Parameter-dependent Stochastic Optimal Control in Finite Discrete Time
Asgar Jamneshan, Michael Kupper, Jos\'e Miguel Zapata
TL;DR
This paper establishes a broad existence theorem for stochastic optimal control in discrete time with controls in conditional metric spaces, utilizing conditional analysis and illustrating with various economic examples.
Contribution
It introduces a novel formalization in conditional metric spaces and applies conditional analysis techniques to stochastic control, extending beyond traditional methods.
Findings
Proves a general existence result for controls in conditional metric spaces.
Demonstrates applications in wealth-dependent utility maximization and risk sharing.
Connects conditional analysis with random set theory.
Abstract
We prove a general existence result in stochastic optimal control in discrete time where controls take values in conditional metric spaces, and depend on the current state and the information of past decisions through the evolution of a recursively defined forward process. The generality of the problem lies beyond the scope of standard techniques in stochastic control theory such as random sets, normal integrands and measurable selection theory. The main novelty is a formalization in conditional metric space and the use of techniques in conditional analysis. We illustrate the existence result by several examples including wealth-dependent utility maximization under risk constraints with bounded and unbounded wealth-dependent control sets, utility maximization with a measurable dimension, and dynamic risk sharing. Finally, we discuss how conditional analysis relates to random set theory.
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Parameter-dependent stochastic optimal control in finite discrete time
Asgar Jamneshan
Department of Mathematics and Statistics, University of Konstanz
,
Michael Kupper
Department of Mathematics and Statistics, University of Konstanz
and
José Miguel Zapata-García
Department of Mathematics, Universidad de Murcia
Abstract.
We prove a general existence result in stochastic optimal control in discrete time where controls take values in conditional metric spaces, and depend on the current state and the information of past decisions through the evolution of a recursively defined forward process. The generality of the problem lies beyond the scope of standard techniques in stochastic control theory such as random sets, normal integrands and measurable selection theory. The main novelty is a formalization in conditional metric space and the use of techniques in conditional analysis. We illustrate the existence result by several examples including wealth-dependent utility maximization under risk constraints with bounded and unbounded wealth-dependent control sets, utility maximization with a measurable dimension, and dynamic risk sharing. Finally, we discuss how conditional analysis relates to random set theory.
2010 Mathematics Subject Classification:
93E20, 28B20, 03E40
The authours would like to thank Ilya Molchanov for discussions on Section 5. The first two authors gratefully acknowledge financial support from DFG-Project KU 2740/2-1. The third author was supported by a grant associated to the project MTM2014-57838-C2-1-P (MINECO)
1. Introduction
The present work investigates parameter-dependent stochastic optimization in finite discrete time with the tools of conditional analysis. In the following, we introduce the mathematical problem and sketch our solution strategy. Given a forward generator , consider a forward process
[TABLE]
the dynamics of which depend on a parameter as a function of earlier decisions and an immediate decision chosen recursively in a state-dependent control set for each . Given a backward generator , the goal is to maximize
[TABLE]
over all possible forward processes initialized at where denotes composition of functions. By the Bellman principle, the global stochastic optimization problem (1.1) is solved by a backward recursion if all the local one-period problems
[TABLE]
attain their maxima.
Given a filtered probability space , we assume that the forward process and the control process assume values in -conditional metric spaces and respectively. An -conditional metric space is a nonempty set endowed with a vector-valued metric satisfying a concatenation property which encodes information at time . An example is the space of strongly -measurable functions with values in a metric space with almost everywhere evaluation of the metric. Intuitively speaking, a conditional metric space is a collection of classical metric spaces , , which are glued together in a measurable way. Instead of arguing in each separately and building on measurable selection lemmas, we directly work in the conditional metric space and build instead on arguments in conditional analysis. This is possible since in conditional metric spaces all basic results 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 be reduced by Bellman’s principle to the local optimization problems (1.2). By backward induction, we show that the value function is upper semi-continuous on the conditional metric space . For this, we assume that the control set is conditionally sequentially compact (for a discussion of the notion of conditional compactness, we refer to [13, Sections 3 and 4] and [16, Sections 3.4 and 4]). Then the existence of an optimizer in (1.2) follows from a conditional version of the fact that a semi-continuous function on a compact space attains its extrema. Moreover, under a regularity condition on the control set - a conditional version of outer semi-continuity in set convergence (see e.g. [48, Chapter 5, Section B]) - it is shown that in (1.2) is upper semi-continuous on . The assumption of conditional compactness on the control set is relaxed in Proposition 4.1 under stronger assumptions on the generators by modifying arguments in [11]. In particular, we additionally require that the backward generators are -sensitive to large losses and increasing in the state variable.
