Stability and Markov Property of Forward Backward Minimal Supersolutions

We show stability and locality of the minimal supersolution of a forward backward stochastic differential equation with respect to the underlying forward process under weak assumptions on the generator. The forward process appears both in the generator and the terminal condition. Painlev\'e-Kuratowski and Convex Epi-convergence are used to establish the stability. For Markovian forward processes the minimal supersolution is shown to have the Markov property. Furthermore, it is related to a time-shifted problem and identified as the unique minimal viscosity supersolution of a corresponding PDE.


Introduction
In this work we study forward backward minimal supersolutions, particularly their stability and locality with respect to the forward process. For the special case of Markovian forward processes, we thereby provide the Markov property of the minimal supersolution and show how the latter is related to viscosity supersolutions of a corresponding PDE. More precisely, given a fixed time horizon, T > 0, measurable functions g and ϕ, a filtered probability space, the filtration of which is generated by a d-dimensional Brownian motion, and a progressive d-dimensional forward process X, we study the minimal supersolution of the decoupled forward backward stochastic differential equation (FBSDE) where 0 ≤ s ≤ t ≤ T . Throughout we work with a standard generator g, that is a positive, lower semicontinous function which is convex in the control variable z, and which in addition is either monotone in y or jointly convex in (y, z). The expression "standard" is justified since the former are, to the best of our knowledge, the mildest assumptions guaranteeing existence and uniqueness of the minimal supersolution (E(X), Z) of ( * ), compare Drapeau et al. [4]. The first novel and main contribution of this paper consists in proving stability of the minimal supersolution as a function of X by combining existing stability results of Drapeau et al. [4] and Gerdes et al. [7] with Painlevé-Kuratowski and Convex epigraphical convergence. This kind of stability generalizes results obtained so far in this direction in that the forward process now affects jointly both the dynamics of the problem through its input on g and the terminal condition. It comes at a small cost in terms of assumptions on the generator, namely at the need of g satisfying the epigraphical lower semi-continuity condition (REC). However, we show that this epigraphical lower semi-continuity condition is met in a significant number of situations using some results about horizon functions, compare Rockafellar and Wets [15], and Paintlevé-Kuratovsky/Convex epigraphical convergence in Aubin and Frankowska [1] and Löhne and Zȃlinescu [9]. Furthermore, we prove that the minimal supersolution is local in the following sense: Given a time t ∈ [0, T ] and a set A ∈ F t it holds E s (X) = 1 A E s (X 1 ) + 1 A c E s (X 2 ) for s ∈ [t, T ] where X 1 and X 2 are two forward processes and X their concatenation. Specifically, this allows to restrict our focus to supersolutions on [t, T ] and forget about the past once we have arrived at time t. Both the results above open the door to the study of supersolutions of Markovian FBSDEs and of their relation to PDE theory, the second part of this work. Supposing X to be the solution to a classical SDE we study under which conditions E is also Markovian in the sense of it being a function of time and the underlying forward process. To this end, we shift the original problem ( * ) in time and introduce the candidate function u(t, x), the value at time zero of the minimal supersolution corresponding to the shifted formulation with a forward process starting in x ∈ R d . Besides proving that x → u(t, x) maintains central features such as lower semicontinuity, we show that E t (X t,x ) = u(t, x) where X t,x is the forward diffusion starting in x at time t, therewith drawing the connection between the original and the time-shifted problem. Furthermore, using X = X t,Xt and approximating X t from below by step functions, we obtain that E t (X) ≥ u(t, X t ) always holds true, with equality if x → u(t, x) is monotone or continuous. For ϕ bounded from below and g jointly convex in (x, y, z) another ansatz to obtain the desired representation E t (X) = u(t, X t ) is to draw on both the convexity of the generator and the relation of Lipschitz BSDEs and PDEs as for instance given in El Karoui et al. [6]. The former allows to approximate g from below by a sequence of Lipschitz generators for which the minimal supersolution coincides with the unique solution of the BSDE, a method first used in Drapeau et al. [5]. The latter in turn then ensures that at each approximation step there is a a one-to-one relation between the (super-)solution and a viscosity solution of the corresponding PDE. Stability of the problem with respect to the generator, compare Drapeau et al. [4], finally allows us to pass to the limit and thereby identify u as a viscosity supersolution of the above PDE. This extends existing results on the connection of BSDEs and PDEs to minimal supersolutions and constitutes the third contribution of this work. Let us briefly discuss the existing literature on related problems. Nonlinear BSDEs were first introduced in Pardoux and Peng [10], whereas their relation to PDEs was extensively studied among other in Pardoux and Peng [11] and Peng [13]. As BSDEs may be ill posed beyond the quadratic case, compare Delbaen et al. [3], minimal supersolutions extend the concept of solutions and were first rigorously studied in Drapeau et al. [4] and then subsequently in Heyne et al. [8], while Drapeau et al. [5] derived their dual representation. In order to keep the presentation neat, we refer the reader to aforementioned works and El Karoui et al. [6] for a broader discussion on the subject. The remainder of this paper is organized as follows. Setting and notations are specified in Section 2, while the central results on stability and locality are given in Section 3. Subsequently, Section 4 covers the study of the Markovian case, whereas the relation between forward backward minimal supersolutions and viscosity supersolutions of PDEs is provided in Section 5. Technical results on epi-convergenge and Painlevé-Kuratowski limits are presented in the appendix.

