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2015, 35(11): 5379-5412. doi: 10.3934/dcds.2015.35.5379

Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations

1. 

Department of Finance and Control Sciences, School of Mathematical Science, Fudan University, Shanghai 200433, China

Received  April 2014 Revised  October 2014 Published  May 2015

The paper is concerned with a semi-linear backward stochastic Schrödinger equation in $\mathbb{R}^d$ or in its bounded domain of a $C^2$ boundary. Galerkin's finite-dimensional approximation method is used and the harmonic role of the Laplacian is shown. The existence, uniqueness and regularity are given for the weak solution of the equation. A more general backward stochastic Hamiltonian partial differential equation is also discussed.
Citation: Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379
References:
[1]

A. Bensoussan, Stochastic maximum principle for distributed parameter systems,, J. Franklin Inst., 315 (1983), 387. doi: 10.1016/0016-0032(83)90059-5.

[2]

A. Bensoussan, Stochastic Control of Partially Observable Systems,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511526503.

[3]

J. Bismut, Conjugate convex functions in optimal stochastic control,, J. Math. Anal. Apl., 44 (1973), 384. doi: 10.1016/0022-247X(73)90066-8.

[4]

J. Bismut, Linear quadradic optimal stochastic control with random coefficients,, SIAM J. Control Optim., 14 (1976), 414. doi: 10.1137/0314028.

[5]

J. Bismut, An introductory approach to duality in optimal stochastic control,, SIAM Rev., 20 (1978), 62. doi: 10.1137/1020004.

[6]

W. Craiq, Transformation theory of Hamiltonian PDE and the problem of water waves,, in Proceedings of the Advanced Study Institute on Hamiltonian Dynamical Systems and Applications, (2008), 67. doi: 10.1007/978-1-4020-6964-2_4.

[7]

A. De Bouard and A. Debussche, A stochastic nonlinear Schrödinger equation with multiplicative noise,, Commun. Math. Phys., 205 (1999), 161. doi: 10.1007/s002200050672.

[8]

K. Du, Backward Stochastic Partial Differential Equations and their Applications,, Ph.D thesis, (2011).

[9]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Mathematical Finance, 7 (1997), 1. doi: 10.1111/1467-9965.00022.

[10]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).

[11]

K. Frieler and C. Knoche, Solutions of Stochastic Differential Equations in Infinite Dimensional Hilbert Spaces and Their Dependence on Initial Data,, Diploma thesis, (2011), 02.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1983). doi: 10.1007/978-3-642-61798-0.

[13]

W. Grecksch and H. Lisei, Stochastic Nonlinear Equations of Schrödinger Type,, Stoch. Ana. Appl., 29 (2011), 631. doi: 10.1080/07362994.2011.581091.

[14]

Y. Hu and S. Peng, Adapted solution of a backward semilinear stochastic evolution equations,, Stoch. Anal. Appl., 9 (1991), 445. doi: 10.1080/07362999108809250.

[15]

N. Krylov and B. Rozovskii, On the Cauchy problem for superparabolic stochastic differential equations,, in Third Soviet-Japanese Sympos on Probability Theory, (1975), 77.

[16]

J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications,, Stoch. Proc. Appl., 70 (1997), 59. doi: 10.1016/S0304-4149(97)00057-4.

[17]

S. Mizohata, On the Cauchy Problem,, Academic Press, (1985).

[18]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation,, Systems Control Lett., 14 (1990), 55. doi: 10.1016/0167-6911(90)90082-6.

[19]

E. Pardoux and S. Peng, Backward stochastic differential equations and quaslinear parabolic partial differential equations,, in Stochastic Partial Differential Equations and their Applications (eds. B. Rozovskii and L. Boris), 176 (1992), 200. doi: 10.1007/BFb0007334.

[20]

S. Tang, Semi-linear systems of backward stochastic partial differential equations in $\mathbbR^n$,, Chin. Ann. Math., 26 (2005), 437. doi: 10.1142/S025295990500035X.

[21]

S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations,, SIAM Journal on Control Optimization, 48 (2009), 2191. doi: 10.1137/050641508.

[22]

S. Tang, The maximum principle for partially observed optimal control of stochastic differential equations,, SIAM J. Control Optim., 36 (1998), 1596. doi: 10.1137/S0363012996313100.

[23]

S. Tang, A new partially observed stochastic maximum principle,, in Proceedings of 37th IEEE Control and Decision Conference, (1998), 2353.

