Representation of Martingales, Quadratic Variation and Applications | Zendy

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Representation of Martingales, Quadratic Variation and Applications

Author(s) -

Eugene Wong

Publication year - 1971

Publication title -

siam journal on control

Language(s) - English

Resource type - Journals

eISSN - 2469-4231

pISSN - 0036-1402

DOI - 10.1137/0309044

Subject(s) - quadratic variation , mathematics , martingale (probability theory) , brownian motion , bounded variation , quadratic equation , local martingale , pure mathematics , bounded function , sigma , mathematical analysis , statistics , geometry , physics , quantum mechanics

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 ifX and Y are sample-continuous local martingales (not necessarily with respect to the same family of a-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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