Random optimization on random sets

Random sets and random preorders naturally appear in financial market modeling with transaction costs. In this paper, we introduce and study a concept of essential minimum for a family of vector-valued random variables, as a set of minimal elements with respect to some random preorder. We provide some conditions under which the essential minimum is not empty and we present two applications in optimisation for mathematical finance and economics.

all financial positions whose liquidation values are non negative at time t. We observe that G t is random as it depends on the future prices only observed at time t. These random solvency sets generate stochastic random preorders t , precisely x t y if x − y ∈ G t . They define the dynamics of the portfolio processes (V t ) t=0,1,...T in discrete time, i.e., V t − V t−1 ∈ −G t (see Kabanov and Safarian 2009, Section 3.1.1), or equivalently V t−1 t V t , t ≥ 1.
Recall that, if ξ is a measurable payoff at time T , the classical super-replication problem is to characterize the set of all portfolio processes (V t ) t=0,1,...,T such that V T T ξ a.s. In the case where the solvency set G is not a cone, classical tools from convex analysis are no more appropriate to solve the problem. Precisely, it is not possible to characterize the super-hedging prices through the dual elements. The latter are the risk-neutral probability measures for frictionless models (see Dalang et al. 1990), and the consistent price systems, i.e., martingales evolving in the positive dual of the solvency sets, for models with proportional transaction costs (see Kabanov and Safarian 2009, Section 3.3).
This is a motivation for new approaches. Techniques based on random preorders and optimization on random sets, are suggested for non conical models and, more generally, for non convex models (see Lépinette and Tran 2016) with fixed costs. Set-valued optimisation may be also used to compute the super-hedging prices of European payoffs when the probability space is finite (Löhne and Rudloff 2014). The same approach is used for set-valued risk measures in presence of proportional transaction costs (Hamel et al. 2011;Feinstein and Rudloff 2015). More recently, a conditional analysis technique is implemented to solve stochastic optimal problems in discrete-time (see Jamneshan et al. 2017), with applications in finance and economics. Similarly, a dynamic programming principle is established in Baptiste et al. (2019). All these works are based on new ideas from conditional analysis applied to the theory of random sets. In particular, the random orders defined by the solvency sets play a crucial role.
For non convex financial market models, we also need to consider new ideas. In the following, we consider the portfolio processes which are minimal with respect to the random preorders. We refer to Lépinette (2013a, b, 2015) for conic models and the concept of essential supremum of a family of vector-valued random variables, with respect to the random preorder defined by a convex cone. In the present paper, we consider a more general setting where the random preorder is either defined by a random set, which is not necessarily convex or is defined by a random countable multi-utility representation. We introduce the notion of essential minimum of a family of vector-valued random variables with respect to the random preorder. Our main contribution is to show the existence of minimal elements, i.e., the essential minimum is not empty, under mild conditions. Finally, we illustrate our main result by two applications. The first one, in mathematical finance, improves a result of Lépinette and Tran (2016) by characterizing the minimal portfolio processes super-replicating a European claim. The second one is a classical problem in economics that we consider in a random environment. Precisely, we minimise a random cost function on a random set.
To do so, we first recall the notions of random set (see Molchanov 2005 for a complete overview) and random preorders with examples. The main result (Theorem 4.1) is then formulated. At last, we present the two applications.

