Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 60J55, 60J60, 60H10, 91G20, 91G80; Secondary: 34F05.

 Citation:

•  [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] L. Andersen, J. Andreasen and D. Eliezer, Static replication of barrier options: Some general results, J. Comput. Finance, 5 (2000), 1-25.doi: 10.2139/ssrn.220010. [3] 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. [4] C. Bender and N. Dokuchaev, A first-order BSPDE for swing option pricing, Mathematical Finance, (2014), in press; web-published.doi: 10.1111/mafi.12067. [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), 183-207.doi: 10.1007/s00245-004-0802-1. [6] P. Carr and A. Chou, Breaking barriers, Risk, 10 (1997), 139-145. [7] A. Chojnowska-Michalik, On processes of Ornstein-Uhlenbeck type in Hilbert space, Stochastics, 21 (1987), 251-286.doi: 10.1080/17442508708833459. [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), 331-356.doi: 10.1007/BF01192465. [9] 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. [10] 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. [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), 662-670.doi: 10.1137/1139051. [12] N. G. Dokuchaev, Estimates for distances between first exit times via parabolic equations in unbounded cylinders, Probability Theory and Related Fields, 129 (2004), 290-314.doi: 10.1007/s00440-004-0341-3. [13] N. G. Dokuchaev, Parabolic Ito equations and second fundamental inequality, Stochastics, 77 (2005), 349-370.doi: 10.1080/17442500500183206. [14] N. Dokuchaev, Estimates for first exit times of non-Markovian Itô processes, Stochastics, 80 (2008), 397-406.doi: 10.1080/17442500701672197. [15] N. Dokuchaev, Parabolic Ito equations with mixed in time conditions, Stochastic Analysis and Applications, 26 (2008), 562-576.doi: 10.1080/07362990802007137. [16] 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. [17] N. Dokuchaev, Representation of functionals of Ito processes in bounded domains, Stochastics, 83 (2011), 45-66.doi: 10.1080/17442508.2010.510907. [18] 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. [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), 255-285.doi: 10.1007/s00440-011-0369-0. [20] 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. [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), 4377-4422.doi: 10.1016/j.jfa.2012.02.024. [22] 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. [23] K. Hamza and F. C. Klebaner, On solutions of first order stochastic partial differential equations, Far East J. Theor. Stat., 20 (2006), 13-25. [24] Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic partial differential equations, Probab. Theory Related Fields, 123 (2002), 381-411.doi: 10.1007/s004400100193. [25] I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer, New York, 1998.doi: 10.1007/b98840. [26] N. V. Krylov, Controlled Diffusion Processes, Shpringer, New York, 1980. [27] 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. [28] O. A. Ladyzhenskaia, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York, 1985.doi: 10.1007/978-1-4757-4317-3. [29] 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. [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), 59-84.doi: 10.1016/S0304-4149(97)00057-4. [31] J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equations, Probability Theory and Related Fields, 113 (1999), 135-170.doi: 10.1007/s004400050205. [32] 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. [33] 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. [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), 1-105.doi: 10.1090/memo/0917. [35] E. Pardoux, Stochastic partial differential equations, a review, Bull. Sci. Math., 117 (1993), 29-47. [36] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, 1999.doi: 10.1007/978-3-662-06400-9. [37] G. O. Roberts and C. F. Shortland, Pricing barrier options with time-dependent coefficients, Mathematical Finance, 7 (1997), 83-93.doi: 10.1111/1467-9965.00024. [38] B. L. Rozovskii, Stochastic Evolution Systems, Linear Theory and Applications to Non-Linear Filtering, Kluwer Academic Publishers Group, Dordrecht, 1990.doi: 10.1007/978-94-011-3830-7. [39] J. B. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Mathematics, 1180 (1986), 265-439.doi: 10.1007/BFb0074920. [40] 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.