Studies On Several Classes Of Backward Stochastic Differential Equations And Their Applications

This thesis is devoted to the studies of several classes of backward stochastic differ-ential equations and their applications.In the first Chapter, we give an overview about BSDEs for the reader's convenience. In particular, recall some well-known results.In Chapter2, we weaken the Lipschitz conditions on generators for anticipated backward stochastic differential equations in Peng and Yang [76]. By using Bihari's inequality, we obtain the existence and uniqueness result for this type of equations and comparison theorems under non-Lipschitz assumptions. Furthermore, we use the more direct Picard's iteration argument to get the Lp solutions compared with the method used in [21].In Chapter3, basing on [76], we study anticipated backward stochastic differential equations with jumps under non-Lipschitz conditions. We get a dual relation between stochastic delay differential equations and this type of equations. Moreover, we also obtain the existence and uniqueness result, Lp solutions for this type of equations, using a different method compared with the method used in Chapter2and with the different assumptions, we get the comparison theorem and its lemma. At last, we study a new type of anticipated backward stochastic differential equations and get the existence and uniqueness result.In Chapter4, combining the generalized anticipated backward stochastic differential equations with doubly reflected backward stochastic differential equations, we study the doubly reflected generalized anticipated backward stochastic differential equations and obtain the existence and uniqueness result as well as stability of solutions. Then, we give several applications for this type of equation. Using the existence and uniqueness result of this type of equations, we study the existence and uniqueness results of doubly reflected generalized anticipated backward stochastic differential equations with two functional barriers, which is applied to stochastic games.In Chapter5, by studying mean field backward stochastic differential equations, backward doubly stochastic differential equations and reflected backward stochastic dif-ferential equations, we study mean field reflected backward doubly stochastic differential equations, and obtain the existence and uniqueness results as well as the comparison theorems. As an application of comparison theorem of this type equations, we get its responding minimal solution and maximal solution. In the general, we often require that the generators should be satisfied the continuous condition and linear growth condition. Compared with the usual conditions, we can obtain similar results, which we don't re-quire the generators satisfy the continuous conditions. In particular, We modify a minor fault in the proof of (V)in Lemma3.1of [28]. By changing the linear condition of f into a similar the linear condition, we can solve this minor fault. If f satisfies the usual linear growth condition, we can not get the the minimal solution and maximal solution results for this type of equations. When changing the linear condition of f into a similar the linear condition, we can use a different method to finish the proof. The concrete method will be described in the main results of this chapter.