Existence Theory and Stability of Solutions for G-SDEs-with the Method of Upper and Lower Solutions.pdf

1、Motivated from the risk measures, superhedging in finance and uncertainties in statistics, the G-Brownian motion {Bt:t≥0}, which is concisely speaking is a continuous process having stationary and independent increments

2、under a given sublinear expectation E[.], was introduced by S. Peng. The theory of G-Brownian motion is really still in its infancy. In this dissertation the existence and stability of solutions for stochastic differenti

3、al equations under G-Brownian motion (G-SDEs) are discussed.
　　The method of upper and lower solutions for G-SDEs is introduced. The exis tence theory for the solutions of G-SDEs with discontinuous coefficients is est

4、ab lished. It is determined that the G-SDEs have more than one solutions if the drift coefficients or the second coefficients are discontinuous functions. The solutions of G-SDEs exist even if first and second coefficien

5、ts are discontinuous functions si multaneously. Comparison theorems are given with the help of upper and lower solutions.
　　The upper and lower solutions method is applied to the backward stochastic dif ferential equ

6、ations under C-Brownish motion (C-BSDEs). The existence of solutions is shown for G-BSDEs, whose coefficients may be discontinuous functions. Differ ent cases are studied considering the first then second and then both c

7、oefficients as discontinuous functions. Some examples are examined by taking the Heaviside and sawtooth functions as the coefficients of G-BSDEs. The existence and uniqueness of solutions for G-SDEs and G-BSDEs in a gene

8、ralized space are also studied.
　　The Lyapunov function method is established for G-SDEs. The p-moment exponential stability for G-SDEs is entrenched. It is validated that under the usual conditions the equilibrium po