Stochastic orderings by nonlinear expectations

Abstract

We study the theory of stochastic order under the nonlinear expectations framework, including g-and G-expectations, which leads to more general concepts of orderings in comparison with the standard linear expectation setting. The g-expectation E g is generated by a solution of Backward Stochastic Differential Equation (BSDE) and it provides a new robust tool for option pricing in finance. In the same manner, we denote by E G a nonlinear expectation generated by a solution of G-Backward Stochastic Differential Equation (G-BSDE), which is defined on G-expectation spaces. In this dissertation, our main contributions consist of four parts as follows:(i) We extend several properties of monotonicity, convexity and continuous dependence for the solutions of Forward-Backward Stochastic Differential Equations (FBSDEs) and associated semilinear parabolic Partial Differential Equations (PDEs). Also, we obtain analogous results for the solutions of G-Forward-Backward Stochastic Differential Equations (G-FBSDEs) and associated G-PDEs (Hamilton-Jacobi-Bellman-type equations). These properties are very crucial to constitute comparison results for the monotonic, convex, increasing convex g-and G-stochastic orderings.(ii) We give some characterizations for a better understanding of g-stochastic orderings which are defined via the g-evaluations and g-expectations E g . We then derive several sufficient conditions for the convex, increasing convex and monotonic g-stochastic orderings of diffusion processes at the terminal time T . Analogous comparison results for g-risk measures have been proposed as consequences, in terms of concave g-stochastic orderings. Applications of the g-stochastic orderings to contingent claim price comparison under different hedging portfolio constraints are also provided. (W t ) t≥0 Standard Brownian motion (B t ) t≥0 G-Brownian motion