An Extension of Kazamaki's Results on BMO Differentials

Abstract

Kazamaki has shown that if $(M^n)_{n\geqq 1}, M$ are BMO martingales with continuous paths and $\lim M^n = M$ in BMO, then $\mathscr{E}(M^n)$ converges in $\mathscr{H}^1$ to $\mathscr{E}(M)$, where $\mathscr{E}(M)$ denotes the stochastic exponential of $M$. While Kazamaki's result does not extend to the right continuous case, it does extend "locally." It is shown here that if $M^n, M$ are semimartingales and $M^n$ converges locally in $\mathscr{H}^\omega$ (a semimartingale BMO-type norm) to $M$ then $X^n$ converges locally in $\mathscr{H}^p (1 \leqq p < \infty)$ to $X$, where $X^n, X$ are respectively solutions of stochastic integral equations with Lipschitz-type coefficients and differentials $dM^n, dM$. (The coefficients are also allowed to vary.) This is a stronger stability than usually holds for solutions of stochastic integral equations, reflecting the strength of the $\mathscr{H}^\omega$ norm.