Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes

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)) _⩽_s_≤_t_⩽_T is a C-evolution family of contractions on a 2-smooth Banach space X and (Wt)t∈[0,T] is a...

Maximal inequalities for stochastic convolutions in 2-smooth Banach spaces and applications to stochastic evolution equations

This paper presents a survey of maximal inequalities for stochastic convolutions in 2-smooth Banach spaces and their applications to stochastic evolution equations. This article is part of the theme issue 'Semigroup applications everywhere'.

On temporal regularity of stochastic convolutions in 2-smooth Banach spaces

We show that paths of solutions to parabolic stochastic differential equations have the same regularity in time as the Wiener process (as of the current state of art). The temporal regularity is considered in the Besov-Orlicz space Bϕ ^1/₂²_∞(0,T;X)where ϕ₂(x)= exp(x²)− 1 and X...

Cylindrical continuous martingales and stochastic integration in infinite dimensions

In this paper we define a new type of quadratic variation for cylindrical continuous local martingales on an infinite dimensional spaces. It is shown that a large class of cylindrical continuous local martingales has such a quadratic variation. For this new class of cylindrical continuous local martingales we develop a stochastic integration...

Maximal ? -regularity

In this paper, we prove maximal regularity estimates in “square function spaces” which are commonly used in harmonic analysis, spectral theory, and stochastic analysis. In particular, they lead to a new class of maximal regularity results for both deterministic and stochastic equations in L p -spaces with 1<p<?. For stochastic equations, the...

A note on maximal estimates for stochastic convolutions

In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces.

Stochastic evolution equations in UMD Banach spaces