Conditional Interior and Conditional Closure of Random Sets

In this paper, we introduce two new types of conditional random set taking values in a Banach space: the conditional interior and the conditional closure. The conditional interior is a version of the conditional core, as introduced by A. Truffert and recently developed by Lépinette and Molchanov, and may be seen as a measurable version of the topological interior. The conditional closure is a generalization of the notion of conditional support of a random variable. These concepts are useful for applications in mathematical finance and conditional optimization.


Introduction
The conditional essential supremum and infimum of a real-valued random variable have been introduced in [1]. A generalization is then proposed in [2] for vector-valued random variables with respect to random preference relations. Actually, these two concepts are related to the notion of conditional core as first introduced in [3] 1 and developed in [4] for random sets in separable Banach spaces, with respect to a complete σ -algebra H. A conditional core of a set-valued mapping Γ (ω), ω ∈ Ω, is defined as the largest H-graph measurable random set Γ (ω) such that Γ (ω) ⊂ Γ (ω). This concept provides a natural conditional risk measure that generalizes the concept of essential infimum for multi-asset portfolios in mathematical finance. Applications are deduced for geometrical market models with transaction costs: see [4,5] and the theory with transaction costs developed in [6].
In this paper, we first introduce the open version of the conditional core as proposed in [3,4]. Precisely, if H is a complete sub-σ -algebra on a probability space, the conditional interior (or open conditional core)of a set-valued mapping Γ (ω), ω ∈ Ω, is defined as the largest H-measurable random open set Γ (ω) such that Γ (ω) ⊂ Γ (ω) P-almost every ω ∈ Ω. It may be seen as a measurable version of the classical interior in topology. One of our main contributions is to show the existence and uniqueness of such a conditional interior for an arbitrary random set in a separable Banach space. Then, the dual concept, the conditional closure, is introduced as a generalization of the conditional support of a real-valued random variable to a family of vector-valued random variables. A numerical application of the latter is deduced in conditional random optimization: We show that an essential supremum is a pointwise supremum on a conditional closure.
The paper is organized as follows. In Sect. 2, we recall the definition and usual properties of measurable random sets in Banach spaces. Then, we introduce the notion of conditional interior and show the existence of such sets in Banach spaces. In Sect. 3, the conditional closure is introduced and an application in conditional optimization is formulated in the next section. At last, we present an application in mathematical finance.

Conditional Interior
In all the paper, we consider a complete probability space (Ω, F, P) and a complete sub-σ -algebra H of F. Set-valued mappings we consider take their values in the family of all subsets of a separable Banach space X equipped with its Borel σ -algebra B(X ). In the following, we first recall the concept of graph measurable random set. We then introduce the concept of conditional interior of a random set. Existence is deduced from the existence of the conditional core of a closed random set, see [4], even if it is not always non-empty.
Recall that a random set Γ (ω), ω ∈ Ω, is a set-valued mapping that assigns to We say that Γ is H-graph measurable, if When the sets Γ (ω) are closed (resp. open) P-almost for all ω ∈ Ω, we shall say that Γ is closed (resp. open). We say that a F-measurable random variable ξ : F) where the set L 0 (X , F) of all X -valued Fmeasurable random variables is equipped with the metric of convergence in probability.

Remark 2.1 If
The following result may be found in [8,Th. 4.4] and, when applied, it is usually mentioned as measurable selection argument, see also [7,Theorem 2.3].

Theorem 2.1 (Measurable selection argument)
If Γ is an F-graph measurable random set which is non-empty and closed a.s., then L 0 (Γ , F) = ∅.
The following result is proven in [4,Proposition 2.7] whenever the random set we consider is closed or not. We denote by cl (X ) the closure of any subset X ⊆ X with respect to the norm topology. The closure of any subset of L 0 (X , F) is taken with respect to the metric topology of L 0 (X , F).
For x ∈ X and r ≥ 0, B(x, r ) denotes the open ball in X of center x and radius r andB(x, r ) is its closure. For any A ⊆ X and λ ∈ R, we use the convention that λ × A = {λa : a ∈ A}. In particular, 0 × A = {0}. Recall the following definition, see the paper by Truffert [3,4].

