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Zbl 0561.60081
Bismut, Jean-Michel
The calculus of boundary processes. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 17, 507-622 (1984). ISSN 0012-9593

A diffusion process $(X\sb t,Z\sb t)$, $t\ge 0$, is considered taking values in the half space $R\sp d\times R\sp+$ with the reflexion and the drift on the boundary and the local time $L\sb t$, $t\ge 0$, on it. By means of probabilistic methods the properties of smoothness of the transition probability function of the strong Markov process $(A\sb t$, $X\sb{A\sb t})$, $t\ge 0$, are investigated, where $A\sb t$, $t\ge 0$, is the right-continuous inverse of $L\sb t$, $t\ge 0.$ \par The techniques of stochastic flows, the calculus of variations on diffusions and jump processes, the Ito's theory of excursions and the stochastic calculus of semimartingales are developed and adapted. The analogous problem in the case of a diffusion process $(X\sb t,Z\sb t)$, $t\ge 0$, taking values in $R\sp d\times R$ with the reflexion and the drift on the two-sided boundary $R\sp d\times \{0\}$ is also discussed.
[B.Grigelionis]

MSC 2000:: *60J60 Diffusion processes
60J50 Boundary theory (probability)
60J55 Additive functionals

Keywords: local time; smoothness of the transition probability function; Ito's theory of excursions; stochastic calculus of semimartingales

Cited in: Zbl 0637.60092 Zbl 0561.60082 Zbl 0566.60057

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