## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 1 (1981), 49-89.

### Stochastic Integration and $L^p$-Theory of Semimartingales

#### Abstract

If $X$ is a bounded left-continuous and piecewise constant process and if $Z$ is an arbitrary process, both adapted, then the stochastic integral $\int X dZ$ is defined as usual so as to conform with the sure case. In order to obtain a reasonable theory one needs to put a restriction on the integrator $Z$. A very modest one suffices; to wit, that $\int X_n dZ$ converge to zero in measure when the $X_n$ converge uniformly or decrease pointwise to zero. Daniell's method then furnishes a stochastic integration theory that yields the usual results, including Ito's formula, local time, martingale inequalities, and solutions to stochastic differential equations. Although a reasonable stochastic integrator $Z$ turns out to be a semimartingale, many of the arguments need no splitting and so save labor. The methods used yield algorithms for the pathwise computation of a large class of stochastic integrals and of solutions to stochastic differential equations.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 1 (1981), 49-89.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994509

**Digital Object Identifier**

doi:10.1214/aop/1176994509

**Mathematical Reviews number (MathSciNet)**

MR606798

**Zentralblatt MATH identifier**

0458.60057

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60H05: Stochastic integrals

**Keywords**

Stochastic integral stochastic differential equations stochastic $H^p$-theory

#### Citation

Bichteler, Klaus. Stochastic Integration and $L^p$-Theory of Semimartingales. Ann. Probab. 9 (1981), no. 1, 49--89. doi:10.1214/aop/1176994509. https://projecteuclid.org/euclid.aop/1176994509