Measurability of random sets
Let R d be the Euclidean space with norm · and the Borel σ -algebra B(R d ). The closure of a set A ⊂ R d is denoted by cl A.
A set-valued mapping ω → X (ω) ⊂ R d from a complete probability space ( , F, P) to the family of all subsets of R d is called F-measurable (graph measurable in Hess (2002) but the two definitions of measurability are equivalent for complete σ -algebras) if its graph In this case, X is said to be a random set. In the same way, the H-measurability of X with respect to a complete sub-σ -algebra H of F is defined. Unless otherwise stated, by the measurability we always understand the measurability with respect to F. The random set X is said to be closed if X (ω) is a closed set for almost all ω. The inclusion between two random sets is understood up to a negligible set.
for almost all ω ∈ is said to be an F-measurable selection of X , L 0 (X , F) denotes the family of all F-measurable selections of X , and L p (X , F) is the family of p-integrable ones.
By Hess (2002, Th. 4.4), an a.s. non-empty random set X has at least one selection, i.e., L 0 (X , F) = ∅ where L 0 (R d , F) is equipped with the topology induced by the convergence in probability. The expression by measurable selection argument will be used in this paper to mention the existence of such a measurable selection when the random set is not empty.
The following result is proved in Lépinette and Molchanov (2019).

Theorem 2.3
Let Ξ be a non-empty subset of L 0 (R d , F). There exists a H-measurable random closed set X such that

if and only if Ξ is H-decomposable and closed.
We deduce the existence of a Castaing representation for any random set (Molchanov 2005, Th. 2.2.3), as stated in the following proposition (see Lépinette and Molchanov 2019).

Proposition 2.4
If X is a random set, then its pointwise closure cl X (ω), ω ∈ , is a random closed set, and L 0 (cl X , F) = cl L 0 (X , F). Furthermore, there exists a countable family (ξ i ) i≥1 of measurable selections of X such that cl X = cl{ξ i , i ≥ 1} a.s.

Random preorders
We recall that a random binary relation (or preference relation), denoted by , is a relation between the elements of R d which is reflexive. The relation is said random as it is supposed to depend on each ω ∈ . A simple example is when the random relation is defined by x y if and only if x − y ∈ G(ω), where G ⊂ R d is a random set containing 0 ∈ R d . In the case where is also transitive, we say that is a random preorder. We extend it to the set of all measurable random variables, i.e., for any γ 1 , γ 2 ∈ L 0 (R d , F), we write γ 1 γ 2 when the set of all ω such that γ 1 (ω) γ 2 (ω) is of full measure. We shall consider two types of random preorders we define below. To do so, we need to introduce the following definitions.
Definition 3.1 Let H be a complete sub σ -algebra of F and let us consider h : × R k → R. The function h is called an H-normal integrand (see Rockafellar and Wets 1998, Definition 14.27) if h is H ⊗ B(R k )-measurable and lower semi-continuous (l.s.c. in the sequel) in x (see Rockafellar and Wets 1998, Corollary 14.34).
Definition 3.2 A random preorder on R d is said of type I if there exists random functions Ł and u such that: (i) The random function Ł is super-additive, 1 satisfies Ł(0) = 0 and, for all

