On solutions to stochastic differential inclusions
This papers studies the optimal weak solutions of a stochastic inclusion with a set valued drift. That is \[ \begin{aligned} d \xi_{t} &\in F(t,\xi_{t})dt+G(t,\xi_{t})dW_{t},\quad t\in [0,T] \\ P^{\xi_{0}} &= \mu \quad \text{(a probability measure on \(\mathbb R^{d}\) with the Borel field \(\beta(\mathbb R^{d})\))} \end{aligned} \] where \(F\), respectively \(G\), are a convex set valued mapping from \([0,T]\times \mathbb R^{d}\) to \(\mathbb R^{d}\), respectively \(\mathbb R^{d\times m}\). These equations provide a theoretical description of stochastic control and the approach used here is to utilize the equivalent martingale problem. The existence of a weak solution for the stochastic inclusion is shown to be equivalent to the existence of a solution to a local martingale problem introduced in the paper. Let \({\mathcal R^{loc}}(F,G,\mu)\) denote the set of all distributions \(Q\) which are solutions of the local martingale problem. These correspond to solutions of the differential inclusion as distributions of \(\xi\) within the space of vector valued continuous functions \({\mathcal C}\) endowed with the Borel \(\sigma\)-field \(\beta ( {\mathcal C})\). That is, \(Q:=P^{\xi}\in {\mathcal M}({\mathcal C})\) with the later set denoting the set of all probability measures on \(({\mathcal C}, \beta ({\mathcal C}))\). The nonemptyness and compactness of \({\mathcal R^{loc}}(F,G,\mu)\) is studied along with the upper semi--continuity of the mapping \(\mu \mapsto {\mathcal R^{loc}}(F,G,\mu)\) viewed as a multifunction from \({\mathcal M}(\mathbb R^{d})\) (the set of all probability measures \(\mu\) on \((\mathbb R^{d},\beta (\mathbb R^{d}))\) equipped with the topology of weak convergence) into the nonempty compact subsets of \({\mathcal M}({\mathcal C})\). Finally the existence is established of an optimal solution \(Q^{\ast}_{\mu} \in {\mathcal R^{loc}}(F,G,\mu)\) in the sense that \[ E_{Q^{\ast}_{\mu}} [\int_{0}^{T} h(t, \pi_{t})dt]=\sup_{Q\in {\mathcal R^{loc}}(F,G,\mu)} E_{Q} [ \int_{0}^{T} h(t,\pi_{t})dt], \] where \(\pi_{t}(x)=x(t)\) for \(x \in {\mathcal C}\), and \(t \in [0,T]\), is the canonical projection process.
weak convergence of probability measures.
scientific article; zbMATH DE number 2046233