Parameter-dependent Stochastic Optimal Control in Finite Discrete Time
aa r X i v : . [ m a t h . O C ] D ec PARAMETER-DEPENDENT STOCHASTIC OPTIMAL CONTROL INFINITE DISCRETE TIME
ASGAR JAMNESHAN, MICHAEL KUPPER, AND JOS´E MIGUEL ZAPATA-GARC´IA
Abstract.
We prove a general existence result in stochastic optimal control in discrete timewhere controls take values in conditional metric spaces, and depend on the current stateand the information of past decisions through the evolution of a recursively defined forwardprocess. The generality of the problem lies beyond the scope of standard techniques instochastic control theory such as random sets, normal integrands and measurable selectiontheory. The main novelty is a formalization in conditional metric space and the use oftechniques in conditional analysis. We illustrate the existence result by several examplesincluding wealth-dependent utility maximization under risk constraints with bounded andunbounded wealth-dependent control sets, utility maximization with a measurable dimension,and dynamic risk sharing. Finally, we discuss how conditional analysis relates to random settheory. Introduction
The present work investigates parameter-dependent stochastic optimization in finite discretetime with the tools of conditional analysis. In the following, we introduce the mathematicalproblem and sketch our solution strategy. Given a forward generator ( v t ) T − t =0 , consider aforward process x t +1 = v t ( x t , z t )the dynamics of which depend on a parameter x t as a function of earlier decisions and animmediate decision z t chosen recursively in a state-dependent control set Θ t ( x t ) for each t = 0 , . . . , T −
1. Given a backward generator ( u t ) Tt =0 , the goal is to maximize u ( x , · , z ) ◦ . . . ◦ u T − ( x T − , · , z T − ) ◦ u T ( x T ) (1.1)over all possible forward processes initialized at x where ◦ denotes composition of functions.By the Bellman principle, the global stochastic optimization problem (1.1) is solved by abackward recursion if all the local one-period problems y t ( x t ) = sup z t ∈ Θ t ( x t ) u t ( x t , y t +1 ( v t ( x t , z t )) , z t ) , t = 0 , . . . , T − , (1.2) y T ( x T ) = u T ( x T )attain their maxima.Given a filtered probability space (Ω , F , ( F t ) Tt =0 , P ), we assume that the forward process x t and the control process z t assume values in F t -conditional metric spaces X t and Z t respectively.An F t -conditional metric space is a nonempty set X endowed with a vector-valued metric d : X × X → L (Ω , F t , P ) satisfying a concatenation property which encodes information attime t . An example is the space of strongly F t -measurable functions with values in a metricspace with almost everywhere evaluation of the metric. Intuitively speaking, a conditionalmetric space is a collection of classical metric spaces X ( ω ), ω ∈ Ω, which are glued together
Mathematics Subject Classification. in a measurable way. Instead of arguing in each X ( ω ) separately and building on measurableselection lemmas, we directly work in the conditional metric space X and build instead onarguments in conditional analysis. This is possible since in conditional metric spaces all basicresults from real metric spaces are true in conditional form, cf. [8, 13, 16].Our main Theorem 2.5 shows that the global supremum in (1.1) is attained and can bereduced by Bellman’s principle to the local optimization problems (1.2). By backward induc-tion, we show that the value function y t is upper semi-continuous on the conditional metricspace X t . For this, we assume that the control set Θ t ( x t ) is conditionally sequentially com-pact (for a discussion of the notion of conditional compactness, we refer to [13, Sections 3and 4] and [16, Sections 3.4 and 4]). Then the existence of an optimizer in (1.2) follows froma conditional version of the fact that a semi-continuous function on a compact space attainsits extrema. Moreover, under a regularity condition on the control set - a conditional versionof outer semi-continuity in set convergence (see e.g. [48, Chapter 5, Section B]) - it is shownthat y t in (1.2) is upper semi-continuous on X t . The assumption of conditional compactnesson the control set is relaxed in Proposition 4.1 under stronger assumptions on the generatorsby modifying arguments in [11]. In particular, we additionally require that the backwardgenerators u t are F t -sensitive to large losses and increasing in the state variable.In Section 3 we provide sufficient conditions for conditional compactness and conditionalouter semi-continuity of the control set. We focus on conditionally finite dimensional controlsets. The results are illustrated with applications in mathematical finance. In Example 3.2 westudy an optimal consumption problem with local risk constraints on the wealth process. Ex-ample 3.5 indicates the importance of conditional Euclidean space with measurable dimensionto model control processes with state-dependent dimension (e.g. the number of traded assetsat time t depends on F t and past decisions). As an application of Proposition 4.1 we deriveoptimal portfolios w.r.t. dynamic risk measures for which the risk aversion coefficient is influ-enced by the current wealth. Moreover, a closed-form solution to a dynamic wealth-dependentrisk sharing problem is obtained which extends the formula of Borch [7].Normal integrands are a widely used tool to investigate parametrized stochastic optimiza-tion, see e.g. [5, 39, 43, 46] and [44] for an introduction. In Section 5, we establish a connectionbetween conditional analysis and random sets, normal integrands and measurable selectiontheory. In Theorem 5.3, we prove a one-to-one correspondence between the set of measurableselections of Effros measurable and closed-valued mappings and stable and sequentially closedsets. This result yields a one-to-one correspondence between normal integrands and stable andsequentially semi-continuous functions. This indicates that control problems formulated in thelanguage of normal integrands and random sets can equally be formulated in the language ofconditional analysis. For a formalization with normal integrands and random sets, measur-able selection lemmas provide the main tool to secure measurability. The use of measurableselection arguments is enforced by a pointwise application of standard results in classical anal-ysis, and relies on topological assumptions such as separability and standard Borel spaces.In this regard, conditional analysis provides a measure-theoretic alternative which does notrely on any topological assumptions, and works as soon as a formalization within its languageis reached which is demonstrated in this article in discrete time stochastic control theory.Conditional analysis approaches measurable functions directly by providing a measurable (orconditional, or stochastic, or random) version of results in classical analysis. The applica-tion of conditional versions of classical theorems preserves measurability, see for example theproofs below in which a measurable version of the Bolzano-Weierstraß theorem, the maximumtheorem and the Heine-Borel theorem are employed. This perspective is implicitly presentin [25, 29], however without a systematic treatment. A conditional version of basic resultsin functional analysis were established in [13, 18, 16], and applied to financial mathematicsin [2, 6, 11, 12, 15, 19, 20, 21, 37]. In [16], conditional versions of classical theorems werestudied systematically and related to a conditional variant of set theory. This naturally raises ARAMETER-DEPENDENT STOCHASTIC OPTIMAL CONTROL IN FINITE DISCRETE TIME 3 connections with mathematical logic, see [3, 8, 35] for related literature. Conditional analysisin L -modules is moreover investigated in e.g. [10, 28, 38]. Related results in randomly normedmodules are studied in e.g. [23, 24, 26].The remainder of this article is organized as follows. In Section 2 we introduce the notionof conditional metric spaces and prove the main existence result. In Section 3 and Section4 we discuss extensions of the main result and provide several examples. The link betweenconditional analysis and random set theory is established in Section 5.2. Main result
Let (Ω , F , P ) be a probability space. Throughout we identify two sets in F whenever theirsymmetric difference is a null set, and identify two functions on Ω if they coincide a.s. (almostsurely). Let G be a sub- σ -algebra of F . Denote by Π G the set of partitions ( A k ) of Ω where A k ∈ G for all k . Let L G , L G ( N ), L G , + , L G , ++ , L G , and ¯ L G denote the spaces of G -measurablerandom variables with values in R , N , [0 , ∞ ), (0 , ∞ ), R ∪ {−∞} and R ∪ {±∞} respectively.Recall that L G with the pointwise a.s. order is a Dedekind complete lattice-ordered ring.The essential supremum and the essential infimum are denoted by sup and inf respectively.Inequalities between random variables with values in an ordered set are always understood inthe pointwise a.s. sense. Definition 2.1. A G - conditional metric on a non-empty set X is a function d : X × X → L G , + such that the following conditions hold:(i) d ( x, y ) = 0 if and only if x = y ,(ii) d ( x, y ) = d ( y, x ) , (iii) d ( x, z ) ≤ d ( x, y ) + d ( y, z ),(iv) for every sequence ( x k ) in X and ( A k ) ∈ Π G there exists exactly one element x ∈ X such that 1 A k d ( x, x k ) = 0 for all k ∈ N .The pair ( X, d ) is called a G - conditional metric space .In the following we call the unique element in (iv) the concatenation of the sequence ( x k )along the partition ( A k ) and denote it by P k A k x k . For a sequence ( x n ) in a conditional metricspace ( X, d ) we write x n → x a.s. whenever d ( x, x n ) → measurable subsequence ( x n k )of ( x n ) is of the form x n k := P j ∈ N { n k = j } x j where ( n k ) is a sequence in L G ( N ) such that n k < n k +1 for all k ∈ N . Definition 2.2.
