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Optional and predictable projections of normal integrands and convex-valued processes

This article studies optional and predictable projections of integrands and convex-valued stochastic processes. The existence and uniqueness are shown under general conditions that are analogous to those for conditional expectations of integrands and random sets. In the convex case, duality correspondences between the projections and projections of epigraphs are given. These results are used to study projections of set-valued integrands. Consistently with the general theory of stochastic processes, projections are not constructed using reference measures on the optional and predictable sigma-algebras.


Introduction
Normal integrands, set-valued integrands, and set-valued mappings have proven to be fundamental concepts both in discrete and continuous time optimization. For applications to discrete time stochastic optimization and mathematical finance, see, e.g. [25,19]. As to the continuous time, these concepts are important in optimal control and calculus of variations [24,17]. In stochastic optimal control, normal integrands are already used in [3] whereas set-valued integrands and set-valued mappings appear in stochastic differential inclusions and set-valued stochastic integrals [15]. In mathematical finance, set-valued mappings are used to model, e.g., portfolio constraints [7] and currency markets [14].
In discrete time, conditional expectations of normal integrands and random sets have received considerable attention [2,5,27,28,6]. In stochastic optimization, the dynamic programming equations can be given in terms of conditional expectations of integrands [10,21]. For applications to ergodic theory and statistics, see [6].
This article extends the analysis to the continuous time setting by studying optional and predictable projections of normal integrands and set-valued processes. We will define these projections so that they correspond to conditional expectation of integrands and random sets. When a set-valued process is single-valued or a normal integrand is simply a stochastic process, our definitions reduce to the ordinary optional and predictable projections. Thus our definitions are consistent with the general theory of stochastic processes. An application to stochastic optimal control is given in [20] whereas financial applications will be studied elsewhere.
In Section 2, we review the definitions and basic properties of normal integrands and set-valued processes; a systematic treatment can be found from [26]. In Section 3, we prove versions of optional and predictable section theorems that involve graphs of optional and predictable set-valued processes. In Sections 4 and 5, we use these section theorems to prove the existence and uniqueness of optional and predictable projections of integrands. In particular, we give general conditions under which projections of normal integrands are again normal integrands. These are similar to those in [6] where conditional expectations of normal integrands were studied. However, when specialized to conditional expectations of integrands on R d , our results slightly extend both [6, Theorem 2.1] and [28,Theorem 1.4]; see Example 4. In Section 6, we prove the existence and uniqueness of optional and predictable projections of convex-valued stochastic processes. The methods in [13] and [29], that deal with conditional expectations of general random sets, use a given probability measure on Ω; here we do not have such measure on Ω × R + at our disposal. Instead, our method is based on conjugacy arguments and on our existence results for projections of integrands. We emphasize that we do not construct projections of set-valued processes as conditional expectations with respect to a given measure on Ω × R + . This is the case, e.g., in the proof of [30,Theorem 3.7].
We finish the article by applying our main results to study projections of set-valued integrands in Section 7. We give existence and uniqueness results for such projections as corollaries of our main theorems. To this end, we assume that the integrands are convex-valued and that they have appropriate inner and outer semicontinuity properties.

