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November  2015, 35(11): 5335-5351. doi: 10.3934/dcds.2015.35.5335

On forward and backward SPDEs with non-local boundary conditions

Nikolai Dokuchaev 1,

1. 

Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, Western Australia, 6845

Received  November 2012 Revised  April 2014 Published  May 2015

We study linear stochastic partial differential equations of parabolic type with non-local in time or mixed in time boundary conditions. The standard Cauchy condition at the terminal time is replaced by a condition that mixes the random values of the solution at different times, including the terminal time, initial time and continuously distributed times. For the case of backward equations, this setting covers almost surely periodicity. Uniqueness, solvability and regularity results for the solutions are obtained. Some possible applications to portfolio selection are discussed.
Keywords: backward SPDEs, SPDEs, portfolio selection., periodic conditions, non-local conditions.
Mathematics Subject Classification: Primary: 60J55, 60J60, 60H10; Secondary: 34F05, 34G1.
Citation: Nikolai Dokuchaev. On forward and backward SPDEs with non-local boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5335-5351. doi: 10.3934/dcds.2015.35.5335
References:
[1]

E. Alós, J. A. León and D. Nualart, Stochastic heat equation with random coefficients, Probability Theory and Related Fields, 115 (1999), 41-94. doi: 10.1007/s004400050236.

[2]

V. Bally, I. Gyongy and E. Pardoux, White noise driven parabolic SPDEs with measurable drift, Journal of Functional Analysis, 120 (1994), 484-510. doi: 10.1006/jfan.1994.1040.

[3]

T. Caraballo, P. E. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207. doi: 10.1007/s00245-004-0802-1.

[4]

A. Chojnowska-Michalik, On processes of Ornstein-Uhlenbeck type in Hilbert space, Stochastics, 21 (1987), 251-286. doi: 10.1080/17442508708833459.

[5]

A. Chojnowska-Michalik, Periodic distributions for linear equations with general additive noise, Bull. Pol. Acad. Sci. Math., 38 (1990), 23-33.

[6]

A. Chojnowska-Michalik and B. Goldys, Existence, uniqueness and invariant measures for stochastic semilinear equations in Hilbert spaces, Probability Theory and Related Fields, 102 (1995), 331-356. doi: 10.1007/BF01192465.

[7]

G. Da Prato and L. Tubaro, Fully nonlinear stochastic partial differential equations, SIAM Journal on Mathematical Analysis, 27 (1996), 40-55. doi: 10.1137/S0036141093256769.

[8]

N. G. Dokuchaev, Boundary value problems for functionals of Ito processes, Theory of Probability and its Applications, 36 (1991), 459-476. doi: 10.1137/1136056.

[9]

N. G. Dokuchaev, Parabolic equations without the Cauchy boundary condition and problems on control over diffusion processes. I, Differential Equations, 30 (1994), 1606-1617; Translation from Differ. Uravn, 30 (1994), 1738-1749.

[10]

N. G. Dokuchaev, Probability distributions of Ito's processes: Estimations for density functions and for conditional expectations of integral functionals, Theory of Probability and its Applications, 39 (1994), 662-670. doi: 10.1137/1139051.

[11]

N. G. Dokuchaev, Estimates for distances between first exit times via parabolic equations in unbounded cylinders, Probability Theory and Related Fields, 129 (2004), 290-314. doi: 10.1007/s00440-004-0341-3.

[12]

N. G. Dokuchaev, Parabolic Ito equations and second fundamental inequality, Stochastics, 77 (2005), 349-370. doi: 10.1080/17442500500183206.

[13]

N. Dokuchaev, Parabolic Ito equations with mixed in time conditions, Stochastic Analysis and Applications, 26 (2008), 562-576. doi: 10.1080/07362990802007137.

[14]

N. Dokuchaev, Duality and semi-group property for backward parabolic Ito equations, Random Operators and Stochastic Equations, 18 (2010), 51-72. doi: 10.1515/ROSE.2010.51.

[15]

N. Dokuchaev, Representation of functionals of Ito processes in bounded domains, Stochastics, 83 (2011), 45-66. doi: 10.1080/17442508.2010.510907.

[16]

N. Dokuchaev, Backward parabolic Ito equations and second fundamental inequality, Random Operators and Stochastic Equations, 20 (2012), 69-102. doi: 10.1515/rose-2012-0003.

[17]

K. Du and S. Tang, Strong solution of backward stochastic partial differential equations in $C^2$ domains, Probability Theory and Related Fields, 154 (2012), 255-285. doi: 10.1007/s00440-011-0369-0.

[18]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380.

[19]

C. Feng and H. Zhao, Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding, Journal of Functional Analysis, 262 (2012), 4377-4422. doi: 10.1016/j.jfa.2012.02.024.

[20]

I. Gyöngy, Existence and uniqueness results for semilinear stochastic partial differential equations, Stochastic Processes and their Applications, 73 (1998), 271-299. doi: 10.1016/S0304-4149(97)00103-8.

[21]

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer, New York, 1998. doi: 10.1007/b98840.

