## On the maximum of a diffusion process in a drifted Brownian environment.(English)Zbl 0791.60071

Azéma, J. (ed.) et al., Séminaire de probabilités XXVII. Berlin: Springer-Verlag. Lect. Notes Math. 1557, 78-85 (1993).
The authors investigate the problem: how fast does $$P\{\max_{t \geq 0} X(t) \geq x\}$$ decay as $$x \to \infty$$, where $$X(t)$$ is a diffusion process with generator $${1 \over 2} \exp \{W(x)+cx\} {d \over dx} (\exp \{-W(x)-cx\} {d \over dx})$$. It turns out that the answer depends on $$c$$ and varies according as $$c>1$$, $$c=1$$, $$0<c<1$$. This problem is a diffusion analog of V. T. Afanas’ev’s problem [Theory Probab. Appl. 35, No. 2, 205-215 (1990); translation from Teor. Veroyatn. Primen. 35, No. 2, 209-219 (1990; Zbl 0714.60054)].
For the entire collection see [Zbl 0780.00013].

### MSC:

 60J65 Brownian motion 60J60 Diffusion processes

### Citations:

Zbl 0725.60074; Zbl 0714.60054
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