An extension of the Girsanov theorem on the change of measures to the case of semi-martingales with jumps

L. I. Gal'čuk

Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 2, pp. 279-294

Résumé

Let $(\Omega,\mathscr F,\mathbf P)$ be a probability space with an increasing family of $\sigma$-algebras $(\mathscr F_t)$, $t\in R_+$, and let $X=(X_t)$ be a semi-martingale, that is $X_t=A_t+M_t$, $\forall t\in R_+$, where $A_t$ is a process with bounded variation and $M_t$ is a martingale. In the paper, under some conditions, a new measure $\widetilde{\mathbf P}(d\omega)=\zeta(\omega)\mathbf P(d\omega)$ is constructed such that, on the new probability space $(\Omega,\mathscr F,\widetilde{\mathbf P})$ with the same family of $\sigma$-algebras $(\mathscr F_t)$, the process $X$ is a process with independent increments.