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.

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

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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