# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021055
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## Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

 School of Mathematics, Shandong University, Jinan 250100, China

* Corresponding author: Guangdong Jing

Received  January 2021 Revised  October 2021 Early access December 2021

Fund Project: The first author is supported by NNSFC grant 11871308; The second author is supported by NNSFC grant 11471189, 11871308

In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in Peng [12] 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.

Citation: Guangdong Jing, Penghui Wang. Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021055
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##### References:
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