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.
Highlights
Introduction and main resultsLet (Ω, F, F, P) be a complete filtered probability space, on which a standard one-dimensionalBrownian motion B = {Bt }t ≥0 is defined, and F = {Ft }t ≥0 is the natural filtration of B augmented by all the P-null sets in F
A real number λ is called an eigenvalue of linear stochastic Hamiltonian system with boundary conditions (1) if there exists a nontrivial solution (x, y, z) of (1)
The eigenfunction space corresponding to each λn is of one dimension. Such a theorem was generalized to the eigenvalue problem of stochastic Hamiltonian system driven Poison process in [6]
Summary
Let (Ω, F , F, P) be a complete filtered probability space, on which a standard one-dimensional. A real number λ is called an eigenvalue of linear stochastic Hamiltonian system with boundary conditions (1) if there exists a nontrivial solution (x, y, z) of (1). This solution is called an eigenfunction corresponding to λ. The eigenfunction space corresponding to each λn is of one dimension Such a theorem was generalized to the eigenvalue problem of stochastic Hamiltonian system driven Poison process in [6]. From the point of view of eigenvalue problem of stochastic Hamiltonian system with boundary conditions, what is different is that some concrete characteristic such as statistic periodicity and stochastic oscillations of solutions of FBSDEs can be given in [4].
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