Definition 2.1
The H-conditional core m(Γ |H), of a set-valued mapping Γ , is the largest H-graph measurable random set Γ such that Γ (ω) ⊆ Γ (ω) a.s. Note that m(Γ |H) is the largest subset in the sense that, ifΓ is another H-graph measurable random set contained in Γ a.s., thenΓ (ω) ⊆ m(Γ |H)(ω) P-almost all ω ∈ Ω. The following result is proved in [4] and is an extension of the primal result of [3]. Recall that, if Γ is a random set, L 0 (Γ , H) ⊆ L 0 (X , H) is the set of all H-measurable random variables γ with values γ (ω) ∈ Γ (ω) a.s.

Proposition 2.2
Suppose that Γ is a closed F-graph measurable set that may be empty. Then, the conditional core m(Γ |H) exists and we have The conditional core plays a role in mathematical finance as it naturally appears when considering the dynamics of a self-financing discrete-time portfolio process where G t is the solvency set, see [6], i.e., V t−1 ∈ m(V t + G t |F t−1 ), see [5]. When the σ -algebra is trivial, the conditional core becomes the set of fixed points of X ; it is also related to the essential intersection considered in [9]. Notice that the H-measurable conditional core is mainly independent of F, i.e., instead of completing the underlying measurable space with respect to a single (or a dominated family of) probability measure(s), one may pass to the universal completion, and hence, avoid postulating the existence of any reference measure.
We now introduce an open version of the conditional core: Therefore, the random sets As H n is H-graph measurable and closed, H n ⊆ m(F n |H) for all n. We deduce We then deduce the general case: The following lemma is recalled for the sake of completeness. It is used in the proof above.

Lemma 2.1 Let O be an open set in a normed space. For every
As the sequence (z n ) n is bounded, we deduce by a compactness argument that for a subsequence z n → z as n → ∞. Then, z − x = r * hence z ∈ cl O. In the case where z ∈ O, z n ∈ O for n large enough since O is open, which yields a contradiction. Therefore, z ∈ ∂O. This implies that r * = d(x, z) ≥ d(x, ∂O) and finally r * = d(x, ∂O).

Conditional Closure
We now introduce the concept of conditional closure. The existence is proved in the following theorem which is the second main contribution of this paper. As previously, F and H are supposed to be complete with respect to some probability measure P. Proof The first part is a direct consequence of Corollary 2.3. Indeed, we first observe that Γ ⊆ X \o(X \Γ |H). Moreover, if Γ H is a closed-valued H-graph measurable set containing Γ , then X \Γ H ⊆ X \Γ . We deduce that X \Γ H ⊆ o(X \Γ |H) and, finally, X \o(X \Γ |H) ⊆ Γ H . Suppose that P({Γ ∩ B(γ , ε) = ∅} ∩ H ) = 0 for some H ∈ H and ε ∈ L 0 ((0, ∞), H). Therefore, by definition of the conditional closure as a smallest set, we have Indeed, the H-graph measurable set in the r.h.s. above contains Γ by assumption; hence, it contains cl (Γ |H). We get a contradiction since γ ∈ L 0 (cl (Γ |H), H) a.s. by assumption. Uniqueness is clear as the conditional closure of Γ is the smallest closed set, up to a negligible set, in the sense that it is included in any other H-measurable set containing Γ .
Remark 3. 1 We may deduce the conditional support of a random variable X ∈ L 0 (X , F). Precisely, there exists a smallest H-graph measurable random closed set denoted by supp H (X ) such that P(X ∈ supp H (X )) = 1. It is given by Notice that the conditional support is necessarily non-empty a.s. hence, it admits a measurable selection. In the following, we adopt the notation k A = {ka : a ∈ A} for any subset A ⊆ X and k ∈ A.
Consider now the case where ess sup H r = ∞. We need to show that Γ H = X . In the contrary case, on a nonnull set, we may construct a H-measurable selection ζ of X \Γ H . Moreover, on a smaller non-null set, r ≥ ζ − γ as ess sup H r = ∞. It follows that ζ ∈ Γ hence ζ ∈ Γ H , i.e., a contradiction.

