Iterated logarithm law for anticipating stochastic differential equations

TL;DR

This paper establishes a law of iterated logarithm for a class of anticipating stochastic differential equations involving generalized Stratonovich integrals, extending classical results to non-adapted initial conditions.

Contribution

It introduces a functional law of iterated logarithm for anticipating SDEs with non-adapted initial data and generalized Stratonovich integrals, a novel extension of classical LIL results.

Findings

Proves a functional law of iterated logarithm for anticipating SDEs.

Extends classical LIL to equations with anticipating initial conditions.

Handles generalized Stratonovich integrals with bounded derivatives.

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 $u>e$, $W=\{(W_t^1,...,W_t^k), 0\le t\le 1\}$ is a standard $k$-dimensional Wiener process, $A_0^u,A_1^u,..., A_k^u:\mathbb{R}^d\longrightarrow \mathbb{R}^d$ 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 .