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Weak Local Linear Discretizations for Stochastic Differential Equations with Jumps

Part of: Markov processes Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

F. Carbonell  and
J. C. Jimenez
F. Carbonell*
Affiliation:
Instituto de Cibernética
J. C. Jimenez*
Affiliation:
Instituto de Cibernética
*
∗Current address: The Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 2K6. Email address: [email protected]
∗∗Postal address: Departamento de Matemática Interdisciplinaria, Instituto de Cibernética, Matemática y Física, Calle 15, No. 551, e/C y D, Vedado, La Habana 4, C.P. 10400, Cuba.
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Abstract

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Weak local linear approximations have played a prominent role in the construction of effective inference methods and numerical integrators for stochastic differential equations. In this note two weak local linear approximations for stochastic differential equations with jumps are introduced as a generalization of previous ones. Their respective order of convergence is obtained as well.

Keywords

Jump diffusion processstochastic differential equationnumerical integrationlocal linearizationweak convergence

MSC classification

Secondary: 60J75: Jump processes 60H35: Computational methods for stochastic equations 60J60: Diffusion processes
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 201 - 210
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Bruti-Liberati, N. (2007). Numerical solution of stochastic differential equations with Jumps in finance. , School of Finance and Economics, University of Technology, Sydney.Google Scholar
[2] Bruti-Liberati, N. and Platen, E. (2007). On weak predictor-corrector schemes for Jump-diffusion processes in finance. To appear in Numerical Methods in Finance (Financial Math. Ser. 8), Chapman & Hall, Boca Raton, FL.Google Scholar
[3] Carbonell, F., Jimenez, J. C. and Biscay, R. J. (2006). Weak local linear discretizations for stochastic differential equations: convergence and numerical schemes. J. Comput. Appl. Math. 197, 578–596.Google Scholar
[4] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall, Boca Raton, FL.Google Scholar
[5] Glasserman, P. and Merener, N. (2003). Numerical solution of Jump-diffusion LIBOR market models. Finance Stoch. 7, 1–27.Google Scholar
[6] Higham, D. and Kloeden, P. E. (2006). Convergence and stability of implicit methods for Jump-diffusion systems. Internat. J. Numer. Anal. Model. 3, 125–140.Google Scholar
[7] Jimenez, J. C. and Ozaki, T. (2003). Local linearization filters for nonlinear continuous-discrete state space models with multiplicative noise. Internat. J. Control 76, 1159–1170.Google Scholar
[8] Jimenez, J. C. and Ozaki, T. (2006). An approximate innovation method for the estimation of diffusion processes from discrete data. J. Time Ser. Anal. 27, 77–97.Google Scholar
[9] Jimenez, J. C., Biscay, R. and Ozaki, T. (2006). Inference methods for discretely observed continuous-time stochastic volatility models: a commented overview. Asian-Pacific Financial Markets 12, 109–141.Google Scholar
[10] Kloeden, P. E. and Platen, E. (1995). Numerical Solution of Stochastic Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
[11] Kubilius, K. and Platen, E. (2002). Rate of weak convergenece of the Euler approximation for diffusion processes with Jumps. Monte Carlo Methods Appl. 8, 83–96.Google Scholar
[12] Liu, X. Q. and Li, C. W. (2000). Weak approximations and extrapolations of stochastic differential equations with Jumps. SIAM J. Numerical Anal. 37, 1747–1767.CrossRefGoogle Scholar
[13] Mikulevicius, R. and Platen, E. (1998). Time discrete Taylor approximations for Itô processes with Jump component. Math. Nachr. 138, 93–104.Google Scholar
[14] Mora, C. M. (2005). Weak exponential schemes for stochastic differential equations with additive noise. IMA J. Numerical Anal. 25, 486–506.Google Scholar
[15] Øksendal, B. and Sulem, A. (2005). Applied Stochastic Control of Jump Diffusions. Springer, Berlin.Google Scholar
[16] Ozaki, T. (1992). A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: a local linearization approach. Statistica Sinica 2, 113–135.Google Scholar
[17] Ozaki, T. (1994). The local linearization filter with application to nonlinear system identification. In Proc. First US/Japan Conf. Frontiers Statist. Modeling: An Informational Approach, ed. Bozdogan, H., Kluwer, Dordrecht, pp. 217–240.Google Scholar
[18] Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, New York.Google Scholar
[19] Protter, P. and Talay, D. (1997). The Euler scheme for Levy driven stochastic differential equations. Ann. Prob. 25, 393–423.Google Scholar
[20] Shoji, I. and Ozaki, T. (1997). Comparative study of estimation methods for continuous time stochastic processes. J. Time Ser. Anal. 18, 485–506.Google Scholar
[21] Shoji, I. and Ozaki, T. (1998). A statistical method of estimation and simulation for systems of stochastic differential equations. Biometrika 85, 240–243.Google Scholar
[22] Shoji, I. and Ozaki, T. (1998). Estimation for nonlinear stochastic differential equations by a local linearization method. Stoch. Anal. Appl. 16, 733–752.CrossRefGoogle Scholar
[23] Tuckwell, H. C. (1989). Stochastic Processes in Neurosciences. SIAM, Philadelphia, PA.Google Scholar