Iterated logarithm law for anticipating stochastic differential equations

Abstract

We prove a functional law of iterated logarithm for the following kind of anticipating stochastic differential equations $${\xi }_t^u=X_0^u+\frac{1}{\sqrt{\log \log u}}\sum _{j=1}^k{\int }_0^t{A}_j^u({\xi }_s^u)\circ d{W}_s^j+{\int }_0^t{A}_0^u({\xi }_s^u)ds,$$ where $u>e$, $W=\{(W_t^1,...,W_t^k),0\le t\le 1\}$ is a standard $k$-dimensional Wiener process, $A_0^u,A_1^u,...,A_k^u:{\mathbb{R}}^d⟶{\mathbb{R}}^d$ 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 .