˆk 1 sds, u2 = R T t ˆk 2 sds for some ˆk 1 , ˆk 2 ∈ S 2 F (Kˆ ). Thus u 1 +u 2 = −X + R T t ( ˆk 1 + ˆk 2 )sds where ˆk 1 + ˆk 2 ∈ S 2 F (Kˆ ), and u 1 + u 2 ∈ Rt(X). 5. Multi-portfolio time-consistency: Since (Rt)0≤t<T is a normalized set-valued dynamic risk measure, then multi-portfolio time-consistency is equivalent to At = At,r + Ar, for any t < r. Note that At := {X ∈ L 2 d (FT ) : 0 ∈ Rt(X)} = n X ∈ L 2 d (FT ) : ∃ˆk ∈ S 2 F (Kˆ ), Z T t ˆksds = X o , At,r := L 2 d (Fr) ∩ At = n X ∈ L 2 d (Fr) : ∃ˆk ∈ S 2 F (Kˆ ), Z T t ˆksds = X o . Let X1 ∈ At,r. Then X1 ∈ L 2 d (Fr) and X1 = R T t ˆk 1 sds for some ˆk 1 ∈ S 2 F (Kˆ ). Let X2 ∈ Ar. Then X2 ∈ L 2 d (FT ) and X2 = R T r ˆk 2 sds for some ˆk 2 ∈ S 2 F (Kˆ ). Thus X := X1 + X2 ∈ L 2 d (FT ) and X = R T t ˆk 1 sds + R T r ˆk 2 sds = R T t ( ˆk 1 + I[r,T] ˆk 2 )sds, and therefore X ∈ At . Conversely, let X ∈ At . Then X ∈ L 2 d (FT ) and X = R T t ˆksds for some ˆk ∈ S 2 F (Kˆ ). Take X1 := R r t ˆksds and X2 := R T r ˆksds. Then X1 ∈ At,r, X2 ∈ Ar and X = X1 + X2 , hence the result. Remark 5.13. Since the set-valued functions Rt : L 2 d (FT ) → P L 2 d (Ft); (L 2 d (Ft))+ 6 defined as Rt(X) := Γ1 t (−X) is a normalized multi-portfolio time-consistent set-valued dynamic risk measure, then Rt(X) = S Z∈Rr(X) Rt(−Z), for any r > t. Thus, Γ 1 t (X) = Rt(−X) = [ Z∈Rr(−X) Rt(−Z) = [ Z∈Γ1 r (X) Γ 1 t (Z). Theorem 5.14. For r > t, we have Γ 1 t (X) = L 2 d (Ft) ∩ Γ 1 r (X) + n Z r t ˆksds : ˆk ∈ S 2 F (Kˆ ) o. Proof. By multi-portfolio time-consistency of Rt(X) := Γ1 t (−X), we have for r > t Γ 1 t (X) = ∪Z∈Γ1 r (X)Γ 1 t (Z). Thus Γ 1 t (X) = ∪Z∈Γ1 r (X) n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), −Z + u = Z T t ˆksdso = n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), u = Z T t ˆksds + Z, Z ∈ Γ 1 r (X) o . Clearly, n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), u = Z r t ˆksds + Z, Z ∈ Γ 1 r (X) o ⊆ n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), u = Z T t ˆksds + Z, Z ∈ Γ 1 r (X) o . To show that n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), u = Z r t ˆksds + Z, Z ∈ Γ 1 r (X) o ⊇ n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), u = Z T t ˆksds + Z, Z ∈ Γ 1 r (X) o , (5.21) 69 let u ∈ L 2 d (Ft) such that u = R T t ˆksds + Z for some ˆk ∈ S 2 F (Kˆ ) and Z ∈ Γ 1 r (X). Then Z ∈ L 2 d (Fr) and Z = R T r ˜ksds + X for some ˜k ∈ S 2 F (Kˆ ). Thus u = Z T t ˆksds + Z T r ˜ksds + X = Z r t ˆksds + Z T r ( ˆk + ˜k)sds + X. Since u ∈ L 2 d (Ft) ⊆ L 2 d (Fr) and R r t ˆksds ∈ L 2 d (Fr), then Z˜ := R T r ( ˆk + ˜k)sds + X ∈ L 2 d (Fr) and therefore Z˜ ∈ Γ 1 r (X). Thus u = R r t ˆksds+Z, ˜ Z˜ ∈ Γ 1 