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April, 1975 On the Existence and Path Properties of Stochastic Integrals
Olav Kallenberg
Ann. Probab. 3(2): 262-280 (April, 1975). DOI: 10.1214/aop/1176996397


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.


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Olav Kallenberg. "On the Existence and Path Properties of Stochastic Integrals." Ann. Probab. 3 (2) 262 - 280, April, 1975.


Published: April, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0307.60050
MathSciNet: MR400391
Digital Object Identifier: 10.1214/aop/1176996397

Primary: 60H05
Secondary: 60G17 , 60G45 , 60J30

Keywords: existence and path properties , Martingales , processes with stationary independent increments , stochastic integrals

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 2 • April, 1975
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