Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes
Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes
Abstract We prove a new Burkholder–Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if $$(S(t,s))_{0\leqslant s\le t\leqslant T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>⩽</mml:mo> <mml:mi>s</mml:mi> <mml:mo>≤</mml:mo> …