An operator approach for Markov chain weak approximations with an application to infinite activity Lévy driven SDEs

An operator approach for Markov chain weak approximations with an application to infinite activity Lévy driven SDEs

Hideyuki Tanaka and Arturo Kohatsu-Higa

Source: Ann. Appl. Probab. Volume 19, Number 3 (2009), 1026-1062.

Abstract

Weak approximations have been developed to calculate the expectation value of functionals of stochastic differential equations, and various numerical discretization schemes (Euler, Milshtein) have been studied by many authors. We present a general framework based on semigroup expansions for the construction of higher-order discretization schemes and analyze its rate of convergence. We also apply it to approximate general Lévy driven stochastic differential equations.

Primary Subjects: 60H35, 60J75, 65C05, 60H10

Secondary Subjects: 65C30, 60J22

Keywords: Stochastic differential equations; jump processes; weak approximation

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References

[1] Alfonsi, A. (2008). High-order discretization schemes for the CIR process: Application to affine term structure and Heston models. Preprint.

[2] Asmussen, S. and Rosiński, J. (2001). Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38 482–493.

Mathematical Reviews (MathSciNet): MR1834755

Digital Object Identifier: doi:10.1239/jap/996986757

Project Euclid: euclid.jap/996986757

[3] Bally, V. and Talay, D. (1996). The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Related Fields 104 43–60.

Mathematical Reviews (MathSciNet): MR1367666

Digital Object Identifier: doi:10.1007/BF01303802

[4] Butcher, J. C. (1987). The Numerical Analysis of Ordinary Differential Equations. Wiley, Chichester.

Mathematical Reviews (MathSciNet): MR878564

[5] Fujiwara, T. (2006). Approximation of Expectations of Jump-Diffusion Processes (in Japanese). Master thesis, Univ. Tokyo.

[6] Fujiwara, T. (2006). Sixth-order methods of Kusuoka approximation. Preprint.

[7] Jacod, J., Kurtz, T. G., Méléard, S. and Protter, P. (2005). The approximate Euler method for Lévy driven stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist. 41 523–558.

Mathematical Reviews (MathSciNet): MR2139032

Digital Object Identifier: doi:10.1016/j.anihpb.2004.01.007

[8] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.

Mathematical Reviews (MathSciNet): MR1617049

Digital Object Identifier: doi:10.1214/aop/1022855419

Project Euclid: euclid.aop/1022855419

[9] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1214374

[10] Kunita, H. (1984). Stochastic differential equations and stochastic flows of diffeomorphisms. In École D’été de Probabilités de Saint-Flour, XII—1982. Lecture Notes in Mathematics 1097 143–303. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR876080

Digital Object Identifier: doi:10.1007/BFb0099433

[11] Kusuoka, S. (2004). Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus. In Advances in Mathematical Economics. Adv. Math. Econ. 6 69–83. Springer, Tokyo.

Mathematical Reviews (MathSciNet): MR2079333

[12] Kusuoka, S., Ninomiya, M. and Ninomiya, S. (2007). A new weak approximation scheme of stochastic differential equations by using the Runge–Kutta method. Preprint.

[13] Lyons, T. and Victoir, N. (2004). Cubature on Wiener space. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 169–198.

Mathematical Reviews (MathSciNet): MR2052260

Digital Object Identifier: doi:10.1098/rspa.2003.1239

JSTOR: links.jstor.org

[14] Mordecki, E., Szepessy, A., Tempone, R. and Zouraris, G. E. (2008). Adaptive weak approximation of diffusions with jumps. SIAM J. Numer. Anal. 46 1732–1768.

Mathematical Reviews (MathSciNet): MR2399393

Digital Object Identifier: doi:10.1137/060669632

[15] Ninomiya, M. and Ninomiya, S. (2008). A new weak approximation scheme of stochastic differential equations by using the Runge–Kutta method. Preprint.

[16] Ninomiya, S. and Victoir, N. (2008). Weak approximation of stochastic differential equations and application to derivative pricing. Appl. Math. Finance 15 107–121.

Mathematical Reviews (MathSciNet): MR2409419

Digital Object Identifier: doi:10.1080/13504860701413958

[17] Protter, P. E. (2005). Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability 21. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR2273672

[18] Protter, P. and Talay, D. (1997). The Euler scheme for Lévy driven stochastic differential equations. Ann. Probab. 25 393–423.

Mathematical Reviews (MathSciNet): MR1428514

Digital Object Identifier: doi:10.1214/aop/1024404293

Project Euclid: euclid.aop/1024404293

[19] Talay, D. and Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl. 8 483–509 (1991).

Mathematical Reviews (MathSciNet): MR1091544

Digital Object Identifier: doi:10.1080/07362999008809220

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