Definition 3.3
The random preference relation is said of type II if there exists a countable family of random functions (u i ) i∈N which are F-normal integrands and such that, Note that the random preference of type I is not trivial if the set {Ł = 0} is not reduced to {0} or R d . Similarly, if it is of type II, it is not trivial as soon as u i (x) = u i (y) for at least one i ∈ N when x = y. We only consider countable multi-utility representations to obtain some measurability properties. In the case where the multiutility representation is reduced to a singleton, the points of R d are all comparable. This does not hold if the binary relation is represented by at least two utility functions. An example is the natural order on R 2 .
s. In mathematical finance, G is interpreted as the set of all financial positions which are solvent, i.e., that can be liquidated without any debt. Let us consider In finance, Ł(x) is the liquidation value of the financial position x.
Let us define the binary relation x y if x − y ∈ G. By the assumptions on G, this is a random preorder and x y if and only if Ł( In the case where |Ł(x n )| → +∞ for a subsequence, we use the normalizationx n := x n /(1 + |Ł(x n )|) → 0, and we get that −e 1 ∈ K as n → ∞ hence a contradiction. We deduce that lim sup n Ł(x n ) < ∞ and, as n → ∞, x − lim sup n Ł(x n )e 1 ∈ G. We conclude that the inequality Ł( Finally, we suppose that it is possible to separate the points of the boundary ∂G by a bounded F-normal integrand u in the sense that x − y ∈ ∂G implies that u(x) > u(y) if x = y. In that case, is of type I since {Ł = 0} ⊂ ∂G. This condition is satisfied by the financial model we present further.
, is of type II. Example 3. 6 We propose an example of random preorder which is of type II but not of type I. Let ξ be a positive random variable defined on a complete probability space ( , F, P). Let us define the random utility function This function is l.s.c. and measurable with respect to F ⊗ B(R). We consider the binary relation on R defined by x y if u(x) ≥ u(y). We may verify that the relation defines an order on R. In particular, it is of type II but it is not of type I. Indeed, in this example, the set Observe that it is not trivial to show whether a random preorder of type I is of type II or not. In Kabanov and Lépinette (2015, Proposition 2.8), a condition is given for a preorder to admit a countable family of utility functions.
The following concepts are introduced to solve minimization problems with respect to random preorders. See also Kabanov and Lépinette (2013a;2013b; where similar notions are considered.
In the following, we introduce the concept of essential minimum as proposed in Kabanov and Lépinette (2013a) and other papers. We follow the initial idea using the adjective essential to define the notion of essential infimum, which is a measurable minimum of a non necessarily countable family of random variables, see Definition 6.2.
Definition 3.8 Under the conditions of Lemma 3.7, we denote by Essmin H (X ) the unique setX satisfying Conditions (i) and (ii) above when it exists.
A trivial example on the real line R is to consider its natural order and the family , see Definition 6.2. In the following example, the essential minimum is not closed.
Example 3. 9 We consider in R 2 a deterministic polyhedral closed cone defined by G := cone (i 1 , i 2 ). We suppose that it is generated by two unit vectors i 1 , i 2 ∈ int R 2 + such that (i 1 , i 2 ) = +π/3. In the following, we denote by  C. We deduce that Essmin(X ) is not closed.
We may only consider random preorders if we only consider a weak version of the essential minimum. To do so, we write γ 1 ∼ γ 2 if γ 1 γ 2 and γ 2 γ 1 . In the case of a preorder, the equivalence classes are not necessarily reduced to singletons. H) is not empty. There exists at most one subsetX ⊂ X H denoted byX = Essmin w H (X ) satisfying the following properties:
Proof It suffices to repeat the proof of Lemma 3.7 where we use (iii) instead of the antisymmetry condition.

Definition 3.11
Under the conditions of Lemma 3.7, we denote by Essmin w H (X ) the unique setX satisfying Conditions (i), (ii) and (iii) above when it exists.
When necessary, we denote by Essmin H (X ) or Essmin w, H (X ) the essential minimum Essmin H (X ) or Essmin w H (X ) to specify the preorder. Example 3.12 (Minimization on a random set) Let D be a random F-measurable closed set we interpret as the domain of control variables x ∈ R d satisfying some random contraints. Let c be a random cost function we suppose to be an F-normal integrand. Our goal is to study the problem of minimizing c over D, i.e., to solve min x∈D(ω) c(ω, x), for each ω ∈ . We propose to construct measurable minimizers of the problem with values in D. This is why we introduce the following random preorder of type II: x y if c(x) c(y) a.s. We show that min x∈D(ω) c(ω, x) = c(γ ) a.s. whatever γ ∈ Essmin w F (L 0 (D, F)) = ∅, see Theorem 6.1.

Main results
The goal of this section is to provide conditions under which the essential minimum X = Essmin H (X ) (resp. Essmin w H (X )) of some family of random variables X ⊂ L 0 (R d , F) is not empty. H) is not empty, is H-decomposable and closed in L 0 . Assume that for any γ ∈ X H and any sequence (γ n ) n∈N ∈ X H such that γ n γ , we have lim inf n |γ n | < ∞ a.s. If the random preorder is of type I, thenX = Essmin H (X ) = ∅.

