A note on "Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions"

Abstract

The eigenvalue problem of stochastic Hamiltonian systems with boundary conditions was studied by Peng \cite{peng} in 2000. For one-dimensional case, denoting by $\{{\lambda }_n{\}}_{n=1}^{\infty }$ all the eigenvalues of such an eigenvalue problem, Peng proved that ${\lambda }_n\to +\infty$. In this short note, we prove that the growth order of ${\lambda }_n$ is the same as $n^2$ as $n\to +\infty$. Apart from the interesting of its own, by this result, the statistic period of solutions of FBSDEs can be estimated directly by corresponding coefficients and time duration.

7 pages