In Section 3 we provide sufficient conditions for conditional compactness and conditional outer semi-continuity of the control set. We focus on conditionally finite dimensional control sets. The results are illustrated with applications in mathematical finance. In Example 3.2 we study an optimal consumption problem with local risk constraints on the wealth process. Example 3.5 indicates the importance of conditional Euclidean space with measurable dimension to model control processes with state-dependent dimension (e.g. the number of traded assets at time depends on and past decisions). As an application of Proposition 4.1 we derive optimal portfolios w.r.t. dynamic risk measures for which the risk aversion coefficient is influenced by the current wealth. Moreover, a closed-form solution to a dynamic wealth-dependent risk sharing problem is obtained which extends the formula of Borch [7].
Normal integrands are a widely used tool to investigate parametrized stochastic optimization, see e.g. [5, 39, 43, 46] and [44] for an introduction. In Section 5, we establish a connection between conditional analysis and random sets, normal integrands and measurable selection theory. In Theorem 5.3, we prove a one-to-one correspondence between the set of measurable selections of Effros measurable and closed-valued mappings and stable and sequentially closed sets. This result yields a one-to-one correspondence between normal integrands and stable and sequentially semi-continuous functions. This indicates that control problems formulated in the language of normal integrands and random sets can equally be formulated in the language of conditional analysis. For a formalization with normal integrands and random sets, measurable selection lemmas provide the main tool to secure measurability. The use of measurable selection arguments is enforced by a pointwise application of standard results in classical analysis, and relies on topological assumptions such as separability and standard Borel spaces. In this regard, conditional analysis provides a measure-theoretic alternative which does not rely on any topological assumptions, and works as soon as a formalization within its language is reached which is demonstrated in this article in discrete time stochastic control theory. Conditional analysis approaches measurable functions directly by providing a measurable (or conditional, or stochastic, or random) version of results in classical analysis. The application of conditional versions of classical theorems preserves measurability, see for example the proofs below in which a measurable version of the Bolzano-Weierstraß theorem, the maximum theorem and the Heine-Borel theorem are employed. This perspective is implicitly present in [25, 29], however without a systematic treatment. A conditional version of basic results in functional analysis were established in [13, 18, 16], and applied to financial mathematics in [2, 6, 11, 12, 15, 19, 20, 21, 37]. In [16], conditional versions of classical theorems were studied systematically and related to a conditional variant of set theory. This naturally raises connections with mathematical logic, see [3, 8, 35] for related literature. Conditional analysis in -modules is moreover investigated in e.g. [10, 28, 38]. Related results in randomly normed modules are studied in e.g. [23, 24, 26].
The remainder of this article is organized as follows. In Section 2 we introduce the notion of conditional metric spaces and prove the main existence result. In Section 3 and Section 4 we discuss extensions of the main result and provide several examples. The link between conditional analysis and random set theory is established in Section 5.
2. Main result
Let be a probability space. Throughout we identify two sets in whenever their symmetric difference is a null set, and identify two functions on if they coincide a.s. (almost surely). Let be a sub--algebra of . Denote by the set of partitions of where for all . Let , , , , , and denote the spaces of -measurable random variables with values in , , , , and respectively. Recall that with the pointwise a.s. order is a Dedekind complete lattice-ordered ring. The essential supremum and the essential infimum are denoted by and respectively. Inequalities between random variables with values in an ordered set are always understood in the pointwise a.s. sense.
Definition 2.1**.**
A -conditional metric on a non-empty set is a function such that the following conditions hold:
- (i)
if and only if ,
- (ii)
- (iii)
,
- (iv)
for every sequence in and there exists exactly one element such that for all .
The pair is called a -conditional metric space.
In the following we call the unique element in (iv) the concatenation of the sequence along the partition and denote it by . For a sequence in a conditional metric space we write a.s. whenever a.s. A measurable subsequence of is of the form where is a sequence in such that for all .
Definition 2.2**.**
Let and be -conditional metric spaces, and and subsets of and , respectively. We call
- •
-stable if and for all and every sequence in ,
- •
sequentially closed if contains every such that there is a sequence in with a.s.
