Multi-valued Backward Stochastic Differential Equations Driven By G-Brownian Motion

In this thesis, we aim to consider a class of multi-valued backward stochastic differ-ential equations driven by G-Brownian motion (G-MBSDE, in short). This work includes three parts. In the first part, the uniqueness and existence of a solution for the following G-MBSDE are proved by means of the Moreau-Yosida approximation methodwhere K. is a decreasing G-martingale. B. is a G-Brownian motion, <B>. is the quadratic variation process associated to B., (?)? is a lower and semicontinuous function on Rd,which is a subdifferential operator concerned to ?, ? is called as the terminal condition.Moreover, we will give a probabilistic interpretation for the viscosity solutions of a class of nonlinear variational inequalities. In details, we consider the following coupled forward-backward G-SDEWe aim to show the function defined by u(t, x) = Yt,xt is the viscosity solution of the following nonlinear variational inequality whereIn the second part, we establish the large deviation principle (LDP, in short) of the solution for G-BSDE. To do so, we consider the following decoupled small perturbation of the G-SDE and G-BSDE with deterministic terminal time, for 0 ? t ? T,We prove the solution (Xx,?, Yx,?, Zx,?) of the above decoupled forward-backward G-SDEs(1.3) converges, as ? goes to 0, to the solution (Xx,Yx,Zx) of the following decoupled forward-backward equation, for 0 < t < T,Also, we will establish the LDP for the law of Ytx,?.In the third part, we study the LDP of the solution for G-MBSDE. To do so, we consider the following small perturbation of the G-MBSDE, for 0 ? t ? T,We aim to get asymptotic behavior of the family (Yx,?t)?>0 as ? goes to 0, converge to following system and satisfies a LDP.