An extended stochastic calculus for Poisson random measures. (English. Ukrainian original) Zbl 0923.60071

Kaminskij, A. B.

Theory Probab. Math. Stat. 55, 93-105 (1997); translation from Teor. Jmovirn. Mat. Stat. 55, 90-102 (1996).

The author introduces the notion of extended stochastic integral with respect to the Poisson random measure. This integral is a generalization of the Skorokhod extended stochastic integral. The author obtains the integration by parts formula for this integral. Moreover, the author finds solutions of some stochastic differential equations of the extended type with respect to the Poisson random measure. For example, under some assumptions the stochastic process \[ \begin{split} X(t) = \exp \biggl( \int_0^t \alpha(s) d\int_{R^d} y d\Pi(s,y)\biggl) G \diamondsuit\\ \diamondsuit\exp\biggl(\int_0^t \int_R \ln(1+\beta(s)y) d\nu(s,y) - \int_0^t \beta(s) d\int_R y\Pi(s,y)\biggl)\end{split} \] is a solution of the equation \[ X(t)= G + \int_0^t \int_R \alpha(s) yX(s) d\Pi(s,y) + \delta\bigl( {\mathbf 1}_{[0,t]}(s) \beta(s)yX(s)\bigl). \]

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05	Stochastic integrals

Keywords:

extended stochastic integral; Poisson random measure; chaotic calculus; Malliavin calculus; stochastic differential equation