Normal integrands and set-valued processes
We assume throughout that the filtered probability space (Ω, F , (F t ) t≥0 , P ) satisfies the usual hypotheses, that is, (F t ) t≥0 is right continuous and F 0 contains all the P -null sets. We denote the predictable and optional σ-algebras on Ω×R + by P and O. The set of [0, ∞]-valued stopping times is denoted by T and its subset of predictable times by T p . We use the common notations Standard references for these basic concepts are, e.g., [8,9,11].
Let S be a σ-algebra on Ω×R + . We say that an extended real-valued function h : We assume throughout that all integrands are F ⊗ B(R + )-integrands on R d unless stated otherwise. An integrand h can be viewed as a measurable stochastic process (ω, t) → h t (x, ω) depending on the parameter x ∈ R d , or, as a functionvalued stochastic process.
Recall that a set on Ω × R + is evanescent if its projection to Ω is a P -null set. For integrands h 1 and h 2 , we denote is evanescent and we say that h 1 and h 2 are indistinguishable if h 1 = h 2 . Later, we will not distinguish two indistinguishable integrands, i.e., they are regarded as the same. Accordingly, all properties of integrands are understood to be satisfied outside an evanescent set. For example, we say that an integrand h is convex if h t (·, ω) is a convex function outside an evanescent set. Likewise, all equalities involving (at most countable number of) integrands are understood to hold everywhere outside some evanescent set. Note that, since F 0 is complete, such evanescent set can always be assumed to be predictable. Indeed, for evanescent E ⊆ Ω × R + , we have E ⊆ π(E) × R + and π(E) × R + ∈ P by [11,Theorem III.21]. Here π denotes the projection from Ω × R + onto Ω. Throughout, for an integrand h and an arbitrary w : Ω×R + → R d , we denote by h(w) the process (ω, t) → h t (w t (ω), ω). Note that h(w) is S-measurable whenever h is an S-integrand and w is S-measurable. For measurable φ : Ω → R, we use the convention that the expectation is +∞ unless the positive part is integrable. In particular, sums of extended real numbers are defined as +∞ if any of the terms equals +∞.
We call set-valued mappings from Ω × R + to R d set-valued stochastic processes. A set-valued stochastic process Γ is S-measurable if the inverse image of every open O ⊆ R d belongs to S. In particular, the domain mapping is known as the graph of Γ. Throughout, set-valued stochastic processes are considered to be equal if they coincide outside an evanescent set. We call a process w a selection of Γ if w t (ω) ∈ Γ t (ω) outside an evanescent set. We denote the set of S-measurable selections of Γ by L 0 (S; Γ). When S = F ⊗ B(R + ), we write simply L 0 (Γ).
When S is complete with respect to some σ-finite measure, then an Sintegrand h for which h t (·, ω) is lower semicontinuous for all (ω, t), is a normal S-integrand [26,Corollary 14.34]. Some authors take these properties as the definition of a normal integrand, but since we will work with incomplete σ-algebras, we use the more precise concept given in terms of the measurability of the epigraphical mapping. Indeed, this is the case with, e.g., the optional and predictable σ-algebras; see, e.g., [22].

Optional and predictable section theorems
For a set A, we denote ½ A (x) = 1 if x ∈ A and ½ A (x) = 0 otherwise, whereas δ A (x) = 0 if x ∈ A and δ A (x) = +∞ otherwise. For a set A in Ω × R + , we denote by 1 A the stochastic process (1 A ) t (ω) = ½ A (ω, t). Motivated by the notion of P -discretely dense set (see [28,Section 1.2]), a set H of S-measurable processes on Ω × R + is said to be S-discretely dense if, for every S-measurable process w, there exists an S-measurable covering (A ν ) ∞ ν=1 of Ω × R + and a sequence (w ν ) ∞ ν=1 in H such that 1 A ν w = 1 A ν w ν for every ν. For example, the set of bounded optional processes is O-discretely dense. Recall that a stochastic process w is said to be bounded if there is an M > 0 such that |w t (ω)| ≤ M outside an evanescent set.
By the optional section theorem, there exists τ ∈ T such that gph τ ⊂ (A ν ∩C) and P ({τ < ∞}) ≥ P (π(A ν ∩C)) − ǫ/4. We have gph τ ⊆ dom Γ, w τ ∈ Γ τ on {τ < ∞} and Theorem 2. Assume that gph Γ of a set-valued process Γ is P⊗B(R d )-measurable and that H is a P-discretely dense set. For any ǫ > 0, there exists τ ∈ T p and w ∈ H such that Proof. This can be proved like Theorem 1 by invoking the predictable cross section theorem [8,Theorem IV.85] instead of the optional cross section theorem.
Given set-valued processes Γ andΓ, Γ is said to be smaller thanΓ if Γ t (ω) ⊆ Γ t (ω) outside an evanescent set, in which case we denote Γ ⊆Γ. For set-valued mappingsS and S from Ω to R d , we denoteS ⊆ S ifS(ω) ⊆ S(ω) almost surely. Such mappings are called random sets. It follows from the Castaing representation [26,Theorem 14.5], that if Γ is optional, then Γ τ is an F τmeasurable random closed set for each τ ∈ T . Likewise, if Γ is predictable, then Γ τ is F τ − -measurable random closed set for each τ ∈ T p .
The following result gives the set-valued analog of the fundamental result (see, e.g., [11,Theorem 4.10]) that optional (resp. predictable) stochastic processes v 1 and v 2 satisfy v 1 ≤ v 2 if and only v 1 τ ≤ v 2 τ almost surely for every bounded τ ∈ T (resp. for every bounded τ ∈ T p ). A stopping time τ is said to be bounded if there is an M > 0 such that τ (ω) ≤ M almost surely. Lemma 3. LetΓ and Γ be optional set-valued processes. ThenΓ ⊆ Γ if and only ifΓ τ ⊆ Γ τ for every bounded τ ∈ T . IfΓ and Γ are predictable, then it is sufficient thatΓ τ ⊆ Γ τ for every bounded τ ∈ T p .

