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 $(S(t,s){)}_{0\le s\le T}$ is a $C_0$-evolution family of contractions on a $2$-smooth Banach space $X$ and $(W_t{)}_{t\in [0,T]}$ is a cylindrical Brownian motion on a probability space $(\Omega ,P)$, then for every $0<p<\infty$ there exists a constant $C_{p,X}$ such that for all progressively measurable processes $g:[0,T]\times \Omega \to X$ the process $({\int }_0^{t}S(t,s)g_{s}d{W}_s{)}_{t\in [0,T]}$ has a continuous modification and $$E\underset{t\in [0,T]}{\sup }\Vert {\int }_0^{t}S(t,s)g_{s}d{W}_s{\Vert }^p\le C_{p,X}^p\mathbb{E}({\int }_0^T\Vert g_t{\Vert }_{\gamma (H,X)}^{2}dt{)}^{p/2}.$$ Moreover, for $2\le p<\infty$ one may take $C_{p,X}=10D\sqrt{p},$ where $D$ is the constant in the definition of $2$-smoothness for $X$. Our result improves and unifies several existing maximal estimates and is even new in case $X$ is a Hilbert space. Similar results are obtained if the driving martingale $g_{t}d{W}_t$ is replaced by more general $X$-valued martingales $d{M}_t$. Moreover, our methods allow for random evolution systems, a setting which appears to be completely new as far as maximal inequalities are concerned. As a second application, for a large class of time discretisation schemes we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs $$d{u}_t=A(t)u_{t}dt+g_{t}d{W}_t,u_0=0,$$ Under spatial smoothness assumptions on the inhomogeneity $g$, contractivity is not needed and explicit decay rates are obtained. In the parabolic setting this sharpens several know estimates in the literature; beyond the parabolic setting this seems to provide the first systematic approach to pathwise uniform convergence to time discretisation schemes.

Dedication added. Online first in Stochastics and Partial Differential Equations: Analysis and Computations

References in corpus (4)

• Rate of Convergence of Space Time Approximations for stochastic evolution equations
• Maximal estimates for stochastic convolutions in 2-smooth Banach spaces and applications to stochastic evolution equations
• Strong martingale type and uniform smoothness
• Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence

Cited by in corpus (2)

• Maximal estimates for stochastic convolutions in 2-smooth Banach spaces and applications to stochastic evolution equations
• Temporal approximation of stochastic evolution equations with irregular nonlinearities