Let (
X, d X ) and ( Z, d Z ) be G - conditional metric spaces , and H and G subsetsof X and Z , respectively. We call H • G -stable if H = ∅ and P k A k x k ∈ H for all ( A k ) ∈ Π G and every sequence ( x k ) in H , • sequentially closed if H contains every x ∈ X such that there is a sequence ( x k ) in H with x k → x a.s.A function f : H → G is said to be • G -stable if f ( P k A k x k ) = P k A k f ( x k ) for all ( A k ) ∈ Π G and every sequence ( x k ) in H , where H and G are assumed to be G -stable, • sequentially continuous whenever f ( x ) = lim k f ( x k ) if x k → x a.s. in H ,and if G = ¯ L F , then f is said to be • sequentially lower semi-continuous if f ( x ) ≤ lim inf f ( x k ) if x k → x a.s. in H , • sequentially upper semi-continuous if lim sup f ( x k ) ≤ f ( x ) if x k → x a.s. in H . Remark 2.3. 1.
If (
X, d ) is a G -conditional metric space then the metric d : X × X → L G , + is G -stable, i.e. d ( P k A k x k , P k A k y k ) = P k A k d ( x k , y k ) for every sequences ( x k ) and ( y k ) ASGAR JAMNESHAN, MICHAEL KUPPER, AND JOS´E MIGUEL ZAPATA-GARC´IA in X and ( A k ) ∈ Π G . Indeed, denoting by x = P k A k x k and y = P k A k y k the respectiveconcatenations, it follows from the triangular inequality that1 A k d ( x, y ) ≤ A k d ( x, x k ) + 1 A k d ( x k , y k ) + 1 A k d ( y k , y ) = 1 A k d ( x k , y k ) ≤ A k d ( x k , x ) + 1 A k d ( x, y ) + 1 A k d ( y k , y ) = 1 A k d ( x, y )which shows that 1 A k d ( x, y ) = 1 A k d ( P k A k x k , P k A k y k ) = 1 A k d ( x k , y k ) for all k ∈ N .Summing up over all k yields the desired G -stability. Let (
X, d X ) and ( Y, d Y ) be two G -conditional metric spaces. Then its product X × Y endowed with the G -conditional metric d X × Y (( x, y ) , ( x ′ , y ′ ) = max { d X ( x, x ′ ) , d Y ( y, y ′ ) } is a G -conditional metric space. In the following all products of conditional metric spaces areendowed with this conditional metric. Let (
X, d X ) be a G -conditional metric spaces. Then the set X of all pairs ( x, A ) ∈ X × G ,where ( x, A ) and ( y, B ) are identified if A = B and 1 A d ( x, y ) = 0 is a conditional set . Ingeneral, a conditional set Y is an abstraction of this example, and can be viewed as a set-likestructure on which G acts such that Y is closed w.r.t. countable concatenations of its elementsalong partitions in Π G . Conditional set theory is investigated in [16] and does not require ametric structure as in Definition 2.1. For further results on conditional metric spaces in thecontext of conditional set theory we refer to [16, Section 4].We next introduce the parameter-dependent stochastic optimal control problem for con-ditional metric spaces. For a fixed finite time horizon T ∈ N , we consider a filtration F ⊂ F ⊂ . . . ⊂ F T = F . For simplicity, we often abbreviate the index F t by t , and writefor instance L t for L F t . For each t = 0 , . . . , T , let ( X t , d X t ) and ( Z t , d Y t ) be F t -conditionalmetric spaces. Our aim is to study control problems for which the control set Θ t depends on F t , but also on a state parameter x ∈ X t . For every t = 0 , . . . , T −
1, we assume that the state-dependent control set Θ t satisfies(c1) ∅ 6 = Θ t ( x ) ⊂ Z t for all x ∈ X t ,(c2) Θ t is F t -stable, i.e.Θ t (cid:16) X k A k x k (cid:17) = X k A k Θ t ( x k ) := n X k A k z k : z k ∈ Θ t ( x k ) for all k o for all ( A k ) ∈ Π t and every sequence ( x k ) in X t ,(c3) for every x ∈ X t , the set Θ t ( x ) is conditionally sequentially compact , i.e. for everysequence ( z n ) in Θ t ( x ) there exists a measurable subsequence n < n < · · · with n k ∈ L t ( N ) such that z n k → z ∈ Θ t ( x ) a.s.,(c4) for every sequence ( x n ) in X t such that x n → x ∈ X t a.s. and every sequence ( z n ) inΘ t ( x n ) there exists a measurable subsequence n < n < · · · with n k ∈ L t ( N ) and asequence ( z ′ k ) in Θ t ( x ) such that d Z t ( z n k , z ′ k ) → F t -stability of Θ t implies that Θ t ( x ) is F t -stable for all x ∈ X t .We consider forward generators v t : X t × Z t → X t +1 , t = 0 , . . . , T − , which are(v1) F t -stable,(v2) sequentially continuous.For every x t ∈ X t we consider the set C t ( x t ) := n(cid:0) ( x s ) Ts = t +1 , ( z s ) T − s = t (cid:1) : x s +1 = v s ( x s , z s ) , z s ∈ Θ s ( x s ) for all s = t, . . . , T − o of all parameter processes ( x s ) Ts = t which can be realized by the state-dependent controls z s ∈ Θ t ( x s ) for s = t, . . . , T − ARAMETER-DEPENDENT STOCHASTIC OPTIMAL CONTROL IN FINITE DISCRETE TIME 5
As for the objective function, we consider backward generators u t : X t × L t +1 × Z t → L t , t = 0 , , . . . , T − , which are(u1) F t -stable,(u2) increasing in the second component,(u3) sequentially upper semi-continuous.We assume that u T : X T → L T is F T -stable and sequentially upper semi-continuous. Givensuch a family ( u t ) Tt =0 of backward generators, our goal is to maximize y t ( x t ) := sup (( x s ) Ts = t +1 , ( z s ) T − s = t ) ∈ C t ( x t ) u t ( x t , · , z t ) ◦ · · · ◦ u T − ( x T − , · , z T − ) ◦ u T ( x T ) (2.1)over all realizable state processes initialized at x t ∈ X t . Remark 2.4.
The objective function in the stochastic control problem (2.1) is recursivelydefined. Its generators (aggregators) are functions between conditional metric spaces whichsatisfy monotonicity and semi-continuity. The aggregators are not necessarily (conditional)expected utilities. In case of (conditional) expected utility, the generators are closely relatedwith dynamic and conditional risk measures, see [1, 6, 14, 19, 20]. The preferences whichunderly conditional expected utility functionals were studied in [15] under the name of condi-tional preference orders.In decision theory, there is an extensive literature on recursive utilities starting with the sem-inal work [30, 31]. The preferences therein are defined on sets of temporal lotteries (probabilitytrees), and follow a kind of Bellman recursive structure which is similar to the constructionabove on a formal level (see [30, Theorem 1]). This was later extended in [17] where non-expected utilities were incorporated as well, and established under the name of Epstein-Zinutilities. See also [34] for a survey on non-expected utility theory. With the techniques ofconditional analysis and based on results in BSDE theory, [12] solves a utility maximizationproblem in continuous time for Epstein-Zin utilities.The following result shows that the global supremum in (2.1) is attained and can be reducedto local optimization problems by the following Bellman’s principle.
Theorem 2.5.
Suppose that (c1)–(c4), (v1)–(v2), and (u1)–(u3) are fulfilled. Then the func-tions y t : X t → L t are F t -stable and sequentially upper semi-continuous for all t = 0 , . . . , T ,and can be computed by backward recursion y T ( x T ) = u T ( x T ) y t ( x t ) = max z t ∈ Θ t ( x t ) u t ( x t , y t +1 ( v t ( x t , z t )) , z t ) , t = 0 , . . . , T − . Moreover, for every x t ∈ X t the process (( x ∗ s ) Ts = t , ( z ∗ s ) T − s = t ) given by x ∗ t = x t and the forwardrecursion x ∗ s +1 = v s ( x ∗ s , z ∗ s ) where z ∗ s ∈ argmax z s ∈ Θ s ( x ∗ s ) u s (cid:0) x ∗ s , y s +1 ( v t ( x ∗ s , z s )) , z s (cid:1) , s = t, . . . T − , (2.2) satisfies (( x ∗ s ) Ts = t +1 , ( z ∗ s ) T − s = t ) ∈ C t ( x t ) and y t ( x t ) = u t ( x t , · , z ∗ t ) ◦ · · · ◦ u T − ( x ∗ T − , · , z ∗ T − ) ◦ u T ( x ∗ T ) . Proof.
