Abstract
Numerical methods for the Dirichlet problem for linear parabolic stochastic partial differential equations are constructed. The methods are based on the averaging-over-characteristic formula and the weak-sense numerical integration of ordinary stochastic differential equations in bounded domains. Their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. The Monte Carlo technique is used for practical realization of the methods. Results of some numerical experiments are presented.
AMS 2000 Subject Classification: 65C30, 60H15, 60H35, 60G35
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References
Allen, E.J., Novosel, S.J., Zhang, Z.: Finite element and difference approximation of some linear stochastic partial differential equations. Stoch. Stoch. Rep. 64, 117–142 (1998)
Bensoussan, A., Glowinski, R., Răşcanu, A.: Approximation of some stochastic differential equations by the splitting up method. Appl. Math. Optim. 25, 81–106 (1992)
Bonaccorsi, S., Guatteri, G.: Stochastic partial differential equations in bounded domains with Dirichlet boundary conditions. Stoch. Stoch. Rep. 74, 349–370 (2002)
Bonaccorsi, S., Guatteri, G.: Classical solutions for SPDEs with Dirichlet boundary conditions. Prog. Probab. 52, Birkhäuser, 33–44 (2002)
de Bouard, A., Debussche, A.: Weak and strong order of convergence of a semi discrete scheme for the stochastic nonlinear Schrö dinger equation. J. Appl. Maths Optim. 54, 369–399 (2006)
Brzezniak, Z.: Stochastic partial differential equations in M-type 2 Banach spaces. Poten. Anal. 4, 1–45 (1995)
Clark, J.M.C.: The design of robust approximation to the stochastic differential equations of nonlinear filtering. In: J.K. Skwirzynski (ed.) Communication Systems and Random Process Theory, NATO Advanced Study Institute series E Applied Sciences25, pp. 721–734. Sijthoff and Noordhoff, (1978)
Chow, P.L.: Stochastic partial differential equations. Chapman and Hall/CRC (2007)
Chow, P.L., Jiang, J.L., Menaldi, J.L.: Pathwise convergence of approximate solutions to Zakai’s equation in a bounded domain. In: Stochastic partial differential equations and applications (Trento, 1990), Pitman Res. Notes Math. Ser. 268, pp. 111–123. Longman Sci. Tech., Harlow (1992)
Crisan, D.: Particle approximations for a class of stochastic partial differential equations. J. Appl. Math. Optim. 54, 293–314 (2006)
Crisan, D., Lyons, T.: A particle approximation of the solution of the Kushner-Stratonovich equation. Prob. Theory Rel. Fields 115, 549–578 (1999)
Crisan, D., Del Moral, P., Lyons, T.: Interacting particle systems approximations of the Kushner-Stratonovitch equation. Adv. Appl. Probab. 31, 819–838 (1999)
Cruzeiro, A.B., Malliavin, P., Thalmaier, A.: Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation. Acad. Sci. Paris Ser. I 338, 481–486 (2004)
Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Cambridge University Press (1992)
Debussche, A.: Weak approximation of stochastic partial differential equations: the nonlinear case. Math. Comp. (2010), in print
Dynkin, E.B.: Markov processes. Springer (1965)
Flandoli, F.: Regularity theory and stochastic flows for parabolic SPDEs. Gordon and Breach (1995)
Flandoli, F., Schaumlöffel, K.-U.: Stochastic parabolic equations in bounded domains: random evolution operators and Lyapunov exponents. Stoch. Stoch. Rep. 29, 461–485 (1990)
Freidlin, M.I.: Functional Integration and Partial Differential Equations. Princeton University Press (1985)
Fridman, A.: Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs (1964)
Germani, A., Piccioni, M.: Semi-discretization of stochastic partial differential equations on R d by a finite-element technique. Stochastics 23, 131–148 (1988)
Gobet, E., Pages, G., Pham, H., Printems, J.: Discretization and simulation of Zakai equation. SIAM J. Num. Anal. 44, 2505–2538 (2006)
Grecksch, W., Kloeden, P. E.: Time-discretised Galerkin approximations of parabolic stochastic PDEs. Bull. Austral. Math. Soc. 54, 79–85 (1996)
Gyöngy, I.: Approximations of stochastic partial differential equations. In: Stochastic partial differential equations and applications (Trento, 2002), Lecture Notes in Pure and Appl. Math. 227, pp. 287–307. Dekker (2002)
Gyöngy, I., Krylov, N.: On the splitting-up method and stochastic partial differential equations. Ann. Appl. Probab. 31, 564–591 (2003)
Hou, T., Luo, W., Rozovskii, B.L., Zhou, H.-M.: Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics. J. Comput. Phys. 216, 687–706 (2006)
Jentzen,A., Kloeden, P. E.: The numerical approximation of stochastic partial differential equations. Milan J. Math. 77, 205–244 (2009)