A function is said to be
- •
-stable if for all and every sequence in , where and are assumed to be -stable,
- •
sequentially continuous whenever if a.s. in ,
and if , then is said to be
- •
sequentially lower semi-continuous if if a.s. in ,
- •
sequentially upper semi-continuous if if a.s. in .
Remark 2.3**.**
1. If is a -conditional metric space then the metric is -stable, i.e. for every sequences and in and . Indeed, denoting by and the respective concatenations, it follows from the triangular inequality that
[TABLE]
which shows that for all . Summing up over all yields the desired -stability.
2. Let and be two -conditional metric spaces. Then its product endowed with the -conditional metric is a -conditional metric space. In the following all products of conditional metric spaces are endowed with this conditional metric.
3. Let be a -conditional metric spaces. Then the set of all pairs , where and are identified if and is a conditional set. In general, a conditional set is an abstraction of this example, and can be viewed as a set-like structure on which acts such that is closed w.r.t. countable concatenations of its elements along partitions in . Conditional set theory is investigated in [16] and does not require a metric structure as in Definition 2.1. For further results on conditional metric spaces in the context of conditional set theory we refer to [16, Section 4].
We next introduce the parameter-dependent stochastic optimal control problem for conditional metric spaces. For a fixed finite time horizon , we consider a filtration . For simplicity, we often abbreviate the index by , and write for instance for . For each , let and be -conditional metric spaces. Our aim is to study control problems for which the control set depends on , but also on a state parameter . For every , we assume that the state-dependent control set satisfies
- (c1)
for all , 2. (c2)
is -stable, i.e.
[TABLE]
for all and every sequence in , 3. (c3)
for every , the set is conditionally sequentially compact, i.e. for every sequence in there exists a measurable subsequence with such that a.s., 4. (c4)
for every sequence in such that a.s. and every sequence in there exists a measurable subsequence with and a sequence in such that a.s.
Note that -stability of implies that is -stable for all .
We consider forward generators
[TABLE]
which are
- (v1)
-stable, 2. (v2)
sequentially continuous.
For every we consider the set
[TABLE]
of all parameter processes which can be realized by the state-dependent controls for .
As for the objective function, we consider backward generators
[TABLE]
which are
- (u1)
-stable, 2. (u2)
increasing in the second component, 3. (u3)
sequentially upper semi-continuous.
We assume that is -stable and sequentially upper semi-continuous. Given such a family of backward generators, our goal is to maximize
[TABLE]
over all realizable state processes initialized at .
Remark 2.4**.**
The objective function in the stochastic control problem (2.1) is recursively defined. Its generators (aggregators) are functions between conditional metric spaces which satisfy monotonicity and semi-continuity. The aggregators are not necessarily (conditional) expected utilities. In case of (conditional) expected utility, the generators are closely related with dynamic and conditional risk measures, see [1, 6, 14, 19, 20]. The preferences which underly conditional expected utility functionals were studied in [15] under the name of conditional preference orders.
In decision theory, there is an extensive literature on recursive utilities starting with the seminal work [30, 31]. The preferences therein are defined on sets of temporal lotteries (probability trees), and follow a kind of Bellman recursive structure which is similar to the construction above 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-Zin utilities. See also [34] for a survey on non-expected utility theory. With the techniques of conditional analysis and based on results in BSDE theory, [12] solves a utility maximization problem in continuous time for Epstein-Zin utilities.
The following result shows that the global supremum in (2.1) is attained and can be reduced to 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 functions are -stable and sequentially upper semi-continuous for all , and can be computed by backward recursion
[TABLE]
Moreover, for every the process given by and the forward recursion
[TABLE]
satisfies and
[TABLE]
Proof.
The proof is by backward induction. For it follows from (2.1) that which by assumption is an -stable and sequentially upper semi-continuous function from to .
