# American Institute of Mathematical Sciences

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

## On forward and backward SPDEs with non-local boundary conditions

 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.
Citation: Nikolai Dokuchaev. On forward and backward SPDEs with non-local boundary conditions. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1007/s004400050236. Google Scholar [2] V. Bally, I. Gyongy and E. Pardoux, White noise driven parabolic SPDEs with measurable drift,, Journal of Functional Analysis, 120 (1994), 484. doi: 10.1006/jfan.1994.1040. Google Scholar [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. doi: 10.1007/s00245-004-0802-1. Google Scholar [4] A. Chojnowska-Michalik, On processes of Ornstein-Uhlenbeck type in Hilbert space,, Stochastics, 21 (1987), 251. doi: 10.1080/17442508708833459. Google Scholar [5] A. Chojnowska-Michalik, Periodic distributions for linear equations with general additive noise,, Bull. Pol. Acad. Sci. Math., 38 (1990), 23. Google Scholar [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. doi: 10.1007/BF01192465. Google Scholar [7] G. Da Prato and L. Tubaro, Fully nonlinear stochastic partial differential equations,, SIAM Journal on Mathematical Analysis, 27 (1996), 40. doi: 10.1137/S0036141093256769. Google Scholar [8] N. G. Dokuchaev, Boundary value problems for functionals of Ito processes,, Theory of Probability and its Applications, 36 (1991), 459. doi: 10.1137/1136056. Google Scholar [9] N. G. Dokuchaev, Parabolic equations without the Cauchy boundary condition and problems on control over diffusion processes. I,, Differential Equations, 30 (1994), 1606. Google Scholar [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. doi: 10.1137/1139051. Google Scholar [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. doi: 10.1007/s00440-004-0341-3. Google Scholar [12] N. G. Dokuchaev, Parabolic Ito equations and second fundamental inequality,, Stochastics, 77 (2005), 349. doi: 10.1080/17442500500183206. Google Scholar [13] N. Dokuchaev, Parabolic Ito equations with mixed in time conditions,, Stochastic Analysis and Applications, 26 (2008), 562. doi: 10.1080/07362990802007137. Google Scholar [14] N. Dokuchaev, Duality and semi-group property for backward parabolic Ito equations,, Random Operators and Stochastic Equations, 18 (2010), 51. doi: 10.1515/ROSE.2010.51. Google Scholar [15] N. Dokuchaev, Representation of functionals of Ito processes in bounded domains,, Stochastics, 83 (2011), 45. doi: 10.1080/17442508.2010.510907. Google Scholar [16] N. Dokuchaev, Backward parabolic Ito equations and second fundamental inequality,, Random Operators and Stochastic Equations, 20 (2012), 69. doi: 10.1515/rose-2012-0003. Google Scholar [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. doi: 10.1007/s00440-011-0369-0. Google Scholar [18] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations,, Ann. Probab., 31 (2003), 2109. doi: 10.1214/aop/1068646380. Google Scholar [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. doi: 10.1016/j.jfa.2012.02.024. Google Scholar [20] I. Gyöngy, Existence and uniqueness results for semilinear stochastic partial differential equations,, Stochastic Processes and their Applications, 73 (1998), 271. doi: 10.1016/S0304-4149(97)00103-8. Google Scholar [21] I. Karatzas and S. E. Shreve, Methods of Mathematical Finance,, Springer, (1998). doi: 10.1007/b98840. Google Scholar [22] M. Klünger, Periodicity and Sharkovsky's theorem for random dynamical systems,, Stochastic and Dynamics, 1 (2001), 299. doi: 10.1142/S0219493701000199. Google Scholar [23] N. V. Krylov, An analytic approach to SPDEs,, in Stochastic Partial Differential Equations: Six Perspectives, (1999), 185. doi: 10.1090/surv/064/05. Google Scholar [24] O. A. Ladyzhenskaia, The Boundary Value Problems of Mathematical Physics,, Springer-Verlag, (1985). doi: 10.1007/978-1-4757-4317-3. Google Scholar [25] Y. Liu and H. Z. Zhao, Representation of pathwise stationary solutions of stochastic Burgers equations,, Stochactics and