Stochastic Perron for Stochastic Target Problems.

Abstract

This thesis is devoted to the application of stochastic Perron's method in stochastic target problems. In Chapters II-V, we study different stochastic target problems in various setup. For each target problem, stochastic Perron's method produces a viscosity sub-solution and super-solution to its associated Hamilton-Jacobi-Bellman (HJB) equation. We then characterize the value function in each problem as the unique viscosity solution to the associated HJB equation using a comparison result.

In Chapter II, we investigate stochastic target problems in a jump diffusion setup, where the controls are unbounded. Since classical control problems can be analyzed under the framework of stochastic target problems, we use our results to generalize the results of Bayraktar and Sirbu (SIAM J Control Optim 51(6): 4274-4294, 2013) to problems with controlled jumps.

In Chapter III, we study stochastic target problems with a stopper under the setup as in Chapter II. We prove that the target problem with a cooperative stopper (resp. with a non-cooperative stopper) can be expressed in terms of a cooperative controller-stopper problem (resp. a controller-stopper game).

In Chapter IV, we analyze the framework of stochastic target games, in which one player tries to find a strategy such that the state processes reach a given target at a deterministic time no matter which action is chosen by the other player (Nature). Besides obtaining the PDE characterization of the value function, we also prove the dynamic programming principle as a corollary.

In Chapter V, we study two types of stochastic target games with a stopper under the framework of Chapter IV. We show that the value function in each problem is the unique viscosity solution of a variational HJB equation. We also compare the value functions and prove that they coincide when the control set of Nature is a singleton.