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Model Uncertainty Stochastic Mean-Field Control

Oberseminar Finanz- und Versicherungsmathematik

Model Uncertainty Stochastic Mean-Field Control


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
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