Backward Stochastic Differential Equations With Constraints And Related Problems

In 1990, Pardoux-Peng(1990) [67] proposed the general nonlinear backward stochastic differential equations(BSDEs) and provided the existence and uniqueness theorem of their solutions. Based on this equation, Peng(1997) [70] introduced the conception of nonlinear expectation g-expectation, which has a close relationship with risk measurement in finance. Later, El Karoui-Peng-Quenez(1997) [32], ChenEpstein(2002) [19] found that BSDEs were a powerful tool to solve the economic and financial problems. Since then, the theory of BSDEs have become a research hotspot in the fields of stochastic analysis, optimal control and math finance, etc. In this dissertation, we mainly study the basic problems of reflected BSDEs with lower barriers on finite or infinite time horizon the existence, uniqueness and related properties of their solutions, the inverse problem of g-expectation under what conditions, a nonlinear expectation becomes a g-expectation and the viability property for backward stochastic differential equation. Around the three basic issues, we make in-depth exploration and obtain a series of results.In the second chapter, we firstly prove one of our key results, monotonic limit theorem for semimartingales on general time horizon. It can be expressed roughly as: for a monotonic sequence of RCLL semimartingales with respect to a Brownian filtration on general time horizon, its limit is also a RCLL semimartingale. By this tool, we study existence of the smallest g-supersolution of a BSDE with constraints and prove the existence and uniqueness of Lp-solutions for reflected BSDEs with uniformly continuous generators on general time horizon, whose barriers are very irregular. Furthermore, if the barrier is continuous then the equation has a solution with continuous path. For different parameters of the equations, we can also get a comparison theorem. Both of our methods and results are the non-trivial generalization of the ones in Peng-Xu(2005)[72] and Peng(1999) [71].In the third chapter, using the conclusions in the second chapter, we obtain the existence and uniqueness of Lp-solutions for multi-dimensional reflected BSDEs with fixed lower barriers, whose generator satisfies the stochastic Lipschitz condition. And by introducing a new backward stochastic Gronwall inequality, we prove the comparison theorem for the one-dimensional equation. In addition, we also obtain the existence and uniqueness of Lp-solution for multi-dimensional oblique reflected BSDEs on general time horizon and establish a relationship with optimal switching problem. Furthermore, with the aid of this relation, under the condition that g satisfies the general time version of Lipschitz conditions, we can remove the assumption that each component of g only depends on the corresponding component of variable y, and get the existence and uniqueness of its Lp-solutions by constructing a contraction in a suitable space. We also explain how to use the oblique reflection equation to solve an optimal switching problem with stopping. According to the existing literatures, we only know some results about the multi-dimensional equations on finite time horizon with Lipschitz continuous generators, square integrable parameters and continuous barriers.Thus we make great progress in this respect.In the fourth chapter, we study the dynamic consistent nonlinear operator on general time horizon. With the help of monotonic limit theorem in the second chapter, we establish the nonlinear decomposition of Doob-Meyer’s type for RCLL gsupermartingale on the general time interval. And we also solve the inverse problem in the generalized framework: if a filtration-consistent nonlinear expectation with translation invariant property can be dominated by a g-expectation, then it must be a gexpectation. The conclusions are the Lp-integrable version on the general time horizon of the results in Peng(1999) [71] and Coquet-Hu-M′emin-Peng(2002) [23]. In addition, we also get the robust representation of sublinear g-expectation closely related to the consistent risk measurement.In the last chapter, we get rid of the two additional conditions that the generator has the properties of stronger integrability and continuity in Buckdahn-QuincampoixR?as?canu(2000) [11], and provide the representation theorem and a necessary and sufficient condition of viability property for multi-dimensional BSDEs under the basic assumptions. With these results, we can get the(strict) comparison theorem under two different orders(a common partial order and a total order defined by a projection with respect to a unit vector) and for all we know, the strict comparison theorems with respect to the two orders are obtained for the first time. And, we also study the viability for the general stochastic differential equations and established its necessary and sufficient condition and the comparison theorem.