The proof is by backward induction. For t = T it follows from (2.1) that y T = u T which by assumption is an F t -stable and sequentially upper semi-continuous function from X T to L T .As for the induction step, assume that y t +1 : X t +1 → L t +1 is F t +1 -stable and sequentiallyupper semi-continuous, and that for each x t +1 ∈ X t +1 there exists (( x ∗ s ) Ts = t +2 , ( z ∗ s ) T − s = t +1 ) ∈ ASGAR JAMNESHAN, MICHAEL KUPPER, AND JOS´E MIGUEL ZAPATA-GARC´IA C t +1 ( x t +1 ) such that y t +1 ( x t +1 ) = u t +1 ( x t +1 , · , z ∗ t +1 ) ◦ · · · ◦ u T ( x ∗ T ). By (u1) and (v1) thefunction X t × Z t ∋ ( x, z ) u t (cid:0) x, y t +1 ( v t ( x, z )) , z (cid:1) is F t -stable. Moreover, it is sequentially upper semi-continuous. Indeed, let ( x k , z k ) be asequence in X t × Z t such that x k → x ∈ X t a.s. and z k → z ∈ Z t a.s. Since v ( x k , z k ) → v ( x, z )a.s. by (v2) it follows from the induction hypothesis thatlim sup k →∞ y t +1 ( v t ( x k , z k )) ≤ y t +1 ( v ( x, z )) < + ∞ . Since (cid:8) sup k ≥ y t +1 ( v t ( x k , z k )) = + ∞ (cid:9) = \ k ≥ (cid:8) sup k ′ ≥ k y t +1 ( v t ( x k ′ , z k ′ )) = + ∞ (cid:9) = (cid:8) lim sup k →∞ y t +1 ( v t ( x k , z k )) = + ∞ (cid:9) , we have sup k ≥ y t +1 ( v t ( x k , z k )) ∈ L t +1 . Hence, by (u2), (u3) and (v2) we getlim sup k →∞ u t (cid:0) x k , y t +1 ( v t ( x k , z k )) , z k (cid:1) ≤ lim sup k →∞ u t (cid:0) x k , sup k ′ ≥ k y t +1 ( v t ( x k ′ , z k ′ )) , z k (cid:1) ≤ u t (cid:0) x, lim sup k →∞ y t +1 ( v t ( x k , z k )) , z (cid:1) ≤ u t (cid:0) x, y t +1 ( v t ( x, z )) , z (cid:1) (2.3)which shows the desired sequential upper semi-continuity. As a consequence, the supremumin f t ( x t ) := sup z ∈ Θ t ( x t ) u t (cid:0) x t , y t +1 ( v t ( x t , z )) , z (cid:1) (2.4)is attained for each x t ∈ X t . Indeed, since z u t (cid:0) x, y t +1 ( v t ( x, z )) , z (cid:1) and Θ t ( x t ) are F t -stable,it follows from standard properties of the essential supremum that there exists a sequence z n ∈ Θ t ( x t ) such that u t (cid:0) x t , y t +1 ( v t ( x t , z n )) , z n (cid:1) → f t ( x t ) a.s.By (c3) there is a measurable subsequence n < n < · · · with n k ∈ L t ( N ) such that z n k → z ∈ Θ t ( x t ) a.s. Since z u t (cid:0) x, y t +1 ( v t ( x, z )) , z (cid:1) is sequentially upper semi-continuous and F t -stable, it follows that u t (cid:0) x t , y t +1 ( v t ( x t , z )) , z (cid:1) ≥ lim sup k →∞ u t (cid:0) x t , y t +1 ( v t ( x t , z n k )) , z n k (cid:1) = f t ( x t )which shows that the supremum in (2.4) is attained.We next show that f t : X t → L t +1 is sequentially upper semi-continuous. By contradiction,suppose that ( x k ) is a sequence in X t such that x k → x ∈ X t a.s. and f t ( x ) < lim sup k f t ( x k )on some A ∈ F with P ( A ) >
0. Note that f t is F t -stable. Thus, by possibly passing to ameasurable subsequence, we can suppose that there exists r ∈ L t, ++ such that f t ( x ) + r < f t ( x k ) on A, for all k ∈ N . (2.5)Denote by z k ∈ Θ t ( x k ) a respective maximizer of f t ( x k ). By (c4) there exists z ′ k ∈ Θ t ( x )such that d Z t ( z k , z ′ k ) → k < k < · · · with k l ∈ L t ( N ) such that z ′ k l → z ′ ∈ Θ t ( x )a.s. Since d Z t ( z k l , z ′ k l ) → F t -stability of the conditional metric d Z t , it follows fromthe triangular inequality that z k l → z ′ ∈ Θ t ( x ) a.s. By the F t -stability of f t and (c2) it followsthat z k l is in Θ t ( x k l ) and maximizes f t ( x k l ). Hence, it follows from (2.3) thatlim sup l →∞ f t ( x k l ) = lim sup l →∞ u t (cid:0) x k l , y t +1 ( v t ( x k l , z k l )) , z k l (cid:1) ≤ u t (cid:0) x, y t +1 ( v t ( x, z ′ )) , z ′ (cid:1) ≤ sup z ∈ Θ t ( x ) u t (cid:0) x, y t +1 ( v t ( x, z ) , z (cid:1) = f t ( x ) . ARAMETER-DEPENDENT STOCHASTIC OPTIMAL CONTROL IN FINITE DISCRETE TIME 7
Notice that, due to the F t -stability of f t , (2.5) is satisfied for any measurable subsequence of( x k ). Thus, we have that f t ( x )+ r ≤ lim sup l →∞ f t ( x k l ) ≤ f t ( x ) on A , which is a contradiction.We conclude that f t is sequentially upper semi-continuous.Finally, we show that y t = f t . By induction hypothesis, for every x t ∈ X t and z t ∈ Z t thereexists (cid:0) ( x ∗ s ) Ts = t +2 , ( z ∗ s ) T − s = t +1 (cid:1) ∈ C t +1 ( v t ( x t , z t )) such that y t +1 ( v t ( x t , z t )) = u t +1 ( v t ( x t , z t ) , · , z ∗ t +1 ) ◦ · · · ◦ u T − ( x ∗ T − , · , z ∗ T − ) ◦ u T ( z ∗ T ) . In particular, for x t ∈ X t and z ∗ t ∈ Z t being a respective maximizer in (2.4) one has f t ( x t ) = sup z ∈ Θ t ( x t ) u t (cid:0) x t , y t +1 ( v t ( x t , z )) , z (cid:1) = u t (cid:0) x t , y t +1 ( v t ( x t , z ∗ t )) , z ∗ t (cid:1) = u t (cid:0) x t , · , z ∗ t (cid:1) ◦ u t +1 ( v t ( x t , z ∗ t ) , · , z ∗ t +1 ) ◦ · · · ◦ u T − ( x ∗ T − , · , z ∗ T − ) ◦ u T ( z ∗ T )= sup (( x s ) Ts = t +2 , ( z s ) Ts = t +1 ) ∈ C t +1 ( v ( x t ,z ∗ t )) u t (cid:0) x t , · , z ∗ t (cid:1) ◦ u t +1 ( v t ( x t , z t ) , · , z t +1 ) ◦ · · · ◦ ◦ u T ( z T )= sup z t ∈ Θ t ( x t ) sup (( x s ) Ts = t +2 , ( z s ) Ts = t +1 ) ∈ C t +1 ( v ( x t ,z t )) u t (cid:0) x t , · , z t (cid:1) ◦ u t +1 ( v t ( x t , z t ) , · , z t +1 ) ◦ · · · ◦ u T ( z T )= sup (( x s ) Ts = t +1 , ( z s ) Ts = t ) ∈ C t ( x t ) u t (cid:0) x t , · , z t (cid:1) ◦ u t +1 ( v t ( x t , z t ) , · , z t +1 ) ◦ · · · ◦ ◦ u T ( z T )= y t ( x t ) . This shows that (( x ∗ s ) Ts = t +1 , ( z ∗ s ) Ts = t ) ∈ C t ( x t ) is an optimizer of (2.1) whenever it satisfies thelocal optimality criterion z ∗ s ∈ argmax z ∈ Θ t ( x ∗ t ) u s (cid:0) x ∗ s , y s +1 ( v t ( x ∗ s , z s )) , z s (cid:1) and x ∗ s +1 = v s ( x ∗ s , z ∗ s )for all s = t, . . . , T , where x ∗ t = x t . In particular, every process which satisfies the forwardrecursion (2.2) is an optimizer for (2.1). (cid:3) Examples 2.6.
