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

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

where u>e, W={(W ¹ _t ,…,W ^k _t ),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 ^u₀ is a random vector not necessarily adapted and the first integral is a generalized Stratonovich integral.