A representation of some martingales

L. I. Gal'čuk

Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 3, pp. 613-620

Résumé

Let $X_t=(X_t^1,\dots,X_t^n)$, $0\le t\le 1$, be a continuous square integrable martingale for which the processes $\langle X^i,X^j\rangle_t$, $i,j=1,\dots,n$, are deterministic, and let $(Y_t,\mathscr F_t^X)$ be a square integrable martingale where $\mathscr F_t^X=\sigma\{X_s,s\le t\}$. In the paper, the representation $\displaystyle Y_t=Y_0+\int_0^t\sum_{i=1}^nf_{s-}^i\,dX_s^i$ is proved where $f_s^i$ are previsible processes with $\displaystyle\mathbf M\int_0^{\infty}\sum_{i,j=1}^nf_s^if_s^jd\langle X^i,X^j\rangle_s<\infty.$