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Calcul d'Itô étendu

Abstract : Our main results are extensions of the classical stochastic calculus. For a Markov process (X_t) , the problem is to give the explicit decomposition as a Dirichlet process of F (Xt , t) under minimal conditions on F , real-valued deterministic function. We consider the four following cases. In the first case X is a real-valued Lévy process with a Brownian component. In the second case, X is a symmetric Lévy process without Brownian component, but admitting a local time process as a Markov process. In the third case, X is a general symmetric Markov process without condition of existence of local times, but F (x, t) does not depend on t. In the fourth case, we suppress the assumption of symmetry of the third case. In each of the first three cases, we obtain an Itô formula under the only condition that the function F admits locally bounded first order Radon-Nicodym derivatives. Note that under the assumption that X is a general semimartingale, the classical Itˆ formula requires C^2 functions. This is what we have to assume in the fourth case. First case excepted, each of the obtained Itˆ formulas requires the construction of a new stochastic integral with respect to random processes which are not semimartingales.
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Contributor : Alexander Walsh <>
Submitted on : Thursday, September 29, 2011 - 9:06:11 AM
Last modification on : Friday, May 29, 2020 - 3:58:36 PM
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  • HAL Id : tel-00627558, version 1


Alexander Walsh. Calcul d'Itô étendu. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2011. English. ⟨tel-00627558⟩



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