American Institute of Mathematical Sciences

Advanced
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 & 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

[1]

Rafael Abreu, Cristian Morales-Rodrigo, Antonio Suárez. Some eigenvalue problems with non-local boundary conditions and applications. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2465-2474. doi: 10.3934/cpaa.2014.13.2465
[2]

Monica Marras, Nicola Pintus, Giuseppe Viglialoro. On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 2033-2045. doi: 10.3934/dcdss.2020156
[3]

Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239
[4]

Nikolai Dokuchaev. Degenerate backward SPDEs in bounded domains and applications to barrier options. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5317-5334. doi: 10.3934/dcds.2015.35.5317
[5]

Volodymyr O. Kapustyan, Ivan O. Pyshnograiev, Olena A. Kapustian. Quasi-optimal control with a general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with non-local boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1243-1258. doi: 10.3934/dcdsb.2019014
[6]

Yong-Kui Chang, Xiaojing Liu. Time-varying integro-differential inclusions with Clarke sub-differential and non-local initial conditions: existence and approximate controllability. Evolution Equations & Control Theory, 2020, 9 (3) : 845-863. doi: 10.3934/eect.2020036
[7]

Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255
[8]

Wenning Wei. On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5353-5378. doi: 10.3934/dcds.2015.35.5353
[9]

Antoine Tambue, Jean Daniel Mukam. Magnus-type integrator for non-autonomous SPDEs driven by multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4597-4624. doi: 10.3934/dcds.2020194
[10]

Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems & Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036
[11]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511
[12]

Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037
[13]

Shihu Li, Wei Liu, Yingchao Xie. Small time asymptotics for SPDEs with locally monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020127
[14]

Nacira Agram, Astrid Hilbert, Bernt Øksendal. Singular control of SPDEs with space-mean dynamics. Mathematical Control & Related Fields, 2020, 10 (2) : 425-441. doi: 10.3934/mcrf.2020004
[15]

Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417
[16]

Laura Sigalotti. Homogenization of pinning conditions on periodic networks. Networks & Heterogeneous Media, 2012, 7 (3) : 543-582. doi: 10.3934/nhm.2012.7.543
[17]

Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029
[18]

Chiu-Yen Kao, Yuan Lou, Wenxian Shen. Random dispersal vs. non-local dispersal. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 551-596. doi: 10.3934/dcds.2010.26.551
[19]

Matteo Focardi. Vector-valued obstacle problems for non-local energies. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 487-507. doi: 10.3934/dcdsb.2012.17.487
[20]

Hongjie Dong, Doyoon Kim. Schauder estimates for a class of non-local elliptic equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2319-2347. doi: 10.3934/dcds.2013.33.2319

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences