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An extension of the Cameron–Martin result

Part of: Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  14 July 2016

A. I. Yashin
A. I. Yashin*
Affiliation:
Odense University
*
∗ Postal address: Odense Universitet, ISH, Vinsløws Vej 17, 1, DK-5000 Odense C, Denmark. Currently visiting Duke University.
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Abstract

The well-known Cameron–Martin formula allows us to calculate the mathematical expectation where Ws is a Wiener process. This paper extends this result to the case of piecewise continuous martingales. As a particular case the mathematical expectations of a functional of generalized Ornstein– Uhlenbeck processes and pure jump processes are calculated.

Keywords

MARTINGALESEXPONENTIAL FUNCTIONALS

MSC classification

Primary: 60G44: Martingales with continuous parameter
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Short Communications
Information
Journal of Applied Probability , Volume 30 , Issue 1 , March 1993 , pp. 247 - 251
Copyright
Copyright © Applied Probability Trust 1993 

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References

Cameron, R. N. and Martin, W. T. (1945) Evaluation of various Wiener integrals by use of certain Sturm Liouville differential equations. Bull. Amer. Math. Soc. 51, 73–90.Google Scholar
Kabanov, Yu. M., Liptser, R. S. and Shiryaev, A. N. (1978) Absolute continuity and singularity of locally absolute continuous probabilistic distributions. Mat. Sb. 107 (149), 364–415.Google Scholar
Liptser, R. S. and Shiryaev, A. N. (1977) Statistics of Random Processes. Springer-Verlag, New York.Google Scholar
Liptser, R. S. and Shiryaev, A. N. (1989) Theory of Martingales. Kluwer Academic Publishers, Dordrecht.Google Scholar
Myers, L. E. (1981) Survival functions induced by stochastic covariate processes. J. Appl. Prob. 18, 523–529.CrossRefGoogle Scholar
Nagasawa, M. (1989). Transformations of diffusion and Schrödinger processes. Prob. Theory Rel. Fields 82, 106–136.Google Scholar
Revuz, D. and Yor, M. (1991) Continuous Martingales and Brownian Motion. Springer-Verlag, Berlin.Google Scholar
Williams, D. (1976) On a stopped Brownian motion formula of H. M. Taylor. Sem. Prob. XIV, Lecture Notes in Mathematics 511, Springer-Verlag, Berlin.Google Scholar
Yashin, A. I. (1985) Dynamics in survival analysis: conditional Gaussian property versus Cameron-Martin formula. In Statistics and Control of Stochastic Processes, ed. Krylov, N. V., Liptser, R. Sh. and Novikov, A. A. Springer-Verlag, New York.Google Scholar
Yashin, A. I. (1991) How to choose the parametric form of the hazard rate: partially observed covariates. Research Report 91-05-11, CPOP, University of Minnesota, May 1991.Google Scholar
Yor, M. (1980) Remarques sur une formule de P. Levy. Sem. Prob. XIV, Lecture Notes in Mathematics 784, Springer-Verlag, Berlin.Google Scholar