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I cannot find a proper definition of multivariate martingale. If each component is $1$-dimensional martingale is it enough for a $d$-dimensional process to be a martingale?


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up vote 1 down vote accepted

You mean the parameter set is still one-dimensional, and the values are in a vector space like $\mathbb R^n$? Then yes, the definition can be that each component is individually a martingale. Or abstractly that when we apply any linear functional, the result is a scalar-valued martingale.

"Multivariate martingale" might also mean a situation where "time" is replaced by a multi-dimensional parameter set.

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Yes, I was asking about one-dimensional time. Thanks. Just out of curiosity where I read about multi-dimensional parameter set? – learningmath Nov 25 '11 at 20:22
The theory of martingales indexed by $\mathbb R \times \mathbb R$ was initiated in a memoir of Cairoli & Walsh (1975). A 1981 book of Korezliogly, Mazziotto & Szpirglas is a collection of articles on this topic. There is a general notion "block martingale" due to Frangos & Sucheston (1985), and work of Sucheston & Szabo (1991). The last chapter of Edgar & Sucheston has lots of this. – GEdgar Nov 25 '11 at 22:32

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