Setting and notation
We consider the canonical probability space (Ω, ). By W we denote the canonical process, P the Wiener measure and (F t ) the filtration generated by W augmented by the P -null sets of W . For some fixed time horizon T > 0 the sets of F T -measurable random variables are denoted by L 0 , where random variables are identified in the P -almost sure sense. Let furthermore denote L p the set of random variables in L 0 with finite p-norm, for p ∈ [1, +∞]. Inequalities and strict inequalities between any two random variables or processes X 1 , X 2 are understood in the P -almost sure or in the P ⊗ dt-almost everywhere sense, respectively. We denote by S the set of càdlàg progressively measurable processes Y with values in R. We further denote by L the set of R d -valued, progressively measurable processes Z such that T 0 Z 2 s ds < ∞ P -almost surely. For Z ∈ L, the stochastic integral ZdW is well defined and is a continuous local martingale. We define the concatenation ofω, ω ∈ Ω at time t ∈ [0, T ] by (2.1) Given an extended real valued function (x, y, z) → g(x, y, z) defined on a finite dimensional space, we denote domg = {(x, y, z) : g(x, y, z) < ∞} and by a slight abuse of notation, we say that x ∈ domg if g(x, y, z) < ∞ for some y, z. Further, for a sequence (x n ) ⊆ R d we denote by cl{g(x n , ·, ·) : n} the greatest lower semi-continuous function (y, z) → h(y, z) such that h ≤ g(x n , ·, ·) for every n, while clco{g(x n , ·, ·) : n} or clco z {g(x n , ·, ·) : n} is defined likewise with the addition of being jointly convex or convex in z, respectively. This given, we define the Painlevé-Kuratowski and Closed-Convex limit inferior as follows, see Appendix A, Finally, for a lower semi-continuous proper convex function h, we denote by h ∞ the horizon function of h, that is,

Forward backward minimal supersolutions
Throughout we call a jointly measurable function g : Given a generator g, a progressive d-dimensional measurable process X and a measurable function ϕ : R d → R we call a pair (Y, Z) ∈ S × L a supersolution of the decoupled forward backward stochastic differential equation 1 if for every 0 ≤ s ≤ t ≤ T . We call X the forward process, Y the value process and Z its corresponding control process. A control process Z ∈ L is said to be admissible if the continuous local martingale ZdW is a supermartingale and we denote the set collecting all supersolutions by In general, supersolutions are not unique, therefore we define a supersolution (Y, Z) ∈ A(X) to be minimal if Y ≤Ŷ for every (Ŷ ,Ẑ) ∈ A(X). If a minimal supersolution exists, we denote its value process by E(X). If further, A(X) ≡ ∅, we set E(X) = ∞ per convention. Throughout this paper a generator may satisfy (STD) g is positive, lower semicontinuous and z → g(x, y, z) is convex.
Definition 3.1. We say that g is a standard generator if g satisfies (STD) and either (MON) or (CON).

