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Calcul stochastique via régularisation en dimension infinie avec perspectives financières

Abstract : This thesis develops some aspects of stochastic calculus via regularization to Banach valued processes. An original concept of Chi-quadratic variation is introduced, where Chi is a subspace of the dual of a tensor product B⊗B where B is the values space of some process X process. Particular interest is devoted to the case when B is the space of real continuous functions defined on [-τ,0], τ>0. Itô formulae and stability of finite Chi-quadratic variation processes are established. Attention is deserved to a finite real quadratic variation (for instance Dirichlet, weak Dirichlet) process X. The C([-τ,0])-valued process X(•) defined by X_t(y) = X_{t+y}, where y ∈ [-τ,0], is called window process. Let T >0. If X is a finite quadratic variation process such that [X]_t = t and h = H(X_T(•)) where H:C([-T,0])→R is L^{2}([-T,0])-smooth or H non smooth but finitely based it is possible to represent h as a sum of a real H_0 plus a forward integral of type \int_0^T \xi d^-X where H_0 and \xi are explicitly given. This representation result will be strictly linked with a function u:[0,T]x C([-T,0])→R which in general solves an infinite dimensional partial differential equation with the property H_{0}=u(0, X_{0}(•)), \xi_t=Du(t, X_{t}(•))({0}). This decomposition generalizes important aspects of Clark-Ocone formula which is true when X is the standard Brownian motion W. The financial perspective of this work is related to hedging theory of path dependent options without semimartingales.
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https://pastel.archives-ouvertes.fr/tel-00578521
Contributor : Cristina Di Girolami Connect in order to contact the contributor
Submitted on : Monday, March 21, 2011 - 11:52:58 AM
Last modification on : Wednesday, May 11, 2022 - 12:06:05 PM
Long-term archiving on: : Thursday, November 8, 2012 - 12:15:58 PM

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Cristina Di Girolami. Calcul stochastique via régularisation en dimension infinie avec perspectives financières. Mathematics [math]. Université Paris-Nord - Paris XIII, 2010. English. ⟨tel-00578521⟩

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