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Integration by Parts for Point Processes and Monte Carlo Estimation

Part of: Nonparametric inference Stochastic analysis Stochastic processes Special processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Nicolas Privault  and
Xiao Wei
Nicolas Privault
Affiliation:
Université de Poitiers
Xiao Wei*
Affiliation:
Central University of Finance and Economics
*
∗∗Postal address: School of Insurance, Central University of Finance and Economics, Beijing, 100081, P. R. China. Email address: weix@cufe-ins.sina.net
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Abstract

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We develop an integration by parts technique for point processes, with application to the computation of sensitivities via Monte Carlo simulations in stochastic models with jumps. The method is applied to density estimation with respect to the Lebesgue measure via a modified kernel estimator which is less sensitive to variations of the bandwidth parameter than standard kernel estimators. This applies to random variables whose densities are not analytically known and requires the knowledge of the point process jump times.

Keywords

Malliavin calculuspoint processrenewal processsensitivity analysisdensity estimationkernel estimator

MSC classification

Secondary: 62G07: Density estimation 60H07: Stochastic calculus of variations and the Malliavin calculus 65C05: Monte Carlo methods 60G55: Point processes 60K15: Markov renewal processes, semi-Markov processes
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 806 - 823
Copyright
Copyright © Applied Probability Trust 2007 

Footnotes

∗

Current address: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong. Email address: nprivaul@cityu.edu.hk

References

[1] Bally, V., Bavouzet-Morel, M.-P. and Messaoud, M. (2007). Integration by parts formula for locally smooth laws and applications to sensitivity computations. Ann. Appl. Prob. 17, 33–66.Google Scholar
[2] Bavouzet-Morel, M.-P. and Messaoud, M. (2006). Computation of Greeks using Malliavin's calculus in Jump type market models. Electron. J. Prob. 11, 276–300.Google Scholar
[3] Carlen, E. and Pardoux, E. (1990). Differential calculus and integration by parts on Poisson space. In Stochastics, Algebra and Analysis in Classical and Quantum Dynamics (Marseille, 1988; Math. Appl. 59), Kluwer, Dordrecht, pp. 63–73.CrossRefGoogle Scholar
[4] Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, Probability and Its Applications. Springer, New York.Google Scholar
[5] Dash, J. W. (2004). Quantitative Finance and Risk Management. World Scientific, River Edge, NJ.Google Scholar
[6] Davis, M. H. A. and Johansson, M. P. (2006). Malliavin Monte Carlo Greeks for Jump diffusions. Stoch. Process. Appl. 116, 101–129.Google Scholar
[7] Debelley, V. and Privault, N. (2004). Sensitivity analysis of European options in Jump diffusion models via the Malliavin calculus on Wiener space. Preprint, Université de La Rochelle.Google Scholar
[8] El Khatib, Y. and Privault, N. (2004). Computations of Greeks in markets with Jumps via the Malliavin calculus. Finance Stoch. 4, 161–179.Google Scholar
[9] Elliott, R. J. and Tsoi, A. H. (1993). Integration by parts for Poisson processes. J. Multivariate Anal. 44, 179–190.CrossRefGoogle Scholar
[10] Fournié, E., Lasry, J. M., Lebuchoux, J. and Lions, P. L. (2001). Applications of Malliavin calculus to Monte-Carlo methods in finance. II. Finance Stoch. 5, 201–236.Google Scholar
[11] Fournié, E. et al. (1999). Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch. 3, 391–412.Google Scholar
[12] Kohatsu-Higa, A. and Pettersson, R. (2002). Variance reduction methods for simulation of densities on Wiener space. SIAM J. Numer. Anal. 40, 431–450.Google Scholar
[13] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.CrossRefGoogle Scholar
[14] Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Statist. 33, 1065–1076.CrossRefGoogle Scholar
[15] Privault, N. (1994). Chaotic and variational calculus in discrete and continuous time for the Poisson process. Stoch. Stoch. Reports 51, 83–109.Google Scholar
[16] Privault, N. and Wei, X. (2004). A Malliavin calculus approach to sensitivity analysis in insurance. Insurance Math. Econom. 35, 679–690.CrossRefGoogle Scholar
[17] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27, 832–837.Google Scholar
[18] Srinivasan, S. K. (1969). Stochastic Theory and Cascade Processes. American Elsevier, New York.Google Scholar