# Large deviations related to the law of the iterated logarithm for Ito diffusions

## TL;DR

This paper establishes a small-time large deviations principle for the supremum of scaled Ito diffusions, extending classical results on Brownian motion to more general stochastic processes using advanced hitting time analysis.

## Contribution

It extends large deviations results related to the law of the iterated logarithm from Brownian motion to general Ito diffusions, utilizing refined hitting time density techniques.

## Key findings

- Derived a small-time large deviations principle for Ito diffusions
- Extended Strassen's hitting time results to diffusion processes
- Provided quantitative estimates for supremum deviations

## Abstract

When a Brownian motion is scaled according to the law of the iterated logarithm, its supremum converges to one as time tends to zero. Upper large deviations of the supremum process can be quantified by writing the problem in terms of hitting times and applying a result of Strassen (1967) on hitting time densities. We extend this to a small-time large deviations principle for the supremum of scaled Ito diffusions, using as our main tool a refinement of Strassen's result due to Lerche (1986).