Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

Abstract

<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" rid="b12">12</xref>] from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues <inline-formula><tex-math id="M1">\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> and construct corresponding eigenfunctions. Moreover, the order of growth for these <inline-formula><tex-math id="M2">\begin{document}$ \{\lambda_m\} $\end{document}</tex-math></inline-formula> are obtained: <inline-formula><tex-math id="M3">\begin{document}$ \lambda_m\sim m^2 $\end{document}</tex-math></inline-formula>, as <inline-formula><tex-math id="M4">\begin{document}$ 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>