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 ItôLévy process controlled by an agent. The performance functional is expressed as the Qexpectation 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 forwardbackward 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 zerosum case (finding necessary and sufficient conditions for saddle points), and for the nonzero 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)
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