Publication Cover
Stochastics
An International Journal of Probability and Stochastic Processes
Volume 86, 2014 - Issue 3
182
Views
1
CrossRef citations to date
0
Altmetric
Original Articles

Numerical solution for a class of SPDEs over bounded domains

&
Pages 450-472 | Received 01 Aug 2011, Accepted 18 Jun 2013, Published online: 02 Sep 2013

References

  • M.S.Arulampalam, S.Maskell, N.Gordon, and T.Clapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Trans. Signal Process.50(2) (2002), pp. 174–188.
  • A.Bain and D.Crisan, Fundamentals of Stochastic Filtering, Stochastic Modelling and Applied Probability, Vol. 60, Springer, New York, 2009.
  • M.Bossy, E.Gobet, and D.Talay, A symmetrized Euler scheme for an efficient approximation of reflected diffusions, J. Appl. Probab.41(3) (2004), pp. 877–889.
  • J.Carpenter, P.Clifford, and P.Fearnhead, Improved particle filter for nonlinear problems, IEE Proc. Radar Sonar Navigation146(1) (1999), pp. 2–7.
  • J.M.C.Clark and D.Crisan, On a robust version of the integral representation formula of nonlinear filtering, Probab. Theory Relat. Fields133(1) (2005), pp. 43–56.
  • D.Crisan, Exact rates of convergence for a branching particle approximation to the solution of the Zakai equation, Ann. Probab.31(2) (2003), pp. 693–718.
  • D.Crisan, Discretizing the continuous time filtering problem, in The Oxford Handbook of Non-Linear Filtering, Oxford University Press, Oxford,2011, pp. 572–597.
  • D.Crisan, P.Del Moral, and T.Lyons, Interacting particle systems approximations of the Kushner–Stratonovitch equation, Adv. Appl. Probab.31(3) (1999), pp. 819–838.
  • D.Crisan and T.Lyons, A particle approximation of the solution of the Kushner–Stratonovitch equation, Probab. Theory Relat. Fields115(4) (1999), pp. 549–578.
  • D.Crisan and B.Rozovsky, The Oxford Handbook of Nonlinear Filtering, Oxford University Press, Oxford, 2011.
  • D.Crisan and J.Xiong, Approximate McKean–Vlasov representations for a class of SPDEs, Stoch. Int. J. Probab. Stoch. Process.82(1) (2010), pp. 53–68.
  • P.Del Moral, Non-linear filtering: Interacting particle solution, Markov Process. Relat. Fields2 (1996), pp. 555–580.
  • P.Del Moral, Non-linear filtering using random particles, Theory Probab. Appl.40(4) (1996), pp. 690–701.
  • P.Del Moral, Feynman–Kac Formulae. Genealogical and Interacting Particle Systems with Applications, Springer, New York, 2004.
  • A.Doucet and A.M.Johansen, A tutorial on particle filtering and smoothing: Fifteen years later, in The Oxford Handbook of Nonlinear Filtering, D.Crisan and B.Rozovsky, eds., Oxford University Press, Oxford, 2011, pp. 656–704.
  • S.Fang and T.Zhang, A study of a class of stochastic differential equations with non-Lipschitzian coefficients, Probab. Theory Relat. Fields132 (2005), pp. 356–390.
  • A.Friedman, Stochastic Differential Equations and Applications, Vol. 1, Academic Press, New York, 1975.
  • N.J.Gordon, D.J.Salmond, and A.F.M.Smith, Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEE Proc. Radar Signal Process. F140 (1993), pp. 107–113.
  • G.Kitagawa, Non-Gaussian state-space modeling of nonstationary time series, J. Am. Stat. Assoc.82 (1987), pp. 1032–1063.
  • T.Kurtz and J.Xiong, Particle representations for a class of nonlinear SPDEs, Stoch. Process. Appl.83 (1999), pp. 103–126.
  • T.Kurtz and J.Xiong, Numerical solutions for a class of SPDEs with application to filtering, in Stochastics in Finite and Infinite Dimension: In Honor of Gopinath Kallianpur, T.Hida, R.Karandikar, H.Kunita, B.Rajput, S.Watanabe, and J.Xiong, eds., Trends in Mathematics, Birkhauser, Boston, MA, 2000, pp. 233–258.
  • T.Kurtz and J.Xiong, A stochastic evolution equation arising from the fluctuation of a class of interacting particle systems, Commun. Math. Sci.2 (2004), pp. 325–358.
  • O.A.Ladyženskaja, V.A.Solonnikov, and N.N.Uralćeva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Translated from the Russian by S. SmithTranslations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1967.
  • P.L.Lions and A.S.Sznitman, Stochastic differential equations with reflecting boundary conditions, Commun. Pure Appl. Math.37(4) (1984), pp. 511–537.
  • U.Orguner and F.Gustafsson, Risk-sensitive particle filters for mitigating sample impoverishment, IEEE Trans. Signal Process. Part 256(10) (2008), pp. 5001–5012.
  • R.Pettersson, Approximations for stochastic differential equations with reflecting convex boundaries, Stoch. Process. Appl.59(2) (1995), pp. 295–308.
  • R.Pettersson, Penalization schemes for reflecting stochastic differential equations, Bernoulli3(4) (1997), pp. 403–414.
  • S. Reich, A Gaussian mixture ensemble transform filter, preprint, to appear in the Q.J.R. Meteorol. Soc. Available at http://arxiv.org/abs/1102.3089.
  • Y.Saisho, Stochastic differential equations for multidimensional domain with reflecting boundary, Probab. Theory Relat. Fields74(3) (1987), pp. 455–477.
  • L.Slominski, Euler's approximations of solutions of SDEs with reflecting boundary, Stoch. Process. Appl.94(2) (2001), pp. 317–337.
  • H.Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J.9(1) (1979), pp. 163–177.
  • J.Xiong, An Introduction to Stochastic Filtering Theory, Oxford Graduate Texts in Mathematics, Vol. 18, Oxford University Press, Oxford, 2008.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.