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Calcul des variations sur un Brownien subordonné. (Calculus of variations on a subordinated Brownian). (French) Zbl 0663.60062
Séminaire de probabilités XXII, Strasbourg/France, Lect. Notes Math. 1321, 414-433 (1988).
[For the entire collection see Zbl 0635.00013.]
The main ideas in the article are concerned with methods of solving stochastic differential equations for pure jump processes, by means of solving equations involving Brownian motion. Namely, let \(A_ t\) be the right-continuous inverse of the local time \(L_ t\) corresponding to the Brownian motion. It is shown that \(x_{A_ t}\), which is a pure jump process, arises as a solution to a stochastic differential equation for a pure jump process and, more importantly, this solution can be recovered from solving another equation \[ dx_ t=Ito\quad differential (x_{\cdot},L_ t,\quad Brownian\quad motion) \] by taking \(t=A_ t\). The second part of the paper deals with Malliavin calculus. It is shown, under some assumptions, that if \(\phi \in H^{\infty}\), then \(\phi\) has a density \(f\in C^{\infty}\), and the upper bound for \[ \sup_{x\in R}| (d^ r/dx^ r)f(x)|,\quad r=1,2,... \] is derived. As an application, a sufficient condition for the existence of \(C^{\infty}\) densities for a certain class of processes is given.
Reviewer: A.Korzeniowski

60J60 Diffusion processes
58J65 Diffusion processes and stochastic analysis on manifolds
60J75 Jump processes (MSC2010)
60H05 Stochastic integrals
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