r (X), and hence (5.21) holds. Thus Γ 1 t (X) = n u ∈ L 2 d (Ft) : ∃ˆk ∈ S 2 F (Kˆ ), u = Z r t ˆksds + Z, Z ∈ Γ 1 r (X) o = L 2 d (Ft) ∩ Γ 1 r (X) + n Z r t ˆksds : ˆk ∈ S 2 F (Kˆ ) o. The following notion of set-valued conditional expectation will be useful for the next refinement of the definition of Γt(X). Definition 5.15. [51] Let X be a set-valued random variable such that S p F (X) ̸= ∅ and let G ⊆ F be a sub-σ-algebra. The set-valued conditional expectation of X relative to G, denoted by E[X|G], is the G-measurable set-valued random variable that satisfies S p G (E[X|G]) = decG{E[x|G] : x ∈ S p F (X)}. By [Corollary 3.4.1.,[51]], the set {E[x|G] : x ∈ S p F (X)} is G-decomposable. Thus, S p G (E[X|G]) = cl{E[x|G] : x ∈ S p F (X)}. By the usual “tower” property of conditional expectation, we have the following Proposition. 70 Proposition 5.16. Let X be a set-valued random variable such that S p F (X) ̸= ∅ and let H ⊆ G ⊆ F. Then S p H(E[X|H]) = S p H(E[S p G (E[X|G])|H]). Proposition 5.17. If X is a convex FT -measurable set-valued random variable. Then for any r < T, we have E[decFT S p Fr (X)|Fr] = S p Fr (X). Proof. Clearly, S p Fr (X) = E[S p Fr (X)|Fr] ⊆ E[decFT S p Fr (X)|Fr]. To show that E[decFT S p Fr (X)|Fr] ⊆ S p Fr (X), let u ∈ decFT S p Fr (X). Then u = Pm i=1 aiIAi , for some (ai) m i=1 ⊆ S p Fr (X) and FT -measurable partition {Ai} m i=1 of Ω. Since X is convex, then E[u|Fr] = Xm i=1 aiP(Ai |Fr) ∈ S p Fr (X). For any u ∈ decFT S p Fr (X), there exists (u n ) ∞ n=1 ⊆ decFT S p Fr (X) such that (u n ) ∞ n=1 converges to u in L p . Since E[u n |Fr] ∈ S p Fr (X), and S p Fr (X) is closed in L p , then E[u|Fr] ∈ S p Fr (X). Corollary 5.18. If X is a convex Fr-measurable set-valued random variable, then for any r < T, we have E[S p FT (X)|Fr] = S p Fr (X). In particular, E h S p FT Z r t Kˆ sds |Fr i = S p Fr Z r t Kˆ sds . Proof. Since X is an Fr-measurable set-valued random variable, then S p FT (X) = 71 decFT S p Fr (X). Thus by Proposition 5.17, we have E[S p FT (X)|Fr] = S p Fr (X). The second part then immediately follows since R r t Kˆ sds ∈ A p Fr (Ω, C (R d )). Proposition 5.19. Let M1 , M2 ⊆ L p d (FT ). Then L p d (Ft) ∩ (M1 ⊕ M2 ) ⊆ E[M1 |Ft ] ⊕ E[M2 |Ft ]. Proof. Let u ∈ L p d (Ft) ∩ (M1 ⊕ M2 ). Then there exist (u n,1 ) ∞ n=1 ⊆ M1 and (u n,2 ) ∞ n=1 ⊆ M2 such that u n,1 + u n,2 converges to u in L p . Thus, E[u n,1 + u n,2 |Ft ] converges to E[u|Ft ] = u in L p . Since E[M1 |Ft ] ⊕ E[M2 |Ft ] = cl{E[u 1 |Ft ] : u 1 ∈ M1 } ⊕ cl{E[u 2 |Ft ] : u 2 ∈ M2 } = cl{E[u 1 + u 2 |Ft ] : u 1 ∈ M1 , u2 ∈ M2 }, then u ∈ E[M1 |Ft ] ⊕ E[M2 |Ft ], and therefore L p d (Ft) ∩ (M1 ⊕ M2 ) ⊆ E[M1 |Ft ] ⊕ E[M2 |Ft ]. Now, consider the superset Γ2 t (X) ⊇ Γt(X) defined as Γ 2 t (X) := S p Ft E h X + Z T t Kˆ sds|Ft i. Lemma 5.20. The set-valued functions Rt : L p d (FT ) → P L p d (Ft); (L p d (Ft))+ defined as Rt(X) := Γ2 t (−X) is a normalized set-valued dynamic risk measure. Proof. Clearly, Rt(X) ⊆ Rt(X) + (L p d (Ft))+. To see that Rt(X) + (L p d (Ft))+ ⊆ Rt(X), let u ∈ Rt(X) and v ∈ (L p d (Ft))+. Then u = E[−X|Ft ] + ˜u, for some u˜ ∈ S p Ft E h R T t Kˆ sds|Ft i. Since u + v = E[−X|Ft ] + ˜u + v and u˜ + v ∈ S p Ft E h Z T t Kˆ sds|Ft i + S p Ft (R d +) ⊆ S p Ft E h Z T t Kˆ sds|Ft i, 7 then Rt(X) + (L p d (Ft))+ ⊆ Rt(X). Next, we check the properties of set-valued risk measures: 1. L p d (Ft)-translative: Let mt ∈ L p d (Ft). Then Rt(X + mt) = S p Ft E h − X − mt + Z T t Kˆ sds|Ft i = S p Ft E h − X + Z T t Kˆ sds|Ft i − mt = S p Ft E h − X + Z T t Kˆ sds|Ft i − mt = Rt(X) − mt . 2. L p d (FT )+-monotone: Let X, Y ∈ L p d (FT ) and let Z := Y − X ∈ (L p d (FT ))+. Then Rt(X) = S p Ft E h − X + Z T t Kˆ sds|Ft i = S p Ft E h Z − Y + Z T t Kˆ sds|Ft i = S p Ft E[Z|Ft ] + E h − Y + Z T t Kˆ sds|Ft i = E[Z|Ft ] + S p Ft E h − Y + Z T t Kˆ sds|Ft i ⊆ S p Ft (R d +) + S p Ft E h − Y + Z T t Kˆ sds|Ft i = S p Ft E h − Y + Z T t Kˆ sds|Ft i = Rt(Y ). 3. Finite at zero: Follows by the fact that Kˆ is a cone and Rt(0) = S p Ft E h Z T t Kˆ sds|Ft i. 73 4. Normalized: Since 0 ∈ Rt(0), then Rt(X) ⊆ Rt(X) + Rt(0). To see that Rt(X) + Rt(0) ⊆ Rt(X), note that Rt(X) + Rt(0) = E[−X|Ft ] + S p Ft E h Z T t Kˆ sds|Ft i + S p Ft E h Z T t Kˆ sds|Ft i ⊆ E[−X|Ft ] + S p Ft E h Z T t Kˆ sds|Ft i = Rt(X). Moreover, we have the following Proposition: Proposition 5.21. The dynamic risk measure Rt(X) := Γ2 t (−X) satisfies the following properties: 1. Additive: Rt(X + Y ) = Rt(X) + Rt(Y ); 2. Conditionally positive homogeneous: Rt(λX) = λRt(X), λ ∈ L ∞(Ft)++; 3. Local: IARt(IAX) = IARt(X), for an Ft-measurable set A; 4. Multi-portfolio time-consistent: Rt(X) = ∪Z∈Rr(X)Rt(−Z) for r > t. Proof. 1. This follows immediately since Rt(X + Y ) = E[−X − Y |Ft ] + S p Ft E h Z T t Kˆ sds|Ft i = E[−X|Ft ] + E[−Y |Ft ] + S p Ft E h Z T t Kˆ sds|Ft i + S p Ft E h Z T t Kˆ sds|Ft i = Rt(X) + Rt(Y ). 74 2. Let λ ∈ L ∞(Ft)++. Since S p Ft E h R T t Kˆ sds|Ft i is a convex cone, then λSp Ft E h Z T t Kˆ sds|Ft i = S p Ft E h Z T t Kˆ sds|Ft i. Thus Rt(λX) = E[−λX] + S p Ft E h Z T t Kˆ sds|Ft i = λE[−X] + λSp Ft E h Z T t Kˆ sds|Ft i = λRt(X). 3. Let A be an Ft-measurable set. Then IARt(IAX) = IAE[−IAX|Ft ] + IAS p Ft E h Z T t Kˆ sds|Ft i = IAE[−X|Ft ] + IAS p Ft E h Z T t Kˆ sds|Ft i = IARt(X). 