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Absolute continuity for some one-dimensional processes. (English) Zbl 1248.60062
Summary: We introduce an elementary method for proving the absolute continuity of 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 the method 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 is in general not Malliavin differentiable.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G30 Continuity and singularity of induced measures
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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