Open Access
February, 1980 On a Stopped Doob's Inequality and General Stochastic Equations
M. Metivier, J. Pellaumail
Ann. Probab. 8(1): 96-114 (February, 1980). DOI: 10.1214/aop/1176994827


An upper bound for $E(\sup_{0\leqslant s\leqslant\tau}\|M_s\|^2)$, where $M$ is a square integrable martingale and $\tau$ a stopping time is given in terms of $\lbrack M\rbrack_{\tau^-}$ and $\langle M\rangle_{\tau^-}$. Counter examples show that $4E(\langle M\rangle_{\tau^-})$, which is easily derived as an upper bound from a classical Doob's inequality, when $\tau$ is predictable or totally unaccessible, is no longer an upper bound in general. The obtained majoration is used to prove existence and uniqueness of strong solutions of a stochastic equation $dX_t = a(t, X) dZ_t$, where $a$ is a functional, depending possibly on the whole past of $X$ before $t$, and $Z$ is a semimartingale. Our result thus extends to systems "with memory" recent results by Protter, Kazamaki, Doleans-Dade and Meyer.


Download Citation

M. Metivier. J. Pellaumail. "On a Stopped Doob's Inequality and General Stochastic Equations." Ann. Probab. 8 (1) 96 - 114, February, 1980.


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

zbMATH: 0426.60059
MathSciNet: MR556417
Digital Object Identifier: 10.1214/aop/1176994827

Primary: 60H20
Secondary: 60G45

Keywords: inequalities for martingales , Martingales , Semimartingales , stochastic equations

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 1 • February, 1980
Back to Top