no. 1
Équations différentielles stochastiques
A note on “Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions”
Jing, Guangdong1 ; Wang, Penghui2  
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, Tome 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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.103
Classification : 34L15, 60H10
Jing, Guangdong 1 ; Wang, Penghui 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. 
@article{CRMATH_2021__359_1_99_0,
     author = {Jing, Guangdong and Wang, Penghui},
     title = {A note on {{\textquotedblleft}Problem} of eigenvalues of stochastic {Hamiltonian} systems with boundary conditions{\textquotedblright}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {99--104},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {1},
     year = {2021},
     doi = {10.5802/crmath.103},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.103/}
}
TY  - JOUR
AU  - Jing, Guangdong
AU  - Wang, Penghui
TI  - A note on “Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions”
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 99
EP  - 104
VL  - 359
IS  - 1
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.103/
DO  - 10.5802/crmath.103
LA  - en
ID  - CRMATH_2021__359_1_99_0
ER  - 
%0 Journal Article
%A Jing, Guangdong
%A Wang, Penghui
%T A note on “Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions”
%J Comptes Rendus. Mathématique
%D 2021
%P 99-104
%V 359
%N 1
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.103/
%R 10.5802/crmath.103
%G en
%F CRMATH_2021__359_1_99_0
Jing, Guangdong; Wang, Penghui. A note on “Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions”. Comptes Rendus. Mathématique, Tome 359 (2021) no. 1, pp. 99-104. doi : 10.5802/crmath.103. http://www.numdam.org/articles/10.5802/crmath.103/

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

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

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

Cité par Sources :