As for the induction step, assume that is -stable and sequentially upper semi-continuous, and that for each there exists such that . By (u1) and (v1) the function
[TABLE]
is -stable. Moreover, it is sequentially upper semi-continuous. Indeed, let be a sequence in such that a.s. and a.s. Since a.s. by (v2) it follows from the induction hypothesis that
[TABLE]
Since
[TABLE]
we have . Hence, by (u2), (u3) and (v2) we get
[TABLE]
which shows the desired sequential upper semi-continuity. As a consequence, the supremum in
[TABLE]
is attained for each . Indeed, since z\mapsto u_{t}\big{(}x,y_{t+1}(v_{t}(x,z)),z\big{)} and are -stable, it follows from standard properties of the essential supremum that there exists a sequence such that
[TABLE]
By (c3) there is a measurable subsequence with such that a.s. Since z\mapsto u_{t}\big{(}x,y_{t+1}(v_{t}(x,z)),z\big{)} is sequentially upper semi-continuous and -stable, it follows that
[TABLE]
which shows that the supremum in (2.4) is attained.
We next show that is sequentially upper semi-continuous. By contradiction, suppose that is a sequence in such that a.s. and on some with . Note that is -stable. Thus, by possibly passing to a measurable subsequence, we can suppose that there exists such that
[TABLE]
Denote by a respective maximizer of . By (c4) there exists such that a.s. by possibly passing to a measurable subsequence. By (c3) there exists a measurable subsequence with such that a.s. Since a.s. by -stability of the conditional metric , it follows from the triangular inequality that a.s. By the -stability of and (c2) it follows that is in and maximizes . Hence, it follows from (2.3) that
[TABLE]
Notice that, due to the -stability of , (2.5) is satisfied for any measurable subsequence of . Thus, we have that on , which is a contradiction. We conclude that is sequentially upper semi-continuous.
Finally, we show that . By induction hypothesis, for every and there exists \big{(}(x^{\ast}_{s})_{s=t+2}^{T},(z^{\ast}_{s})_{s=t+1}^{T-1}\big{)}\in C_{t+1}(v_{t}(x_{t},z_{t})) such that
[TABLE]
In particular, for and being a respective maximizer in (2.4) one has
[TABLE]
This shows that is an optimizer of (2.1) whenever it satisfies the local optimality criterion
[TABLE]
for all , where . In particular, every process which satisfies the forward recursion (2.2) is an optimizer for (2.1). ∎
Examples 2.6**.**
As for the illustration we provide examples of -conditional metric spaces which are of interest for the control and parameter spaces in Theorem 2.5.
1. Given a nonempty metric space , denote by the set of all strongly -measurable functions , i.e. the set of those for which there exists a sequence of countable simple functions with and , such that for a.a. . Notice that the metric extends from to with values in by defining
[TABLE]
where and are sequences of countable simple functions such that and for a.a. , and
[TABLE]
Notice that does not depend on the choice of approximating sequences and . Then is a -conditional metric space. For instance, if is a sequence in such that is the limit of the countable sequence and then the concatenation is the unique element in given as the limit of the countable simple functions for .
2. The conditional Euclidean space with measurable dimension is defined as
[TABLE]
The -conditional metric on is defined by
[TABLE]
where and . Here, denotes the -conditional metric on which extends the Euclidean metric on as defined in the previous example. Straightforward verification shows that is a -conditional metric space.
3. For , we define the conditional -space
[TABLE]
with -conditional metric . By definition, is a -conditional metric space.
3. Compactness condition for the control set
3.1. The finite dimensional case
Suppose that . As shown in Example 2.6 the Euclidean metric of extends to the -conditional metric .
Proposition 3.1**.**
Suppose that for each , the control set satisfies (c1), (c2) and the following conditions:
- (i)
\big{\{}(x,z)\in X_{t}\times L^{0}_{t}(\mathbb{R}^{d}):z\in\Theta_{t}(x)\big{\}}* is sequentially closed,*
- (ii)
for every sequence in with a.s. there exists such that for all .
Then, the control set satisfies (c1)-(c4).
Proof.
Let in be a sequence such that a.s., and . Since by assumption for some , the conditional Bolzano-Weierstrass theorem [13, Theorem 3.8] implies a measurable subsequence with such that a.s. for some . Since satisfies (c2) one has , and therefore by (i). This shows (c4) and (c3) follows by considering the constant sequence for all . ∎
Example 3.2**.**
Let be a -dimensional -adapted price process. Given an initial investment , we consider the wealth process
[TABLE]
where the control consists of an investment strategy and a consumption . The forward generator satisfies (v1) and (v2). We assume that the wealth process is regulated by
[TABLE]
i.e. is acceptable w.r.t. a -conditional convex risk measure for all . Recall that a -conditional convex risk measure is
- •
normalized, i.e. ,
- •
monotone, i.e. for all with ,
- •
-translation invariant, i.e. for all and ,
- •
-convex, i.e. for all and with .
