Large deviations related to the law of the iterated logarithm for Itô diffusions. (English) Zbl 1434.60088

Summary: 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 V. Strassen [in: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability. Vol. II. Part 1. Berkeley, CA: University of California Press. 315–343 (1967; Zbl 0201.49903)] 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 H. R. Lerche [Boundary crossing of Brownian motion. Its relation to the law of the iterated logarithm and to sequential analysis. Berlin: Springer (1986; Zbl 0604.62075)].


60F10 Large deviations
60J60 Diffusion processes
60J65 Brownian motion


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