Application in finance
On a complete stochastic basis ( , (F t ) t=0,...,T , P), we consider a financial market model defined by a sequence of random sets (G t ) t=0,...,T such that G t is F t -measurable at any instant t = 0, . . . , T and T ≥ 1 is an horizon date. We suppose that, for any t, G t satisfies the conditions of Example 3.4. In particular, the liquidation value process is given by so that G t = {x : Ł t (x) ≥ 0} is the set of all solvent financial portfolio positions which can be liquidated without any debt. In the following, we define the associated random orders t by x t y if and only if x − y ∈ G t . Finally, we suppose that the points of the boundary ∂G may be separated by a bounded F-normal integrand u in the sense that x − y ∈ ∂G implies that u(x) > u(y) if x = y. In that case, is of type I since {Ł = 0} ⊂ ∂G.
A classical 2-dimensional model is defined by a discounted bond S 0 t = 1 and a discounted risky asset defined by bid and ask prices S b and S a such that S b ≤ S a . Recall that S b is the price we get when selling one unit of risky asset and S a is the price we pay when buying one unit of risky asset. In that case, Ł t ((x, y)) = x + y + S b t − y − S a t where x + := max(x, 0) and x − := − min(x, 0). Moreover, G t is a convex cone containing R 2 + whose positive dual is the smallest convex cone containing {1} × [S b t , S a t ]. In Lépinette and Tran (2016), there is also a model where a fixed cost is charged for every transaction. In that case, the solvency sets (G t ) t=0,...,T are no more convex but still define orders satisfying the conditions of Example 3.4. In the following, the solvency sets (G t ) t=0,...,T are not necessarily convex.

Definition 5.1 A portfolio process is a stochastic process
(5.1) The interpretation of the dynamics (5.1) is simple: we may write V t−1 = V t + (− V t ) so that it is possible to change V t−1 into V t since we may liquidate the residual term (− V t ) without any debt. We may also interpret (5.1) as the payment of transaction costs when changing V t−1 into V t (see Kabanov and Safarian 2009, Section 3). Notice that (5.1) may be reformulated as V t−1 t V t for all t ≤ T .
The classical problem in finance is to characterise the set of all portfolio processes whose terminal values are larger than some given contingent claim ξ ∈ L 0 (R d , F T ) of a financial derivative, i.e., some wealth ξ which is delivered at the maturity date upon payment at time 0. The mathematical problem is to estimate the prices of ξ at any time t, i.e., the values V t of portfolio processes super-hedging ξ , i.e., V T ≥ ξ .
In general, in particular for models without transaction costs, a no-arbitrage condition is imposed to characterize the prices of a contingent claim ξ (see Kabanov and Safarian (2009), Proposition 2.1.9). In the following, we suppose that the no-arbitrage condition called NA2, 2 initially introduced by Ràsonyi (2009), is satisfied. This condition means that it is not possible to construct a solvent terminal portfolio when starting from an initial capital which is not solvent. In Lépinette and Tran (2016), it is proven that NA2 is equivalent to L 0 (G t+1 , F t ) ⊂ L 0 (G t , F t ) for all t ≤ T −1. We may reformulate this equivalent condition as m(G t+1 |F t ) ⊂ G t a.s. for all t ≤ T − 1, where m(G t+1 |F t ) is the largest F t -measurable set contained in G t+1 , called conditional core (see Lépinette and Molchanov 2019). Under NA2, we propose to characterize the minimal portfolio processes super-hedging ξ . To do so, let us denote by H ξ the set of all portfolio processes super-hedging ξ .
Definition 5.2 A portfolio process (V t ) t=0,...,T of H ξ is said minimal if whatever V ∈ H ξ , the conditionV t t V t implies thatV = V .
Notice thatV t t V t means thatV t = V t + g t where g t ∈ L 0 (G t , F t ), i.e., we need to add the extra position g t to V t to obtainV t . In particular, Ł t (V t ) ≥ Ł t (V t ). Let us introduce the minimal cost function i.e., C t (z) is the minimal amount of cash we need to get the position z. We may show i.e., g t = 0. Therefore, if g t = 0, C t (g t ) > 0, i.e., it is expensive to change V t intoV t . In other words, V t is cheaper thanV t . We deduce that the minimal portfolios in Definition 5.2 are the cheapest ones. Knowing these minimal portfolios is sufficient as any portfolio of H ξ may be reduced to a minimal one.
The natural problem is to study the existence of minimal portfolio processes and to obtain a characterization of them. We denote by H ξ min the set of all minimal portfolio processes super-hedging ξ and by H ξ min (t) their values at time t. The following result improves (Lépinette and Tran 2016, Proposition 4.9), which is only formulated for a random preorder defined by G t+1 instead of G t here. We provide a backward characterization of H ξ min , which allows to compute the minimal portfolio processes.