[24]

X. Zhou, A duality analysis on stochastic partial differential equations,, Journal of Functional Analysis, 103 (1992), 275. doi: 10.1016/0022-1236(92)90122-Y.

[25]

X. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations,, SIAM J. Control Optim., 31 (1993), 1462. doi: 10.1137/0331068.

show all references

References:
[1]

A. Bensoussan, Stochastic maximum principle for distributed parameter systems,, J. Franklin Inst., 315 (1983), 387. doi: 10.1016/0016-0032(83)90059-5.

[2]

A. Bensoussan, Stochastic Control of Partially Observable Systems,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511526503.

[3]

J. Bismut, Conjugate convex functions in optimal stochastic control,, J. Math. Anal. Apl., 44 (1973), 384. doi: 10.1016/0022-247X(73)90066-8.

[4]

J. Bismut, Linear quadradic optimal stochastic control with random coefficients,, SIAM J. Control Optim., 14 (1976), 414. doi: 10.1137/0314028.

[5]

J. Bismut, An introductory approach to duality in optimal stochastic control,, SIAM Rev., 20 (1978), 62. doi: 10.1137/1020004.

[6]

W. Craiq, Transformation theory of Hamiltonian PDE and the problem of water waves,, in Proceedings of the Advanced Study Institute on Hamiltonian Dynamical Systems and Applications, (2008), 67. doi: 10.1007/978-1-4020-6964-2_4.

[7]

A. De Bouard and A. Debussche, A stochastic nonlinear Schrödinger equation with multiplicative noise,, Commun. Math. Phys., 205 (1999), 161. doi: 10.1007/s002200050672.

[8]

K. Du, Backward Stochastic Partial Differential Equations and their Applications,, Ph.D thesis, (2011).

[9]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Mathematical Finance, 7 (1997), 1. doi: 10.1111/1467-9965.00022.

[10]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).

[11]

K. Frieler and C. Knoche, Solutions of Stochastic Differential Equations in Infinite Dimensional Hilbert Spaces and Their Dependence on Initial Data,, Diploma thesis, (2011), 02.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1983). doi: 10.1007/978-3-642-61798-0.

[13]

W. Grecksch and H. Lisei, Stochastic Nonlinear Equations of Schrödinger Type,, Stoch. Ana. Appl., 29 (2011), 631. doi: 10.1080/07362994.2011.581091.

[14]

Y. Hu and S. Peng, Adapted solution of a backward semilinear stochastic evolution equations,, Stoch. Anal. Appl., 9 (1991), 445. doi: 10.1080/07362999108809250.

[15]

N. Krylov and B. Rozovskii, On the Cauchy problem for superparabolic stochastic differential equations,, in Third Soviet-Japanese Sympos on Probability Theory, (1975), 77.

[16]

J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications,, Stoch. Proc. Appl., 70 (1997), 59. doi: 10.1016/S0304-4149(97)00057-4.

[17]

S. Mizohata, On the Cauchy Problem,, Academic Press, (1985).

[18]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation,, Systems Control Lett., 14 (1990), 55. doi: 10.1016/0167-6911(90)90082-6.

[19]

E. Pardoux and S. Peng, Backward stochastic differential equations and quaslinear parabolic partial differential equations,, in Stochastic Partial Differential Equations and their Applications (eds. B. Rozovskii and L. Boris), 176 (1992), 200. doi: 10.1007/BFb0007334.

[20]

S. Tang, Semi-linear systems of backward stochastic partial differential equations in $\mathbbR^n$,, Chin. Ann. Math., 26 (2005), 437. doi: 10.1142/S025295990500035X.

[21]

S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations,, SIAM Journal on Control Optimization, 48 (2009), 2191. doi: 10.1137/050641508.

[22]

S. Tang, The maximum principle for partially observed optimal control of stochastic differential equations,, SIAM J. Control Optim., 36 (1998), 1596. doi: 10.1137/S0363012996313100.

[23]

S. Tang, A new partially observed stochastic maximum principle,, in Proceedings of 37th IEEE Control and Decision Conference, (1998), 2353.

[24]

X. Zhou, A duality analysis on stochastic partial differential equations,, Journal of Functional Analysis, 103 (1992), 275. doi: 10.1016/0022-1236(92)90122-Y.

[25]

X. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations,, SIAM J. Control Optim., 31 (1993), 1462. doi: 10.1137/0331068.

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