Construction of Derivation Operators for Spatial Poisson Processes and Stochastic Integral Property

This project aims at the construction of a derivation operator connected to stochastic integration in the framework of the Malliavin calculus for spatial point processes. The Malliavin calculus has brought important advances to the fields of analysis, geometry, stochastic calculus, and mathematical finance, and is based on a stochastic gradient whose adjoint is an extension of the stochastic integral.Although this calculus has been developed in various directions for continuous and jump processes, in the jump case the stochastic integral property is usually connected to finite difference operators which are not practicable for most computations. Derivation operators are also available for point processes, however they are properly connected to stochastic integration only in the case of the standard Poisson process on the half line, thus restricting their applicability. Indeed, the derivation property is of crucial importance for explicit computation in many fields of application.The question whether a derivation operator with the required properties can be constructed for spatial point processes has been the object of a few attempts, however it is still open and this project aims at solving it. More precisely we aim at constructing a derivation operator for spatial Poisson processes, to connect the adjoint of this operator to the Poisson stochastic integral, and to study the relevant applications.