# Model Uncertainty Stochastic Mean-Field Control

#### Oberseminar Finanz- und Versicherungsmathematik

# Model Uncertainty Stochastic Mean-Field Control

### Abstract

We consider the problem of optimal control of a mean-ﬁeld stochastic

diﬀerential equation (SDE) 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 stochastic diﬀerential

game of a mean-ﬁeld related type 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 optimal control problem with

respect to random probability processes µ(t) in a non-Markovian setting is a new

type of stochastic control problems that has not been studied before. By

introducing operator-valued backward stochastic diﬀerential equations (BSDE),

we obtain a suﬃcient and a necessary maximum principle for Nash equilibria for

such games in the general nonzero-sum case, and for saddle points in 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.

Joint with Bernt Oksendal