By -translation invariance it follows that (3.1) is equivalent to . In addition, is -stable since it is -convex (see [13, Lemma 4.3]). Hence we consider the (wealth-dependent) control set
[TABLE]
which is -stable. Suppose that for every one has on and therefore on . Moreover, we assume that for all , and is -sensitive to large losses, i.e. on . Then the control set satisfies (i) and (ii) of Proposition 3.1. Indeed, consider the function defined as , which is -convex and therefore sequentially continuous by [13, Theorem 7.2]. Hence it follows that
[TABLE]
is -convex and sequentially closed, which shows (i). As for (ii) let be a sequence in such that a.s. For one has
[TABLE]
for all . Hence, it remains to show that is -bounded, i.e. there is such that for all . Since contains , by [13, Theorem 3.13] it is enough to show that for each with there exists such that . If this is obvious. Otherwise, one has , in which case on .
By Proposition 3.1 and Theorem 2.5 it follows that for every recursive utility function with backward generators , , satisfying (u1)-(u3) and , there exists a global optimizer of the utility maximization problem (2.1) satisfying the local criterion (2.2).
3.2. Measurable dimension
Suppose that is the conditional Euclidean space with measurable dimension that depends on the parameter (see Example 2.6 for the definition of the conditional Euclidean space with measurable dimension). Let be an -stable and sequentially continuous, where is endowed with the -conditional metric which extends the discrete metric on . The control set is chosen such that
- (c1)
for all , 2. (c2)
is -stable, i.e.
[TABLE]
for all and every sequence in ,
are satisfied.
Remark 3.3**.**
Since depends on the state we are in a more general setting as the main Theorem 2.5. However, since is endowed with the conditional discrete metric, for every sequence in such that there exists such that for all . In particular, for all and Theorem 2.5 still holds true by exploring the arguments on for sequences a.s. in the conditional space .
A variant of Proposition 3.1 for control sets with measurable dimension can be formulated as follows.
Proposition 3.4**.**
Suppose that for each , the control set satisfies (c1), (c2) and the following conditions:
- (i)
\big{\{}(x,z)\in X_{t}\times L^{0}_{t}(\mathbb{R})^{d_{t}(x)}:z\in\Theta_{t}(x)\big{\}}* is sequentially closed.*
- (ii)
For every sequence in with a.s. there exists and a measurable subsequence in such that for every for some such that for all .
Then, the control set satisfies (c1)-(c4).
Proof.
Let in be a sequence such that a.s., and . By Remark 3.3 there exists a measurable subsequence with such that for all . Hence, we can argue similar as in the proof of Proposition 3.1. ∎
Example 3.5**.**
Consider a portfolio maximization problem, where the number of traded assets depends on past decision. More precisely, given a portfolio chosen at time (with initial value ), the investor can rebalance the portfolio at time to
[TABLE]
Here, the state spaces and the control spaces both depend on the past decision . In line with Remark 3.3 the convergence is understood as a.s. in the conditional metric space , since for all for some . Suppose that the control set satisfies (c1), (c2) as well as (i) and (ii) of Proposition 3.4. Then, along the same argumentation as in Proposition 3.4 it follows that satisfies (c1)–(c4). Since satisfies (v1) and (v2), Theorem 2.5 is applicable whenever the backward generators satisfy (u1)-(u3).
The measurable dimension depending on past decisions allows for instance to add new assets at time (i.e. ) which are traded at . Notice that denotes the set of all attainable portfolios at time . For instance, let be a price process with fixed . Without frictions and short-selling constraints one has which satisfies (c1)-(c4). Transaction costs can be included into the model by considering for a solvency region , see e.g. [40] for a discussion of different market models. Notice that the solvency regions can be modeled state-dependently with measurable dimension .
4. Unbounded control sets
In this section we consider unbounded control sets and do not assume constraints on the controls, but derive (c3) and (c4) for upper-level sets of as a result of stronger assumptions on the forward and backward generators. In particular, we additionally need that the backward generators are -sensitive to large losses and increasing in the first argument, see (u5) and (u2’) below. Suppose that the forward generators
[TABLE]
- (v3)
is increasing in the first component, 2. (v4)
for all , and with , 3. (v5)
on for all and , 4. (v6)
for all .
