## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60H05: Stochastic integrals

Secondary: 60G17: Sample path properties 60G45 60J30

**Keywords**

Stochastic integrals existence and path properties processes with stationary independent increments martingales

#### Citation

Kallenberg, Olav. On the Existence and Path Properties of Stochastic Integrals. Ann. Probab. 3 (1975), no. 2, 262--280. doi:10.1214/aop/1176996397. https://projecteuclid.org/euclid.aop/1176996397