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 Itô diffusions, using as our main tool a refinement of Strassen’s result due to Lerche (1986).
"Large deviations related to the law of the iterated logarithm for Itô diffusions." Electron. Commun. Probab. 25 1 - 11, 2020. https://doi.org/10.1214/20-ECP297