Uniform Decay and Equicontinuity for Normalized, Parameter Dependent, Ito Integrals

Dependent, Ito Integrals David Levanony, Adam Shwartz and Ofer Zeitouni Department of Electrical Engineering Technion Israel Institute of Technology Haifa 32000, Israel Abstract Let fMt( ); t 0g 2IRd be a collection of continuous, continuous-time martingales such that for all t > 0, the associated increasing processes satisfy < M( ) >t!1 as k k ! 1. We show that if < M( ) >t grows with k k su ciently fast, then Mt( )= < M( ) >t! 0 as k k ! 1 uniformly in t 2 [t0;1); t0 > 0. An equicontinuity property for normalized, parameter dependent stochastic integrals follows. These results serve in the study of the maximum likelihood estimation problem, over unbounded sets, for di usion processes. 18 December 1991