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Iterated Logarithm Law for Anticipating Stochastic Differential Equations

Authors:

David Márquez-Carreras Carles Rovira

Institution:

(1) Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

Abstract:

We prove a functional law of iterated logarithm for the following kind of anticipating stochastic differential equations

$\xi^{u}_{t}=X_{0}^{u}+\frac{1}{\sqrt{\log\log u}}\sum_{j=1}^{k}\int_{0}^{t}A_{j}^{u}(\xi^{u}_{s})\circ dW_{s}^{j}+\int_{0}^{t}A_{0}^{u}(\xi^{u}_{s})ds,$

where u>e, W={(W _t¹,…,W _t^k),0≤t≤1} is a standard k-dimensional Wiener process, $A_{0}^{u},A_{1}^{u},\dots,A_{k}^{u}:\mathbb{R}^{d}\longrightarrow \mathbb{R}^{d}$ are functions of class $\mathcal{C}^{2}$ with bounded partial derivatives up to order 2, X ₀^u is a random vector not necessarily adapted and the first integral is a generalized Stratonovich integral. The work is partially supported by DGES grant BFM2003-01345.

Keywords:

Iterated logarithm law Stochastic differential equations Anticipative calculus

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