Open Access
May 2010 Absolute continuity for some one-dimensional processes
Nicolas Fournier, Jacques Printems
Bernoulli 16(2): 343-360 (May 2010). DOI: 10.3150/09-BEJ215


We introduce an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the one-step Euler approximation of the underlying process. We obtain some absolute continuity results for stochastic differential equations with Hölder continuous coefficients. Furthermore, we allow such coefficients to be random and to depend on the whole path of the solution. We also show how it can be extended to some stochastic partial differential equations and to some Lévy-driven stochastic differential equations. In the cases under study, the Malliavin calculus cannot be used, because the solution in generally not Malliavin differentiable.


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Nicolas Fournier. Jacques Printems. "Absolute continuity for some one-dimensional processes." Bernoulli 16 (2) 343 - 360, May 2010.


Published: May 2010
First available in Project Euclid: 25 May 2010

zbMATH: 1248.60062
MathSciNet: MR2668905
Digital Object Identifier: 10.3150/09-BEJ215

Keywords: Absolute continuity , Hölder coefficients , Lévy processes , random coefficients , Stochastic differential equations , Stochastic partial differential equations

Rights: Copyright © 2010 Bernoulli Society for Mathematical Statistics and Probability

Vol.16 • No. 2 • May 2010
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