On Stochastic Integrals up to Infinity and Predictable Criteria for Integrability

Abstract

The first goal of this paper is to give an adequate definition of the stochastic integral

$$\int_0^\infty {{H_s}} {\text{d}}{X_s},(*)$$

where $H = (H_t)_{t\ge0}$ is a predictable process and $X = (X_t)_{t\ge0}$ is a semimartingale. We consider two different definitions of (*): as a stochastic integral up to infinity and as an improper stochastic integral.

The second goal of the paper is to give necessary and sufficient conditions for the existence of the stochastic integral

$$\int_0^t {{H_s}d{X_s}} , t \geqslant 0$$

and for the existence of the stochastic integral up to infinity (*). These conditions are expressed in predictable terms, i.e. in terms of the predictable characteristics of X.

Moreover, we recall the notion of a semimartingale up to infinity (martingale up to infinity, etc.) and show its connection with the existence of the stochastic integral up to infinity. We also introduce the notion of $\gamma$-localization.