Representation Theorems For Generators Of BSDEs And Their Applications To PDEs

In 1990,Pardoux-Peng [115] proposed the nonlinear Backward Stochastic Differential Equations(BSDEs for abbreviation),and obtained the existence and uniqueness for solutions.Since then,the BSDE theory attracted many researchers in interest and importance due to its many applications to various fields,such as stochastic analysis,Partial Differential Equations(PDEs for short),stochastic control and mathematical finance.This dissertation primarily studies the existence and uniqueness for solutions and the representation theorem for generators of BSDEs,the nonlinear Doob-Meyer's decomposition for continuous g-supermartingals by the existence and uniqueness result,second order nonlinear PDEs with obstacle or boundary problems by the representation theorem,stochastic differential games with state constraints,and a series of results.The major achievement of this dissertation is “The probabilistic interpretation problem for viscosity solutions of PDEs(Feynman-Kac formula)can be boiled down to a representation problem for a generator of a certain BSDE”.In Chapter 2,we firstly prove an existence and uniqueness result for solutions of general time interval(0 ? T ? ?)multidimensional BSDEs by a global domain truncation method involved with a convolution approximation technique,assuming the generator g satisfies weak monotonicity and general growth conditions with respect to y,and a Lipschitz condition with respect to z;secondly,we show the existence and uniqueness for solutions of BSDEs with terminal time being an unbounded stopping time by the method of truncating the time interval using the stopping time;thirdly,under the same conditions we prove a comparison theorem for solutions of stochastic interval one-dimensional BSDEs;finally,with the help of these two results of stochastic interval cases,we prove a nonlinear Doob-Meyer's decomposition theorem for continuous g-supermartingales by postulating an additional one-side growth condition.The framework of T = ? and the general growth condition leads to the lack of weak compactness for stochastic process sequences,which make the classical weak convergence method invalid.This difficulty is successfully overcome by studying the weak sequential compactness of the sequence in a new space.In Chapter 3,we propose a unified approach — the representation theorem for generators of BSDEs — to prove the probabilistic interpretation for viscosity solutions of second order PDE of semilinear,quasilinear and HJB types.We first show the probabilistic interpretation for quasilinear parabolic PDEs with Cauchy problems utilizing the representation theorem,where b(t,x,y,z)= b(t,x,y)and ?(t,x,y,z)= ?(t,x,y),i.e.,both of the drift and diffusion coefficients are independent of z.We discuss the case of b and ? being dependent on z in Chapter 4.In order to illustrate the advantage of representation theorem approach,we prove a representation theorem under a general growth condition,and then apply it to show the probabilistic interpretation for semilinear parabolic PDEs with Cauchy problems.Finally,under the classical Lipschitz conditions we present the probabilistic interpretation for parabolic HJB equations.To summarize,the probabilistic interpretation problems for viscosity solutions of nonlinear PDEs with Cauchy problems can be boiled down to representation problems for generators of some BSDEs.In Chapter 4,we first prove the existence and uniqueness for solutions of a fully coupled Forward-Backward Stochastic Differential Equation with Reflections(FBSDER for short)by the method of contraction mapping under the monotonicity and Lipschitz conditions.Then,inspired by the representation theorem approach in Chapter 3,we prove the existence for viscosity solutions of a second order quasilinear parabolic PDE with obstacle and Neumann problems by the unique solution of the FBSDER.This existence enjoys the following features: the spatial variable lives in a bounded connected closed region without convex constraints;the second order coefficient of PDEs depends on the gradient of the solution;the algebra equation,which is required when ? depending on z,is solved only under Lipschitz condition,and the monotonicity condition is eliminated.Finally,we prove a comparison principle for viscosity supersolutions and subsolutions in the case of ?(t,x,y,z)= ?(t,x),i.e.,? being independent of y and z,which leads to the uniqueness result of viscosity solutions.In Chapter 5,we first employ a random time change method to prove the representation theorem for generators of a Generalized BSDE(GBSDE for short).Due to the presence of a random measure d Arin the GBSDE,the existing methods to the classical case cease to work.Fortunately,the random measure can be transformed to a Lebesgue measure dr by the time change,which results in an equivalent between the GBSDE and a BSDE driven by martingales.Then the representation theorem for generators of GBSDEs can be transformed to the counterpart of BSDEs driven by martingales.After that,we study a two-player zero-sum stochastic differential game problem with state constraints,in which the state process is governed by a reflected stochastic differential equation and lives in a bounded connected closed region,and the cost functional is described by a GBSDE.We obtain the strong dynamic programming principle and regularity property of the value function.Then we adopt the representation theorem for generators of GBSDEs to characterize the value function as a viscosity solution of an Isaacs equation with nonlinear Neumann boundary problems.The uniqueness is suggested by the comparison principle for viscosity supersolutions and subsolutions.Taking into account the results of Chapters 3 – 5,it is the representation theorem for generators of BSDEs that we utilize to prove the probabilistic interpretation,in viscosity sense,for second order parabolic semilinear,quasilinear PDEs,HJB and Isaacs equations which are fully nonlinear PDEs,and the corresponding Cauchy initial value,Neumann boundary and obstacle problems.These results interpret that all the probabilistic interpretation for these usual types of PDEs(nonlinear Feynman-Kac formula)can be transformed to the representation problem for generators of BSDEs.Therefore,we call the representation theorem method a unified approach to the probabilistic interpretation problems.