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Iterated logarithm law for anticipating stochastic differential equations
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Abstract
We prove a functional law of iterated logarithm for the following kind of anticipating stochastic differential equations
$$
\xi_t^u=X_0^u+\frac{1}{\sqrt{\log \log u}} \sum_{j=1}^k \int_0^t A_j^u\left(\xi_s^u\right) \circ d W_s^j+\int_0^t A_0^u\left(\xi_s^u\right) d s
$$
where $u>e, W=\left\{\left(W_t^1, \ldots, W_t^k\right), 0 \leq t \leq 1\right\}$ is a standard $k$ dimensional Wiener process, $A_0^u, A_1^u, \ldots, A_k^u: \mathbb{R}^d \longrightarrow \mathbb{R}^d$ 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 .
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MÁRQUEZ, David (Márquez Carreras) and ROVIRA ESCOFET, Carles. Iterated logarithm law for anticipating stochastic differential equations. Journal of Theoretical Probability. 2007. Vol. 21, num. 3, pags. 650-659. ISSN 0894-9840. [consulted: 19 of August of 2026]. Available at: https://hdl.handle.net/2445/216550