Projections of integrands
If an R d -valued stochastic process w is T -integrable in the sense that ½ {τ <∞} |w τ | is integrable for every τ ∈ T , then there exists an R d -valued optional process o w such that which is unique up to indistinguishability [11,Theorems 5.3]. The processes o w and p w are known as the optional projection and the predictable projection of v, respectively.
Likewise, when v is an extended real-valued nonnegative stochastic process, there exists an optional process o v and a predictable process p v such that which are unique up to indistinguishability. For nonnegative real-valued processes, this is a classical fact [9, Theorem VI.43] which extends to the extended nonnegative real-valued case using the monotone convergence theorem. For an extended real-valued stochastic process The following definitions extend these basic concepts to integrands. We assume throughout that h is an integrand on R d . We define h + = max{h, 0} and h − = max{−h, 0}, and denote by for any process w, and we use these notations interchangeably depending on which one we find more natural in the context.
Thus the definition of an optional projection of h reduces to that of an optional projection of the stochastic process v.
The following lemma shows that optional and predictable projections of h are characterized by any O or P-discretely dense subset H of Λ O h and Λ P h , respectively.

Lemma 4. Let h be an integrand. If there exists an
thenh is an optional projection of h. If there exists a P-discretely dense setĤ contained in Λ O h and a predictable integrandĥ such that thenĥ is a predictable projection of h.

Proof. Assume for a contradiction thath(w) =
which is a contradiction. The predictable case is proved similarly using Theorem 2.
The following lemma states that the projections preserve order. Choosing h 1 = h 2 in the lemma, one gets uniqueness of the projections as soon as Λ O h contains an O-discretely dense set. The example after the proof shows that this is not the case if such O-discretely dense set does not exist.
We prove only the optional case, the claim for p h 1 ≤ p h 2 follows similarly from Theorem 2. We assume for a contradiction that dom Γ is not evanescent which is a contradiction with h 1 ≤ h 2 . Therefore, dom Γ is evanescent, and Example 2. Let η be a measurable process for which η + and η − are not Tintegrable, and let h t (x, ω) = η t (ω).
Here Λ O h = ∅, so every optional integrand is an optional projection of h. The following result is a monotone convergence theorem for projections of integrands. If each p h ν exists and there exists a P-discretely dense subset of predictable processes w for which h 1 (w) − is T p -integrable, then p h exists and For any w ∈ H, τ ∈ T and A ∈ F τ , the monotone convergence and Lemma 5 imply that Thus sup ν o h ν = o h by Lemma 4. The predictable case is similar.
Now we are ready to prove our first main result.
h contains a P-discretely dense set, then h has a unique predictable projection.
Proof. The uniqueness follows from Lemma 5, so it suffices to prove the existence. We use a monotone class argument together with Lemma 4.
Let A ∈ B(R d ) and B ∈ F ⊗ B(R + ). Then every τ ∈ T , F ∈ F τ and optional process w, we get from [11,Corollary 3.23] Let (h ν ) ∞ ν=1 be a nondecreasing nonnegative sequence which converges pointwise to some bounded h and such that o h ν exists for every ν. By Theorem 6, sup ν o h ν t is the optional projection of h. Evidently, when a bounded h has an optional projection, then −h has an optional projection as well, so, by the monotone class theorem [11,Theorem 1.4], every bounded integrand admits an optional projection.
Any nonnegative integrand is a point-wise limit of a nondecreasing sequence of nonnegative bounded integrands, so, the existence of an optional projection for such integrand follows from Theorem 6.
Assume that H is an O-discretely dense set contained in Λ O h . For w ∈ H, so the optional projection of h is given by The predictable case is proved similarly.
We finish this section by relating the optional and predictable projection to the conditional expectation. Let f : where Λ G f is the set of G-measurable random variables η for which f (η) + or f (η) − is integrable. Here and in what follows, E G denotes the conditional expectation with respect to G. For results on conditional expectations, we refer to [2,5,27,28,6].   ]. This can be seen from the following example which does not satisfy the assumptions of either theorem. Indeed, f is not bounded from below by an integrable random variable on any set that intersects R − , whereas, for α > 0, f is not lower semicontinuous.
Example 4. Assume that d = 1, G is the trivial (and complete) σ-algebra, and that where η is F -measurable, nonnegative, and non-integrable, and α ∈ R. Here it is easy to verify that the assumptions of Corollary 8 are met and that

