## The Annals of Probability

### On the Existence and Path Properties of Stochastic Integrals

Olav Kallenberg

#### Abstract

We study stochastic integrals of the form $Y(t) = \int^t_0 V d X, t \geqq 0$, where $X$ is a process with stationary independent increments while $V$ is an adapted previsible process, thus continuing the work of Ito and Millar. In the case of vanishing Brownian component, we obtain conditions for existence which are considerably weaker than the classical requirement that $V^2$ be a.s. integrable. We also examine the asymptotic behavior of $Y(t)$ for large and small $t$, and we consider the variation with respect to suitable functions $f$. The latter leads us to investigate nonlinear integrals of the form $\int f(V dX)$. The whole work is based on extensions of two general martingale-type inequalities, due to Esseen and von Bahr and to Dubins and Savage respectively, and on a super-martingale which was discovered and explored in a special case by Dubins and Freedman.

#### Article information

Source
Ann. Probab. Volume 3, Number 2 (1975), 262-280.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996397

Digital Object Identifier
doi:10.1214/aop/1176996397

Mathematical Reviews number (MathSciNet)
MR400391

Zentralblatt MATH identifier
0307.60050

JSTOR