Optimization of small deviation for mixed fractional Brownian motion with trend

TL;DR

This paper derives bounds for the small deviation probabilities of mixed fractional Brownian motion with trends, optimizing the trend split to improve probability estimates, and analyzes asymptotic behaviors.

Contribution

It introduces a method to optimize the trend split in mixed fractional Brownian motion for better small deviation probability bounds, including solving a Fredholm integral equation.

Findings

Optimal trend split maximizes the lower bound of probability.

Upper bounds depend on the same optimal split.

Asymptotic behavior analyzed for zero trend and specific upper limits.

Abstract

In this paper, we investigate two-sided bounds for the small ball probability of a mixed fractional Brownian motion with a general deterministic trend function, in terms of respective small ball probability of a mixed fractional Brownian motion without trend. To maximize the lower bound, we consider various ways to split the trend function between the components of the mixed fractional Brownian motion for the application of Girsanov theorem, and we show that the optimal split is the solution of a Fredholm integral equation. We find that the upper bound for the probability is also a function of this optimal split. The asymptotic behaviour of the probability as the ball becomes small is analyzed for zero trend function and for the particular choice of the upper limiting function.