Existence And Uniqueness Of Solutions And Representation Theorems Of Generators For A Class Of Stochastic Dynamical Systems-backward Stochastic Differential Equations

General speaking, stochastic dynamic systems can be modeled by stochastic differen-tial equations or stochastic partial differential equations, which describe the evolution ofstochastic dynamic systems. If a target in future is given for a stochastic dynamic system,then what strategies can we adopted to reach this target? Under many circumstances, this isa question of solving a backward stochastic differential equation (BSDE in short). Today,the BSDE theory has become one of hot directions in stochastic analysis and probabilityfields.Chapter 1 of this thesis is the introduction, which brie?y introduces the backgroundand some preliminaries of stochastic dynamic systems and BSDEs together with mainresults of this thesis. From Chapter 2 of this thesis, two fundamental problems in theBSDE theory are investigated in depth and some obvious advances are obtained.The first problem is about the existence, the existence and uniqueness and the com-parison theorem of solutions for one-dimensional and multidimensional BSDEs with finiteor infinite time horizons. This part contains four chapters from Chapter 2 to Chapter 5.The contents of these four chapters are introduced as follows: Chapter 2 includes threeworks. First, it is proved that if the generator g is continuous in (y,z) and has a lineargrowth in (y,z) which is not uniform on t, then there exists a unique minimal solution inL2 for the one-dimensional BSDE with a finite or an infinite horizon (see Theorem 2.1),which improves the result in Lepeltier-San Martin [1997]. Then, under the conditions thatthe generators satisfy the one-sided Osgood condition in y and the uniform continuity con-dition in z, both of which are not uniform on t, a general comparison theorem for solutionsin L2 of one-dimensional BSDEs with finite and infinite time horizons is established (seeTheorem 2.2). It generalizes four classical comparison theorems (see Briand-Hu [2006],Cao-Yan [1999], Chen-Wang [2000] and El Karoui-Peng-Quenez [1997]). Finally, an ex-istence and uniqueness result for solutions in L2 of one-dimensional BSDEs with finiteand infinite time horizons is obtained when the generators satisfy the Osgood condition iny and the uniform continuity condition in z, both of which are not uniform on t (see The-orem 2.3). This generalizes some corresponding results in Chen-Wang [2000], Constantin[2001], Hamade`ne [2003], Jia [2008b], Mao [1995] and Pardoux-Peng [1990] in the one-dimensional setting. In Chapter 3, a general existence and uniqueness result for solutions inLp (p > 1) of multidimensional BSDEs with finite or infinite time horizons is establishedwhen the generators satisfy a non-Lipschitz condition in y and a Lipschitz condition in z,both of which are not uniform in t (see Theorem 3.1). This includes and improves some corresponding results in Chen [1997], Chen-Wang [2000], Constantin [2001], Mao [1995],Pardoux-Peng [1990], Wang-Huang [2009] and Wang-Wang [2003]. In Chapter 4, underthe conditions that the generator g is monotonic in y andα?Ho¨lder (0 <α< 1) continu-ous in z, an existence and uniqueness result of solutions in L1 for one-dimensional BSDEswith finite time horizons is obtained (see Theorem 4.1). It can be regarded, in some sense,as a supplement and improvement of Theorem 6.2 and Theorem 6.3 in Briand-Delyon-Hu-Pardoux-Stoica [2003]. Chapter 5 proves the existence and uniqueness of solutions inLp (p > 1) for a multidimensional BSDE with a finite time horizon when g is uniformlycontinuous in (y,z) and the ith component gi of g only depends the ith row of the matrixz (see Theorem 5.2). This solves a problem in Hamade`ne [2003] on the uniqueness ofsolutions for multidimensional BSDEs under the uniform continuity condition of g withrespect to z.The second problem investigated by this thesis is about the representation theorem ofgenerators for one-dimensional BSDEs with finite time horizons and polynomial-growthgenerators in y. This part is placed in Chapter 6. All known representation theorems dealtwith the case that the generator g is of linear growth in y. As far as we know, this thesisis the first time to consider the case that g is of polynomial growth in y. More precisely,on basis of the existence and uniqueness result of the minimal and maximal solutions forone-dimensional BSDEs obtained in Briand-Lepeltier-San Martin [2007], Chapter 6 ofthis thesis establishes a new representation theorem in the space of processes, where thegenerator g is continuous in (y,z) and monotonic in y, and it has a polynomial growthin y and a linear growth in z (see Theorem 6.1). This representation theorem generalizesthe corresponding results in Fan [2006], Fan [2007a] and Fan-Hu [2008]. Furthermore,this chapter also establishes a representation theorem in the space of processes, where thegenerator g is continuous and of linear growth in (y,z) (see Theorem 6.6).Finally, it should be mentioned that in order to prove our main results, many technicalresults are established in this thesis, such as Lemma 2.3, Lemma 2.6, Remark 2.7, Remark3.4, Lemma 3.5, Proposition 4.3, Proposition 5.1, Proposition 6.3 and Lemma 6.4, etc.Many of them are purely analytic. They seem to be irrelevant to the probability theory andstochastic processes, however it is just owe to them that many key problems are overcome.