Projections of normal integrands
The aim of this section is to give general conditions under which the projections of a normal integrand exist and they are normal integrands. Choosing α ≤ 0 in Example 4, we see that this is not the case in general.
The following definition is motivated by the results in [6], where conditional expectations of normal integrands were studied.
If m i can be chosen T p -integrable, then h is of class (C p ).
We remark that each B i in the definition of class (C) and (C p ) can be chosen bounded and the sequence (B i ) ∞ i=1 increasing. For an integrand of class (C), the set of bounded optional process is an O-discretely dense set contained in Λ O h . Similarly, when h is of class (C p ), the set of bounded predictable processes is a P-discretely dense set contained in Λ P h . The following lemma follows directly from Lemma 4 and Lemma 5.

Lemma 9.
If h is an integrand of class (C) and there exists an optional integrandh such thath for all bounded optional processes w, thenh is the unique optional projection of h. If h is an integrand of class (C p ) and there exists a predictable integrandĥ such thatĥ for all bounded predictable processes w, thenĥ is the unique predictable projection of h.
Given K > 0, an integrand h is said to be K-Lipschitz if, for all (ω, t) for which h t (·, ω) takes a value less than +∞ somewhere, we have that h t (·, ω) is finite everywhere and Proof. The existence and uniqueness of the projections follow from Theorem 7. Let h be a K-Lipschitz integrand of class (C). Since the optional projection of a T -integrable m is T -integrable, o h is of class (C). Assume that the domain of is nonevanescent for someα ∈ R. Then, by Theorem 1, there existsτ ∈ T and bounded optional processesṽ andw such that which is a contradiction. Thus, if o h t (·, ω) takes a finite value somewhere, o h t (·, ω) is finite everywhere. Now, assume that the domain of is nonevanescent for some α ∈ R. Then, by Theorem 1, there exists τ ∈ T and bounded optional processes v and w such that which is a contradiction. The claim for the predictable projection follows similarly from Theorem 2.
The following theorem is our main result on projections of normal integrands. It is a direct analog of the result in [6] on conditional expectations of normal integrands.
Theorem 11. The optional projection of a normal integrand of class (C) is a normal integrand of class (C). The predictable projection of a normal integrand of class (C p ) is a normal integrand of class (C p ).
Proof. Let h be a normal integrand of class (C) and (B i ) ∞ i=1 the sequence of increasing open sets from the definition of class (C). We define is the optional projection of h.
Using Theorem 6, we get Since the sequence k i is nonincreasing, we thus see that ½ B i k = ½ B i k i and that k t (·, ω) is lower semicontinuous which in conjunction with [26,Proposition 14.44] implies that k is a normal integrand, k is of class (C), and that k = o h. The predictable case is proved similarly.
Recall that a real-valued function h on Ω × R + × R d is called Carathéodory integrand if h t (·, ω) is continuous outside an evanescent set and (ω, t) → h t (x, ω) is measurable for all x ∈ R d . By [26,Example 14.29], a Carathéodory integrand is a normal integrand. We say that an integrand h is of class (C ′ ) if h and −h are of class (C). Likewise, we say that an integrand h is of class (C ′ p ) if h and −h are of class (C p ). The following corollary extends Theorem 11 beyond the class (C).