As for the illustration we provide examples of F t -conditional metric spaceswhich are of interest for the control and parameter spaces in Theorem 2.5. Given a nonempty metric space (
X, d ), denote by L t ( X ) the set of all strongly F t -measurable functions x : Ω → X , i.e. the set of those x for which there exists a sequence( x n ) of countable simple functions x n = P k A nk x nk with x nk ∈ X and ( A nk ) ∈ Π t , such that d ( x ( ω ) , x n ( ω )) → ω ∈ Ω. Notice that the metric d extends from X to L t ( X ) withvalues in L t, + by defining d L t ( X ) ( x, ¯ x ) := lim n →∞ d ( x n , ¯ x n )where x n = P k A nk x nk and ¯ x n = P k ¯ A nk ¯ x nk are sequences of countable simple functions suchthat d ( x ( ω ) , x n ( ω )) → d (¯ x ( ω ) , ¯ x n ( ω )) → ω ∈ Ω, and d ( x n , ¯ x n ) := X k,k ′ A nk ∩ ¯ A nk ′ d ( x nk , ¯ x nk ′ ) . Notice that d L t ( X ) ( x, ¯ x ) does not depend on the choice of approximating sequences ( x n ) and(¯ x n ). Then ( L t ( X ) , d L t ( X ) ) is a F t -conditional metric space. For instance, if ( x k ) is a sequencein L t ( X ) such that x k is the limit of the countable sequence ( x nk ) and ( A k ) ∈ Π t then theconcatenation P k A k x k is the unique element in L t ( X ) given as the limit of the countablesimple functions P k A k x nk for n → ∞ . The conditional Euclidean space with measurable dimension n = P k A k n k ∈ L t ( N ) isdefined as L t ( R ) n = X k A k L t ( R n k ) := n X k A k x k : x k ∈ L t ( R n k ) for all k o . ASGAR JAMNESHAN, MICHAEL KUPPER, AND JOS´E MIGUEL ZAPATA-GARC´IA
The F t -conditional metric on L t ( R ) n is defined by d L t ( R ) n ( x, ¯ x ) := X k A k d L t ( R nk ) ( x k , ¯ x k ) , where x = P k A k x k and ¯ x = P k A k ¯ x k . Here, d L t ( R nk ) denotes the F t -conditional metricon L t ( R n k ) which extends the Euclidean metric on R n k as defined in the previous example.Straightforward verification shows that ( L t ( R ) n , d L t ( R ) n ) is a F t -conditional metric space. For 1 ≤ p < ∞ , we define the conditional L p -space L pt := { x ∈ L T : E [ | x | p |F t ] < + ∞ a.s. } with F t -conditional metric d L pt ( x, ¯ x ) := E [ | x − ¯ x | p |F t ] /p . By definition, ( L pt , d L pt ) is a F t -conditional metric space.3. Compactness condition for the control set
The finite dimensional case.
Suppose that Z t = L t ( R d ). As shown in Example 2.6the Euclidean metric of R d extends to the F t -conditional metric d L t ( R d ) : L t ( R d ) → L t, + . Proposition 3.1.
Suppose that for each t = 0 , . . . , T − , the control set Θ t satisfies (c1),(c2) and the following conditions:(i) (cid:8) ( x, z ) ∈ X t × L t ( R d ) : z ∈ Θ t ( x ) (cid:9) is sequentially closed,(ii) for every sequence ( x n ) in X t with x n → x ∈ X t a.s. there exists M ∈ L t, + such that d L t ( R d ) ( z, ≤ M for all z ∈ S n Θ t ( x n ) .Then, the control set Θ t satisfies (c1)-(c4).Proof. Let ( x n ) in X t be a sequence such that x n → x ∈ X t a.s., and z n ∈ Θ t ( x n ). Sinceby assumption d L t ( R d ) ( z n , ≤ M for some M ∈ L t, + , the conditional Bolzano-Weierstrasstheorem [13, Theorem 3.8] implies a measurable subsequence n < n < · · · with n k ∈ L t ( N )such that d L t ( R d ) ( z n k , z ) → z ∈ L t ( R d ). Since Θ t satisfies (c2) one has z n k ∈ Θ t ( x n k ), and therefore z ∈ Θ t ( x ) by (i). This shows (c4) and (c3) follows by consideringthe constant sequence x n = x for all n ∈ N . (cid:3) Example 3.2.
Let ( S t ) Tt =0 be a d -dimensional ( F t )-adapted price process. Given an initialinvestment x >
0, we consider the wealth process x t +1 = v t ( x t , z t ) := x t + ϑ t · ∆ S t +1 − c t , where the control z t = ( ϑ t , c t ) ∈ L t ( R d ) × L consists of an investment strategy ϑ t ∈ L t ( R )and a consumption c t ∈ L t, + . The forward generator v t : L t × L t ( R d × R + ) → L t +1 satisfies(v1) and (v2). We assume that the wealth process is regulated by ρ t ( x t +1 ) ≤ , (3.1)i.e. x t +1 is acceptable w.r.t. a F t -conditional convex risk measure ρ t : L t +1 → ¯ L t for all t = 0 , . . . , T −
1. Recall that a F t -conditional convex risk measure is • normalized , i.e. ρ t (0) = 0, • monotone , i.e. ρ t ( x ) ≤ ρ t ( y ) for all x, y ∈ L t +1 with x ≥ y , • F t -translation invariant , i.e. ρ t ( x + m ) = ρ ( x ) − m for all x ∈ L t +1 and m ∈ L t , • F t -convex , i.e. ρ t ( λx + (1 − λ ) y ) ≤ λρ t ( x ) + (1 − λ ) ρ t ( y ) for all x, y ∈ L t +1 and λ ∈ L t with 0 ≤ λ ≤ F t -translation invariance it follows that (3.1) is equivalent to ρ t ( ϑ t · ∆ S t +1 ) ≤ x t − c t . Inaddition, ρ t is F t -stable since it is F t -convex (see [13, Lemma 4.3]). Hence we consider the(wealth-dependent) control setΘ t ( x t ) := n z t = ( ϑ t , c t ) ∈ L t ( R d × R + ) : ρ t ( ϑ t · ∆ S t +1 ) ≤ x t − c t and 0 ≤ c t ≤ x t o , ARAMETER-DEPENDENT STOCHASTIC OPTIMAL CONTROL IN FINITE DISCRETE TIME 9 which is F t -stable. Suppose that for every ϑ ∈ L t ( R d ) one has P ( ϑ · ∆ S t +1 < | F t ) > { ϑ = 0 } and therefore P ( ϑ · ∆ S t +1 > | F t ) > { ϑ = 0 } . Moreover, we assume that ρ t ( ϑ · ∆ S t +1 ) ∈ L t for all ϑ ∈ L t ( R d ), and ρ t is F t - sensitive to large losses , i.e. lim m →∞ ρ t ( my ) =+ ∞ on { P ( y < | F t ) > } . Then the control set Θ t satisfies (i) and (ii) of Proposition3.1. Indeed, consider the function f t : L t ( R d × R + ) × L t → L t defined as f t ( ϑ, c, x ) := ρ t ( ϑ · ∆ S t +1 )+ c − x , which is F t -convex and therefore sequentially continuous by [13, Theorem7.2]. Hence it follows that (cid:8) ( x, ϑ, c ) ∈ L t × L t ( R d × R + ) : ( ϑ, c ) ∈ Θ t ( x ) (cid:9) = (cid:8) ( x, ϑ, c ) ∈ L t × L t ( R d × R + ) : f t ( ϑ, c, x ) ≤ (cid:9) is F t -convex and sequentially closed, which shows (i). As for (ii) let ( x n ) be a sequence in L t such that x n → x ∈ L t a.s. For ¯ x := sup n x n ∈ L t one hasΘ t ( x n ) ⊂ Θ t (¯ x )for all n ∈ N . Hence, it remains to show that Θ t (¯ x ) is F t -bounded, i.e. there is M ∈ L t, + suchthat d L t ( R d ) ( ϑ,
0) + c ≤ M for all ( ϑ, c ) ∈ Θ t (¯ x ). Since Θ t (¯ x ) contains (0 , ∈ L t ( R d × R + ),by [13, Theorem 3.13] it is enough to show that for each ( ϑ, c ) ∈ Θ t (¯ x ) with ( ϑ, c ) = (0 , k ∈ N such that k ( ϑ, c ) / ∈ Θ t (¯ x ). If c = 0 this is obvious. Otherwise, one has P ( ϑ = 0) >
0, in which case lim m →∞ ρ t ( mϑ · ∆ S t +1 ) = + ∞ on { ϑ = 0 } .By Proposition 3.1 and Theorem 2.5 it follows that for every recursive utility function withbackward generators ( u t ), t = 1 , . . . , T , satisfying (u1)-(u3) and x >
0, there exists a globaloptimizer (( x ∗ s ) Ts =1 , ( ϑ ∗ s , c ∗ s ) T − s =0 ) ∈ C ( x ) of the utility maximization problem (2.1) satisfyingthe local criterion (2.2).3.2. Measurable dimension.
Suppose that Z t is the conditional Euclidean space L t ( R ) d t with measurable dimension d t = d t ( x ) ∈ L t ( N ) that depends on the parameter x ∈ X t (seeExample 2.6 for the definition of the conditional Euclidean space with measurable dimension).Let d t : X t → L t ( N ) be an F t -stable and sequentially continuous, where L t ( N ) is endowed withthe F t -conditional metric which extends the discrete metric on N . The control set Θ t is chosensuch that(c1) ∅ 6 = Θ t ( x ) ⊂ L t ( R ) d t ( x ) for all x ∈ X t ,(c2) Θ t is F t -stable, i.e.Θ t (cid:16) X k A k x k (cid:17) = X k A k Θ t ( x k ) ⊂ L t ( R ) d t ( P k Ak x k ) for all ( A k ) ∈ Π t and every sequence ( x k ) in X t ,are satisfied. Remark 3.3.
