Multivalued Stochastic Differential Equations

There has been an increasing interest in studying multivalued stochastic differential equations in the last twenty years. We consider a particular type of the multivalued stochastic differential equations in this thesis, that is multivalued stochastic differential equations with maximal monotone operators. An example of these maximal monotone operators is the subdifferential of a closed convex function. Therefore, the multivalued stochastic differential equations we considered contain a class of SDEs with reflection in a convex domain. On the other hand, the multivalued stochastic differential equations have closed relation with stochastic variational inequalities.On basis of Cépa's result, we considered some properties of the multivalued stochastic differential equations.(1) Applying Doss's method, we can get explicit solutions of the multivalued stochastic differential equations in one dimensional. That is to say, we can get the solution by solving an ordinary differential equation and a multivalued ordinary differential equation. But there is no existence and uniqueness result to this multivalued ordinary equation available. So we have to prove such a result firstly. Finally, we give a comparison theorem to the solutions of two multivalued stochastic differential equations with the same diffusion coefficient as an example.(2) We consider the limit theorem for the multivalued stochastic differential equations in multidimensional, and transfer properties from ordinary differential equations to the multivalued stochastic differential equations. We mainly use Meyer-Zheng topology, Kurtz lemma and the equivalence of weak solutions and martingale problems. We also give a new proof of Cépa's existence and uniqueness result.(3) Using Ikeda and Watanabe's method, we get the Denjoy's approximate continuity of solutions to the multivaled stochastic differential equations.(4) We prove the existence and uniqueness of solutions to the multivalued stochastic differential equations with non-Lipschitz coefficients. For the uniqueness, we use the Tanaka's formula and Le Gall's method. Besides, we get a bicontinuous modification for the solution in short time. Finally, using Ren and Zhang's method, we get a bicontinuous modification for the solution in long time.(5) Using the contraction principle, Cépa only considered the large deviation principle for the multivalued stochastic differential equations in one dimensional case based on the explicit construciton of the solutions. Multidimensional case is still open. Based on recently well developed weak convergence method due to Dupis and Ellis, we get a large deviation principle of Freidlin-Wentzell's type for the multivalued stochastic differential equations with monotone drifts.