[22]

M. Klünger, Periodicity and Sharkovsky's theorem for random dynamical systems, Stochastic and Dynamics, 1 (2001), 299-338. doi: 10.1142/S0219493701000199.

[23]

N. V. Krylov, An analytic approach to SPDEs, in Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, 64, AMS., Providence, RI, 1999, 185-242. doi: 10.1090/surv/064/05.

[24]

O. A. Ladyzhenskaia, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4757-4317-3.

[25]

Y. Liu and H. Z. Zhao, Representation of pathwise stationary solutions of stochastic Burgers equations, Stochactics and Dynamics, 9 (2009), 613-634. doi: 10.1142/S0219493709002798.

[26]

J. Ma and J. Yong, Adapted solution of a class of degenerate backward stochastic partial differential equations, with applications, Stochastic Processes and Their Applications, 70 (1997), 59-84. doi: 10.1016/S0304-4149(97)00057-4.

[27]

B. Maslowski, Stability of semilinear equations with boundary and pointwise noise, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (4), 22 (1995), 55-93.

[28]

J. Mattingly, Ergodicity of 2D Navier-Stokes equations with random forcing and large viscosity, Comm. Math. Phys., 206 (1999), 273-288. doi: 10.1007/s002200050706.

[29]

S.-E. A. Mohammed, T. Zhang and H. Z. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc., 196 (2008), 1-105. doi: 10.1090/memo/0917.

[30]

T. Morozan, Periodic solutions of affine stochastic differential equations, Stoch. Anal. Appl., 4 (1986), 87-110. doi: 10.1080/07362998608809081.

[31]

E. Pardoux, Stochastic partial differential equations, a review, Bulletin des Sciences Mathematiques, 2e Serie, 117 (1993), 29-47.

[32]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.

[33]

A. E. Rodkina, On solutions of stochastic equations with almost surely periodic trajectories, (in Russian) Differ. Uravn, 28 (1992), 534-536.

[34]

A. Rodkina, N. Dokuchaev and J. Appleby, On limit periodicity of discrete time stochastic processes, Stochastic and Dynamics, 14 (2014), 1450011, 8pp. doi: 10.1142/S0219493714500117.

[35]

B. L. Rozovskii, Stochastic Evolution Systems,Linear Theory and Applications to Non-Linear Filtering, Kluwer Academic Publishers, Dordrecht-Boston-London, 1990. doi: 10.1007/978-94-011-3830-7.

[36]

Ya. Sinai, Burgers system driven by a periodic stochastic flows, in Ito's Stochastic Calculus and Probability Theory, Springer, Tokyo, 1996, 347-353.

[37]

J. B. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Mathematics, 1180 (1986), 265-439. doi: 10.1007/BFb0074920.

[38]

J. Yong and X. Y. Zhou, Stochastic controls: Hamiltonian systems and HJB equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[39]

X. Y. Zhou, A duality analysis on stochastic partial differential equations, Journal of Functional Analysis, 103 (1992), 275-293. doi: 10.1016/0022-1236(92)90122-Y.

show all references

References:
[1]

E. Alós, J. A. León and D. Nualart, Stochastic heat equation with random coefficients, Probability Theory and Related Fields, 115 (1999), 41-94. doi: 10.1007/s004400050236.

[2]

V. Bally, I. Gyongy and E. Pardoux, White noise driven parabolic SPDEs with measurable drift, Journal of Functional Analysis, 120 (1994), 484-510. doi: 10.1006/jfan.1994.1040.

[3]

T. Caraballo, P. E. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207. doi: 10.1007/s00245-004-0802-1.

[4]

A. Chojnowska-Michalik, On processes of Ornstein-Uhlenbeck type in Hilbert space, Stochastics, 21 (1987), 251-286. doi: 10.1080/17442508708833459.

[5]

A. Chojnowska-Michalik, Periodic distributions for linear equations with general additive noise, Bull. Pol. Acad. Sci. Math., 38 (1990), 23-33.

[6]

A. Chojnowska-Michalik and B. Goldys, Existence, uniqueness and invariant measures for stochastic semilinear equations in Hilbert spaces, Probability Theory and Related Fields, 102 (1995), 331-356. doi: 10.1007/BF01192465.

[7]

G. Da Prato and L. Tubaro, Fully nonlinear stochastic partial differential equations, SIAM Journal on Mathematical Analysis, 27 (1996), 40-55. doi: 10.1137/S0036141093256769.

[8]

N. G. Dokuchaev, Boundary value problems for functionals of Ito processes, Theory of Probability and its Applications, 36 (1991), 459-476. doi: 10.1137/1136056.

[9]

N. G. Dokuchaev, Parabolic equations without the Cauchy boundary condition and problems on control over diffusion processes. I, Differential Equations, 30 (1994), 1606-1617; Translation from Differ. Uravn, 30 (1994), 1738-1749.

[10]

N. G. Dokuchaev, Probability distributions of Ito's processes: Estimations for density functions and for conditional expectations of integral functionals, Theory of Probability and its Applications, 39 (1994), 662-670. doi: 10.1137/1139051.

