The local principle of large deviations for solutions of Itô stochastic equations with quick drift

Abstract

The solution of the stochastic equation X(t) = x₀ + b∫₀^tsign(X(s))|X(s)|^γds + w(t); where w(t) is the Wiener process, the constant b ≠ 0, and γ ∈ (0; 1]; is considered. The local principle of large deviations for the sequence of processes \( {X}_n(t)=\frac{X(nt)}{n^{\alpha }},\alpha >1/2 \), is proved. The form of the rate function is found.