Absolute continuity for some one-dimensional processes

Details Absolute continuity for some one-dimensional processes Absolute continuity for some one-dimensional processes

2008-04-18

arxiv.org/abs/0804.3037v3

Bernoulli 2010, Vol. 16, No. 2, 343-360

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/09-BEJ215 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

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 H\"{o}lder 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 L\'{e}vy-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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