Applications Of Backward Stochastic Differential Equation And Nonlinear Expectations: Risk Measure, Option Pricing And Estimation Of Evaluation Mechanism

The linear Backward Stochastic Differential Equation (BSDE) was first introduced by Bismut (1973). Pardoux & Peng (1990) proved the existence and uniqueness theorem of the solution of nonlinear BSDE under Lipschitz condition. Duffie & Epstein (1992b) also proposed a type of BSDE independently to characterize the stochastic differential utility (SDU). From then on, BSDE is further studied and applied widely in stochastic control, partial differential equation (PDE), mathematical finance and economics.The traditional expectation is a linear functional. There is a 1-1 correspondence between the traditional linear expectations and additive probability measures, but this 1-1 correspondence fails in nonlinear situations. In general, given a nonlinear expectation, one can still derive a nonadditive probability measure, but there exist an infinite number of nonlinear expectations satisfying the same relation. Thus in nonlinear situations the notion of expectation is more characteristic than that of non additive measures.Choquet (1954) extended the probability measure to capacity, and obtained definition of Choquet expectation. Choquet expectations have been applied in statistics, economics, finance and physics. But it is difficult to define conditional Choquet expectation in term of Choquet expectations. Peng (1997) introduced a kind of nonlinear expectation: g-expectation via a particular backward stochastic differential equation. Using Peng's g-expectation, it is easy to define conditional expectations. But g-expectation is a quasi linear expectation, i.e. the fully nonlinear situation can not be covered, Peng (2005b) proposed more general F_t consistent nonlinear expectations and nonlinear Markovian chains, Peng (2006a,b) proposed the notation and theory of G-expectation.Coherent risk measure was introduce by Artzner, Delbaen, Eber & Heath (1997, 1999), and it is an axiomatic tool able to quantify riskiness of financial positions. Convex risk measures were firstly studied by Heath (2000), later, in general probability...