Since Z t = L t ( R ) d t ( x ) depends on the state x ∈ X t we are in a more generalsetting as the main Theorem 2.5. However, since L t ( N ) is endowed with the conditionaldiscrete metric, for every sequence ( x n ) in X t such that x n → x ∈ X t there exists n ∈ L t ( N )such that d t ( x n ) = d t ( x ) for all n ≥ n . In particular, L t ( R ) d t ( x n ) = L t ( R ) d t ( x ) for all n ≥ n and Theorem 2.5 still holds true by exploring the arguments on z n ∈ Θ t ( x n ) for sequences x n → x a.s. in the conditional space L t ( R ) d t ( x ) .A variant of Proposition 3.1 for control sets with measurable dimension can be formulatedas follows. Proposition 3.4.
Suppose that for each t = 0 , . . . , T − , the control set Θ t satisfies (c1),(c2) and the following conditions:(i) (cid:8) ( x, z ) ∈ X t × L t ( R ) d t ( x ) : z ∈ Θ t ( x ) (cid:9) is sequentially closed. (ii) For every sequence ( x n ) in X t with x n → x ∈ X t a.s. there exists M ∈ L t, + and ameasurable subsequence n < n < · · · in L t ( N ) such that d L t ( R dt ( x ) ) ( z, ≤ M forevery z ∈ S k ≥ k Θ t ( x n k ) for some k ∈ L t ( N ) such that Θ t ( x n k ) ⊂ Z t ( d t ( x )) for all k ≥ k .Then, the control set Θ t satisfies (c1)-(c4).Proof. Let ( x n ) in X t be a sequence such that x n → x ∈ X t a.s., and z n ∈ Θ t ( x n ). ByRemark 3.3 there exists a measurable subsequence n < n < · · · with n k ∈ L t ( N ) such that z n k ∈ Θ t ( x n k ) ⊂ L t ( R ) d t ( x ) for all k . Hence, we can argue similar as in the proof of Proposition3.1. (cid:3) Example 3.5.
Consider a portfolio maximization problem, where the number of traded as-sets depends on past decision. More precisely, given a portfolio x t = z t − = ( ϑ t − , d t − ) ∈ L t − ( R ) d t − × L t − ( N ) chosen at time t − x − = ( ϑ − , d − ) ∈ R d − × N ),the investor can rebalance the portfolio at time t to x t +1 = z t = ( ϑ t , d t ) ∈ Θ t ( x t ) ⊂ L t ( R ) d t − × L t ( N ) . Here, the state spaces and the control spaces X t +1 = Z t = L t ( R ) d t − × L t ( N ) both dependon the past decision d t − . In line with Remark 3.3 the convergence x nt = ( ϑ nt − , d nt − ) → x t =( ϑ t − , d t − ) is understood as ϑ nt − → ϑ t − a.s. in the conditional metric space L t ( R ) d t − , since d nt − = d t − for all n ≥ n for some n ∈ L t − ( N ). Suppose that the control set Θ t satisfies(c1), (c2) as well as (i) and (ii) of Proposition 3.4. Then, along the same argumentation asin Proposition 3.4 it follows that Θ t satisfies (c1)–(c4). Since v t ( x t , z t ) := z t satisfies (v1) and(v2), Theorem 2.5 is applicable whenever the backward generators u t satisfy (u1)-(u3).The measurable dimension depending on past decisions allows for instance to add new assetsat time t (i.e. d t > d t − ) which are traded at t + 1. Notice that Θ t ( ϑ t − , d t − ) denotes theset of all attainable portfolios at time t . For instance, let S t ∈ L t, ++ ( R d ) be a price processwith fixed d ∈ N . Without frictions and short-selling constraints one has Θ t ( ϑ t − ) := { ϑ t ∈ L t, + ( R d ) : ϑ t · S t = ϑ t − · S t } which satisfies (c1)-(c4). Transaction costs can be included intothe model by considering Θ t ( ϑ t − ) := { ϑ t ∈ L t, + ( R d ) : ϑ t − ϑ t − ∈ C t } for a solvency region C t ⊂ L t ( R d ), see e.g. [40] for a discussion of different market models. Notice that the solvencyregions can be modeled state-dependently with measurable dimension d t ∈ L t ( N ).4. Unbounded control sets
In this section we consider unbounded control sets Θ t ≡ L t ( R d ) and do not assume con-straints on the controls, but derive (c3) and (c4) for upper-level sets of y t as a result of strongerassumptions on the forward and backward generators. In particular, we additionally need thatthe backward generators are F t -sensitive to large losses and increasing in the first argument,see (u5) and (u2’) below. Suppose that the forward generators v t : L t × L t ( R d ) → L t +1 , t = 0 , , . . . , T − , satisfy (v1), (v2) and(v3) v t is increasing in the first component,(v4) v t ( x, λz + (1 − λ ) z ′ ) ≥ λv t ( x, z ) + (1 − λ ) v t ( x, z ′ ) for all x ∈ L t , z, z ′ ∈ L t ( R d ) and λ ∈ L t with 0 ≤ λ ≤ P ( v t ( x, z ) < x | F t ) > { z = 0 } for all x ∈ L t and z ∈ L t ( R d ),(v6) v t ( x,
0) = x for all x ∈ L t .As for the backward generators, let u T : L T → L T be the identity mapping, and u t : L t × L t +1 × L t ( R d ) → L t , t = 0 , . . . , T − , satisfy (u1) and (u3) as well as ARAMETER-DEPENDENT STOCHASTIC OPTIMAL CONTROL IN FINITE DISCRETE TIME 11 (u2’) increasing in the first and second component,(u4) u t ( x, λy + (1 − λ ) y ′ , λz + (1 − λ ) z ′ ) ≥ min { u t ( x, y, z ) , u t ( x, y ′ , z ′ ) } for all x ∈ L t ( I ), y, y ′ ∈ L t +1 , z, z ′ ∈ L t ( R d ) and λ ∈ L t with 0 ≤ λ ≤ u t ( x, y + c, z ) = u t ( x, y, z ) + c for all x ∈ L t , y ∈ L t +1 , z ∈ L t ( R d ) and c ∈ L t ,(u6) lim m →∞ u t ( x, my, mz ) = −∞ a.s. on { P ( y < | F t ) > } for every z ∈ L t ( R d ) and y ∈ L t +1 ,(u7) u t ( x, ,
0) = 0 for all x ∈ L t .Let y t : L t → L t be given as in (2.1) where C t ( x t ) := n(cid:0) ( x s ) Ts = t +1 , ( z s ) T − s = t (cid:1) : x s +1 = v s ( x s , z s ) , z s ∈ L t ( R d ) for all s = t, . . . , T − o . Then the following variant of Theorem 2.5 holds.
Proposition 4.1.
Suppose that (v1)–(v6) and (u1),(u2’),(u3)–(u7) are fulfilled, and thereexists a constant
K > such that sup z ∈ L t ( R d ) u t ( x, v t ( x, z ) , z ) − x ≤ K for all t = 0 , . . . , T − and x ∈ L t . Then the functions y t : L t → L t are F t -stable, increasingand sequentially upper semi-continuous for all t = 0 , . . . , T , and can be computed by backwardrecursion y T ( x T ) = u T ( x T ) = x T y t ( x t ) = max z t ∈ L t ( R d ) u t ( x t , y t +1 ( v t ( x t , z t )) , z t ) , t = 0 , . . . , T − . Moreover, for every x t ∈ L t the process (( x ∗ s ) Ts = t , ( z ∗ s ) T − s = t ) given by x ∗ t = x t and forwardrecursion x ∗ s +1 = v s ( x ∗ s , z ∗ s ) where z ∗ s ∈ argmax z s ∈ L t ( R d ) u s (cid:0) x ∗ s , y s +1 ( v t ( x ∗ s , z s )) , z s (cid:1) , s = t, . . . T − , (4.1) satisfies (( x ∗ s ) Ts = t +1 , ( z ∗ s ) T − s = t ) ∈ C t ( x t ) and y t ( x t ) = u t ( x t , · , z ∗ t ) ◦ · · · ◦ u T − ( x ∗ T − , · , z ∗ T − ) ◦ u T ( x ∗ T ) . Proof.
