A note on “Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions”

Guangdong Jing; Penghui Wang

doi:10.5802/crmath.103

Équations différentielles stochastiques

A note on “Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions”

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.

Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 99-104.

Résumé

The eigenvalue problem of stochastic Hamiltonian systems with boundary conditions was studied by Peng [4] in 2000. For the one-dimensional case, denoting by ${λ_{n}}_{n = 1}^{\infty}$ all the eigenvalues of such an eigenvalue problem, Peng proved that $λ_{n} \to + \infty$ as $n \to \infty$ . 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.

Reçu le : 2019-10-23
Révisé le : 2020-04-17
Accepté le : 2020-07-28
Publié le : 2021-03-01

DOI : 10.5802/crmath.103

Classification : 34L15, 60H10

Affiliations des auteurs :

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.

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {A note on {{\textquotedblleft}Problem} of eigenvalues of stochastic {Hamiltonian} systems with boundary conditions{\textquotedblright}},
     journal = {Comptes Rendus. Math\'ematique},
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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/

Bibliographie
Cité par

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | DOI | MR | Zbl

[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 | Zbl

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