[11]

N. G. Dokuchaev, Estimates for distances between first exit times via parabolic equations in unbounded cylinders, Probability Theory and Related Fields, 129 (2004), 290-314. doi: 10.1007/s00440-004-0341-3.

[12]

N. G. Dokuchaev, Parabolic Ito equations and second fundamental inequality, Stochastics, 77 (2005), 349-370. doi: 10.1080/17442500500183206.

[13]

N. Dokuchaev, Parabolic Ito equations with mixed in time conditions, Stochastic Analysis and Applications, 26 (2008), 562-576. doi: 10.1080/07362990802007137.

[14]

N. Dokuchaev, Duality and semi-group property for backward parabolic Ito equations, Random Operators and Stochastic Equations, 18 (2010), 51-72. doi: 10.1515/ROSE.2010.51.

[15]

N. Dokuchaev, Representation of functionals of Ito processes in bounded domains, Stochastics, 83 (2011), 45-66. doi: 10.1080/17442508.2010.510907.

[16]

N. Dokuchaev, Backward parabolic Ito equations and second fundamental inequality, Random Operators and Stochastic Equations, 20 (2012), 69-102. doi: 10.1515/rose-2012-0003.

[17]

K. Du and S. Tang, Strong solution of backward stochastic partial differential equations in $C^2$ domains, Probability Theory and Related Fields, 154 (2012), 255-285. doi: 10.1007/s00440-011-0369-0.

[18]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380.

[19]

C. Feng and H. Zhao, Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding, Journal of Functional Analysis, 262 (2012), 4377-4422. doi: 10.1016/j.jfa.2012.02.024.

[20]

I. Gyöngy, Existence and uniqueness results for semilinear stochastic partial differential equations, Stochastic Processes and their Applications, 73 (1998), 271-299. doi: 10.1016/S0304-4149(97)00103-8.

[21]

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer, New York, 1998. doi: 10.1007/b98840.

[22]

M. Klünger, Periodicity and Sharkovsky's theorem for random dynamical systems, Stochastic and Dynamics, 1 (2001), 299-338. doi: 10.1142/S0219493701000199.

[23]

N. V. Krylov, An analytic approach to SPDEs, in Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, 64, AMS., Providence, RI, 1999, 185-242. doi: 10.1090/surv/064/05.

[24]

O. A. Ladyzhenskaia, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4757-4317-3.

[25]

Y. Liu and H. Z. Zhao, Representation of pathwise stationary solutions of stochastic Burgers equations, Stochactics and Dynamics, 9 (2009), 613-634. doi: 10.1142/S0219493709002798.

[26]

J. Ma and J. Yong, Adapted solution of a class of degenerate backward stochastic partial differential equations, with applications, Stochastic Processes and Their Applications, 70 (1997), 59-84. doi: 10.1016/S0304-4149(97)00057-4.

[27]

B. Maslowski, Stability of semilinear equations with boundary and pointwise noise, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (4), 22 (1995), 55-93.

[28]

J. Mattingly, Ergodicity of 2D Navier-Stokes equations with random forcing and large viscosity, Comm. Math. Phys., 206 (1999), 273-288. doi: 10.1007/s002200050706.

[29]

S.-E. A. Mohammed, T. Zhang and H. Z. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc., 196 (2008), 1-105. doi: 10.1090/memo/0917.

[30]

T. Morozan, Periodic solutions of affine stochastic differential equations, Stoch. Anal. Appl., 4 (1986), 87-110. doi: 10.1080/07362998608809081.

[31]

E. Pardoux, Stochastic partial differential equations, a review, Bulletin des Sciences Mathematiques, 2e Serie, 117 (1993), 29-47.

[32]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.

[33]

A. E. Rodkina, On solutions of stochastic equations with almost surely periodic trajectories, (in Russian) Differ. Uravn, 28 (1992), 534-536.

[34]

A. Rodkina, N. Dokuchaev and J. Appleby, On limit periodicity of discrete time stochastic processes, Stochastic and Dynamics, 14 (2014), 1450011, 8pp. doi: 10.1142/S0219493714500117.

[35]

B. L. Rozovskii, Stochastic Evolution Systems,Linear Theory and Applications to Non-Linear Filtering, Kluwer Academic Publishers, Dordrecht-Boston-London, 1990. doi: 10.1007/978-94-011-3830-7.

[36]

Ya. Sinai, Burgers system driven by a periodic stochastic flows, in Ito's Stochastic Calculus and Probability Theory, Springer, Tokyo, 1996, 347-353.

[37]

J. B. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Mathematics, 1180 (1986), 265-439. doi: 10.1007/BFb0074920.

[38]

J. Yong and X. Y. Zhou, Stochastic controls: Hamiltonian systems and HJB equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[39]

X. Y. Zhou, A duality analysis on stochastic partial differential equations, Journal of Functional Analysis, 103 (1992), 275-293. doi: 10.1016/0022-1236(92)90122-Y.

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