The proof is similar to Theorem 2.5. However, since the control set is not compact wehave to argue differently to show the existence of (2.4), i.e. that the supremum in y t ( x t ) := sup z ∈ L t ( R d ) u t (cid:0) x t , y t +1 ( v t ( x t , z )) , z (cid:1) , x t ∈ L t is attained. To do so, we first show that0 ≤ y t ( x ) − x ≤ K t for all x ∈ L t , (4.2)where K t := ( T − t ) K for all t = 0 , , . . . , T . For t = T , one has y T ( x ) − x = 0. By inductionsuppose that y t +1 ( x ) − x ≤ ( T − t ) K t +1 . Then, by (u2’) and (u5) for every z ∈ L t ( R d ) onehas u t (cid:0) x, y t +1 ( v t ( x, z ) (cid:1) , z ) − x = u t (cid:0) x, y t +1 ( v t ( x, z )) − v t ( x, z ) + v t ( x, z ) , z (cid:1) − x ≤ u t ( x, v t ( x, z ) , z ) − x + K t +1 ≤ K + K t +1 = K t so that y t ( x ) − x ≤ K t . As for the lower bound, suppose by induction that x ≤ y t +1 ( x ). By(v6), (u2’), (u5) and (u7) it follows that y t ( x ) ≥ u t (cid:0) x, y t +1 ( v t ( x, , (cid:1) ≥ u t (cid:0) x, y t +1 ( x ) , (cid:1) ≥ u t (cid:0) x, x, (cid:1) = u t (cid:0) x, , (cid:1) + x = x. Fix x ∈ L t . For every z ∈ L t ( R d ) which satisfies u t ( x, y t +1 ( v t ( x, , ≤ u t ( x, y t +1 ( v t ( x, z )) , z ),it follows from (4.2) (u5), (u7) and (v6) that x = u t ( x, x, ≤ u t ( x, y t +1 ( x ) , ≤ u t (cid:0) x, y t +1 ( v t ( x, , (cid:1) ≤ u t (cid:0) x, y t +1 ( v t ( x, z )) , z (cid:1) ≤ u t (cid:0) x, v t ( x, z ) , z (cid:1) + K t +1 . This shows that y t ( x ) = sup z ∈ Θ t ( x ) u t (cid:0) x, y t +1 ( v t ( x, z )) , z (cid:1) for the F t -stable setΘ t ( x ) := n z ∈ L t ( R d ) : u t ( x, v t ( x, z ) , z ) ≥ x − K t +1 o . It remains to show that Θ t satisfies (c1)-(c4). To that end, we verify (i) and (ii) of Proposition3.1. By (u3) and (v2) it follows that the set (cid:8) ( x, z ) ∈ L t × L t ( R d ) : z ∈ Θ t ( x ) (cid:9) is sequentially closed, which shows (i) of Proposition 3.1. As for (ii) of Proposition 3.1 let( x n ) be a sequence in L t such that x n → x ∈ L t a.s. Defining x := inf n x n ∈ L t as well as¯ x := sup n x n ∈ L t , it follows from (u2’) and (v3) thatΘ t ( x n ) ⊂ n z ∈ L t ( R d ) : u t (¯ x, v t (¯ x, z ) , z ) ≥ x − K t +1 o =: Θ t ( x, ¯ x )for all n ∈ N . Moreover, by (u4) and (v4) the set Θ t ( x, ¯ x ) is F t -convex. It remains to show thatthere exists M ∈ L t such that d L t ( R d ) ( z, ≤ M for all z ∈ Θ t ( x, ¯ x ). This L t -boundedness ofΘ t ( x, ¯ x ) would follow from [13, Theorem 3.13], if for all z ∈ Θ t ( x, ¯ x ) with z = 0, there exists A ∈ F t with P ( A ) > m →∞ u t (¯ x, v t (¯ x, mz ) , mz ) = −∞ a.s. on A. (4.3)Indeed, since by (v5) one has P ( v t (¯ x, z ) < ¯ x | F t ) > { z = 0 } , there exists l ∈ N such that A := (cid:8) P (cid:0) | ¯ x | + l ( v t (¯ x, z ) − ¯ x ) < | F t (cid:1) > (cid:9) ∈ F t satisfies P ( A ) >
0. By (v4) it follows that v t (¯ x, z ) ≥ m v t (¯ x, mz ) + m − m v t (¯ x, m (cid:0) v t (¯ x, z ) − ¯ x (cid:1) ≥ v t (¯ x, mz ) − ¯ x for all m ∈ N . This shows that u t (¯ x, v t (¯ x, mz ) , mz ) ≤ u t (¯ x, | ¯ x | + v t (¯ x, mz ) − ¯ x, mz ) ≤ u t (cid:16) ¯ x, ml ( | ¯ x | + l ( v t (¯ x, z ) − ¯ x )) , mz (cid:17) for all m ∈ N large enough. Hence, the condition (u6) implies (4.3). (cid:3) Example 4.2.
Let ( S t ) Tt =0 be a R d -valued adapted stochastic process modeling the discountedstock prices of a financial market model. Given a trading strategy ϑ t ∈ L t ( R d ), t = 0 , . . . , T − x ∈ L we define recursively the wealth process x t +1 = v t ( x t , ϑ t ) := x t + ϑ t · ∆ S t +1 , t = 0 , . . . , T − , where ∆ S t +1 := S t +1 − S t denotes the stock price increment. We assume the following no-arbitrage condition (which includes a relevance condition on the market model) ϑ · ∆ S t +1 ≥ ϑ ∈ L t ( R d ) implies ϑ = 0 a.s.for all t = 0 , . . . , T −
1. Then the forward generator v t : L t × L t ( R d ) → L t +1 satisfies (v1)–(v6).As for the backward generators, let u T : L T → L T be the identity and u t : L t × L t +1 → L t , u t ( x, y ) := 1 γ t ( x ) g t ( γ t ( x ) y ) , t = 0 , . . . , T − , ARAMETER-DEPENDENT STOCHASTIC OPTIMAL CONTROL IN FINITE DISCRETE TIME 13 where g t : L t +1 → L t is increasing, F t -concave, F t -translation invariant, sequentially uppersemi-continuous, g t (0) = 0 and lim r g t ( ry ) = −∞ on { P ( y < | F t ) > } . The function γ t : L t → L t, ++ is F t -stable, decreasing and sequentially continuous and models the riskaversion depending on the wealth x t at time t . Then, u t satisfies the conditions (u1),(u2’),(u3)–(u7). We only verify (u2’) and (u3). To prove (u2’) take x ≤ x and y ≤ y . Let β i :=1 /γ t ( x i ), for i = 1 ,
2. By using the monotonicity and F t -concavity of g t , we have g t (cid:16) y β (cid:17) ≥ g t (cid:16) y β (cid:17) ≥ β β g t ( y β ) + β − β β g t (0) = β β g t (cid:16) y β (cid:17) . Multiplying by β we obtain u t ( x , y ) ≥ u t ( x , y ). Due to the monotonicity of u t it sufficesto verify (u3) for decreasing sequences. Indeed, suppose that x k ց x a.s. and y k ց y a.s.Then, by the monotonicity of u t we have g t ( γ t ( x k ) y k ) ≥ γ t ( x k ) γ t ( x ) g t ( γ t ( x ) y ) for all k. Thus, by using that g t is sequentially upper semi-continuous and γ is sequentially continuouswe obtain g t ( γ t ( x ) y ) ≥ lim sup k →∞ g t ( γ t ( x k ) y k ) ≥ lim inf k →∞ g t ( γ t ( x k ) y k ) ≥ lim k →∞ γ t ( x k ) γ t ( x ) g t ( γ t ( x ) y ) = g t ( γ t ( x ) y ) . This shows that lim k →∞ g t ( γ t ( x k ) y k ) = g t ( γ t ( x ) y ), and therefore lim k →∞ u t ( x k , y k ) = u t ( x, y ).Given the wealth process ( x t ) Tt =0 , define the backward process y t ( x t ) = sup (( x s ) Ts = t , ( ϑ s ) T − s = t ) ∈ C t ( x t ) u t ( x t , · ) ◦ . . . ◦ u T − ( x T − , · ) ◦ u T ( x T ) , t = 0 , . . . , T − , where C t ( x t ) := n (( x s ) Ts = t , ( ϑ s ) T − s = t ) : x s +1 = x s + ϑ s +1 · ∆ S s +1 , for s = t, . . . , T − o . By in-duction, one can verify that y t ( x + c ) = y t ( x ) + c for every c ∈ L t − with t = 1 , . . . , T . Supposethere exists K > u t (cid:0) x, v t ( x, ϑ ) , ϑ (cid:1) − x ≤ γ t ( x ) g t (cid:0) γ t ( x ) ϑ ∆ S t +1 (cid:1) ≤ K (4.4)for all t = 0 , . . . , T − ϑ ∈ L t ( R d ), and x ∈ L t . Then, it follows from Proposition 4.1 that y ( x ) = sup (( x t ) Tt =0 , ( ϑ t ) T − t =0 ) ∈ C ( x ) u ( x , · ) ◦ . . . ◦ u T − ( x T − , · ) ◦ u T ( x T )is attained for every x ∈ L . For instance one could think of the dynamic entropic preferencefunctional with generators − γ t log (cid:0) E [exp( − γ t y ) | F t ] (cid:1) where the local risk aversion coefficient γ t = γ t ( x t ) depends on the current wealth x t . Noticethat lim m →∞ − log( E [exp( − my ) | F t ]) = −∞ on { E [ y < | F t ] > } .We conclude this section with a wealth-dependent dynamic risk sharing problem . Let A bea finite set of agents. Each agent a ∈ A is endowed with a wealth process ( H at ) Tt =0 with H at ∈ L t, ++ . The aim is to share optimally the aggregated endowment process H t = P a ∈ A H at , t =0 , . . . , T , with respect to a dynamic wealth-dependent utility. The utilities under considerationare of the form u t : L t, ++ × L t +1 , ++ → L t , u t ( x, y ) := xg t ( y/x )where g t : L t +1 → L t is an F t -concave, increasing and sequentially upper semi-continuousgenerator with g t (0) = 0 for all t = 0 , . . . , T −
1, and u T : L T, ++ → L T is the identity. Proposition 4.3.