4. Let r > t. Then ∪Z∈Rr(X) Rt(−Z) = ∪Z∈Rr(X) E[Z|Ft ] + S p Ft E h Z T t Kˆ sds|Ft i ={E[Z|Ft ] : Z ∈ Rr(X)} + S p Ft E h Z T t Kˆ sds|Ft i = n E[Z|Ft ] : Z ∈ E[−X|Fr] + S p Fr E h Z T r Kˆ sds|Fr io + S p Ft E h Z T t Kˆ sds|Ft i = n E[Z|Ft ] : Z = E[−X|Fr] + Y, Y ∈ S p Fr E h Z T r Kˆ sds|Fr io + S p Ft E h Z T t Kˆ sds|Ft i = n E[E[−X|Fr] + Y |Ft ] : Y ∈ S p Fr E h Z T r Kˆ sds|Fr io + S p Ft E h Z T t Kˆ sds|Ft i =E[−X|Ft ] + n E[Y |Ft ] : Y ∈ S p Fr E h Z T r Kˆ sds|Fr io + S p Ft E h Z T t Kˆ sds|Ft i. Let Y = E[u|Fr], for some u ∈ S p FT ( R T r Kˆ sds) ⊆ S p FT ( R T t Kˆ sds). Then E[Y |Ft ] = 75 E[u|Ft ] ∈ S p Ft E h R T t Kˆ sds|Ft i. Thus n E[Y |Ft ] : Y ∈ S p Fr E h Z T r Kˆ sds|Fr io ⊆ S p Ft E h Z T t Kˆ sds|Ft i, and therefore n E[Y |Ft ] : Y ∈ S p Fr E h Z T r Kˆ sds|Fr io+S p Ft E h Z T t Kˆ sds|Ft i = S p Ft E h Z T t Kˆ sds|Ft i. Hence, ∪Z∈Rr(X)Rt(−Z) = E[−X|Ft ] + S p Ft E h Z T t Kˆ sds|Ft i = Rt(X). Lemma 5.22. For r > t, we have E h Γ 2 r (X)|Ft i = E h Γ 2 r (X)|Ft i ⊕ E h S p FT Z T r Kˆ sds |Ft i . Proof. Clearly, E h Γ 2 r (X)|Ft i ⊆ E h Γ 2 r (X)|Ft i ⊕ E h S p FT R T r Kˆ sds |Ft i . To show that E h Γ 2 r (X)|Ft i ⊕ E h S p FT Z T r Kˆ sds |Ft i ⊆ E h Γ 2 r (X)|Ft i , let u ∈ S p Ft E h Γ 2 r (X)|Ft i and v ∈ S p Ft E h S p FT R T r Kˆ sds |Ft i. For u = E[Y |Ft ] for some Y ∈ Γ 2 r (X) = E[X|Fr] + S p Fr E h R T r Kˆ sds|Fr i, we have u ∈ E[X|Ft ]+S p Ft E h S p Fr E h Z T r Kˆ sds|Fr i|Ft i = E[X|Ft ]+S p Ft E h Z T r Kˆ sds|Ft i. 76 Thus, u + v ∈ E[X|Ft ] + S p Ft E h Z T r Kˆ sds|Ft i + S p Ft E h Z T r Kˆ sds|Ft i = E[X|Ft ] + S p Ft E h Z T r Kˆ sds|Ft i = E[Γ2 r (X)|Ft ]. Corollary 5.23. For r > t, we have Γ 2 t (X) = S p Ft E h Γ 2 r (X) ⊕ S p Fr Z r t Kˆ sds |Ft i. Proof. By multi-portfolio time-consistency of Rt(X) := Γ2 t (−X), we have for r > t Γ 2 t (X) = ∪Z∈Γ2 r (X)Γ 2 t (Z). Γ 2 t (X) = ∪Z∈Γ2 r (X)S p Ft E h Z + Z T t Kˆ sds|Ft i = S p Ft ({E[Z|Ft ] : Z ∈ Γ 2 r (X)}) ⊕ S p Ft E h Z T t Kˆ sds|Ft i = S p Ft (E[Γ2 r (X)|Ft ]) ⊕ S p Ft E h S p FT Z T t Kˆ sds |Ft i = S p Ft (E[Γ2 r (X)|Ft ]) ⊕ S p Ft E h S p FT Z r t Kˆ sds |Ft i ⊕ S p Ft E h S p FT Z T r Kˆ sds |Ft i = S p Ft (E[Γ2 r (X)|Ft ]) ⊕ S p Ft E h S p FT Z r t Kˆ sds |Ft i, where the last equality follows by Lemma 5.22. Moreover, since Kˆ is convex, then by Corollary 5.18, we obtain E h S p FT Z r t Kˆ sds |Ft i = E h E h S p FT Z r t Kˆ sds |Fr i |Ft i = E h S p Fr Z r t Kˆ sds |Ft i . 77 Thus, Γ 2 t (X) = S p Ft (E[Γ2 r (X)|Ft ]) ⊕ S p Ft E h S p Fr Z r t Kˆ sds |Ft i = S p Ft E h Γ 2 r (X) ⊕ S p Fr Z r t Kˆ sds |Ft i. Remark 5.24. 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Abstract (if available)

