  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).