Remark 3.2. Following [4, Section 4.3]
, the positivity assumption in (STD) may be relaxed to g being bounded from below by an affine function of z without violating the validity of our results.
The following is a straightforward application of results in [4,5].
Proof. For a given X ∈ S, setting g X (y, z) := g(X, y, z) and ξ = ϕ(X T ) defines a generator and a terminal condition satisfying the existence and uniqueness assumptions in [4,5], hence the assertion.
Denoting by A(ξ, h) and E(ξ, h) the set of supersolutions and the minimal supersolution, respectively, with terminal condition ξ and generator h(y, z) in the sense of [4,5], it holds The subsequent results of Sections 4 and 5 depend on the stability of the minimal supersolution as a function of X, provided in Theorem 3.4 below. Together with the subsequent Proposition 3.5, it constitutes the first main contribution of this work, generalizes the stability results given in [4] and is partially inspired by driver stability shown in [7]. However, by dependence of the generator on the forward component we obtain a joint stability in the driver and terminal condition. This requires a novel approach and one further assumption on the generator.
(REC) for every bounded sequence (x n ) such that x n → x, it holds • if g satisfies (CON), then g(x, ·, ·) ≤ c-lim inf g(x n , ·, ·); Theorem 3.4. Let g be a standard generator satisfying (REC) and suppose that ϕ is lower semicontinuous. Let (X n ) be a sequence of progressive measurable processes such that X n t → X t almost surely for every t and ϕ(X n T ) ≥ −η where η ∈ L 1 + . Then it holds If furthermore x → g(x, ·, ·), ϕ and (X n ) are increasing, then Proof. We define 3 • if g satisfies (MON): h n := clco z {g(X k , ·, ·); k ≥ n} for which holds that h n is positive, lower semicontinuous, monotone in y and convex in z. Furthermore, it holds h n ≤ h n+1 and h n → h = c z -lim inf g(X n , ·, ·) by definition of c z -lim inf in (2.2).
Define in addition the increasing sequence of terminal conditions ξ n = inf k≥n ϕ(X k T ) for which holds ξ n ≥ −η for every n and ξ := sup ξ n . Given the sequences of terminal conditions (ξ n ) and generators (h n ), both increasing, we adapt the stability proofs in [4] as follows. The monotonicity of the minimal supersolution operator implies there is nothing to prove. Assuming therefore that lim E 0 (ξ n , h n ) < ∞ yields the existence of a non-trivial minimal supersolution for every n . Denote by ((Y n , Z n )) this sequence of minimal supersolutions and define Y = lim Y n since (Y n ) is increasing. The same argumentation as in [4] implies Y being a càdlàg supermartingale and the existence of Z ∈ L together with a sequence (Z n ) in the asymptotic convex hull of (Z n ) such thatZ n → Z P ⊗ dt-almost surely, while Z n dW → ZdW For k fixed, the following holds: • If y → g(x, y, z) is decreasing: Lower semicontinuity, convexity in z, and h k being decreasing in y yield • If y → g(x, y, z) is increasing: Lower semicontinuity, convexity in z, the fact that Y n → Y P ⊗dtalmost everywhere, the function h k being increasing in y, and Y n ≤ Y i for every i = n, . . . , m n , • If (y, z) → g(x, y, z) is jointly convex: thereby h k is jointly convex too. Lower semicontinuity and joint convexity of h k yield In all cases above, for every n greater than k, it follows that As Y T = lim Y n T ≥ lim ξ n = ξ, this shows that (Y, Z) ∈ A(ξ, h). Having identified (Y, Z) as a supersolution with terminal condition ξ and driver h, this implies E 0 (ξ, h) ≤ Y 0 . Since Y n ≤ E(ξ, h) this completes the proof of E 0 (ξ, h) = lim E 0 (ξ n , h n ). Particularly, an inspection of the arguments above yields that, whenever E 0 (ξ, h) < ∞, then E t (ξ n , h n ) increases monotonically to E t (ξ, h) for every t. With this at hand, the monotone assertion (3.3) follows readily by observing h n = g(X n , ·, ·) for every n as well as ξ n = ϕ(X n ). As for the first assertion (3.2), on the one hand, by definition of h n and ξ n for every n it holds h n ≤ g(X n , ·, ·) and ξ n ≤ ϕ(X n T ).
As the preceding proof exhibits, the stability depends heavily on the generator g satisfying (REC). The following proposition shows that this assumption is indeed fulfilled in many circumstances. The main part of its proof, being of convex analytical nature, is addressed in Appendix A. (i) g(x, y, z) = g 1 (x) + g 2 (y, z) with g 1 lower semi-continuous and g 2 a standard generator; (ii) g satisfies (CON) and f ∞ = g ∞ (x n , ·, ·) for every n where f = clco{g(x n , ·, ·) : n}; (iii) g satisfies (CON) and for every γ the level set ∪ n {(y, z) : g(x n , y, z) ≤ γ} is relatively compact; (iv) g satisfies (MON) and for every y and γ the level set ∪ n {z : g(x n , y, z) ≤ γ} is relatively compact.
The same argumentation is valid in the case where (MON) is satisfied by considering the convex hull solely in z. The cases (ii) and (iv) are subjects of the Proposition A.1 and A.2 in the Appendix A. Finally, a slight modification of Proposition A.2 in the jointly convex case yields (iii). Remark 3.6. Note that assumption (ii) is satisfied if g(x, y, z) ≥ h(y, z) for some lower semi-continuous and convex function such that h ∞ = g ∞ (x, ·, ·) for every x. In particular if h is coercive in which case (iii) also holds. Assumption (iv) is fulfilled if g(x, y, z) ≥ c(y) |z| for some c(y) > 0.
We conclude this section by a further central property of forward backward minimal supersolutions, namely their locality with respect to the underlying forward process. Proposition 3.7. For t ∈ [0, t] fixed, let X 1 , X 2 be two forward processes and A ∈ F t . Define the forward process Proof. Let us denote by Reversely, if A(X) = ∅, then equality holds. Indeed, an application of Theorem 3.3 restricted to [t, T ] yields the existence ofẐ ∈ S |[t,T ] such that Hence, by stability of supersolutions with respect to pasting, compare [4, Lemma 3.1], the pair defined byỸ belongs to A(X). However, this impliesỸ s = I t s ≥ E s (X) for t ≤ s ≤ T and thus With this at hand, under the assumption A t (X 1 ), A t (X 2 ), A t (X) = ∅ it is straightforward to check that for every t ≤ s ≤ T . In combination with (3.6) the former yields the proof is done.

