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Comptes Rendus. Mathématique
Stochastic differential equations
A note on “Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions”
Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 99-104.

The eigenvalue problem of stochastic Hamiltonian systems with boundary conditions was studied by Peng [4] in 2000. For the one-dimensional case, denoting by {λ n } n=1 all the eigenvalues of such an eigenvalue problem, Peng proved that λ n + as n. In this short note, we prove that the growth order of λ n is the same as n 2 . Apart from the interest of this result in itself, the statistic periodicity of solutions of FBSDEs can be estimated directly by corresponding coefficients and time duration.

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DOI: https://doi.org/10.5802/crmath.103
Classification: 34L15,  60H10
Guangdong Jing 1; Penghui Wang 2

1. School of Mathematics, Shandong University Jinan, Shandong 250100, The People’s Republic of China
2. School of Mathematics, Shandong University Jinan, Shandong 250100, The People’s Republic of China.
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Guangdong Jing; Penghui Wang. A note on “Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions”. Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 99-104. doi : 10.5802/crmath.103. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.103/

[1] Ying Hu; Shige Peng Solution of forward-backward stochastic differential equations, Probab. Theory Relat. Fields, Volume 103 (1995) no. 2, pp. 273-283 | MR 1355060 | Zbl 0831.60065

[2] Jin Ma; Philip Protter; Jiongmin Yong Solving forward-backward stochastic differential equations explicitly — a four step scheme, Probab. Theory Relat. Fields, Volume 98 (1994) no. 3, pp. 339-359 | MR 1262970 | Zbl 0794.60056

[3] Jin Ma; Zhen Wu; Detao Zhang; Jianfeng Zhang On well-posedness of forward-backward SDEs — a unified approach, Ann. Appl. Probab., Volume 25 (2015) no. 4, pp. 2168-2214 | MR 3349005 | Zbl 1319.60132

[4] Shige Peng Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions, Stoch. Proc. Appl., Volume 88 (2000) no. 2, pp. 259-290 | Article | MR 1767848 | Zbl 1045.60061

[5] Shige Peng; Zhen Wu Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optimization, Volume 37 (1999) no. 3, pp. 825-843 | Article | MR 1675098 | Zbl 0931.60048

[6] Haiyang Wang; Zhen Wu Eigenvalues of stochastic Hamiltonian systems driven by Poisson process with boundary conditions, Bound. Value Probl., Volume 2017 (2017), 164 | MR 3722278 | Zbl 1382.34026

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