One of the aftereffects of the financial crisis is the increased awareness of the need for more advanced modeling in mathematical finance, and a focus of attention is on the problem of model uncertainty. This presentation is motivated by a topic of this type.
We consider a stochastic system described by a general Itô-Lévy process controlled by an agent. The performance functional is expressed as the Q-expectation of an integrated profit rate plus a terminal payoff, where Q is a probability measure absolutely continuous with respect to the original probability measure P. We may regard Q as a scenario measure controlled by the market or the environment. If Q = P the problem becomes a classical stochastic control problem
.If Q is uncertain, however, the agent might seek the strategy which maximizes the performance in the worst possible choice of Q.
This leads to a stochastic differential game between the agent and the market.
Our approach is the following:
We write the performance functional as the value at time t = 0 of the solution of an associated backward stochastic differential equation (BSDE). Thus we arrive at a (zero sum) stochastic differential game of a system of forward-backward SDEs (FBSDEs), which we study
by a maximum principle approach.
We then apply the above results to study optimal portfolio and consumption problems under model uncertainty. We establish a connection between market viability under uncertainty and martingale measures.
Finally we give explicit formulas for the optimal portfolio and the optimal scenario parameter in some special cases.
The presentation is based on recent joint work with Agnès Sulem, INRIA, Paris.|