Markovian minimal supersolutions
For the remainder, the forward process X is given by the solution of the stochastic differential equation where X 0 ∈ R n and µ : [0, T ] × R n → R n and σ : [0, T ] × R n → R n×d are jointly measurable functions satisfying the usual assumptions of SDE theory, namely (SDE) µ · (0) and σ · (0) belong to L 2 ; σ and µ are uniformly Lipschitz and of linear growth in their second component.
The goal of the current section goal is to show that in this case E t (X) = u(t, X t ) where u is a function defined on [0, T ] × R n . To this end, given t ∈ [0, T ], we first define for every ξ ∈ L 2 (F t ) the process X t,ξ as the unique solution of Notice that X t,ξ is well defined and uniquely determined. Indeed, it is the unique solution of an SDE with Lipschitz coefficients between t and T and initial value ξ ∈ L 2 (F t ) and the unique solution of the Lipschitz BSDE with driver µ between 0 and t and terminal condition ξ. It is furthermore continuous and adapted. In particular, it holds X = X t,Xt Next, we need to consider the t-shifted problem. More precisely, let W t := W t+· − W t be the Brownian motion on [0, T − t] together with the corresponding filtration F t s := σ(W t r : 0 ≤ r ≤ s). Accordingly, for each x ∈ R n defineX t,x as the solution of the stochastic differential equatioñ Similarly, t-shifted supersolutions are those pairs (Y, and we collect all t-shifted supersolutions on [0, T − t] in the set