Theorem 5.3 Suppose that Condition NA2 holds. Suppose that
Let us now verify that, for any γ ∈ L 0 (R d , F T −1 ) and for all sequence (γ n ) n∈N of X F T −1 such that γ n T −1 γ , we have lim inf n |γ n | < ∞ a.s. To see it, observe that ξ T γ n T −1 γ . On F T −1 = {lim inf n |γ n | = ∞} ∈ F T −1 , divide the previous inequality by 1 + |γ n | and we may assume by (Kabanov and Safarian 2009, Lemma) thatγ n := γ n /(1 + |γ n |) →γ on F T −1 such that |γ | = 1 and 0 Tγ T −1 0. In particular,γ ∈ L 0 (G T , . By the previous reasoning and Theorem 4.1, we get that We then reiterate the construction. From H ξ min (T ) = ξ , we generate a minimal portfolio. Precisely, starting fromV t+1 ∈ H ξ min (t + 1), it suffices to consider Reciprocally, by construction, (0) is the initial value of some minimal portfolio processV witĥ V t ∈ H ξ min (t) for all t ≤ T . Moreover, it is clear by construction that, for any V ∈ H ξ , there existsV ∈ H ξ min such that V t tVt , for all t ≤ T , which also proves that the sets H ξ min (t), t ≤ T , generate H ξ min = ∅.

Random minimisation on a random set
In this section, we develop Example 3.12. As announced in the example, we are interested in Essmin w F (X ) where X = L 0 (D, F) is a non empty set of measurable controls. Any γ ∈ L 0 (D, F) may be reduced into a cheaper control γ ∈ Essmin w F (X ) ⊂ L 0 (D, F), i.e., such that c(γ ) ≥ c(γ ). We claim that anŷ γ ∈ Essmin w F (X ) ⊂ L 0 (D, F) satisfies min x∈D(ω) c(ω, x) = c(γ ). The following result may be compared to Rockafellar and Wets (1998, Theorem 14.37), which is a result formulated in the deterministic case and under the condition that the function to minimize admits bounded lower level sets. In that case, it is proven that the argmin is nonempty. Our contribution is to extend this result by providing a measurable minimizer to the corresponding random problem. To prove this result, we need to recall the notion of real-valued essential infinimum of a family of random variables (see Kabanov and Safarian 2009, Section 5.3): Proposition 6.2 Let (γ i ) i∈I be a family of F-measurable random variables with values in [−∞, ∞] on a complete probability space ( , F, P). There exists a unique (up to a negligible set) F-measurable random variableγ ∈ L 0 ([−∞, ∞], F), called essential infinimum, such that: (i)γ ≤ γ i a.s. for every i ∈ I . (ii) If γ ∈ L 0 ([−∞, ∞], F) satisfies γ ≤ γ i a.s. for every i ∈ I , then γ ≤γ a.s.