Problem Of Eigenvalues For Stochastic Hamiltonian System With Boundary Conditions

In this paper,we study the eigenvalue problem of stochastic Hamiltonian systems with boundary conditions,which is a kind of parameterized two points boundary value problem.Eigenvalue problems are elementary in both basic and applied mathematics,and the eigenvalue problems for differential equations play important roles in,for example,the stability of solutions for equations.Compared with the extensive study for eigenvalue problems in deterministic setting,the stochastic counterpart just sets out.Peng in 2000[25]studied the eigenvalue problems for stochastic differential equations with constant coefficients and boundary conditions utilizing the blow-up time for solutions to Riccati equations,through which he further studied the statistic periodicity for solutions of stochastic differential equations.Besides,Wu and Wang[29]studied stochastic eigenvalue problems for autonomous Hamiltonian systems driven by Poisson jumps with boundary conditions.Based on these results,the present paper study the eigenvalue problems of stochastic Hamiltonian systems with boundary conditions.Stochastic Hamiltonian systems were put out originally as necessary conditions for stochastic optimal control problems,and is a kind of coupled forward backward stochastic differential equations.Since the breakthrough of Peng and Pardoux in 1990[21]for nonlinear backward stochastic differential equations,the theory for general coupled forward backward stochastic differential equations has developed rapidly[1,5,9,17,19,22,26,34,36,38,39],which still attracts much attention at present[4,6,7,8,10,11,15,20,30,32,40].The method for studying these eigenvalue problems is firstly transforming the original Hamiltonian systems into dual Hamiltonian systems by Legendre transformation.Then deriving Riccati equations by decoupling methods for forward backward stochastic differential equations,and then studying the characterization of solutions to these Riccati equations.Besides,several comparison theorems and continuously dependence of solutions with respect to coefficients,initial conditions,and parameters also play important roles among the main proofs.The specific arrangement of this paper is as follows:In the first section,we introduce some definitions and set the problem studied afterwards.We also illustrate,on one hand,the essential difference between the stochastic eigenvalue problems and deterministic counterparts,and on the other hand,the particular difficulties for non-autonomous stochastic eigenvalue problems compared with autonomous analogues,through several concrete examples.Besides,some hypothesis conditions and lemmas are presented.In the second section,for a kind of eigenvalue problem for stochastic Hamiltonian system with boundary conditions on the basis of the existence results in[25],we characterize the eigenvalues' increasing order.The third section is concerned with a kind of eigenvalue problem for nonautonomous one dimensional stochastic Hamiltonian systems with boundary conditions.We firstly prove the existence of a series of eigenvalues and construct related eigenfunctions,as well as estimate the dimension of eigenspaces.Then we present a mild condition ensuring identifying every eigenvalue.What's more,by constructing proper auxiliary problems we give an estimation for the increasing order of the series of eigenvalues,and this result is further utilized to characterize the statistic periods of corresponding forward backward stochastic differential equations directly by coefficients.The fourth section is devoted to studying eigenvalue problems for high dimensional non-autonomous stochastic Hamiltonian systems with boundary conditions,confirming the existence of eigenvalues,constructing the eigenfunctions,and estimating the dimension of eigenspaces.Another kind of problem which also cannot be included in Peng's functional framework is also studied.The fifth section deals with similar problems but with systems driven by Poisson jumps and Brownian diffusions.