.3) holds and
ZdW t is a supermartingale .
Analogously, we denote byẼ(X t,x ) the t-shifted minimal supersolution operator and define our candidate The reader should keep in mind that for the sequel a "tilde" appearing in the notation of expressions always indicates a relation to the t-shifted problem on [0, T − t] above.
The ensuing theorem provides the second contribution of this work by collecting important properties of u and drawing the connection between the original problem, the t-shifted one and the function u.
Theorem 4.1. We suppose that g is a generator satisfying (STD) and (REC), µ and σ satisfy (SDE), and ϕ is lower semicontinuous and linearly bounded from below. Then the following assertions hold true: x) is lower semicontinuous, either identically ∞ or proper for every t ∈ [0, T ]. If furthermore g, ϕ, µ and σ are convex, then x → u(t, x) is convex.
In particular, E t (X t,x ) is a real number corresponding to the infimum of the t-shifted minimal solution problem.
with equality if A(X) = ∅ and x → u(t, x) is continuous or monotone.
Proof. For the remainder of the proof, we fix t ∈ [0, T ].

Point (i):
For x n → x, up to a subsequence it holds limX t,xn s =X t,x s for every s and inf nX t,xn T ∈ L 2 , both as a consequence of [16,Theorem 2.4]. Since ϕ is lower semicontinuous and linearly bounded from below, it follows that inf n ϕ(X t,xn T ) − ∈ L 1 and therefore the stability Proposition 3.4 yields lim inf u(t, x n ) = lim infẼ 0 (X t,xn ) ≥Ẽ 0 (X t,x ) = u(t, x).
Finally, it holds thatẼ 0 (X t,x ) ≥ E[ϕ(X t,x T )] > −∞, by which we deduce that u is either proper or uniformly equal to ∞. The proof of the convexity property is goes along the lines of the argumentation in [4,Proposition 3.3

.(4)].
Point (ii): First, let X t,x be defined as in (4.1). The locality result of Proposition 3.7 yields where I t,x t = ess inf{Y t : (Y, Z) ∈ A t (X t,x )} and A t (X t,x ) is defined analogously to (3.5). It remains to show the equality I t,x t = u(t, x). In other terms, we need to establish the relation between the set A t (X t,x ) of supersolutions between [t, T ] with forward process X t,x and the setÃ(X t,x ) of t-shifted supersolutions on [0, T −t] with forward processX t,x . Clearly, for every (Y, Z) ∈Ã(X t,x ), the observa- . Together with (4.4) this implies E t (X t,x ) ≤ u(t, x). Reciprocally, since A(X t,x ) is non-empty, neither is A t (X t,x ) and thus there exists a control Z t,x corresponding to the [t, T ]-minimal supersolution I t,x . Observe that for almost allω ∈ Ω ω → (Yω s , Zω s ) : is a t-shifted supersolution with forward processX t,x , that is, an element ofÃ(X t,x ). Indeed, it is measurable by definition and defines a pair of a càdlàg and a progressive process on [0, T −t]. In addition, this pair is adapted to F t . This follows from it being a functional ofω ⊗ t ω and thus by means of (2.1) of (ω t+s − ω t ) s∈[0,T −t] . The fact that it satisfies (4.3) follows fromX t,x s = X t,x t+s and the generator g not depending on ω. Hence, (Yω s , Zω s ) s∈[0,T −t] ∈Ã(X t,x ) and therefore, for almost allω ∈ Ω, it holds Yω 0 ≥Ẽ 0 (X t,x ) = u(t, x). Using the definition of Yω in combination with (4.4) we obtain proving Point (ii).

Point (iii):
The inequality E t (X) ≥ u(t, X t ) is obtained by the path-wise argumentation of the previous point. Suppose now that x → u(t, x) is continuous or increasing. Since x → u(t, x) is lower semicontinuous, in both cases, for every increasing sequence of random variable (X n t ) ⊆ L 2 (F t ) converging to X t , it holds lim u(t, X n t ) = u(t, X). We thus approximate X t from below by step functions, that is X n t ր X t where for each n we have X n t = n k=1 1 A n k x n k . Using (X n t ) we define the family of terminal values (X n T ) by X n T := X t, n k=1 1 A n k x n k T which, by means of (4.2), satisfy It clearly holds X n s → X s for every s ≥ t and X n t ր X t . The function x → u(t, x) being either increasing or continuous yields lim inf u(t, X n t ) = lim u(t, X n t ) = u(t, X t ). Furthermore, by locality of E, see Proposition 3.7, we have Finally, the stability result of Theorem 3.4 together with relations (4.5) and (4.6) yields showing the reverse inequality and thereby completing the proof.

