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A Girsanov transformation for birth and death on a Brownian flow

Published online by Cambridge University Press:  14 July 2016

Michael J. Phelan
Michael J. Phelan*
Affiliation:
University of Pennsylvania
*
∗ Postal address: Department of Statistics, The Wharton School of the University of Pennsylvania, 3000 Steinberg Hall-Dietrich Hall, Philadelphia, PA 19104–6302, USA.
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Abstract

We consider a system of particles in birth and death on a Brownian flow. The system includes a particle process tracking the spatial configuration of live particles on the flow. The particle process is a Markov process on the space of counting measures. The system depends on a handful of parameters including a rate of drift, diffusion, birth, and killing. We exhibit a Girsanov formula for the absolutely continuous change of particle system by way of a change of drift, birth, and killing rates. We can only allow for a restrictive change of drift on the flow, but for fairly unrestrictive change of birth and death rates. The result is therefore of some interest in problems of statistical inference from passive transport of transient tracers.

Keywords

BROWNIAN FLOWPOISSON PROCESSGIRSANOV FORMULA

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996 , pp. 88 - 100
Copyright
Copyright © Applied Probability Trust 1996 

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References

ÇlInlar, E. and Jacod, J. (1980) Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. Seminar in Stochastic Processes, ed Çlinlar, E., pp. 159–242. Birkhauser, Boston.Google Scholar
ÇlInlar, E. and Kao, J. S. (1992a) Particle systems on flows. Applied Stochastic Models and Data Analysis. Wiley, New York.Google Scholar
ÇlInlar, E. and Kao, J. S. (1992b) Birth and death on flows. Diffusion Processes and Related Problems in Analysis, II, Stochastic Flows, ed. Pinsky, M. A. and Wihstutz, V. pp. 121–137. Birkhauser, Boston.Google Scholar
Conway, J. B. (1985) A Course in Functional Analysis. Springer, New York.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York.Google Scholar
Jacod, J. and Shiryaev, A. N. (1987) Limit Theorems for Stochastic Processes. Springer, New York.Google Scholar
Kunita, H. (1990) Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge.Google Scholar
Métivier, M. (1982) Semimartingales: A Course in Stochastic Processes. Walter de Gruyter, New York.Google Scholar
Phelan, M. J. (1994a) A quasi likelihood for integral data on birth and death on flows. Stoch. Proc. Appl. 53, 379–392.Google Scholar
Phelan, M. J. (1994b) Transient tracers and birth and death on Brownian flows: parametric estimation of rates of drift, injection, and decay. J. Stat. Plan. and Inf. (to appear).Google Scholar
Phelan, M. J. (1996) Birth and death on a Brownian flow: A Feller semigroup and its generator and a martingale problem. Adv. Appl. Prob. 33, 71–87.Google Scholar