On Stochastic Derivative.

Abstract

The stochastic integral representation for an arbitrary random variable in a standard $L_2$-space is considered in a case of a general $L_2$-continuous martingale as integrator. In relation to this, a certain stochastic derivative is defined. Through this derivative it can be seen whether the random variable admits the above type integral representation. In any case, it is shown that this derivative determines the integrand in the stochastic integral which serves as the best $L_2$-approximation to the random variable considered. For a general Levy process as integrator some specification of the suggested stochastic derivative is given; in this way, for Wiener process, the known Clark-Ocone formula is derived.

Key-words: non-anticipating integration, stochastic derivative, integral representation, Levy processes, Clark-Ocone formula.