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Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

  • * Corresponding author: Guangdong Jing

    * Corresponding author: Guangdong Jing 

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

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  • 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.

    Mathematics Subject Classification: Primary: 60H10, 34B99; Secondary: 34F05, 34L15.


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