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Calcul stochastique via régularisation et applications financières

Abstract : In the first part of this thesis we apply stochastic calculus via regularization to model financial markets when the price of the risky asset is not a semimartingale. The lack of the semimartingale property is justified if arbitrage is possible among simple predictable strategies. That is the case if the investor is an insider or the class A of admissible strategies is restricted. We assume that prices only have finite quadratic variation. That assumption is verified if the risky asset price itself has to be admitted in the class of all admissible strategies. We provide examples of self-financing strategies and we introduce the notion of A-martingale process. A calculus with respect those processes is developed. We show that the no-arbitrage condition is recovered if the price process is an A-martingale. We face some problems such as viability, hedging and utility maximization.
The second part of the thesis is devoted to the study of a one-dimensional stochastic differential equation driven by a strong cubic variation process and a semimartingale. The implementation of our method leads us to improve some results about stochastic calculus via regularization. In particular an Ito-Wentzell type formula related to finite cubic variation processes is
established and the structure of weak Dirichlet processes is clarified when the underlying filtration is Brownian.
Our approach applies to prove existence and uniqueness when the driven process is Hölder continuous in space. Using a Ito formula for reversible semimartingale we prove existence of a solution when the equation is driven by a Brownian motion and the diffusion coefficient is only continuous
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Contributor : Rosanna Coviello <>
Submitted on : Thursday, January 11, 2007 - 2:34:09 PM
Last modification on : Tuesday, October 20, 2020 - 3:56:28 PM
Long-term archiving on: : Tuesday, April 6, 2010 - 9:22:50 PM



  • HAL Id : tel-00121525, version 1


Rosanna Coviello. Calcul stochastique via régularisation et applications financières. Mathématiques [math]. Université Paris-Nord - Paris XIII; Scuola Normale Superiore, 2006. Français. ⟨tel-00121525⟩



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