Malliavin calculus for processes with jumps. (English) Zbl 0706.60057

Stochastics Monographs, 2. New York etc.: Gordon and Breach Science Publishers. ix, 161 p. $ 36.00 (1987).
The monograph is devoted to an extension of the Malliavin calculus to the class of solutions of stochastic differential equations with jumps. Malliavin and Bismut approaches are exposed. Existence and smoothness of densities are established via both approaches under nondegeneracy conditions which include diffusion coefficients and measures of jumps. Hörmander type conditions are not considered. The smoothness in initial data is investigated for SDE’s with jumps as a necessary auxiliary step.
The monograph is clearly written and has a rather small volume. It will be useful for specialists in random processes theory and for postgraduate studies.
Reviewer: A.Yu.Veretennikov


60H07 Stochastic calculus of variations and the Malliavin calculus
60G30 Continuity and singularity of induced measures
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J75 Jump processes (MSC2010)