## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 39, Number 2 (1968), 372-376.

### Transforms of Stochastic Processes

#### Abstract

In this note, the notion of an optimal transform of a (discrete parameter) stochastic process is introduced. Such transforms are shown to exist in certain cases, and a relationship to optimal stopping times is discussed. These ideas lead naturally to the representation of any given stochastic process as the transform of a submartingale. This type of representation theorem is extended to continuous parameter processes, where it is shown that in certain cases a quasi-martingale can be represented as a stochastic integral with respect to a submartingale.

#### Article information

**Source**

Ann. Math. Statist., Volume 39, Number 2 (1968), 372-376.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177698398

**Digital Object Identifier**

doi:10.1214/aoms/1177698398

**Mathematical Reviews number (MathSciNet)**

MR225372

**Zentralblatt MATH identifier**

0155.23604

**JSTOR**

links.jstor.org

#### Citation

Millar, P. Warwick. Transforms of Stochastic Processes. Ann. Math. Statist. 39 (1968), no. 2, 372--376. doi:10.1214/aoms/1177698398. https://projecteuclid.org/euclid.aoms/1177698398