Short Communications

Stable subspaces and a theorem on a decomposition of martingales

L. I. Gal'čuk

Moscow

Abstract: Let $m=(m_t)$, $t\in R_+$, be an $n$-dimensional continuous local martingale, $\mu(\omega,dt,dx)$ be an integervalued random measure on a $R_+\times E$ and $\nu(\omega,dt,dx)$ be its dual predictable projection. We prove that every martingale $X\in H^q$, $q\in[1,\infty[$, possesses a unique decomposition of the form
$$ X_t-X_0=\int_0^tf(s)\,dm_s+\int_0^t\int_Eg(s,x)(\mu-\nu)(ds,dx)+\int_0^t\int_Eh(s,x)\mu(ds,dx)+X_t'. $$
All additive terms of the rigth hand side belong to the space $H^q$ and the process $X'$ is orthogonal to $m$ and hasn't jumps on the support of $\mu$.

Received: 23.12.1977

 English version: Theory of Probability and its Applications, 1981, 25:2, 366–370

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