Abstract In this dissertation, we focus on the set-valued (stochastic) analysis on the space of convex, closed, but possibly unbounded sets. By establishing a new theoretical framework on such sets, which is beyond the existing theory of set-valued analysis, we shall study the set-valued SDEs (SV-SDEs) with unbounded coefficients, and their applications in the super-hedging problem of a continuous-time model with transac- tion costs in finance. The space that we will be focusing on is convex, closed sets that are “generated” by a given cone with certain constraints. We shall argue that, for such a special class of unbounded sets, the cancellation law could still be valid, and many algebraic and topological properties of the existing theory of set-valued analysis on compact sets and standard techniques for studying SV-SDEs can be ex- tended to the case with unbounded (drift) coefficients. In the super-hedging problem of discrete-time models with transaction costs, the set of self-financing portfolios are often described by the (unbounded) “solvency cone”. Our study of unbounded sets is therefore crucial in extending the theory to the continuous-time model. In the model with transaction costs and vector-valued contingent claims, the set of super-hedging positions is inherently a closed convex unbounded set. We shall argue that the (dynamic) super-hedging set can be expressed as set-valued integrals of the solvency cones, and define a set-valued dynamic risk measure. Finally, after some refinement, we show that the dynamic super-hedging sets satisfy a recursive relation which can be considered as a geometric dynamic programming principle (DPP).

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Creator Almuzaini, Atiqah Hamoud S (author)

Core Title Set-valued stochastic differential equations with unbounded coefficients and applications

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School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Degree Conferral Date 2024-05

Publication Date 04/08/2025

Defense Date 03/06/2024

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag LC-space,OAI-PMH Harvest,set-valued stochastic analysis on unbounded sets,set-valued stochastic differential equations,solvency cones,super-hedging.,unbounded coefficients

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Advisor Ma, Jin (committee chair), Xu, Renyuan (committee member), Zhang, Jianfeng (committee member)

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