existencetheoryandstabilityofsolutionsforgsdeswiththemethodofupperandlowersolutions

1、Existence Theory and Stability ofSolutions for G-SDEs:with the Method of Upper and Lower SolutionsbyFaizullah FaizSupervised byProf. Dr. Daxiong PiaoCollege of Physical and Environmental Oceanography,Ocean University of

2、China, Qingdao 266071, PR ChinaA thesis submitted for the degree ofDoctor of Philosophyc ? Faiz 2012AbstractMotivated from the risk measures, superhedging in finance and uncertainties instatistics, the G-Brownian motion

3、{Bt : t ≥ 0}, which is concisely speaking is acontinuous process having stationary and independent increments under a givensublinear expectation E[.], was introduced by S. Peng. The theory of G-Brownianmotion is really s

4、till in its infancy. In this dissertation the existence and stability ofsolutions for stochastic differential equations under G-Brownian motion (G-SDEs)are discussed.The method of upper and lower solutions for G-SDEs is

5、introduced. The exis-tence theory for the solutions of G-SDEs with discontinuous coefficients is estab-lished. It is determined that the G-SDEs have more than one solutions if the driftcoefficients or the second coeffici

6、ents are discontinuous functions. The solutions ofG-SDEs exist even if first and second coefficients are discontinuous functions si-multaneously. Comparison theorems are given with the help of upper and lowersolutions.Th

7、e upper and lower solutions method is applied to the backward stochastic dif-ferential equations under G-Brownian motion (G-BSDEs). The existence of solutionsis shown for G-BSDEs, whose coefficients may be discontinuous

8、functions. Differ-ent cases are studied considering the first then second and then both coefficients asdiscontinuous functions. Some examples are examined by taking the Heaviside andsawtooth functions as the coefficients