Dynamics, 9 (2009), 613. doi: 10.1142/S0219493709002798. Google Scholar [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. doi: 10.1016/S0304-4149(97)00057-4. Google Scholar [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. Google Scholar [28] J. Mattingly, Ergodicity of 2D Navier-Stokes equations with random forcing and large viscosity,, Comm. Math. Phys., 206 (1999), 273. doi: 10.1007/s002200050706. Google Scholar [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. doi: 10.1090/memo/0917. Google Scholar [30] T. Morozan, Periodic solutions of affine stochastic differential equations,, Stoch. Anal. Appl., 4 (1986), 87. doi: 10.1080/07362998608809081. Google Scholar [31] E. Pardoux, Stochastic partial differential equations, a review,, Bulletin des Sciences Mathematiques, 117 (1993), 29. Google Scholar [32] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion,, Springer-Verlag, (1999). doi: 10.1007/978-3-662-06400-9. Google Scholar [33] A. E. Rodkina, On solutions of stochastic equations with almost surely periodic trajectories,, (in Russian) Differ. Uravn, 28 (1992), 534. Google Scholar [34] A. Rodkina, N. Dokuchaev and J. Appleby, On limit periodicity of discrete time stochastic processes,, Stochastic and Dynamics, 14 (2014). doi: 10.1142/S0219493714500117. Google Scholar [35] B. L. Rozovskii, Stochastic Evolution Systems,Linear Theory and Applications to Non-Linear Filtering,, Kluwer Academic Publishers, (1990). doi: 10.1007/978-94-011-3830-7. Google Scholar [36] Ya. Sinai, Burgers system driven by a periodic stochastic flows,, in Ito's Stochastic Calculus and Probability Theory, (1996), 347. Google Scholar [37] J. B. Walsh, An introduction to stochastic partial differential equations,, Lecture Notes in Mathematics, 1180 (1986), 265. doi: 10.1007/BFb0074920. Google Scholar [38] J. Yong and X. Y. Zhou, Stochastic controls: Hamiltonian systems and HJB equations,, Springer-Verlag, (1999). doi: 10.1007/978-1-4612-1466-3. Google Scholar [39] X. Y. Zhou, A duality analysis on stochastic partial differential equations,, Journal of Functional Analysis, 103 (1992), 275. doi: 10.1016/0022-1236(92)90122-Y. Google Scholar

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. doi: 10.1007/s004400050236. Google Scholar [2] V. Bally, I. Gyongy and E. Pardoux, White noise driven parabolic SPDEs with measurable drift,, Journal of Functional Analysis, 120 (1994), 484. doi: 10.1006/jfan.1994.1040. Google Scholar [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. doi: 10.1007/s00245-004-0802-1. Google Scholar [4] A. Chojnowska-Michalik, On processes of Ornstein-Uhlenbeck type in Hilbert space,, Stochastics, 21 (1987), 251. doi: 10.1080/17442508708833459. Google Scholar [5] A. Chojnowska-Michalik, Periodic distributions for linear equations with general additive noise,, Bull. Pol. Acad. Sci. Math., 38 (1990), 23. Google Scholar [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. doi: 10.1007/BF01192465. Google Scholar [7] G. Da Prato and L. Tubaro, Fully nonlinear stochastic partial differential equations,, SIAM Journal on Mathematical Analysis, 27 (1996), 40. doi: 10.1137/S0036141093256769. Google Scholar [8] N. G. Dokuchaev, Boundary value problems for functionals of Ito processes,, Theory of Probability and its Applications, 36 (1991), 459. doi: 10.1137/1136056. Google Scholar [9] N. G. Dokuchaev, Parabolic equations without the Cauchy boundary condition and problems on control over diffusion processes. I,, Differential Equations, 30 (1994), 1606. Google Scholar [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. doi: 10.1137/1139051. Google Scholar [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. doi: 10.1007/s00440-004-0341-3. Google Scholar [12] N. G. Dokuchaev, Parabolic Ito equations and second fundamental inequality,, Stochastics, 77 (2005), 349. doi: 10.1080/17442500500183206. Google Scholar [13] N. Dokuchaev, Parabolic Ito equations with mixed in time conditions,, Stochastic Analysis and Applications, 26 (2008), 562. doi: 10.1080/07362990802007137. Google Scholar [14] N. Dokuchaev, Duality and semi-group property for backward parabolic Ito equations,, Random Operators and Stochastic Equations, 18 (2010), 51. doi: 10.1515/ROSE.2010.51. Google Scholar [15] N. Dokuchaev, Representation of functionals of Ito processes in bounded domains,, Stochastics, 83 (2011), 45. doi: 10.1080/17442508.2010.510907. Google Scholar [16] N. Dokuchaev, Backward parabolic Ito equations and second fundamental inequality,, Random Operators and Stochastic Equations, 20 (2012), 69. doi: 10.1515/rose-2012-0003. Google Scholar [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. doi: 10.1007/s00440-011-0369-0. Google Scholar [18] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations,, Ann. Probab., 31 (2003), 2109. doi: 10.1214/aop/1068646380. Google Scholar [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. doi: 10.1016/j.jfa.2012.02.024. Google Scholar [20] I. Gyöngy, Existence and uniqueness results for semilinear stochastic partial differential equations,, Stochastic Processes and their Applications, 73 (1998), 271. doi: 10.1016/S0304-4149(97)00103-8. Google Scholar [21] I. Karatzas and S. E. Shreve, Methods of Mathematical Finance,, Springer, (1998). doi: 10.1007/b98840. Google Scholar [22] M. Klünger, Periodicity and Sharkovsky's theorem for random dynamical systems,, Stochastic and Dynamics, 1 (2001), 299. doi: 10.1142/S0219493701000199. Google Scholar [23] N. V. Krylov, An analytic approach to SPDEs,, in Stochastic Partial Differential Equations: Six Perspectives, (1999), 185. doi: 10.1090/surv/064/05. Google Scholar [24] O. A. Ladyzhenskaia, The Boundary Value Problems of Mathematical Physics,, Springer-Verlag, (1985). doi: 10.1007/978-1-4757-4317-3. Google Scholar [25] Y. Liu and H. Z. Zhao, Representation of pathwise stationary solutions of stochastic Burgers equations,, Stochactics and Dynamics, 9 (2009), 613. doi: 10.1142/S0219493709002798. Google Scholar [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. doi: 10.1016/S0304-4149(97)00057-4. Google Scholar [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. Google Scholar [28] J. Mattingly, Ergodicity of 2D Navier-Stokes equations with random forcing and large viscosity,, Comm. Math. Phys., 206 (1999), 273. doi: 10.1007/s002200050706. Google Scholar [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. doi: 10.1090/memo/0917. Google Scholar [30] T. Morozan, Periodic solutions of affine stochastic differential equations,, Stoch. Anal. Appl., 4 (1986), 87. doi: 10.1080/07362998608809081. Google Scholar [31] E. Pardoux, Stochastic partial differential equations, a review,, Bulletin des Sciences Mathematiques, 117 (1993), 29. Google Scholar [32] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion,, Springer-Verlag, (1999). doi: 10.1007/978-3-662-06400-9. Google Scholar [33] A. E. Rodkina, On solutions of stochastic equations with almost surely periodic trajectories,, (in Russian) Differ. Uravn, 28 (1992), 534. Google Scholar [34] A. Rodkina, N. Dokuchaev and J. Appleby, On limit periodicity of discrete time stochastic processes,, Stochastic and Dynamics, 14 (2014). doi: 10.1142/S0219493714500117. Google Scholar [35] B. L. Rozovskii, Stochastic Evolution Systems,Linear Theory and Applications to Non-Linear Filtering,, Kluwer Academic Publishers, (1990). doi: 10.1007/978-94-011-3830-7. Google Scholar [36] Ya. Sinai, Burgers system driven by a periodic stochastic flows,, in Ito's Stochastic Calculus and Probability Theory, (1996), 347. Google Scholar [37] J. B. Walsh, An introduction to stochastic partial differential equations,, Lecture Notes in Mathematics, 1180 (1986), 265. doi: 10.1007/BFb0074920. Google Scholar [38] J. Yong and X. Y. Zhou, Stochastic controls: Hamiltonian systems and HJB equations,, Springer-Verlag, (1999). doi: 10.1007/978-1-4612-1466-3. Google Scholar [39] X. Y. Zhou, A duality analysis on stochastic partial differential equations,, Journal of Functional Analysis, 103 (1992), 275. doi: 10.1016/0022-1236(92)90122-Y. Google Scholar
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