As for the backward generators, let be the identity mapping, and
[TABLE]
satisfy (u1) and (u3) as well as
- (u2’)
increasing in the first and second component, 2. (u4)
for all , , and with , 3. (u5)
for all , , and , 4. (u6)
a.s. on for every and , 5. (u7)
for all .
Let be given as in (2.1) where
[TABLE]
Then the following variant of Theorem 2.5 holds.
Proposition 4.1**.**
Suppose that (v1)–(v6) and (u1),(u2’),(u3)–(u7) are fulfilled, and there exists a constant such that
[TABLE]
for all and . Then the functions are -stable, increasing and sequentially upper semi-continuous for all , and can be computed by backward recursion
[TABLE]
Moreover, for every the process given by and forward recursion
[TABLE]
satisfies and
[TABLE]
Proof.
The proof is similar to Theorem 2.5. However, since the control set is not compact we have to argue differently to show the existence of (2.4), i.e. that the supremum in
[TABLE]
is attained. To do so, we first show that
[TABLE]
where for all . For , one has . By induction suppose that . Then, by (u2’) and (u5) for every one has
[TABLE]
so that . As for the lower bound, suppose by induction that . By (v6), (u2’), (u5) and (u7) it follows that
[TABLE]
Fix . For every which satisfies , it follows from (4.2) (u5), (u7) and (v6) that
[TABLE]
This shows that
[TABLE]
for the -stable set
[TABLE]
It remains to show that satisfies (c1)-(c4). To that end, we verify (i) and (ii) of Proposition 3.1. By (u3) and (v2) it follows that the set
[TABLE]
is sequentially closed, which shows (i) of Proposition 3.1. As for (ii) of Proposition 3.1 let be a sequence in such that a.s. Defining as well as , it follows from (u2’) and (v3) that
[TABLE]
for all . Moreover, by (u4) and (v4) the set is -convex. It remains to show that there exists such that for all . This -boundedness of would follow from [13, Theorem 3.13], if for all with , there exists with such that
[TABLE]
Indeed, since by (v5) one has on , there exists such that
[TABLE]
satisfies . By (v4) it follows that
[TABLE]
which by (v6) implies m\big{(}v_{t}(\bar{x},z)-\bar{x}\big{)}\geq v_{t}(\bar{x},mz)-\bar{x} for all . This shows that
[TABLE]
for all large enough. Hence, the condition (u6) implies (4.3).
∎
Example 4.2**.**
Let be a -valued adapted stochastic process modeling the discounted stock prices of a financial market model. Given a trading strategy , , and an initial investment we define recursively the wealth process
[TABLE]
where denotes the stock price increment. We assume the following no-arbitrage condition (which includes a relevance condition on the market model)
[TABLE]
for all . Then the forward generator satisfies (v1)–(v6). As for the backward generators, let be the identity and
[TABLE]
where is increasing, -concave, -translation invariant, sequentially upper semi-continuous, and on . The function is -stable, decreasing and sequentially continuous and models the risk aversion depending on the wealth at time . Then, satisfies the conditions (u1),(u2’),(u3)–(u7). We only verify (u2’) and (u3). To prove (u2’) take and . Let , for . By using the monotonicity and -concavity of , we have
[TABLE]
Multiplying by we obtain . Due to the monotonicity of it suffices to verify (u3) for decreasing sequences. Indeed, suppose that a.s. and a.s. Then, by the monotonicity of we have
[TABLE]
Thus, by using that is sequentially upper semi-continuous and is sequentially continuous we obtain
[TABLE]
This shows that , and therefore .
Given the wealth process , define the backward process
[TABLE]
where . By induction, one can verify that for every with . Suppose there exists such that
[TABLE]
for all , , and . Then, it follows from Proposition 4.1 that
[TABLE]
is attained for every . For instance one could think of the dynamic entropic preference functional with generators
[TABLE]
where the local risk aversion coefficient depends on the current wealth . Notice that on .
We conclude this section with a wealth-dependent dynamic risk sharing problem. Let be a finite set of agents. Each agent is endowed with a wealth process with . The aim is to share optimally the aggregated endowment process , , with respect to a dynamic wealth-dependent utility. The utilities under consideration are of the form
[TABLE]
where is an -concave, increasing and sequentially upper semi-continuous generator with for all , and is the identity.