Krylov, N.V.: A W 2 n-theory of the Dirichlet problem for SPDEs in general smooth domains. Probab. Theory Relat. Fields 98, 389–421 (1994)
Krylov, N.V., Rozovskii, B.L.: On the characteristics of degenerate second order parabolic Ito equations. J. Soviet Math. 32, 336–348 (1986)
Kunita, H.: Stochastic flows and stochastic differential equations. Cambridge University Press (1990)
Kurtz, T.G., Xiong, J.: Numerical solutions for a class of SPDEs with application to filtering. In: Stochastics in finite and infinite dimensions, Trends Math., pp. 233–258. Birkhäuser, Boston (2001)
Le Gland, F.: Splitting-up approximation for SPDEs and SDEs with application to nonlinear filtering. Lecture Notes in Control and Inform. Sc. 176, pp. 177–187. Springer (1992)
Liptser, R.S., Shiryaev, A.N.: Statistics of random processes. Springer, I (1977), II (1978)
Lototsky, S.V.: Dirichlet problem for stochastic parabolic equations in smooth domains. Stoch. Stoch. Rep. 68, 145–175 (2000)
Lototsky, S.V.: Linear stochastic parabolic equations, degenerating on the boundary of a domain. Electr. J. Prob. 6, Paper No. 24, 1–14 (2001)
Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems. Birkhauser (1995)
Mikulevicius, R., Pragarauskas, H.: On Cauchy-Dirichlet problem for parabolic quasilinear SPDEs. Poten. Anal. 25, 37–75 (2006)
Mikulevicius, R., Pragarauskas, H., Sonnadara, N.: On the Cauchy-Dirichlet problem in the half space for parabolic sPDEs in weighted Hölder spaces. Acta Appl. Math. 97, 129–149 (2007)
Mikulevicius, R., Rozovskii, B.L.: Linear parabolic stochastic PDEs and Wiener chaos. SIAM J. Math. Anal. 29, 452–480 (1998)
Milstein, G.N.: Solution of the first boundary value problem for equations of parabolic type by means of the integration of stochastic differential equations. Theory Probab. Appl. 40, 556–563 (1995)
Milstein, G.N.: Application of the numerical integration of stochastic equations for the solution of boundary value problems with Neumann boundary conditions. Theory Prob. Appl. 41, 170–177 (1996)
Milstein, G.N., Tretyakov, M.V.: The simplest random walks for the Dirichlet problem. Theory Probab. Appl. 47, 53–68 (2002)
Milstein, G.N., Tretyakov, M.V.: Stochastic numerics for mathematical physics. Springer (2004)
Milstein, G.N., Tretyakov, M.V.: Monte Carlo algorithms for backward equations in nonlinear filtering. Adv. Appl. Probab. 41, 63–100 (2009)
Milstein, G.N., Tretyakov, M.V.: Solving parabolic stochastic partial differential equations via averaging over characteristics. Math. Comp. 78, 2075–2106 (2009)
Milstein, G.N., Tretyakov, M.V.: Averaging over characteristics with innovation approach in nonlinear filtering. In: Crisan, D., Rozovskii, B.L. (eds.) Handbook on Nonlinear Filtering. Oxford University Press (2010), to appear
Pardoux, E.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3, 127–167 (1979)
Pardoux, E.: Filtrage de diffusions avec conditions frontieres: caracterisation de la densite conditionnelle. In: Dacumba-Castelle, D., Van Cutsem, B. (eds.) Journees de Statistique dans les Processus Stochastiques, Lecture Notes in Mathematics 636 (Proceedings, Grenoble, 1977), pp. 163–188. Springer (1978)
Pardoux, E.: Stochastic partial differential equation for the density of the conditional law of a diffusion process with boundary. In: Stochastic analysis (Proceeings of International Conference, Northwestern University, Evanston, Ill., 1978), pp. 239–269. Academic (1978)
Picard, J.: Approximation of nonlinear filtering problems and order of convergence. Lecture Notes in Contr. Inform. Sc. 61, pp. 219–236. Springer (1984)
Rozovskii, B.L.: Stochastic evolution systems, linear theory and application to nonlinear filtering. Kluwer, Dordrecht (1991)
Shardlow, T.: Numerical methods for stochastic parabolic PDEs. Numer. Func. Anal. Optimiz. 20, 121–145 (1999)
Yan, Y.: Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Num. Anal. 43, 1363–1384 (2005)
Yoo, H.: Semi-discretzation of stochastic partial differential equations on R 1 by a finite-difference method. Math. Comp. 69, 653–666 (2000)
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The work was partially supported by the UK EPSRC Research Grant EP/D049792/1.
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Stanciulescu, V.N., Tretyakov, M.V. (2011). Numerical Solution of the Dirichlet Problem for Linear Parabolic SPDEs Based on Averaging over Characteristics. In: Crisan, D. (eds) Stochastic Analysis 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15358-7_9
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DOI: https://doi.org/10.1007/978-3-642-15358-7_9
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