Viscosity supersolutions
The last relation of Theorem 4.1, namely E t (X) = u(t, X t ), holds in the special cases of monotonicity or continuity. The current and final section shows that it is also valid as soon as g is jointly convex, and even more, in this case the minimal supersolution can be interpreted as a viscosity supersolution of a corresponding PDE.
To begin with, following the notations and definitions in [2], [12] and [16], we consider semilinear parabolic PDEs with terminal conditions of the form Here, S(d) denotes the set of symmetric d × d matrices, while F is supposed to be lower semicontinuous. Further, Dv and D 2 v corresponds to the gradient vector and matrix of second partial derivatives of v, respectively. In the case under consideration F is of the form Note that as σ t (x) is positive semi-definite, F is degenerate elliptic. where P −(1,2) u(t, x) are the semi-jets of u at (t, x), that is those (a, p, M ) ∈ R × R d × S(d) satisfying

Theorem 5.2.
Assume that the assumptions of Theorem 4.1 are fulfilled and g is convex. If in addition ϕ is bounded from below, that is ϕ ≥ C for some C ∈ R, and A(X) = ∅, then it holds Furthermore, u is a lower semicontinuous viscosity supersolution of the PDE (5.1) Proof. Note that if A(X) = ∅, then g is proper. As in [5], for each n define g n (x, y, z) := sup |α|∨|β|∨|γ|≤n {αx + βy + γz − g * (α, β, γ)} and ϕ n (x) = ϕ(x) ∧ n where g * is the convex conjugate of g. By Fenchel-Moreau, the sequence (g n ) converges pointwise from below to g, while each g n is of linear growth. Being in addition convex, each g n is also Lipschitz continuous. Analogously to Section 3, we define E n (X) as the minimal supersolution of the FBSDE with generator g n , forward process X and terminal function ϕ n . As g n is Lipschitz and ϕ n is bounded, it follows from [5,Remark 3.6] that the minimal supersolution E n (X) corresponds to the unique solution of the Lipschitz BSDE with generator g n and terminal condition ϕ n (X T ).
Hence, a well-established result connecting Lipschitz BSDEs and semilinear PDEs, compare for instance [16,Proposition 10.8], yields u n : where u n is a continuous solution of the PDE (5.1) with F n and ϕ n instead of F and ϕ respectively. Note that in addition, for each t ∈ [0, T ] the function u n (t, ·) corresponds exactly to the t-shifted problem with generator g n used in the proof of Theorem 4.1. More precisely, with the notation analogous to above and n indicating of course that g n is considered instead of g. Using the stability property of minimal supersolutions with respect to increasing drivers, see [4,Theorem 4.14], slightly adapted to in addition having increasing terminal conditions, it follows from E 0 (X) < ∞ that On the other hand, by the same argumentation for the shifted problem we deduce that u n (t, x) ր u(t, x), pointwise which, together with (5.3) yields the desired relation (5.2).
We are left to show that u is a lower semicontinuous viscosity supersolution of the PDE (5.1). By means of [2, Remark 6.3], it follows that is a lower semicontinuous viscosity supersolution of (5.1) with instead of F . However, from g n ր g it follows that F n ր F . Since in addition u n ր u, Lemma 5.3 below implies that u * = u and F * = F , completing the proof.
Proof. Fix some z ∈ O. By definition of the limes inferior we may pass to a subsequence, denoted by (n, z n ), satisfying lim n h n (z n ) = h * (z). For a fixed k, the sequence being increasing implies that h n (z n ) ≥ h k (z n ) for all n sufficiently large. Combining the former with the continuity of h k yields implying in turn that h * (t, z) ≥ h(t, z). Conversely, for every ε > 0 there exists k such that for all n ≥ k it holds By sending n to infinity and subsequently using the continuity of h k as well as the definition of h the above yields h * (z) ≤ h k (z) + ε ≤ h(z) + ε. As ε was arbitrary, this finishes the proof.

