Jumping SDEs: absolute continuity using monotonicity

Nicolas Fournier

Stochastic Processes and their Applications, 2002, vol. 98, issue 2, 317-330

Abstract: We study the solution X={Xt}t[set membership, variant][0,T] to a Poisson-driven SDE. This equation is "irregular" in the sense that one of its coefficients contains an indicator function, which allows to generalize the usual situations: the rate of jump of X may depend on X itself. For t>0 fixed, the random variable Xt does not seem to be differentiable (with respect to the alea) in a usual sense (see e.g. Séminaire de Probabilités XVII, Lecture Notes in Mathematics, Vol. 986, Springer, Berlin, 1983, pp. 132-157), and actually not even continuous. We thus introduce a new technique, based on a sort of monotony of the map [omega]|->Xt([omega]), to prove that under quite stringent assumptions, which make possible comparison theorems, the law of Xt admits a density with respect to the Lebesgue measure on .

Keywords: Stochastic; differential; equations; Jump; processes; Stochastic; calculus; of; variations (search for similar items in EconPapers)
Date: 2002
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Citations: View citations in EconPapers (4)

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