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Degenerate backward SPDEs in bounded domains and applications to barrier options
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 |
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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