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Convergence and regularity of probability laws by using an interpolation method

Vlad Bally 1, 2 Lucia Caramellino 3
2 MATHRISK - Mathematical Risk handling
Inria Paris-Rocquencourt, UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech
Abstract : We study the problem of the existence and regularity of a probability density in an abstract framework based on a "balancing" with approximating absolutely contin-uous laws. Typically, the absolutely continuous property for the approximating laws can be proved by standard techniques from Malliavin calculus whereas for the law of interest no Malliavin integration by parts formulas are available. Our results are strongly based on the use of suitable Hermite polynomial series expansions and can be merged into the theory of interpolation spaces. We then apply the results to the solution to a stochastic differential equation with a local Hörmander condition or to the solution to the stochas-tic heat equation, in both cases under weak conditions on the coefficients relaxing the standard Lipschitz or Hölder continuity requests.
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Submitted on : Sunday, January 25, 2015 - 11:27:49 PM
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  • HAL Id : hal-01109276, version 1


Vlad Bally, Lucia Caramellino. Convergence and regularity of probability laws by using an interpolation method. Annals of Probability, Institute of Mathematical Statistics, 2017, 45 (2). ⟨hal-01109276v1⟩



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