Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in Peng [12] from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues \begin{document}$ \{\lambda_m\} $\end{document} and construct corresponding eigenfunctions. Moreover, the order of growth for these \begin{document}$ \{\lambda_m\} $\end{document} are obtained: \begin{document}$ \lambda_m\sim m^2 $\end{document}, as \begin{document}$ m\rightarrow +\infty $\end{document}. 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.