Stochastic Maximum Principle under Sublinear Expectation

Abstract

This thesis extends the famous Pontryagin’s stochastic maximum principle to the case of volatility uncertainty and ambiguity which is modelled by G-Brownian motion (G- SMP) where we present two research topics, the first one is divided into four parts. In the first part, we introduce an optimal control problem where the state equation is driven by G-Brownian motion and the cost functional is given of risk-neutral type. We prove the stability of controlled stochastic differential equations driven by G-Brownian motion (G-SDEs in short) with respect to the control variable by using the convex perturbation method, in which the set of admissible controls is convex. In the second part, we introduce the G-adjoint process and the G-adjoint equation by using the resolvent method and the G-martingale representation theorem. In the third part, we establish necessary and sufficient optimality conditions for this model. Lastly, we illustrate our main result by giving an example of a linear-quadratic problem where we solve the Riccati-type equation. The second topic is characterising the problem of optimal control under a risk-sensitive control model. Both the admissible control and the system dynamics are defined in the same way as those of the first topic. The only difference is the way of defining the performance criterion. Instead of minimizing the direct cost, we aim to minimize a convex disutility function of the cost. As a preliminary step, we clarify the relationship between risk-neutral and risk-sensitive loss functional. Secondly, we are doing a simple reformulation of risk-sensitive problem as a standard risk-neutral problem under G- expectation. Thus, An intermediate G-SMP is obtained by a standard application of risk-neutral result. Thirdly, we prove the equivalence relation between G-expected exponential utility and G-quadratic backward stochastic differential equation. Finally, we deal with the example of Merton’s problem with power utility.