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 [4] in 2000. For the one-dimensional case, denoting by $\lbrace \lambda _n\rbrace _{n=1}^{\infty }$ all the eigenvalues of such an eigenvalue problem, Peng proved that $\lambda _n\rightarrow +\infty $ as $n\rightarrow \infty $. In this short note, we prove that the growth order of $\lambda _n$ is the same as $n^2$. Apart from the interest of this result in itself, the statistic periodicity of solutions of FBSDEs can be estimated directly by corresponding coefficients and time duration.