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

The eigenvalue problem of stochastic Hamiltonian systems with boundary conditions was studied by Peng \cite{peng} in 2000. For one-dimensional case, denoting by $\{λ_n\}_{n=1}^{\infty}$ all the eigenvalues of such an eigenvalue problem, Peng proved that $λ_n\to +\infty$. In this short note, we prove that the growth order of $λ_n$ 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.

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