Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

2021/01/03 by Guangdong Jing, Jing, Guangdong, Penghui Wang +1
Computer Science · Economics, Econometrics and Finance · Mathematics · #34B99 #34F05 #34L15 #60H10 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications #math.CA #math.PR #msc:34B99 #msc:34F05 #msc:34L15 #msc:60H10

paper · pdf · doi:10.48550/arxiv.2101.00572

39 pages

arxiv created 2021/01/03 · openalex publication_date 2021/01/03 · arxiv updated 2021/01/05 · openalex created_date 2022/11/21 · openalex updated_date 2026/07/28

Abstract

In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in S. Peng \citepeng from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues \λ_m\ and construct corresponding eigenfunctions. Moreover, the order of growth for these \λ_m\ are obtained: λ_m∼ m², as m→+∞. 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.

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