# Nacira Agram: Model Uncertainty Stochastic Mean-Field Control

Nacira Agram (University of Oslo) gives a lecture with the title: Model Uncertainty Stochastic Mean-Field Control.

We consider the problem of optimal control of a mean-field stochastic differential equation under model uncertainty. The model uncertainty is represented by ambiguity about the law L(X(t)) of the state X(t) at time t. For example, it could be the law LP(X(t)) of X(t) with respect to the given, underlying probability measure P. This is the classical case when there is no model uncertainty. But it could also be the law LQ(X(t)) with respect to some other probability measure Q or, more generally, any random measure μ(t) on R with total mass 1.

We represent this model uncertainty control problem as a \emph{stochastic differential game} of a mean-field related type stochastic differential equation (SDE) with two players. The control of one of the players, representing the uncertainty of the law of the state, is a measure valued stochastic process μ(t) and the control of the other player is a classical real-valued stochastic process u(t). This control with respect to random probability processes μ(t) on R is a new type of stochastic control problems that has not been studied before. By introducing operator-valued backward stochastic differential equations, we obtain a sufficient maximum principle for Nash equilibria for such games in the general nonzero-sum case, and saddle points for zero-sum games.

As an application we find an explicit solution of the problem of optimal consumption under model uncertainty of a cash flow described by a mean-field related type SDE.