<p style='text-indent:20px;'>In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in Peng [<xref ref-type="bibr" r>12</xref>] from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues <inline-formula><tex-math>\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> and construct corresponding eigenfunctions. Moreover, the order of growth for these <inline-formula><tex-math>\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> are obtained: <inline-formula><tex-math>\begin{document}$ \lambda_m\sim m^2 $\end{document}</tex-math></inline-formula>, as <inline-formula><tex-math>\begin{document}$ m\rightarrow +\infty $\end{document}</tex-math></inline-formula>. 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.</p>

中文翻译：

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带边界条件的随机哈密顿系统的特征值及其应用

<p style='text-indent:20px;'>本文求解带边界条件的随机哈密顿系统的特征值问题。首先，我们将 Peng [<xref ref-type="bibr" r>12</xref>] 中的结果从时间不变的情况扩展到时间相关的情况，证明了一系列特征值的存在 <inline-formula> <tex-math>\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> 并构造对应的特征函数。此外，获得了这些 <inline-formula><tex-math>\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> 的增长顺序: <inline-formula><tex-math>\begin{document}$ \lambda_m\sim m^2 $\end{document}</tex-math></inline-formula>, as <inline-formula>< tex-math>\begin{document}$ m\rightarrow +\infty $\end{document}</tex-math></inline-formula>。作为应用，我们给出了一个关于前向-后向SDE解的统计周期的明确估计公式。此外，通过一个细致的例子，我们展示了时变情况下的微妙情况，即当相关的里卡蒂方程的解没有爆裂时，会出现一些特征值，而这在时不变情况下不会发生。</p>