Tuesday, April 10 at 4:15pm
Frank H.T. Rhodes Hall, 253
This talk is motivated by the problem of optimization under model uncertainty. One way to present model uncertainty mathematically, is by means of a family of probability measures Q which are absolutely continuous with respect to an original reference probability measure P, and by allowing uncertainty regarding which of the measures Q should be taken into account when evaluating performance.
We consider here 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. 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 controlled backward stochastic differential equation (BSDE). Thus we arrive at a (zero sum) stochastic differential game of a system of forward-backward SDEs (FBSDEs) that we study by the maximum principle approach.
We state general stochastic maximum principles for stochastic differential games of FBSDEs with jumps, both in the zero-sum case (finding necessary and sufficient conditions for saddle points), and for the non-zero sum games (finding conditions for Nash equilibria).
We then apply these techniques to study an optimal portfolio problem in a Lévy market, under model uncertainty. We do not assume that the system is Markovian. We establish a connection between market viability under uncertainty and equivalent local martingale measures. Finally we give explicit formulas for the optimal portfolio and the optimal scenario for a class of incomplete markets.
(Joint work with Bernt Øksendal, University of Oslo, Norway)