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

TL;DR

This paper refines the understanding of eigenvalues in stochastic Hamiltonian systems with boundary conditions, showing their growth rate is proportional to n^2, which aids in estimating solution periods of related stochastic differential equations.

Contribution

It establishes that the eigenvalues grow proportionally to n^2, improving previous results on their asymptotic behavior in stochastic Hamiltonian systems.

Findings

Eigenvalues grow as n^2 for large n

Growth order matches that of classical eigenvalue problems

Facilitates estimation of solution periods in FBSDEs

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.