Conditional McKean-Vlasov SDEs and Some Related Stochastic Optimization Problems

We study a class of conditional McKean-Vlasov SDEs (CMVSDEs, for short), in which all dynamics involve both the state and the conditional law of the solutions. We first investigate the (weak) well-posedness of the CMVSDEs in its most general form, and then look at two applications based on particular cases of such equations: a mean-field type stochastic control problems with partial observations and an extended form of the so-called Kyle-Back strategic insider trading equilibrium problem. In the former we prove the corresponding Pontryagin’s Stochastic Maximum Principle, and in the latter we seek a rigorous theoretical basis for the general Kyle-Back strategic insider trading equilibrium model, in the case when the insider is allowed to have dynamic information of the underlying asset rather than only the static one. The theoretical results on CMVSDEs will enable us to tie some loose ends of the heuristic arguments in the literature of this problem.

This talk is based on the joint works with Rainer Buckdahn, Juan Li, Yonghui Zhou, and Rentao Sun.