Proposition 4.3**.**
The wealth-dependent dynamic optimal risk sharing problem
[TABLE]
has the optimal solution
[TABLE]
Moreover, the function is -stable, increasing and sequentially upper semi-continuous for , where denotes the cardinality of .
Proof.
Define , , and for , let
[TABLE]
By backward induction, it can be checked that is -stable, increasing and sequentially upper semi-continuous. We show that
[TABLE]
and
[TABLE]
for all such that , . It would follow from (4.5) and (4.6) that
[TABLE]
is an optimal solution and that . By induction, it can be checked that
[TABLE]
for all and where we put . By summing up (4.7) at over we obtain (4.5). We prove (4.6) by backward induction. It is true at by definition. Let . Then
[TABLE]
where the first inequality follows from -concavity and the last one by monotonicity of and the induction hypothesis. ∎
5. Connection to random set theory
The aim of this section is to discuss methodological similarities and differences of conditional analysis 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 briefly described as follows.
Measurable selections and random set theory are established on the basis of classical analysis. Therefore, they seek a set-valued formalization to which classical theorems can be applied pointwisely. The rôle of measurable selection theorems is then to secure measurability under pointwise application of classical theorems. This is achieved under topological assumptions.
Conditional analysis relies on a measurable version of classical results which can be directly applied to sets of measurable functions. Therefore, the formalization mainly consists in 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 of classical results, that is by construction. A measurable version of classical theorems (more generally, a transfer principle [8]) exists under measure-theoretic assumptions, while the topological restrictions of measurable selection techniques and random set theory can be relaxed.
In the following, we show that conditional analysis extends measurable selections and random set theory. More precisely, we establish a correspondence between basic objects in random set theory and their analogues in conditional analysis under the hypothesis of separability. Unless mentioned otherwise, we fix a measurable space and a Polish space . Recall that a closed-valued map (i.e. is a closed set for all ) is Effros measurable111 There are other measurability concepts besides Effros measurability, see e.g. [36, Section 1.2]. One of them is graph-measurability (i.e. is product-measurable) which is important in applications. For closed-valued mappings, graph measurability is equivalent to Effros measurability whenever the underlying measurable space is complete, cf. e.g. [36, Theorem 2.3]. whenever for all open sets in . We always assume . A measurable selection of is a Borel function such that for all . The following measurable selection theorem is due to Castaing [9], where cl denotes closure.
Theorem 5.1**.**
A closed-valued map is Effros measurable if and only if there exists a countable family of Borel functions such that for each .
As an auxiliary result, a measurable version of the axiom of choice is needed which is adopted to the present setting in what follows, see [16] for the general statement. Let be a stable set. A family of elements of is said to be a stable family if for all and sequences in . A family of subsets of is said to be a stable family of stable sets if each is a stable set and
[TABLE]
for all and sequences in .
Lemma 5.2**.**
Let be a stable family of stable sets in . Then there exists a stable family such that for all .
Proof.
Let
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As each any single-element family with for some is in , and thus . Order by the relation
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It is straightforward to check that is a partially ordered set. Let be a chain in . Define
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By definition is stable. For , put . Since is a stable family, . By construction for all . By Zorn’s lemma, there is a maximal element . By contradiction, suppose there is such that . Let , pick some , and define for . Then is an element of , but . ∎
Fix a probability measure on and complete relative to . We identify two closed-valued Effros measurable maps and whenever a.s. Let denote the set of measurable selections of a set-valued mapping . In the following proposition, we construct a set-valued mapping which is associated to a set in , and which will be denoted by .
Theorem 5.3**.**
Let be a closed-valued and Effros measurable mapping, and let be a stable and sequentially closed set. Then there exist closed-valued and Effros measurable mappings and satisfying the reciprocality relations and respectively.
Proof.
First, is constructed. Second, the reciprocality relations and are established.
- (i)
Let be a countable dense set in . For each and , define the random ball
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Put
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Inspection shows that is a stable subset of which can be identified with a stable set in since is countable. By Lemma 5.2, there is a stable family in such that for each . Next, we construct the largest measurable set restricted to which is conditionally finite, where for , we denote by a random interval of integers which encodes conditional finiteness. For a set in , denote by . Let
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We want to show that . By [22, Lemma 1, Chapter 30], there exists a sequence in such that . Form , each . Then is a sequence of elements in . Indeed, if is a stable bijection and , then defines a stable bijection . Let be a stable bijection. Then defined by is a stable bijection. Thus . By maximality of , there exists a stable bijection since on the conditional index is nowhere conditionally finite. Notice that can be rearranged as where is a sequence of natural numbers and is pairwise disjoint. Now define
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By Theorem 5.1, the map is Effros measurable and closed-valued.