Corollary 13. Assume that h is a normal integrand and let
where B is an optional invertible matrix-valued process and b is an optional R d -valued process. If h is of class (C), then oh is a normal integrand given by If h is of class (C p ) and B and b are predictable, then ph is a normal integrand given by ( Proof. By [26,Proposition 14.45],h is a normal integrand. Let H be the set of optional processes w for which Bw + b is bounded, and letw be an optional process. The set A ν := {|Bw + b| ≤ ν} is nonevanescent for ν large enough, for every optional w ∈ H, so Lemma 4 implies that is the optional projection ofh. Since o h is a normal integrand by Theorem 11,h is a normal integrand by [26,Proposition 14.45]. The predictable case is proved similarly.

Projections of convex-valued processes
Let S : Ω ⇒ R d be a random closed set. We denote the almost sure selections of S by L 0 (S). If there exists a random setŜ that is the smallest G-measurable random closed setS such that E G η ∈ L 0 (S) for every integrable η ∈ L 0 (S), thenŜ is known as the G-conditional expectation of S and it is denoted by E G S. We refer to [5,13,12,29,16] for equivalent formulations. Here we need the fact that if S is integrable in the sense that it has an integrable selection, then the G-conditional expectation of S exists and it is unique up to a P -null set; see, e.g., [29,Theorem 1.4].
Definition 4. Let Γ be a measurable closed-valued stochastic process. If there exists a set-valued stochastic processΓ that is the smallest O-measurable closedvalued stochastic processΓ such that o w ∈ L 0 (Γ) for every T -integrable w ∈ L 0 (Γ), then we callΓ the optional projection of Γ and denote it byΓ = o Γ.
Definition 5. Let Γ be a measurable closed-valued stochastic process. If there exists a set-valued stochastic processΓ that is the smallest P-measurable closedvalued stochastic processΓ such that p w ∈ L 0 (Γ) for every T p -integrable w ∈ L 0 (Γ), then we callΓ the predictable projection of Γ and denote it byΓ = p Γ.
These definitions are consistent with the general theory of stochastic processes in the sense that when a set-valued process is single-valued, they reduce to the ordinary optional and predictable projection. They are also analogous to the single-valued case in the following way. We say that a set-valued process is T -integrable or T p -integrable if it has a T -integrable or T p -integrable selection, respectively.
Proof. To prove the optional case, let w be a T -integrable selection of Γ. It suffices to prove thatΓ is not the optional projection of Γ if and only if there is a bounded τ ∈ T such thatΓ τ is not the F τ -conditional expectation of Γ τ .
For a bounded τ ∈ T , we have that v ∈ L 0 (Γ) and η ∈ L 0 (Γ τ ) are in one-to-one correspondence via the mappings v → v τ , Thus it follows from Lemma 3 and E Fτ v τ = o v τ that (i) is equivalent to that (i') holds for some bounded τ . Similarly, for a bounded τ ∈ T , we have thatΓ in (ii) andS in (ii') are in one-to-one correspondence via the mappings Thus, it follows from Lemma 3 and (2) that (ii) is equivalent to that (ii') holds for some bounded τ . The predictable case is proved similarly.
Let f : Ω × R d → R be a normal F -integrand on R d . The G-conditional expectation of the epigraphical mapping epi f is also an epigraphical mapping of some normal integrand whenever epi f has an integrable selection; see [28, p. 136 and 140]. We define the G-conditional epi-expectation of f as the integrand G f whose epigraphical mapping is the G-conditional expectation of the epigraphical mapping of f . Some authors have called this the conditional expectation of f . In general, conditional epi-expectation and conditional expectation do not coincide (consider, e.g., a trivial σ-algebra G and f (x, ω) = δ {ξ(ω)} (x) for a non-constant integrable random variable ξ), so we introduced a new term, conditional epiexpectation, to distinguish these two notions.
Definition 6. Let h be a normal integrand on R d . If there exists an integrand h whose epigraph is the optional projection of epi h, then we callĥ the optional epi-projection of h and denote it byĥ = O h.

