November  2015, 35(11): 5317-5334. doi: 10.3934/dcds.2015.35.5317

Degenerate backward SPDEs in bounded domains and applications to barrier options

1. 

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

Received  September 2013 Revised  May 2014 Published  May 2015

Backward stochastic partial differential equations of parabolic type in bounded domains are studied in the setting where the coercivity condition is not necessary satisfied. Generalized solutions based on the representation theorem are suggested. Some regularity is derived from the regularity of the first exit times of non-Markov characteristic processes. Uniqueness, solvability and regularity results are obtained. Applications to pricing and hedging of European barrier options are considered.
Citation: 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
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]

L. Andersen, J. Andreasen and D. Eliezer, Static replication of barrier options: Some general results,, J. Comput. Finance, 5 (2000), 1. doi: 10.2139/ssrn.220010. Google Scholar

[3]

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

[4]

C. Bender and N. Dokuchaev, A first-order BSPDE for swing option pricing,, Mathematical Finance, (2014). doi: 10.1111/mafi.12067. Google Scholar

[5]

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

[6]

P. Carr and A. Chou, Breaking barriers,, Risk, 10 (1997), 139. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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 (1995), 662. doi: 10.1137/1139051. Google Scholar

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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

[13]

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

[14]

N. Dokuchaev, Estimates for first exit times of non-Markovian Itô processes,, Stochastics, 80 (2008), 397. doi: 10.1080/17442500701672197. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

K. Hamza and F. C. Klebaner, On solutions of first order stochastic partial differential equations,, Far East J. Theor. Stat., 20 (2006), 13. Google Scholar

[24]

Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic partial differential equations,, Probab. Theory Related Fields, 123 (2002), 381. doi: 10.1007/s004400100193. Google Scholar

[25]

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance,, Springer, (1998). doi: 10.1007/b98840. Google Scholar

[26]

N. V. Krylov, Controlled Diffusion Processes,, Shpringer, (1980). Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[31]

J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equations,, Probability Theory and Related Fields, 113 (1999), 135. doi: 10.1007/s004400050205. Google Scholar

[32]

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

[33]

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

[34]

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

[35]

E. Pardoux, Stochastic partial differential equations, a review,, Bull. Sci. Math., 117 (1993), 29. Google Scholar

[36]

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

[37]

G. O. Roberts and C. F. Shortland, Pricing barrier options with time-dependent coefficients,, Mathematical Finance, 7 (1997), 83. doi: 10.1111/1467-9965.00024. Google Scholar

[38]

B. L. Rozovskii, Stochastic Evolution Systems, Linear Theory and Applications to Non-Linear Filtering,, Kluwer Academic Publishers Group, (1990). doi: 10.1007/978-94-011-3830-7. Google Scholar

[39]

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

[40]

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]

L. Andersen, J. Andreasen and D. Eliezer, Static replication of barrier options: Some general results,, J. Comput. Finance, 5 (2000), 1. doi: 10.2139/ssrn.220010. Google Scholar

[3]

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

[4]

C. Bender and N. Dokuchaev, A first-order BSPDE for swing option pricing,, Mathematical Finance, (2014). doi: 10.1111/mafi.12067. Google Scholar

[5]

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

[6]

P. Carr and A. Chou, Breaking barriers,, Risk, 10 (1997), 139. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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 (1995), 662. doi: 10.1137/1139051. Google Scholar

[12]

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

[13]

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

[14]

N. Dokuchaev, Estimates for first exit times of non-Markovian Itô processes,, Stochastics, 80 (2008), 397. doi: 10.1080/17442500701672197. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

K. Hamza and F. C. Klebaner, On solutions of first order stochastic partial differential equations,, Far East J. Theor. Stat., 20 (2006), 13. Google Scholar

[24]

Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic partial differential equations,, Probab. Theory Related Fields, 123 (2002), 381. doi: 10.1007/s004400100193. Google Scholar

[25]

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance,, Springer, (1998). doi: 10.1007/b98840. Google Scholar

[26]

N. V. Krylov, Controlled Diffusion Processes,, Shpringer, (1980). Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[31]

J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equations,, Probability Theory and Related Fields, 113 (1999), 135. doi: 10.1007/s004400050205. Google Scholar

[32]

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

[33]

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

[34]

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

[35]

E. Pardoux, Stochastic partial differential equations, a review,, Bull. Sci. Math., 117 (1993), 29. Google Scholar

[36]

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

[37]

G. O. Roberts and C. F. Shortland, Pricing barrier options with time-dependent coefficients,, Mathematical Finance, 7 (1997), 83. doi: 10.1111/1467-9965.00024. Google Scholar

[38]

B. L. Rozovskii, Stochastic Evolution Systems, Linear Theory and Applications to Non-Linear Filtering,, Kluwer Academic Publishers Group, (1990). doi: 10.1007/978-94-011-3830-7. Google Scholar

[39]

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

[40]

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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