Related papers: Stochastic Differential Equations: A Wiener Chaos …
We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration. The theory provides a differential structure which describes the infinitesimal evolution of Wiener functionals at very small…
In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify these as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general…
Constructing numerical models of noisy partial differential equations is very delicate. Our long term aim is to use modern dynamical systems theory to derive discretisations of dissipative stochastic partial differential equations. As a…
The spectrum of the evolution Operator associated with a nonlinear stochastic flow with additive noise is evaluated by diagonalization in a polynomial basis. The method works for arbitrary noise strength. In the weak noise limit we…
Numerical simulation of stochastic differential equations over long time intervals poses significant computational challenges. In this paper, we propose a novel recursive polynomial chaos evolution method that achieves model reduction…
A parameter estimation problem is considered for a one-dimensional stochastic wave equation driven by additive space-time Gaussian white noise. The estimator is of spectral type and utilizes a finite number of the spatial Fourier…
We present a method, based on the Keldysh formalism, for deriving stochastic master equations that describe the non-Markovian dynamics of a quantum system coupled to a Gaussian environment. This approach yields a compact expression for the…
For a mixed stochastic differential equation containing both Wiener process and a H\"older continuous process with exponent $\gamma>1/2$, we prove a stochastic viability theorem. As a consequence, we get a result about positivity of…
In this article, we present a general methodology for stochastic control problems driven by the Brownian motion filtration including non-Markovian and non-semimartingale state processes controlled by mutually singular measures. The main…
We consider stochastic differential equations driven by Wiener processes. The vector fields are supposed to satisfy only local Lipschitz conditions. The Lipschitz constants of the drift vector field, valid on balls of radius $R$, are…
In uncertainty quantification, a stochastic modelling is often applied, where parameters are substituted by random variables. We investigate linear dynamical systems of ordinary differential equations with a quantity of interest as output.…
Applications of stochastic models often involve the evaluation of steady-state performance, which requires solving a set of balance equations. In most cases of interest, the number of equations is infinite or even uncountable. As a result,…
An approach to stochastic evolution equations based on a simple generalization of known embedding theorems is presented. It allows for the inclusion of problems which have nonlinear non monotone operators. This is used to discuss the…
Constructing discrete models of stochastic partial differential equations is very delicate. Stochastic centre manifold theory provides novel support for coarse grained, macroscale, spatial discretisations of nonlinear stochastic partial…
Motivated by the probabilistic representation for solutions of the Navier-Stokes equations, we introduce a novel class of stochastic differential equations that depend on the entire flow of its time marginals. We establish the existence and…
A stochastic solution is constructed for a fractional generalization of the KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution uses a fractional generalization of the branching exponential process and propagation processes which…
This paper addresses the Bayesian calibration of dynamic models with parametric and structural uncertainties, in particular where the uncertain parameters are unknown/poorly known spatio-temporally varying subsystem models. Independent…
Stochastic optimization methods have been hugely successful in making large-scale optimization problems feasible when computing the full gradient is computationally prohibitive. Using the theory of modified equations for numerical…
In this article we develop a new methodology to prove weak approximation results for general stochastic differential equations. Instead of using a partial differential equation approach as is usually done for diffusions, the approach…
It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high-dimensional random inputs, which are typically caused by the large number of stochastic basis functions…