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Asymptotic results for exit probabilities of stochastic processes governed by an integral type rate function

Abundo M., Macci C., Stabile G.

Probability and Mathematical Statistics

2012

Vol. 32, Fasc. 1

25--39

In this paper we present asymptotic results for exit probabilities of stochastic processes in the fashion of large deviations. The main result concerns stochastic processes which satisfy the large deviation principle with an integral type rate function. We also present results for exit probabilities of linear diffusions and particular growth processes, and we give two examples.

Some remarks on the maximum of a one-dimensional diffusion process

Abundo M.

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 1

107--120

For a certain class of one-dimensional diffusions X(t); we study the distribution of maxtЄ[0;T] X(t) and the distribution of the first instant at whichX(t) attains the maximum by reducingX(t) to Brownian motion. Moreover, for T fixed or random, we study the asymptotics of threshold crossing probability, i.e. the rate of decay of P(maxsЄ[0;T ] X(s) > z ) as z goes to infinity. Some examples are also reported.

On first-passage times for one-dimensional jump-diffusion processes

Abundo M.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 2

399--423

Some problems of first-crossing times over two time-dependent boundaries for one-dimensional jump-diffusion processes are considered. The moments of the first-crossing times over each boundary are shown to be the solutions of certain partial differential-difference equations with suitable outer conditions. An approach based on the Laplace transform allows us to compare the moments of the first-crossing times of the jump-diffusion process with those of the corresponding simple-diffusion without jumps. For some examples where the boundaries are constant, the results are illustrated graphically.