Conditional Optimization
The second main contribution is the following. It allows one to compute numerically an essential supremum as a pointwise supremum on the conditional closure. This result is a generalization of [10, Proposition 2.9] and is useful in robust finance dynamic programming, see our example below. We recall that an integrand h(ω, x), (ω, x) ∈ Ω × X , is a jointly measurable function, which is lower semi-continuous in x and takes values in the extended real line R ∪ {+∞}, see [11,Corollary 14.34].
Proof As cl (Γ |H) is H-graph measurable and closed-valued, it admits a Castaing representation cl (Γ |H)(ω) = cl{γ n (ω) : n ∈ N} a.s. where, for all n, γ n ∈ L 0 (cl (Γ |H), H), by Proposition 2.1. Notice that we may adjust the values on a set of measure zero, and therefore assume that the equality holds everywhere on Ω, see the proof of [6,Proposition 5.4.4].
As P(Λ n |H) > 0 a.s., we deduce by (1) that for every n ≥ 1. Therefore, SinceB(γ , ε n ) is a.s. compact and h is a.s. lower semi-continuous, we deduce that inf z∈B(γ ,ε n ) h(z) = h(z n ) where z n ∈B(γ , ε n ) converges pointwise to γ as n → ∞. We finally deduce by lower semi-continuity that This inequality holds for any selection γ of cl(Γ |H). Therefore, we get that The conclusion of the lemma follows.
Recall that a set Λ of measurable random variables is said F-decomposable if for any finite partition (F i ) n i=1 ⊆ F of Ω, and for every family Decomposability was initially introduced by Rockafellar: see also [12]. In the following, we denote by Σ(Λ) the F-decomposable envelope of Λ, i.e., the smallest F-decomposable family containing Λ. Notice that The closure Σ(Λ) in probability of Σ(Λ) is decomposable even if Λ is not decomposable. By [6,Proposition 5.4.3], there exists an F-graph measurable closed random set σ (Λ) such that Σ(Λ) coincides with L 0 (σ (Λ), F), the set of all measurable selectors of σ (Λ).

Theorem 4.2 Let h(ω, x), x ∈ X , be an H ⊗ B(X )-measurable integrand. Let us consider a family Λ of measurable random variables so that
is the set of all measurable selectors of some F-graph measurable random closed set σ (Λ). Then, Proof Notice that for any finite partition (F i ) n i=1 ⊆ F of Ω, n ≥ 1, and for every family Therefore, as ess sup H {h(γ ) : γ ∈ Λ} ≥ h(γ ) a.s. for any γ ∈ Λ, we deduce that ess sup H {h(γ ) : γ ∈ Λ} ≥ h(γ ) a.s. for any γ ∈ Σ(Λ). Since h is l.s.c. and any γ ∈ Σ(Λ) is a limit of elements of Σ(Λ), we get that the inequality also holds for any γ ∈ Σ(Λ). Taking the essential supremum overall γ ∈ Σ(Λ), we deduce that and, finally, the equality holds since Λ ⊆ Σ(Λ). The last equality of the corollary is deduced from Theorem 4.1.

