Hostname: page-component-7dd5485656-s6l46 Total loading time: 0 Render date: 2025-10-31T09:48:54.062Z Has data issue: false hasContentIssue false
  • English
  • Français

Local Linear Approximations of Jump Diffusion Processes

Part of: Stochastic analysis Markov processes

Published online by Cambridge University Press:  14 July 2016

J. C. Jimenez  and
F. Carbonell
J. C. Jimenez*
Affiliation:
Instituto de Cibernética, Matemática y Física
F. Carbonell*
Affiliation:
Instituto de Cibernética, Matemática y Física
*
∗ Postal address: Departamento de Sistemas Adaptativos, Instituto de Cibernética, Matemática y Física, Calle 15, No. 551, Vedado, La Habana 4, C.P. 10400, Cuba.
∗ Postal address: Departamento de Sistemas Adaptativos, Instituto de Cibernética, Matemática y Física, Calle 15, No. 551, Vedado, La Habana 4, C.P. 10400, Cuba.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Local linear approximations have been the main component in the construction of a class of effective numerical integrators and inference methods for diffusion processes. In this note, two local linear approximations of jump diffusion processes are introduced as a generalization of the usual ones. Their rate of uniform strong convergence is also studied.

Keywords

Jump diffusion processstochastic differential equationnumerical integrationlocal linearization methodpseudo-likelihood method

MSC classification

Primary: 60J75: Jump processes 60J60: Diffusion processes 60H35: Computational methods for stochastic equations

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 1 , March 2006 , pp. 185 - 194
Copyright
© Applied Probability Trust 2006 

Footnotes

Partially supported by the research grant 03-059 RG/MATHS/LA from the ThirdWorld Academy of Science.

References

Biscay, R., Jimenez, J. C., Riera, J. and Valdes, P. (1996). Local linearization method for the numerical solution of stochastic differential equations. Ann. Inst. Statist. Math. 48, 631–644.Google Scholar
Jimenez, J. C. (2002). A simple algebraic expression to evaluate the local linearization schemes for stochastic differential equations. Appl. Math. Lett. 15, 775–780.Google Scholar
Jimenez, J. C. and Biscay, R. (2002). Approximation of continuous time stochastic processes by the local linearization method revisited. Stoch. Anal. Appl. 20, 105–121.Google Scholar
Jimenez, J. C., Shoji, I. and Ozaki, T. (1999). Simulation of stochastic differential equations through the local linearization method. A comparative study. J. Statist. Phys. 94, 587–602.Google Scholar
Liu, X. Q. and Li, C. W. (1999). Product expansion for stochastic Jump diffusions and its application to numerical approximation. J. Comput. Appl. Math. 108, 1–17.CrossRefGoogle Scholar
Maghsoodi, Y. (1998). Exact solutions and doubly efficient approximations of Jump-diffusion Itô equations. Stoch. Anal. Appl. 16, 1049–1072.CrossRefGoogle Scholar
Mikulevičius, R. and Platen, E. (1988). Time discrete Taylor approximations for Itô processes with Jump component. Math. Nachr. 138, 93–104.Google Scholar
Ozaki, T. (1985). Nonlinear time series models and dynamical systems. In Time Series in the Time Domain (Handbook Statist. 5), eds Hannan, E. J. et al., North-Holland, Amsterdam, pp. 25–83.Google Scholar
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
Prakasa-Rao, B. L. S. (1999). Statistical Inference for Diffussion Type Processes. Oxford University Press.Google Scholar
Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, Berlin.Google Scholar
Schurz, H. (2002). Numerical analysis of stochastic differential equations without tears. In Handbook of Stochastic Analysis and Applications (Statist. Textbook Monogr. 163), eds Kannan, D. and Lakahmikamtham, V., Marcel Dekker, New York, pp. 237–359.Google Scholar
Shoji, I. and Ozaki, T. (1997). Comparative study of estimation methods for continuous time stochastic processes. J. Time Series Anal. 18, 485–506.Google Scholar
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
Shoji, I. and Ozaki, T. (1998). Estimation for nonlinear stochastic differential equations by a local linearization method. Stoch. Anal. Appl. 16, 733–752.Google Scholar