A. Epi-convergence: technical results
Throughout, let X, Y, Z denote three finite dimensional euclidean real vector spaces. We denote by cl(C) and clco(C) the closure and closure of the convex hull of a set C, respectively. For a sequence of sets (C n ), we define the Painlevé-Kuratowski limit superior and the Closed Convex limit superior by e-lim sup C n = ∩ n cl ∪ k≥n C k and c-lim sup C n = ∩ n clco ∪ k≥n C k , respectively, see [15,Chapter 4] and [9]. For a sequence (f n ) of functions, we define e-lim inf f n or c-lim inf f n as the function the epigraph of which corresponds to the Painlevé-Kuratowsky or Closed-Convex limit superior of the epigraphs of (f n ), respectively, see [ We denote by C ∞ := {x : λ n x n → x for some (x n ) ⊆ C and λ n ↓ 0} the horizon cone of a set C. Given a proper closed convex function f , we denote by f ∞ its horizon function, that is the function the epigraph of which corresponds to the horizon cone of the epigraph of f . Proof. If f n ≡ ∞ except for finitely many n, then the inequality is trivially satisfied. Without loss of generality we may thus assume f n to be proper for every n ∈ N. By lower semicontinuity of f and [15,

Proposition 7.2] it follows that
e-lim inf f n (z) = min {α ∈ R : lim inf f (x n , z n ) = α for some z n → z} ≥ f (x, z), and since f is lower semicontinuous and convex in z, we deduce f (x, z) ≤ clco{e-lim inf f n }(z), z ∈ Z.
Let now C n = epif n . By assumption, C n is non-empty, closed and convex for every n ∈ N. Furthermore, as C := epi(h) = clco(∪ n C n ), it holds that (C n ) ∞ = C ∞ for every n. In the following, we consider the convex hull only with respect to certain dimensions which notation-wise is stressed by means of an index. For instance, the convex hull in the second variable z of a set C ⊆ Y ×Z is denoted by co z (C).
Proposition A.2. Let f : X × Y × Z →] − ∞, ∞] be a proper lower semicontinuous function that is convex in z and monotone in y. Suppose that for every bounded sequence (x n ) ⊆ X, y ∈ Y and γ ∈ R the set ∪ n {z : f (x n , y, z) ≤ γ} is contained in a compact set. Then, denoting f n := f (x n , ·), for every (x n ) ⊆ X with x n → x it holds f (x, y, z) ≤ clco z {elim inf f n } (y, z) = c z -lim inf f n (y, z), y, z ∈ Y × Z.

1.9]
Proof. An argumentation analogous to the proof of Proposition A.1 allows to assume that f n is proper for every n and it holds f (x, y, z) ≤ clco z {elim inf f n } (y, z), y, z ∈ Y × Z.
Furthermore, the relation c z -lim inf f n (y, z) ≤ clco z {elim inf f n } (y, z) y, z ∈ Y × Z. (A.1) is naturally satisfied. Let γ ∈ R and define C n γ := {(y, z) : f n (y, z) ≤ γ}. To show the reverse inequality in (A.1), it is sufficient to show that clco z (e-lim sup C n γ ) = c z -lim sup C n γ for every γ. Let (y, z) ∈ c z -lim sup C n γ and with d = dim Z denote by ∆ the d + 1-dimensional simplex. By Caratheodory's Theorem, there exist sequences (y n ), (z i n ) i=1,...,d+1 , (λ n ) such that (y n , z i n ) ∈ ∪ k≥n C k γ , λ n ∈ ∆, y n → y and i λ i n z i n → z. Up to a subsequence, we may assume that λ n → λ ∈ ∆. Furthermore, for every i it holds (z i n ) ⊆ ∪ n {z : f (x n ,ỹ, z) ≤ γ} is contained in some compact set, since (x n ) ⊆ X is bounded and whereỹ = sup y n orỹ = inf y n depending on f being increasing or decreasing in y. Hence, up to yet another subsequence, z i n → z i holds for every i. In particular, (y, z i ) ∈ ∩ n cl(∪ k≥n C k γ ). Thus, (y, z) = lim(y n , i λ i n z i n ) = λ i (y, z i ) ∈ cl(co z (∩ n cl(∪ k≥n C n γ )) = cl(co z (e-lim sup C n γ )) which ends the proof.