Application in Finance: Robust Super-Hedging of an European or Asian Option
We consider a financial market in discrete time defined by a complete stochastic basis (Ω, (F t ) T t=0 , P). We suppose that there is a non-risky asset whose price is S 0 = 1, without loss of generality. The (discounted) prices are modeled by a vector-valued stochastic process (S t ) T t=0 adapted to the filtration (F t ) T t=0 with values in R d , d ≥ 1. We consider the one step super-hedging problem between two dates t − 1 and t with t ≥ 1. We suppose that after time t − 1 but strictly before time t the portfolio manager observes the price S t−1 , as a consequence of her/his order. More precisely, the portfolio manager knows (S u ) u≤t−2 at time t − 1 and sends an order at time t − 1 which is executed with a delay so that the executed price S t−1 is only observed strictly after t − 1.
Let us consider, for each t ≤ T , Λ t ⊆ L 0 (R d + , F t ) an F t -measurable random set representing the possible prices for the risky assets at time t. We suppose that, at time t, the set Λ t may depend on the observed prices before time t − 1, i.e., to each vector of prices (S u ) u≤t−1 , we associate a set Λ t = Λ t ((S u ) u≤t−1 ) representing the possible next prices at time t given that we have observed the executed prices (S u ) u≤t−1 . Therefore: for all t = 1, · · · , T and S −1 ∈ R d is given .
Note that S t represents the prices (S 1 t , · · · , S d t ) of the risky assets proposed by the market to the portfolio manager when selling or buying. A typical case could be Λ t = L 0 (I t , F t ) with where (S bj ) d j=1 and (S aj ) d j=1 are, respectively, the bid and the ask price processes observed in the market at time t that may depend on (S u ) u≤t−1 . They are not necessary the best bid/ask prices as, in practice, the real transaction price may be a convex combination of bid and ask prices. Indeed, a transaction is the result of an agreement between sellers and buyers, but it also depends on the traded volume. Clearly, the portfolio manager does not benefit from the last price observed in the market when sending an order. On the contrary, he should face an uncertain price S t which depends on the type of order (which may be not executed), but it also depends on some random events he does not control, e.g., slippage. A simple way to model this phenomenon is to suppose that the executed prices obtained by the manager belong to random intervals.
Another interesting case could be when Λ t coincides with a parametrized family {S θ t : θ ∈ Θ} of random variables. For instance, consider fixed processes (ξ u ) u≤T and (m u ) u≤T adapted to (F t ) T t=0 and independent of F t−1 . Let C be a compact set and suppose that S −1 is given. We define recursively In this model, there is an uncertainty on prices because of the unknown parameter (e.g., volatility) σ .
In the following, we consider the σ -algebra F t−1 = σ (S u : u ≤ t − 1) for all t ≥ 1. Let us consider a random function g t defined on R t , t ≥ 1. We assume that the mapping (ω, z) → g t (S 0 (ω), · · · , S t−1 (ω), z) is F t−1 × B(R)-measurable and z → g t (S 0 , S 1 , · · · , S t−1 , z) is lower-semicontinuous (l.s.c.) almost surely whatever the price process (S u ) u≤t−1 . Our goal is to characterize the set P t−1 of all V t−1 ∈ L 0 (R, F t−1 ) such that for some θ t−1 ∈ L 0 (R d , F t−1 ). We observe that, by lower-semicontinuity, (2) holds if and only if Recall that Σ(Λ t ((S u ) u≤t−1 )) is defined in the previous section. This means that we may suppose w.l.o.g. that Σ(Λ t ((S u ) u≤t−1 )) = Λ t ((S u ) u≤t−1 ). In the following, we denote by I t ((S u ) u≤t−1 ) the F t -measurable closed random set such that In the formula above, f * t−1 (y) = sup z∈R d (yz − f t−1 (z)) is the Fenchel-Legendre conjugate function of f t−1 defined as , ∞} is infinite on the complimentary of cl (I t ((S u ) u≤t−1 )|F t−1 ) and 0 otherwise. Notice that f * t−1 is convex and l.s.c. as a supremum (on cl (I t ((S u ) u≤t−1 )|F t−1 )) of convex and l.s.c. functions. Moreover, by Theorem 4.1, We deduce that the F t−1 -measurable prices at time t − 1 are given by The second step is to determine the infimum super-hedging price as p t−1 ((S u ) u≤t−1 ) = ess inf F t−1 P t−1 ((S u ) u≤t−1 ).
In this section, we have solved the super-hedging problem without any no-arbitrage condition, contrarily to what it is usual to do.

Conclusions
In Sect. 4.2, we have solved the general super-hedging problem in one step for an Asian option. The next step is to repeat the whole procedure to deduce backwardly the infimum prices and the associated super-hedging strategy from the maturity date T to the starting date t = 0. This is a current project, that shows the relevance of the conditional closure.
The conditional closure could be also useful more generally for robust finance dynamic programming as in the paper [13]. We conjecture that it is possible to solve a discrete-time stochastic control problem through random set conditioning.
At last, some interesting problems leave open in the direction of conditional topologies: see [14]. A deeper study of the basic properties of the conditional closure and interior of random sets may be interesting with a comparison to the classical results of topology but also with the paper by Truffert [3]. This also allows to consider new types of martingales, see [12], and, in continuous time, new problems should arise.