Some remarks on the numerical approximation of stochastic differential equations
Authors:
Brian D. Ewald and Roger Témam
Journal:
Quart. Appl. Math. 66 (2008), 595-605
MSC (2000):
Primary 65C20
DOI:
https://doi.org/10.1090/S0033-569X-08-01128-1
Published electronically:
July 2, 2008
MathSciNet review:
2445530
Full-text PDF Free Access
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Abstract: The aim of this article is to discuss the convergence of some numerical stochastic schemes in geophysical fluid dynamics (GFD) and to make some remarks on the numerical analysis of stochastic differential equations (SDEs).
References
- Brian D. Ewald and Roger Témam, Numerical analysis of stochastic schemes in geophysics, SIAM J. Numer. Anal. 42 (2005), no. 6, 2257–2276. MR 2139392, DOI https://doi.org/10.1137/S0036142902418333
- B. D. Ewald and R. Témam, Analysis of stochastic numerical schemes for the evolution equations of geophysics, Appl. Math. Lett. 16 (2003), no. 8, 1223–1229. MR 2015716, DOI https://doi.org/10.1016/S0893-9659%2803%2990121-2
- Ewald, B.D.; Penland, C.; Témam, R., Accurate integration of stochastic climate models with application to El Niño, Monthly Weather Review, 132, 2004, 154-164.
- Brian D. Ewald, Error and convergence of two numerical schemes for stochastic differential equations, Numer. Methods Partial Differential Equations 22 (2006), no. 5, 1247–1253. MR 2248048, DOI https://doi.org/10.1002/num.20165
- Peter Friz and Nicolas Victoir, Approximations of the Brownian rough path with applications to stochastic analysis, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 4, 703–724 (English, with English and French summaries). MR 2144230, DOI https://doi.org/10.1016/j.anihpb.2004.05.003
- Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR 1214374
- Antoine Lejay and Nicolas Victoir, On $(p,q)$-rough paths, J. Differential Equations 225 (2006), no. 1, 103–133. MR 2228694, DOI https://doi.org/10.1016/j.jde.2006.01.018
- Terry Lyons and Nicolas Victoir, Cubature on Wiener space, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2041, 169–198. Stochastic analysis with applications to mathematical finance. MR 2052260, DOI https://doi.org/10.1098/rspa.2003.1239
- G. N. Milstein, Numerical integration of stochastic differential equations, Mathematics and its Applications, vol. 313, Kluwer Academic Publishers Group, Dordrecht, 1995. Translated and revised from the 1988 Russian original. MR 1335454
- W. Rümelin, Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal. 19 (1982), no. 3, 604–613. MR 656474, DOI https://doi.org/10.1137/0719041
- Denis Talay, Probabilistic numerical methods for partial differential equations: elements of analysis, Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995) Lecture Notes in Math., vol. 1627, Springer, Berlin, 1996, pp. 148–196. MR 1431302, DOI https://doi.org/10.1007/BFb0093180
- Denis Talay, Discretization of stochastic differential equations. Application to simulation. Stochastic numerical methods for partial differential equations, Numerical mathematics and advanced applications (Jyväskylä, 1999) World Sci. Publ., River Edge, NJ, 2000, pp. 72–86. MR 1936171
References
- Ewald, Brian D.; Témam, Roger, Numerical analysis of stochastic schemes in geophysics, SIAM J. Numer. Anal. 42 (2005), no. 6, 2257-2276. MR 2139392 (2006b:65006)
- Ewald, B.D.; Témam, R., Analysis of stochastic numerical schemes for the evolution equations of geophysics, Appl. Math. Lett. 16 (2003), no. 8, 1223-1229. MR 2015716
- Ewald, B.D.; Penland, C.; Témam, R., Accurate integration of stochastic climate models with application to El Niño, Monthly Weather Review, 132, 2004, 154-164.
- Ewald, B.D., Error and convergence of two numerical schemes for stochastic differential equations, Numer. Methods Partial Differential Equations 22 (2006), 1247–1253. MR 2248048 (2007e:65008)
- Friz, Peter; Victoir, Nicolas, Approximations of the Brownian rough path with applications to stochastic analysis, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 4, 703–724. MR 2144230 (2007e:60018)
- Kloeden, Peter E.; Platen, Eckhard, Numerical solution of stochastic differential equations. Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. MR 1214374 (94b:60069)
- Lejay, Antoine; Victoir, Nicolas, On $(p,q)$-rough paths, J. Differential Equations 225 (2006), no. 1, 103–133. MR 2228694 (2007i:60069)
- Lyons, Terry; Victoir, Nicolas, Cubature on Wiener space, Stochastic analysis with applications to mathematical finance. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2041, 169–198. MR 2052260 (2005b:35306)
- Milstein, G.N., Numerical integration of stochastic differential equations. Translated and revised from the 1988 Russian original. Mathematics and its Applications, 313, Kluwer Academic Publishers Group, Dordrecht, 1995. MR 1335454 (96e:65003)
- Rümelin, W., Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal. 19 (1982), no. 3, 604-613. MR 656474 (83i:60075)
- Talay, Denis, Probabilistic numerical methods for partial differential equations: Elements of analysis. Probabilistic models for nonlinear partial differential equations (Montecatini Terme 1995), 148-196, Lecture Notes in Math., 1627, Springer, Berlin, 1996. MR 1431302 (98j:60092)
- Talay, Denis, Discretization of stochastic differential equations. Application to simulation. Stochastic numerical methods for partial differential equations. Numerical Mathematics and Advanced Applications (Jyväskä 1999), 72-86. World Sci. Publ., River Edge, NJ, 2000. MR 1936171 (2003i:65010)
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Additional Information
Brian D. Ewald
Affiliation:
Department of Mathematics, Florida State University, Room 205C, 1017 Academic Way, Tallahassee, Florida 32306-4510
Roger Témam
Affiliation:
Institute for Scientific Computing and Applied Mathematics, Indiana University, Rawles Hall, Bloomington, Indiana 47405-5701
MR Author ID:
171480
Received by editor(s):
May 15, 2007
Published electronically:
July 2, 2008
Article copyright:
© Copyright 2008
Brown University