Optional and Predictable Projections of Normal Integrands and Convex-Valued Processes

Abstract

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.

On a Stochastic Representation Theorem for Meyer-measurable Processes and its Applications in Stochastic Optimal Control and Optimal Stopping

“…The next result uses Kiiski and Perkkiö [2017]. Specifically, we will use Kiiski and Perkkiö [2017], Section 5, Corollary 2, p.324 for d = 1, but for general Meyer-σ-fields rather than just for the optional and predictable-σ-field.…”

Convex integral functionals of cadlag processes

“…implies that o z exists and belongs to D. The process z satisfies ( o z) − = p (z − ), by [24,Lemma 4]. Moreover, by [18,Corollary 4], o z ∈ D(S). Let (σ j,ν ) be an announcing sequence for τ j , where we may assume that σ j+1,ν ≥ τ j for every j and ν.…”

“…where in the first equality we used the fact that optional projections of bounded càdlàg processes are bounded and càdlàg [13, Theorem VI.47] and that optional projections of selections are again selections; see [18,Corollary 4]. In the first inequality we applied Jensen's inequality Lemma 22 and Jensen's inequality for optional set-valued mappings [18,Theorem 9]. Assume now that EI h is finite for arbitrary ȳ ∈ D(S).…”

“…where the first equality follows from [32, Theorem 14.60], since the process identically equal to zero belongs to L ∞ (Ξ, Ø, θ). For a stopping time τ , h τ is a normal integrand; see the discussion after Theorem 2 in [18]. Assume that θ( τ ) = 0.…”

Convex integral functionals of regular processes

“…The definition involves the notion of the optional projection of a normal integrand that we now recall; see [KP16].…”