- (ii)
Inspection shows that . Suppose there exists such that . Then there is such that for all . This contradicts the construction of . Hence . By the previous, . It follows from Theorem 5.1 that as well.
∎
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 this remark, we discuss how Theorem 5.3 relates to these results. Let be a separable Banach space, and let be the Bochner space of all -integrable functions for . For a set-valued mapping , denote by the set of -integrable selections of . Let be norm-closed. By [36, Theorem 2.1.6], for an Effros measurable closed-valued mapping if and only if is finitely decomposable222The property “stability under (countable) gluings” is known under the name “decomposability” in measurable 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.. An extension of [36, Theorem 2.1.6] to the case , when is endowed with the metric of convergence in probability, can be found in [32, Theorem 2.3]. In both cases ( and ) it can be verified that if a set is decomposable and closed, then it is also infinitely decomposable, that is stable. Therefore, the previous proposition extends the aforementioned results to the case that is a Polish space and convergence in norm or probability is replaced by almost sure convergence.
A frequently employed concept in stochastic optimal control is a normal integrand (see e.g. [41, 47] and the references therein), that is a function whose epigraphical mapping , , is closed-valued and Effros measurable. A consequence of normality of an integrand is that is measurable in whenever is a measurable function. Moreover, a normal integrand is measurable in for fixed and lower semi-continuous in for fixed (cf. [48, Proposition 14.28]). We obtain the following “functional” version of Theorem 5.3, where two normal integrands and are considered as identical if their epigraphical mappings coincide a.s.
Corollary 5.5**.**
Let be stable and sequentially lower semi-continuous and let be a normal integrand. Then there exist a stable and sequentially lower semi-continuous function and a normal integrand such that and .
Proof.
Due to normality, given by is well defined. Direct inspection shows that is stable and sequentially lower semi-continuous. Conversely, put . By assumption, is a stable and sequentially closed subset of . By Proposition 5.3, there exist an Effros measurable and closed-valued map corresponding to . Thus defined by a.s. is a normal integrand where denotes the -section of . It follows from the reciprocality relations in Proposition 5.3 that and . ∎
We compare the assumptions which underly conditional analysis and measurable selections. Conditional analysis is applicable under the following two purely measure-theoretic hypotheses:
- •
A probability measure on needs to be fixed a priori in order to identify333Actually, one needs to fix a -ideal of such that the quotient is a complete Boolean algebra. The ideal of null sets of a -finite measure is one such example, see [16] and the references therein for more examples. sets, functions, relations, etc.
- •
One consequently works in the context of conditional sets [16]. In particular, all involved 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 prevalent in measurable selections and random set theory:
- •
standard Borel space444For example, for purposes of a dynamical programming principle in finite discrete time stochastic optimal control, more precisely the existence of disintegration of measure, the underlying measure space is additionally assumed to be standard Borel in [4]., measure completeness, closed-valued mappings and Polish spaces.
The established connections in Theorem 5.3 and Corollary 5.5 suggest that a stochastic control problem 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 technicalities in the proofs, see also [27, 33, 44] and the references therein for basic studies on the relations of (conditional) expectations and integrands. The main results in Section 2 are established for general utilities which are not necessarily in the form of expected utilities, and no integrability assumptions are required. Moreover, the following features of our control sets distinguish us from the existing literature.
- •
We introduce a new notion of conditional compactness which works in finite and infinite dimensional settings thanks to a conditional version of the Heine-Borel theorem [16, Theorem 4.6]. Conditional compactness extends the notion of compact-valued and Effros measurable mappings, see [28] where it is proved that conditional compactness uniquely corresponds to compact-valued and Effros measurable mappings in the finite dimensional case.
- •
The control sets work in any conditional metric space. This involves many examples which are out of reach of the existing technology, for example conditional -spaces on general probability spaces, with a measurable dimension, and where is a non-separable metric space. Another example are conditional weak topologies which are not included in this article for which conditional analysis offers extensive tools as well, see e.g. [16, 28, 49].
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