Conditional tail probabilities in continuous-time martingale LLN with application to parameter estimation in diffusions

Let M be a continuous martingale,h:+-->+ continuous and increasing such that M(t)/h(F t --> 0 (a.s.) as t --> [infinity]. It is shown that w.p.l, large deviations type limits exist for a class of conditional probabilities which are induced on (C([0, [infinity]),||·[infinity]) by the tail processes yt(·) = M(t + ·)/h( t+.). This is obtained via a simple use of the Borell inequality for Gaussian processes, combined with a random time change argument. Results are applied to obtain convergence rates for the (conditional) tail probabilities of consistent parameter estimators in diffusion processes. This is followed by the derivation of efficient stopping rules. Finally, unconditional large deviations lower bounds for the tails of consistent estimators in diffusions are investigated via an extension of a well known direct method.