Sonja Cox

Bio: Sonja Cox is an academic researcher from University of Amsterdam. The author has contributed to research in topics: Banach space & Stochastic partial differential equation. The author has an hindex of 10, co-authored 29 publications receiving 479 citations. Previous affiliations of Sonja Cox include Delft University of Technology & ETH Zurich.

Stochastic integration in quasi-Banach spaces

Abstract: In this paper we develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent $p\in (0,1]$. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic partial differential equations.

Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations

Abstract: Recently, Hairer et. al [14] showed that there exist SDEs with infinitely often differentiable and globally bounded coefficient functions whose solutions fail to be locally Lipschitz continuous in the strong L p -sense with respect to the initial value for every p ∈ [1,∞]. In this article we provide sufficient conditions on the coefficien t functions of the SDE and on p ∈ (0,∞] which ensure local Lipschitz continuity in the strong L p -sense with respect to the initial value and we establish explicit estimates for the local Lipschitz continuity constants. In particular, we prove local Lipschitz continuity in the initial value for several nonlinear SDEs from the literature such as the stochastic van der Pol oscillator, Brownian dynamics, the Cox-Ingersoll-Ross processes and the Cahn-Hilliard-Cook equation. As an application of our estimates, we obtain strong completeness for several nonlinear SDEs.

Vector-valued decoupling and the Burkholder–Davis–Gundy inequality

Abstract: Let X be a (quasi-)Banach space. Let d = (dn)n 1 be an X-valued sequence of random variables adapted to a ltration ( Fn)n 1 on a probability space ( ;A;P), dene F1 := (Fn : n 1) and let e = (en)n 1 be aF1-conditionally independent sequence on ( ;A;P) such that L(dnjFn 1) =L(enjF1) for all n 1 (F0 =f ;?g). If there exists a p2 (0;1) and a constant Dp independent of d and e such that one has, for all n 1, X k=1 dk p D pE n X k=1 ek p

Pathwise Hölder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise

Abstract: In this article we prove pathwise Holder convergence with optimal rates of the implicit Euler scheme for the abstract stochastic Cauchy problem $$\begin{aligned} \left\{ \begin{aligned} dU(t)&= AU(t)\,dt + F(t,U(t))\,dt + G(t,U(t))\,dW_H(t);\quad t\in [0,T],\\ U(0)&=x_0. \end{aligned}\right. \end{aligned}$$ Here $$A$$ is the generator of an analytic $$C_0$$ -semigroup on a umd Banach space $$X,\,W_H$$ is a cylindrical Brownian motion in a Hilbert space $$H$$ , and the functions $$F:[0,T]\times X\rightarrow X_{\theta _F}$$ and $$G:[0,T]\times X\rightarrow {\fancyscript{L}}(H,X_{\theta _G})$$ satisfy appropriate (local) Lipschitz conditions. The results are applied to a class of second order parabolic SPDEs driven by multiplicative space-time white noise.

Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions

Abstract: We show that if a sequence of piecewise affine linear processes converges in the strong sense with a positive rate to a stochastic process that is strongly Holder continuous in time, then this sequence converges in the strong sense even with respect to much stronger Holder norms and the convergence rate is essentially reduced by the Holder exponent. Our first application hereof establishes pathwise convergence rates for spectral Galerkin approximations of stochastic partial differential equations. Our second application derives strong convergence rates of multilevel Monte Carlo approximations of expectations of Banach-space-valued stochastic processes.

An Introduction to Computational Stochastic PDEs

Abstract: Part I. Deterministic Differential Equations: 1. Linear analysis 2. Galerkin approximation and finite elements 3. Time-dependent differential equations Part II. Stochastic Processes and Random Fields: 4. Probability theory 5. Stochastic processes 6. Stationary Gaussian processes 7. Random fields Part III. Stochastic Differential Equations: 8. Stochastic ordinary differential equations (SODEs) 9. Elliptic PDEs with random data 10. Semilinear stochastic PDEs.