Model Uncertainty in Finance and Second Order Backward Stochastic Differential Equations.

Summary The main objective of this thesis is to study some financial mathematics problems in an incomplete market with model uncertainty. Recently, the theory of second order backward stochastic differential equations (2EDSRs) has been developed by Soner, Touzi and Zhang on this topic. In this thesis, we adopt their viewpoint. This thesis contains four parts in the area of 2EDSRs. We start by generalizing the theory of 2EDSRs initially introduced in the case of continuous lipschitzian generators to the case of quadratically growing generators. This new class of 2EDSRs will then allow us to study the robust utility maximization problem in non-dominated models. In the second part, we study this problem for three utility functions. In each case, we give a characterization of the value function and of an optimal investment strategy via the solution of a 2EDSR. In the third part, we also provide an existence and uniqueness theory for second-order reflected EDSRs with lower obstacles and lipschitzian generators. We then apply this result to the study of the American option pricing problem in a financial model with uncertain volatility. In the fourth part, we study 2EDSRs with jumps. In particular, we prove the existence of a unique solution in an appropriate space. As an application of these results, we study a robust exponential utility maximization problem with model uncertainty. The uncertainty affects both the volatility process and the measurement of jumps.