On Distribution Dependent Stochastic Differential Equations Driven By G-Brownian Motion

Distribution dependent stochastic differential equations have been a very hot direction with extensive applications in fields such as statistical physics,random control,chemical kinetics and so on.Scholars have studied the existence and uniqueness of solutions,as well as many important properties.On the other hand,motivated by uncertainty problems,risk measures and super-hedging in finance,Academician Peng Shige invented the Gexpectation framework,under which he introduced stochastic differential equations driven by G-Brownian motion(in short form,G-SDEs),and proved the existence and uniqueness of solutions to G-SDEs under the Lipschitz condition.Since then,scholars have studied the existence and uniqueness of G-SDEs solutions,moment estimates of the solutions,and the Euler-Maruyama approximation under different non-Lipschitz conditions.Based on these studies,we introduce the distribution dependent stochastic differential equations driven by G-Brownian motion(in short form,distribution dependent G-SDEs),we are mainly concerned with the well-posedness of the distribution dependent G-SDEs.In Chapter 3 we construct functional spaces where the distributions of random vectors are located and functional process spaces where the distributions of random processes are located under the sublinear expectation framework,and introduce the metrics on the spaces respectively,proving that the metric spaces we give are the complete metric spaces.In Chapter 4 we establish the existence and uniqueness of the solutions of distribution dependent G-SDEs under the Lipschitz condition by utilising fix point argument.Subsequently,we prove the moment estimates for the distribution dependent G-SDEs solutions directly with the help of inequalities on G-random integrals under the Lipschitz condition,then give a different proof using the G-It? formula.In Chapter 5 we introduce the distribution dependent non-Lipschitz condition under the G-framework and prove the uniqueness and existence of the solutions of distribution-dependent G-SDEs under the non-Lipschitz condition using Picard’s iterative method with the help of Bihari’s inequality,and then give moment estimates for distribution-dependent G-SDEs solutions under the non-Lipschitz condition using the G-Ito formula.Finally,we study the Euler-Maruyama approximation scheme for distribution dependent G-SDEs,and give the Euler-Maruyama convergence theorem under the non-Lipschitz condition.