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December, 1980 An Extension of Kazamaki's Results on BMO Differentials
Philip Protter
Ann. Probab. 8(6): 1107-1118 (December, 1980). DOI: 10.1214/aop/1176994572


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.


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Philip Protter. "An Extension of Kazamaki's Results on BMO Differentials." Ann. Probab. 8 (6) 1107 - 1118, December, 1980.


Published: December, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0446.60042
MathSciNet: MR602384
Digital Object Identifier: 10.1214/aop/1176994572

Primary: 60H10
Secondary: 60G45 , 60H20

Keywords: BMO , Martingales , Semimartingales , Stochastic differential equations

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 6 • December, 1980
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