The wealth-dependent dynamic optimal risk sharing problem y t (( H at ) a ∈ A )= sup (X a ∈ A u t ( H at , · ) ◦ . . . ◦ u T − ( x aT − , x aT ) : X a ∈ A x as = H s , x as ∈ L s, ++ , s = t + 1 , . . . , T ) has the optimal solution x a, ∗ s = H at H t H s , a ∈ A , s = t + 1 , . . . , T. Moreover, the function y t : L t (cid:0) (0 , ∞ ) | A | (cid:1) → L t is F t -stable, increasing and sequentially uppersemi-continuous for t = 0 , . . . , T , where | A | denotes the cardinality of A .Proof. Define ¯ y T : L T (cid:0) (0 , ∞ ) | A | (cid:1) → L T , ¯ y T (( x a ) a ∈ A ) = P a ∈ A x a , and for t = 0 , . . . , T −
1, let¯ y t : L t (cid:16) (0 , ∞ ) | A | (cid:17) → L t , ¯ y t (cid:0) ( x a ) a ∈ A (cid:1) = X a ∈ A x a ! g t ¯ y t +1 (cid:0) ( H at +1 ) a ∈ A (cid:1)P a ∈ A x a ! . By backward induction, it can be checked that ¯ y t is F t -stable, increasing and sequentiallyupper semi-continuous. We show that X a ∈ A u t ( H at , · ) ◦ u t +1 ( x a, ∗ t +1 , · ) ◦ . . . ◦ u T − ( x a, ∗ T − , x a, ∗ T ) = ¯ y t (( H at ) a ∈ A ) (4.5)and X a ∈ A u t ( H at , · ) ◦ u t +1 ( x at +1 , · ) ◦ . . . ◦ u T − ( x aT − , x aT ) ≤ ¯ y t (( H at ) a ∈ A ) (4.6)for all ( x as ) a ∈ A such that P a ∈ A x as = H s , s = t + 1 , . . . , T . It would follow from (4.5) and (4.6)that x a, ∗ s = H at H t H s , a ∈ A , s = t + 1 , . . . , T is an optimal solution and that y t (( H at ) a ∈ A ) = ¯ y t (( H at ) a ∈ A ). By induction, it can be checkedthat u s ( x a, ∗ s , · ) ◦ u s +1 ( x a, ∗ s +1 , · ) ◦ . . . ◦ u T − ( x a, ∗ T − , x a, ∗ T ) = x a, ∗ s ¯ y s (( H as ) a ∈ A ) H s (4.7)for all a ∈ A and s = t, . . . , T where we put x a, ∗ t = H at . By summing up (4.7) at s = t over A we obtain (4.5). We prove (4.6) by backward induction. It is true at T by definition. Let t ≤ s < T . Then X a ∈ A u s ( x as , · ) ◦ u s +1 ( x as +1 , · ) ◦ . . . ◦ u T − ( x aT − , x aT )= H s X a ∈ A x as H s g s (cid:18) x as u s +1 ( x as +1 , · ) ◦ . . . ◦ u T − ( x aT − , x aT ) (cid:19) ≤ H s g s X a ∈ A x as H s x as u s +1 ( x as +1 , · ) ◦ . . . ◦ u T − ( x aT − , x aT ) ! ≤ H s g s (cid:18) y s +1 (( H as +1 ) a ∈ A ) H s (cid:19) = y s (( H as ) a ∈ A )where the first inequality follows from F s -concavity and the last one by monotonicity of g s and the induction hypothesis. (cid:3) ARAMETER-DEPENDENT STOCHASTIC OPTIMAL CONTROL IN FINITE DISCRETE TIME 15 Connection to random set theory
The aim of this section is to discuss methodological similarities and differences of conditionalanalysis on the one hand and measurable selections and random set theory on the other hand.Both techniques were developed to deal with measurability. The two approaches can be brieflydescribed as follows.Measurable selections and random set theory are established on the basis of classical analy-sis. Therefore, they seek a set-valued formalization to which classical theorems can be appliedpointwisely. The rˆole of measurable selection theorems is then to secure measurability underpointwise application of classical theorems. This is achieved under topological assumptions.Conditional analysis relies on a measurable version of classical results which can be di-rectly applied to sets of measurable functions. Therefore, the formalization mainly consistsin describing those sets for which a measurable version of classical results can be proved [16].Measurability is then systematically preserved by the application of a conditional version ofclassical results, that is by construction. A measurable version of classical theorems (moregenerally, a transfer principle [8]) exists under measure-theoretic assumptions, while the topo-logical restrictions of measurable selection techniques and random set theory can be relaxed.In the following, we show that conditional analysis extends measurable selections and ran-dom set theory. More precisely, we establish a correspondence between basic objects in randomset theory and their analogues in conditional analysis under the hypothesis of separability. Un-less mentioned otherwise, we fix a measurable space (Ω , F ) and a Polish space E . Recall thata closed-valued map S : Ω ⇒ E (i.e. S ( ω ) ⊂ E is a closed set for all ω ∈ Ω) is Effros measur-able whenever S − ( O ) := { ω ∈ Ω : S ( ω ) ∩ O = ∅} ∈ F for all open sets O in E . We alwaysassume S − ( E ) = Ω. A measurable selection of S is a Borel function x : Ω → E such that x ( ω ) ∈ S ( ω ) for all ω ∈ Ω. The following measurable selection theorem is due to Castaing [9],where cl denotes closure.
Theorem 5.1.
A closed-valued map S : Ω ⇒ E is Effros measurable if and only if there existsa countable family of Borel functions x n : Ω → E such that S ( ω ) = cl { x n ( ω ) : n ∈ N } for each ω ∈ Ω . As an auxiliary result, a measurable version of the axiom of choice is needed which isadopted to the present setting in what follows, see [16] for the general statement. Let I ⊂ L ( N ) be a stable set. A family ( x i ) i ∈ I of elements of L ( E ) is said to be a stable family if x P k Ak i k = P k A k x i k for all ( A k ) ∈ Π F and sequences ( i k ) in I . A family ( H i ) i ∈ I of subsetsof L ( E ) is said to be a stable family of stable sets if each H i is a stable set and H P k Ak i k = X k A k H i k := n X k A k z k : z k ∈ H i k o for all ( A k ) ∈ Π F and sequences ( i k ) in I . Lemma 5.2.
Let ( H i ) i ∈ I be a stable family of stable sets in L ( E ) . Then there exists a stablefamily ( x i ) i ∈ I such that x i ∈ H i for all i ∈ I .Proof. Let H = { ( x j ) j ∈ J : x j ∈ H j for all j ∈ J, J ⊂ I stable } . As each H i = ∅ any single-element family { x j } with x j ∈ H j for some { j } ⊂ I is in H , andthus H = ∅ . Order H by the relation( x j ) j ∈ J ≤ (¯ x j ) j ∈ ¯ J if and only if J ⊂ ¯ J and x j = ¯ x j for all j ∈ J. There are other measurability concepts besides Effros measurability, see e.g. [36, Section 1.2]. One ofthem is graph-measurability (i.e. { ( ω, x ) ∈ Ω × E : x ∈ S ( ω ) } is product-measurable) which is important inapplications. For closed-valued mappings, graph measurability is equivalent to Effros measurability wheneverthe underlying measurable space is complete, cf. e.g. [36, Theorem 2.3]. It is straightforward to check that ( H , ≤ ) is a partially ordered set. Let ( x αj ) j ∈ J α be a chainin H . Define J := n X k A k j k : ( A k ) ∈ Π F , j k ∈ ∪ α J α for each k o . By definition J ⊂ I is stable. For j ∈ J , put x j = P k A k x αj k . Since ( H i ) is a stablefamily, ( x j ) j ∈ J ∈ H . By construction ( x αj ) j ∈ J α ≤ ( x j ) j ∈ J for all α . By Zorn’s lemma, thereis a maximal element ( x ∗ j ) j ∈ J ∗ ∈ H . By contradiction, suppose there is i ∈ I such thatsup j ∈ J ∗ { i = j } 6 = Ω. Let ˆ J = { A i + 1 A c j : j ∈ J ∗ , A ∈ F } , pick some x i ∈ H i , and defineˆ x ˆ j = 1 A x i + 1 A c x ∗ j for ˆ j ∈ ˆ J . Then (ˆ x j ) j ∈ ˆ J is an element of H , but ( x ∗ j ) j ∈ J ∗ < (ˆ x j ) j ∈ ˆ J . (cid:3) Fix a probability measure P on (Ω , F ) and complete F relative to P . We identify twoclosed-valued Effros measurable maps S and S whenever S ( ω ) = S ( ω ) a.s. Let X S denotethe set of measurable selections of a set-valued mapping S . In the following proposition, weconstruct a set-valued mapping which is associated to a set X in L ( E ), and which will bedenoted by S X . Theorem 5.3.
