On the martingale property for generalized stochastic processes

Abstract

In the recent years, several groups have studied stochastic equations (e.g. SDE's, SPDE's, stochastic Volterra equations) outside the framework of the Ito calculus. Often, this led to solutions in spaces of generalized random processes or fields. It is therefore of interest to study the probabilistic properties of generalized stochastic processes, and the present paper makes some (rather naive) first steps into this direction. If we think of a generalized process as a mapping from the real line (or an interval) into aspace of generalized random variables (with some additional properties), then there is a wide range of choices for the latter: e.g., the space (S)* of Hida distributions (e.g. [HKPS]), the space (S)-1 of Kondratiev distributions (e.g. [KLS]), the Sobolev type space D* which is used within the Malliavin calculus and so on. Often, the generalized processes coming up in applications have a chaos decomposition with kernels which belong to L2(IRn), and in this paper we shall focus our attention on this situation. It will be convenient to work with a space G* which is larger than D*. This space has already been considered by several authors, cf. e.g. [PT] and the references quoted there. It turns out, that basic notions from the theory of stochastic processes like conditional expectation, martingales, sub- (super-) martingales etc. have a rather natural generalization to mappings from the realline into G* . The paper is organized as follows. In Section 2 we shall give a construction of the Ito integral (with respect to Brownian motion) of generalized stochastic processes, and compare it with the Hitsuda-Skorokhod integral (e.g. [HKPS]). In Section 3 we define generalized martingales and derive a number of properties. In particular, we prove that the generalized Ito integrals of Section 2 are indeed generalized martingales. Moreover, a representation of generalized martingales (in analogy with the Clark-Haussmann formula) will be shown, and we prove that the Wick product of two generalized martingales is again one. Finally, in Section 4 we define the notion of a generalized semimartingale and give a class of examples. In the remainder of this Introduction we provide the necessary background.