# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021055
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## Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

 School of Mathematics, Shandong University, Jinan 250100, China

* Corresponding author: Guangdong Jing

Received  January 2021 Revised  October 2021 Early access December 2021

Fund Project: The first author is supported by NNSFC grant 11871308; The second author is supported by NNSFC grant 11471189, 11871308

In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in Peng [12] from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues $\{\lambda_m\}$ and construct corresponding eigenfunctions. Moreover, the order of growth for these $\{\lambda_m\}$ are obtained: $\lambda_m\sim m^2$, as $m\rightarrow +\infty$. 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.

Citation: Guangdong Jing, Penghui Wang. Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021055
##### References:
 [1] F. Antonelli, Backward-forward stochastic differential equations, Ann. Appl. Probab., 3 (1993), 777-793. [2] A. Bensoussan, Lectures on stochastic control, Nonlinear Filtering and Stochastic Control (Cortona, 1981), 1–62, Lecture Notes in Math., 972, Springer, Berlin-New York, 1982. [3] J.-M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44 (1973), 384-404.  doi: 10.1016/0022-247X(73)90066-8. [4] F. Delarue, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Process. Appl., 99 (2002), 209-286.  doi: 10.1016/S0304-4149(02)00085-6. [5] Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Probab. Theory Related Fields, 103 (1995), 273-283.  doi: 10.1007/BF01204218. [6] G. Jing and P. Wang, A note on "Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions", C. R. Math. Acad. Sci. Paris, 359 (2021), 99-104.  doi: 10.5802/crmath.103. [7] J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly – a four step scheme, Probab. Theory Related Fields, 98 (1994), 339-359.  doi: 10.1007/BF01192258. [8] J. Ma, Z. Wu, D. Zhang and J. Zhang, On well-posedness of forward-backward SDEs – a unified approach, Ann. Appl. Probab., 25 (2015), 2168-2214.  doi: 10.1214/14-AAP1046. [9] J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, , Springer-Verlag, Berlin, 1999. [10] E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Related Fields, 114 (1999), 123-150.  doi: 10.1007/s004409970001. [11] S. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27 (1993), 125-144.  doi: 10.1007/BF01195978. [12] S. Peng, Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions, Stochastic Process. Appl., 88 (2000), 259-290.  doi: 10.1016/S0304-4149(00)00005-3. [13] S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim., 37 (1999), 825-843.  doi: 10.1137/S0363012996313549. [14] H. Wang and Z. Wu, Eigenvalues of stochastic Hamiltonian systems driven by Poisson process with boundary conditions, Bound. Value Probl., 2017, Paper No. 164, 20 pp. doi: 10.1186/s13661-017-0896-4. [15] J. Yong, Forward-backward stochastic differential equations with mixed initial-terminal conditions, Trans. Amer. Math. Soc., 362 (2010), 1047-1096.  doi: 10.1090/S0002-9947-09-04896-X. [16] J. Yong, Linear forward-backward stochastic differential equations, Appl. Math. Optim., 39 (1999), 93-119.  doi: 10.1007/s002459900100. [17] J. Yong, Linear forward-backward stochastic differential equations with random coefficients, Probab. Theory Related Fields, 135 (2006), 53-83.  doi: 10.1007/s00440-005-0452-5. [18] J. Zhang, Backward Stochastic Differential Equations. From Linear to Fully Nonlinear Theory, Springer, New York, 2017. doi: 10.1007/978-1-4939-7256-2. [19] J. Zhang, The wellposedness of FBSDEs, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 927-940.  doi: 10.3934/dcdsb.2006.6.927.

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##### References:
 [1] F. Antonelli, Backward-forward stochastic differential equations, Ann. Appl. Probab., 3 (1993), 777-793. [2] A. Bensoussan, Lectures on stochastic control, Nonlinear Filtering and Stochastic Control (Cortona, 1981), 1–62, Lecture Notes in Math., 972, Springer, Berlin-New York, 1982. [3] J.-M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44 (1973), 384-404.  doi: 10.1016/0022-247X(73)90066-8. [4] F. Delarue, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Process. Appl., 99 (2002), 209-286.  doi: 10.1016/S0304-4149(02)00085-6. [5] Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Probab. Theory Related Fields, 103 (1995), 273-283.  doi: 10.1007/BF01204218. [6] G. Jing and P. Wang, A note on "Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions", C. R. Math. Acad. Sci. Paris, 359 (2021), 99-104.  doi: 10.5802/crmath.103. [7] J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly – a four step scheme, Probab. Theory Related Fields, 98 (1994), 339-359.  doi: 10.1007/BF01192258. [8] J. Ma, Z. Wu, D. Zhang and J. Zhang, On well-posedness of forward-backward SDEs – a unified approach, Ann. Appl. Probab., 25 (2015), 2168-2214.  doi: 10.1214/14-AAP1046. [9] J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, , Springer-Verlag, Berlin, 1999. [10] E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Related Fields, 114 (1999), 123-150.  doi: 10.1007/s004409970001. [11] S. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27 (1993), 125-144.  doi: 10.1007/BF01195978. [12] S. Peng, Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions, Stochastic Process. Appl., 88 (2000), 259-290.  doi: 10.1016/S0304-4149(00)00005-3. [13] S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim., 37 (1999), 825-843.  doi: 10.1137/S0363012996313549. [14] H. Wang and Z. Wu, Eigenvalues of stochastic Hamiltonian systems driven by Poisson process with boundary conditions, Bound. Value Probl., 2017, Paper No. 164, 20 pp. doi: 10.1186/s13661-017-0896-4. [15] J. Yong, Forward-backward stochastic differential equations with mixed initial-terminal conditions, Trans. Amer. Math. Soc., 362 (2010), 1047-1096.  doi: 10.1090/S0002-9947-09-04896-X. [16] J. Yong, Linear forward-backward stochastic differential equations, Appl. Math. Optim., 39 (1999), 93-119.  doi: 10.1007/s002459900100. [17] J. Yong, Linear forward-backward stochastic differential equations with random coefficients, Probab. Theory Related Fields, 135 (2006), 53-83.  doi: 10.1007/s00440-005-0452-5. [18] J. Zhang, Backward Stochastic Differential Equations. From Linear to Fully Nonlinear Theory, Springer, New York, 2017. doi: 10.1007/978-1-4939-7256-2. [19] J. Zhang, The wellposedness of FBSDEs, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 927-940.  doi: 10.3934/dcdsb.2006.6.927.
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