Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

Abstract: In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in S. Peng \cite{peng} from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues ${\lambda_m}$ and construct corresponding eigenfunctions. Moreover, the order of growth for these ${\lambda_m}$ are obtained: $\lambda_m\sim m^2$, as $m\rightarrow+\infty$. As applications, we give an explicit estimation formula about the statistic period of solutions of Forward-Backward SDEs. Besides, by a meticulous example we show the subtle situation in time-dependent case that some eigenvalues appear when the solution of the associated Riccati equation does not blow-up, which does not happen in time-invariant case.