Selection theorems for stochastic set-valued integrals. (Q2772993)

A number of selection theorems for stochastic set-valued integrals (with respect to the Lebesgue measure) are derived. These integrals are defined by Aumann's proceedure with respect to nonanticipative multi-valued mappings. More specifically let \((\Omega , {\mathcal F}, ({\mathcal F_{t}})_{0\leq t \leq T},P)\) be a filtered probability space. A process \(f_{t}:[0,T]\times \Omega \to \text{\text{ I\!R}}^{n\times m}\) is said to be \({\mathcal F}_{t}\)--nonanticipative when when it is measurable with respect to product \(\sigma\)--algebra \(\beta_{T} \otimes {\mathcal F}\) on the product space \([0,T]\times \Omega\), where \(\beta_{T}\) is the Borel \(\sigma\)--algebra of subsets of \([0,T]\), and \(f_{t}\) is also \(({\mathcal F}_{t}, \beta(\text{\text{ I\!R}}^{n\times m})\)--measurable. Let \({\mathcal M}_{n\times m}^{p} ({\mathcal F}_{t})\) denote the family of equivalence classes of all \(n\times m\)--dimensional \({\mathcal F}_{t}\)--nonanticipative stochastic processes \((f^{t})_{0 \leq t \leq T}\) such that \(\int_{0}^{T} \| f_{t} \|^{p} \, d t < \infty\), P-a.s. For \(p \geq 1\), denote NEWLINE\[NEWLINE {\mathcal L}_{n \times m}^{p} ( {\mathcal F}_{t})= \left\{ (f_{t})_{0 \leq t \leq T} \in {\mathcal M}_{n \times m}^{p} ( {\mathcal F}_{t} ): E \int_{0}^{T} \| f_{t} \| ^{p} \, d t < \infty \right\}, NEWLINE\]NEWLINE where \(E\) is the mean value operator with respect to \(P\). Define \(\int_{0}^{T} f_{t} \, d t\) using the usual Lebesgue integral \(\int _{0}^{T} f_{t} (\omega) \, d t\) for each \(\omega \in \Omega\). Now consider a \({\mathcal F}_{t}\)-- nonanticipative set--valued process defined as a mulitfunction \({\Im }: [0,T]\times \Omega \to Cl(\text{\text{ I\!R}}^{n\times m})\), that are \(\beta_{T}\otimes {\mathcal F}\)--measurable, has closed images and has sections \({\mathcal F}_{t}\) that are \({\mathcal F}_{t}\)--measurable for all \(t \in [0,T]\). The main class of processes considered are denoted \({\mathcal L}_{s-v}^{p} ({\mathcal F}_{t}, \text{\text{ I\!R}}^{n\times m} )\), consisting (of the equivalence class) of all \({\mathcal F}_{t}\)--nonanticipative set--valued processes such that \(E \int_{0}^{T} [\sup \{ | f_{t} | : f_{t} \in {\Im }\}] ^{p} \, d t < \infty\), \(p \geq 1\). Let \(S^{p} (\Im )\) be the set of all \(f \in {\mathcal L}_{m \times n}^{p} ({\mathcal F}_{t})\) such that \(f_{t} (\omega ) \in {\Im}_{t} (\omega )\) for a.e. \((t,\omega) \in [0, T] \times \Omega\) and NEWLINE\[NEWLINE \int_{0}^{T} \Im _{t} \, d t = \left\{ \int_{0}^{T} f_{\tau} \, d t : (f_{\tau})_{0 \leq \tau \leq T} \in S^{p}({\Im}) \right\}. NEWLINE\]NEWLINE Selection theorems of the following type are considered: \newline Let \(\Im \in {\mathcal L}^{1}_{s-v} ({\mathcal F}_{t} , \text{\text{ I\!R}}^{1\times n})\) and \({\mathcal G} \in {\mathcal L}^{4}_{s-v} ({\mathcal F}_{t} , \text{\text{ I\!R}}^{n\times m})\) be convex and diagonally convex, respectively and let \((\varphi_{t})_{0 \leq t \leq T} \in {\mathcal L}_{1 \times n}^{1} (\tilde {\mathcal F}_{t})\) and \((\psi_{t})_{0 \leq t \leq T} \in {\mathcal L}_{n \times m}^{4} (\tilde {\mathcal F}_{t})\). Assume that for \(0 \leq s < t \leq T\) we have \(\tilde E \int_{s}^{t} \varphi_{\tau} \, d \tau \in E \int_{s}^{t} \Im_{\tau} \, d \tau\) and \(\tilde E \int _{s}^{t} \psi_{\tau} \cdot \psi_{\tau}^{T} \, d \tau \in E \int_{s}^{t} D({\mathcal G}_{\tau} ) \, d \tau\), where \(\tilde E\) is the mean valued operator with respect to a measure \(\tilde P\) and \(D({\mathcal G}_{\tau})(\omega) =\{u\cdot u^{T} : u \in {\mathcal G}_{\tau} (\omega ) \}\). Then there are \(f \in S^{1} (\Im )\) and \(g \in S^{4} ({\mathcal G})\) such that \(\tilde E \int_{s}^{t} \varphi_{\tau} \, d \tau = E \int_{s}^{t} f_{\tau} \, d \tau\) and \(\tilde E \int_{s}^{t} \psi_{\tau}\cdot \psi_{\tau}^{T} \, d \tau = E \int_{s}^{t} g_{\tau} \cdot g_{\tau}^{T} \, d \tau\) for every \(0 \leq s < t \leq T\).