Definition 7.
Let h be a normal integrand on R d . If there exists an integrandĥ whose epigraph is the predictable projection of epi h, then we callĥ the predictable epi-projection of h and denote it byĥ = P h.
In this section we study the existence and uniqueness of optional and predictable projections of convex-valued stochastic processes. Our proof is based on conjugacy arguments and on our existence results for projections of normal integrands. We point out that the techniques in [13,29], that are used to obtain the existence of conditional expectations of random nonconvex sets, do not extend to the continuous time setting as such. Briefly, both approaches use the reference measure on the underlying space, but we do not have such measure on Ω × R + . We leave it as an open question how to prove the existence of projections in the nonconvex case.
We assume from now on that h is a convex normal integrand on R d . By [26,Theorem 14.50], the conjugate integrand of h is a convex normal integrand on R d . In other words, h * is an (ω, t)-wise convex conjugate of h.
Theorem 15. Assume that h is a convex normal integrand. If h * (v) + is Tintegrable for some T -integrable v, then h and o h are convex normal integrands of class (C). If h * (v) + is T p -integrable for some T p -integrable v, then h and p h are convex normal integrands of class (C p ).

Proof. The Fenchel inequality
implies that h is of class (C). By Theorem 11, o h exists and it is a normal integrand of class (C). To prove that o h is convex, let α ∈ (0, 1) and Both are normal integrands by [26,Proposition 14.44 and Proposition 14.45]. By Lemma 9, Since h is convex,h ≤h, so Lemma 5 implies that oh ≤ oh . Thus o h is convex as well. The predictable case is proved similarly.
Recall that the recession function of a proper lower semicontinuous convex function g is given by define a nondecreasing sequence of integrands (h λ ) ∞ λ=1 . As in the proof of Corollary 13, the set H of optional processesw, for whichw + w is bounded, is an O-discretely dense set such that h 1 (w) − is T -integrable for eachw ∈ H. Thus the result follows from Theorem 6 and Corollary 13. The predictable case is proved similarly.
whenever there exists ξ ∈ L 1 (F ; R d ) and η ∈ L 0 (G; R d ) such that f (ξ) + ∈ L 1 and f * (η) + ∈ L 1 . Note that (3) means Under analogous assumptions, the next theorem shows that epi-projections and projections of normal integrands are dual operations in the same sense. The conditions in the theorem cannot be omitted in general, as demonstrated after the theorem. As to their role for sufficiency, we refer to the proof of [ If h(w) + and h * (v) + are T p -integrable for some T p -integrable w and some predictable v, then P h is given by For a bounded τ ∈ T , is a convex normal F -integrand on R d . We have that bounded F τ -measurable random variables form an F τ -discretely dense subset of Λ Fτ h * τ and bounded optional processes form an O-discretely dense subset of Λ O h * . Moreover, η is bounded and F τ -measurable if and only if there exists bounded and optional processw such which proves that [ o (h * )] * is the optional epi-projection of h. The predictable case is proved similarly.
The assumptions in Theorem 17 cannot be dropped in general. For example, (4) and (5) where v is real-valued, nonnegative and predictable that is not T p -integrable, and α τ is real-valued, nonnegative, nonintegrable and independent of F τ for each τ ∈ T .
The formulas in the following result reduce to Jensen's inequalities for optional and predictable integrands, since then O h = h and P h = h, respectively, by Theorem 17.
Theorem 18. Assume that h is a convex normal integrand. If h(w) + and h * (v) are T -integrable for some T -integrablew and bounded optionalv, then for every T -integrable w. If h(w) + and h * (v) are T p -integrable for some T pintegrablew and bounded predictablev, then for every T p -integrable w.
Proof. It suffices to show that, for every τ ∈ T and A ∈ F τ , Without loss of generality, we may assume that the right side is not +∞. By [11,Theorem 3.9], τ A := τ +δ A belongs to T . Letw := 1 gph τA w+1 (gph τA) Cw so that (w, h(w)) is a T -integrable selection of epi h. Indeed, h(w) + is T -integrable by assumptions, and h(w) − is T -integrable by the Fenchel inequality and by the assumptions onv.
] by the definition of optional epiprojection, which implies the claim. The predictable case is proved similarly, where, by [11,Theorem 3.29], τ A ∈ T p whenever τ ∈ T p and A ∈ F τ − .
The following theorem is our main result on optional and predictable projections of set-valued processes. Recall that a set-valued process is T -integrable or T p -integrable if it has a T -integrable or T p -integrable selection, respectively.
Theorem 19. Assume that Γ is convex-valued stochastic process. If Γ is Tintegrable, then o Γ is given uniquely by If Γ is T p -integrable, then p Γ is given uniquely by Proof. By Theorem 17 and Theorem 16, is a conjugate of an optional positively homogeneous convex normal integrand, and consequently it is an indicator function of a unique optional closed convex-valued stochastic process o Γ [26,Theorem 8.24]. The predictable case is proved similarly.
The following result is a simple extension of [1, Lemma 3.3] which was formulated for bounded processes.