Let S : Ω ⇒ E be a closed-valued and Effros measurable mapping, and let X ⊂ L ( E ) be a stable and sequentially closed set. Then there exist closed-valued and Effrosmeasurable mappings S X : Ω ⇒ E and S X S : Ω ⇒ E satisfying the reciprocality relations S = S X S and X = X S X respectively.Proof. First, S X is constructed. Second, the reciprocality relations X = X S X and S = S X S are established.(i) Let F = { q , q , . . . } be a countable dense set in E . For each n ∈ L ( N ) and q ∈ L ( F ),define the random ball B /n ( q ) := { x ∈ L ( E ) : d ( x, q ) < /n a.s. } . Put I = { ( n, q ) ∈ L ( N ) × L ( F ) : X ∩ B /n ( q ) = ∅} . Inspection shows that I is a stable subset of L ( N ) × L ( F ) which can be identified with astable set in L ( N ) since F is countable. By Lemma 5.2, there is a stable family ( x i ) in L ( E )such that x i ∈ X ∩ B /n ( q ) for each i = ( n, q ) ∈ I . Next, we construct the largest measurableset A ∈ F restricted to which I is conditionally finite, where for m ∈ L ( N ), we denote by { ≤ k ≤ m } := { k ∈ L ( N ) : 1 ≤ k ≤ m } a random interval of integers which encodesconditional finiteness. For a set N in L ( N ), denote by 1 A N := { A n : n ∈ N } . Let E = { A ∈ F : there are m ∈ L ( N ) and a stable bijection f : 1 A { ≤ k ≤ m } → A I } . We want to show that A ∗ := ∪E ∈ E . By [22, Lemma 1, Chapter 30], there exists a sequence( A n ) in E such that A ∗ = ∪ n A n . Form B n = A n ∩ ( ∪ k ≤ n A ck ), each n . Then ( B n ) is a sequenceof elements in E . Indeed, if f : 1 A { ≤ k ≤ m } → A I is a stable bijection and B ⊂ A , then g (1 B k ) := 1 B f (1 A k ) defines a stable bijection g : 1 B { ≤ k ≤ m } → B I . Let f n : 1 A n { ≤ k ≤ m n } → A n I be a stable bijection. Then f ∗ : 1 A ∗ { ≤ k ≤ P n B n m n } → A ∗ I defined by f ∗ (1 A ∗ k ) := P n B n f n (1 A n k ) is a stable bijection. Thus A ∗ ∈ E . By maximality of A ∗ , thereexists a stable bijection g ∗ : 1 A c ∗ L ( N ) → A c ∗ I since on A ∗ the conditional index I is nowhereconditionally finite. Notice that P n B n m n can be rearranged as P k C k l k where ( l k ) is asequence of natural numbers and ( C k ) is pairwise disjoint. Now define S X ( ω ) := ( cl { x h ( ω ) : h = P k C k h k , ≤ h k ≤ l k } , ω ∈ A ∗ , cl (cid:8) x h ( ω ) : h = h Ω ∈ L ( N ) , h ∈ N (cid:9) , ω ∈ A c ∗ . By Theorem 5.1, the map S is Effros measurable and closed-valued. ARAMETER-DEPENDENT STOCHASTIC OPTIMAL CONTROL IN FINITE DISCRETE TIME 17 (ii) Inspection shows that X S X ⊂ X . Suppose there exists x ∈ X such that x X S X .Then there is n ∈ L ( N ) such that B /m ( x ) ∩ X S X = ∅ for all m ≥ n . This contradictsthe construction of S X . Hence X ⊂ X S X . By the previous, X S ( XS ) = X S . It follows fromTheorem 5.1 that S = S X S as well. (cid:3) Remark 5.4.
There exist characterization results that are related to Theorem 5.3, see e.g. [36,Theorem 2.1.6] and [32, Theorem 2.3] and the references therein for a background. In thisremark, we discuss how Theorem 5.3 relates to these results. Let E be a separable Banachspace, and let L p ( E ) be the Bochner space of all p -integrable functions x : Ω → E for p ∈ [1 , ∞ ].For a set-valued mapping S : Ω ⇒ E , denote by X pS := X S ∩ L p ( E ) the set of p -integrableselections of S . Let X ⊂ L p ( E ) be norm-closed. By [36, Theorem 2.1.6], X = X pS for an Effrosmeasurable closed-valued mapping S : Ω ⇒ E if and only if X is finitely decomposable . Anextension of [36, Theorem 2.1.6] to the case p = 0, when L ( E ) is endowed with the metricof convergence in probability, can be found in [32, Theorem 2.3]. In both cases ( p ∈ [1 , ∞ ]and p = 0) it can be verified that if a set X ⊂ L p ( E ) is decomposable and closed, then itis also infinitely decomposable, that is stable. Therefore, the previous proposition extendsthe aforementioned results to the case that E is a Polish space and convergence in norm orprobability is replaced by almost sure convergence.A frequently employed concept in stochastic optimal control is a normal integrand (seee.g. [41, 47] and the references therein), that is a function f : Ω × E → R whose epigraphicalmapping S f : Ω ⇒ E × R , S f ( ω ) := { ( x, r ) ∈ E × R : f ( ω, x ) ≤ r } , is closed-valued and Effrosmeasurable. A consequence of normality of an integrand is that f ( ω, x ( ω )) is measurable in ω whenever x : Ω → E is a measurable function. Moreover, a normal integrand f ( ω, x ) ismeasurable in ω for fixed x and lower semi-continuous in x for fixed ω (cf. [48, Proposition14.28]). We obtain the following “functional” version of Theorem 5.3, where two normalintegrands f : Ω × E → R and g : Ω × E → R are considered as identical if their epigraphicalmappings coincide a.s. Corollary 5.5.
Let u : L ( E ) → ¯ L be stable and sequentially lower semi-continuous andlet f : Ω × E → R be a normal integrand. Then there exist a stable and sequentially lowersemi-continuous function u f : L ( E ) → ¯ L and a normal integrand f u : Ω × E → R such that u f u = u and f u f = f .Proof. Due to normality, u f : L ( E ) → ¯ L given by x ( ω f ( ω, x ( ω ))) is well defined.Direct inspection shows that u f is stable and sequentially lower semi-continuous. Conversely,put X := { ( x, r ) ∈ L ( E × R ) : u ( x ) ≥ r } . By assumption, X is a stable and sequentiallyclosed subset of L ( E × R ). By Proposition 5.3, there exist an Effros measurable and closed-valued map S X : Ω ⇒ E × R corresponding to X . Thus f u : Ω × E → R ∪ {±∞} defined by f ( ω, x ) := inf S ( ω ) x a.s. is a normal integrand where S ( ω ) x denotes the x -section of S ( ω ). Itfollows from the reciprocality relations in Proposition 5.3 that u f u = u and f u f = f . (cid:3) We compare the assumptions which underly conditional analysis and measurable selections.Conditional analysis is applicable under the following two purely measure-theoretic hypothe-ses: The property “stability under (countable) gluings” is known under the name “decomposability” in mea-surable selections and random set theory, see e.g. [36, 39, 44], where it is usually employed in finite form,i.e. stability w.r.t. gluings along finite partitions. • A probability measure P on (Ω , F ) needs to be fixed a priori in order to identify sets,functions, relations, etc. • One consequently works in the context of conditional sets [16]. In particular, allinvolved sets must satisfy stability w.r.t. countable concatenations (cf. Definition 2.2).Conditional analysis does not rely on the following topological assumptions which are prevalentin measurable selections and random set theory: • standard Borel space , measure completeness, closed-valued mappings and Polish spaces.The established connections in Theorem 5.3 and Corollary 5.5 suggest that a stochastic controlproblem can equally be formalized in the language of conditional set theory. In e.g. [39, 41,42, 45, 47] some form of integrability is always assumed which leads to further technicalitiesin the proofs, see also [27, 33, 44] and the references therein for basic studies on the relationsof (conditional) expectations and integrands. The main results in Section 2 are established forgeneral utilities which are not necessarily in the form of expected utilities, and no integrabilityassumptions are required. Moreover, the following features of our control sets distinguish usfrom the existing literature. • We introduce a new notion of conditional compactness which works in finite and infinitedimensional settings thanks to a conditional version of the Heine-Borel theorem [16,Theorem 4.6]. Conditional compactness extends the notion of compact-valued andEffros measurable mappings, see [28] where it is proved that conditional compactnessuniquely corresponds to compact-valued and Effros measurable mappings in the finitedimensional case. • The control sets work in any conditional metric space. This involves many exampleswhich are out of reach of the existing technology, for example conditional L p -spaceson general probability spaces, L ( R ) n with a measurable dimension, and L ( X ) where X is a non-separable metric space. Another example are conditional weak topologieswhich are not included in this article for which conditional analysis offers extensivetools as well, see e.g. [16, 28, 49]. References [1] B. Acciaio and I. Penner. Dynamic risk measures. In G. D. Nunno and B. ¨Oksendal,
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Department of Mathematics and Statistics, University of Konstanz
E-mail address : [email protected] Department of Mathematics and Statistics, University of Konstanz
E-mail address : [email protected] Department of Mathematics, Universidad de Murcia
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