Electronic Journal of Probability

Concrete Representation of Martingales

Stephen Montgomery-Smith

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Let $(f_n)$ be a mean zero vector valued martingale sequence. Then there exist vector valued functions $(d_n)$ from $[0,1]^n$ such that $\int_0^1 d_n(x_1,\dots,x_n)\,dx_n = 0$ for almost all $x_1,\dots,x_{n-1}$, and such that the law of $(f_n)$ is the same as the law of $(\sum_{k=1}^n d_k(x_1,\dots,x_k))$. Similar results for tangent sequences and sequences satisfying condition (C.I.) are presented. We also present a weaker version of a result of McConnell that provides a Skorohod like representation for vector valued martingales.

Article information

Electron. J. Probab., Volume 3 (1998), paper no. 15, 15 pp.

Accepted: 2 December 1998
First available in Project Euclid: 29 January 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter
Secondary: 60H05: Stochastic integrals

martingale concrete representation tangent sequence condition (C.I.) UMD Skorohod representation

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Montgomery-Smith, Stephen. Concrete Representation of Martingales. Electron. J. Probab. 3 (1998), paper no. 15, 15 pp. doi:10.1214/EJP.v3-37. https://projecteuclid.org/euclid.ejp/1454101775

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