Malliavin Calculus for Stochastic Processes and Random Measures with Independent Increments

Abstract

Malliavin calculus for Poisson processes based on the difference operator or add-one-cost operator is extended to stochastic processes and random measures with independent increments. Our approach is to use a Wiener–Ito chaos expansion, valid for both stochastic processes and random measures with independent increments, to construct a Malliavin derivative and a Skorohod integral. Useful derivation rules for smooth functionals given by Geiss and Laukkarinen (Probab Math Stat 31:1–15, 2011) are proved. In addition, characterizations for processes or random measures with independent increments based on the duality between the Malliavin derivative and the Skorohod integral following an interesting point of view from Murr (Stoch Process Appl 123:1729–1749, 2013) are studied.