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Contents of PMS, Vol. 28, Fasc. 1,
pages 107 - 120
 

SOME REMARKS ON THE MAXIMUM OF A ONE-DIMENSIONAL DIFFUSION PROCESS

Mario Abundo

Abstract: For a certain class of one-dimensional diffusions $X(t),$ we study the distribution of $\max _ (t \in [0,T]) X(t)$ and the distribution of the first instant at which $X(t)$ attains the maximum by reducing $X(t)$ to Brownian motion. Moreover, for $T$ fixed or random, we study the asymptotics of threshold crossing probability, i.e. the rate of decay of $P\Big (\max _(s \in [0,T]) X(s) > z \Big )$ as $z$ goes to infinity. Some examples are also reported.

2000 AMS Mathematics Subject Classification: Primary: 60J60, 60J65; Secondary: 60H10.

Key words and phrases: Diffusion process, Brownian motion, first-crossing time, random time-change.

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