Projections of convex-valued integrands
Let F be a set-valued S-integrand from Ω × R + × R m to R n in the sense that is a normal S-integrand on R m × R n .
Proof. By [4, Lemma A.2], inner semicontinuity of F implies that h t (·, ·, ω) is jointly lsc. We have h t (x, y, ω) = − inf u p t (x, y, u, ω) for Using S-measurability of Γ F , one may verify straight from the definition that epi p is S-measurable mapping from Ω × R + to R m × R n × R n × R. Since the projection mapping from R m × R n × R n × R to R m × R n × R is isc (it is single- Note that F (w) defines an S-measurable set-valued stochastic process for every S-measurable stochastic process w. This follows from [26, Theorem 14.13(a)] by choosing M as the projection from R n × R m to R n and S t (ω) = [{w t (ω)} × R n ] ∩ Γ F t (ω). We denote Λ O F = {w | w is optional and there exists T -integrable u ∈ L 0 (F (w))}, Λ P F = {w | w is predictable and there exists T p -integrable u ∈ L 0 (F (w))}.
Theorem 22. Let F be an inner-semicontinuous closed convex-valued integrand and assume that there exists a stochastic process m such that If m is T -integrable, then there exists a unique optional inner-semicontinuous closed convex-valued integrand o F such that If m is T p -integrable, then there exists a unique predictable inner-semicontinuous closed convex-valued integrand p F such that Proof. We define h t (x, y, ω) = δ * Ft(x,ω) (y). We get from the existence of m that h t (x, y, ω) ≥ −m t (ω)|x||y|, which in conjunction with Lemma 21 implies that h is a normal integrand of class (C). Thus, by Theorem 11, there exists a unique optional projection o h of h that is a normal integrand of class (C).
By [26,Proposition 14.45],h thus defines a normal integrand of class (C). The above lower bound also gives that (w, v) ∈ Λ O h for every bounded optional v. The predictable case is proved similarly.
For set-valued integrands, the analogs of epi-projections are the projections of the associated graphical mappings. Indeed, if F is the epigraphical mapping of a normal integrand, these definitions give exactly the epi-projections. The following theorem is an immediate corollary of Theorem 19.
Theorem 23. Let F be an outer semicontinuous graph-convex integrand. If there exists T -integrable w ∈ Λ O F , then there exists an optional outer semicontinuous graph-convex integrand O F such that Γ O F = o Γ F and it is given uniquely by If there exists T p -integrable w ∈ Λ P F , then there exists a predictable outer semicontinuous graph-convex integrand P F such that Γ P F = p Γ F and it is given uniquely by