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A Technique for Exponential Change of Measure for Markov Processes

Zbigniew Palmowski and Tomasz Rolski
Bernoulli
Vol. 8, No. 6 (Dec., 2002), pp. 767-785
Stable URL: http://www.jstor.org/stable/3318901
Page Count: 19
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A Technique for Exponential Change of Measure for Markov Processes
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Abstract

We consider a Markov process X(t) with extended generator A and domain D( A). Let {Ft} be a right-continuous history filtration and Pt denote the restriction of P to Ft. Let $\tilde{{\Bbb P}}$ be another probability measure on (Ω,F) such that ${\rm d}\tilde{{\Bbb P}}_{t}/{\rm d}{\Bbb P}_{t}=E^{h}(t)$, where Eh(t)=h(X(t))/h(X(0)) exp(-∫0 t( Ah)(X(s))/h(X(s)) ds) is a true martingale for a positive function h∈ D( A). We demonstrate that the process X(t) is a Markov process on the probability space $(\Omega,\scr{F},\{\scr{F}_{t}\},\tilde{{\Bbb P}})$, we find its extended generator $\tilde{{\bf A}}$ and provide sufficient conditions under which $\scr{D}(\tilde{{\bf A}})=\scr{D}({\bf A})$. We apply this result to continuous-time Markov chains, to piecewise deterministic Markov processes and to diffusion processes (in this case a special choice of h yields the classical Cameron-Martin-Girsanov theorem).

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