L^2,2 X-Holomorphic Processes

SYNOPTIC ABSTRACT

The class of processes called holomorphic by Cairoli and Walsh are extended to the class of L^2,2–bounded processes. In particular, the L^2,2 X-holomorphic processes are shown to be Bochner L^2,2–bounded and the integrals of a L^2,2 X-holomorphic process are shown to preserve its L^2,2 X-holomorphicity. Related to L^2,2 X-holomorphic processes are certain stochastic partial derivatives. The stochastic partial derivatives are generalized to L^2,2–bounded processes and the properties of path independence of the line integrals is shown to imply the existence of the derivatives. A Green's theorem is derived for stochastic partial derivatives with respect to L^2,2–bounded processes.

Keywords:

• L^2,2
• holomorphic
• stochastic