Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

TL;DR

This paper extends the eigenvalue analysis of stochastic Hamiltonian systems with boundary conditions from time-invariant to time-dependent cases, providing existence, growth rates, and applications to stochastic differential equations.

Contribution

It generalizes previous results to time-dependent systems, proves eigenvalue existence, growth rates, and explores applications to Forward-Backward SDEs.

Findings

Eigenvalues grow approximately as m^2 for large m.

Explicit estimation formula for the statistic period of solutions.

Identification of subtle eigenvalue behaviors in time-dependent cases.

Abstract

In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in S. Peng \cite{peng} from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues $\{\lambda_m\}$ and construct corresponding eigenfunctions. Moreover, the order of growth for these $\{\lambda_m\}$ are obtained: $\lambda_m\sim m^2$, as $m\rightarrow+\infty$. 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.