Article1971

Representation of Martingales, Quadratic Variation and Applications

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33 citations0/0 files storedSIAM Journal on Control

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Stochastic processes and financial applications Probability and Statistical Research Stochastic processes and statistical mechanics

Abstract

In this paper, we present two related results. First, we shall obtain a sufficient condition under which a second order sample-continuous martingale can be represented as a stochastic integral in terms of a Brownian motion. Secondly, we shall show that if X and Y are sample-continuous local martingales (not necessarily with respect to the same family of ($\sigma $-algebras) and if either $X + Y$ or $X - Y$ is almost surely of bounded variation, then the quadratic variations of the two martingales are equal. This rather simple result has some surprising consequences.

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References

1967On Square Integrable Martingales 1951On a Formula Concerning Stochastic Differentials 1962A decomposition theorem for supermartingales 1965Quasi-Martingales

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2020Zero-sum path-dependent stochastic differential games in weak formulation

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