Mixingale Estimation Function for SPDEs with Random Sampling

We study the mixingale estimation function estimator of the drift parameter in the stochastic partial differential equation when the process is observed at the arrival times of a Poisson process. We use a two stage estimation procedure. We first estimate the intensity of the Poisson process. Then we substitute this estimate in the estimation function to estimate the drift parameter. We obtain the strong consistency and the asymptotic normality of the mixingale estimation function estimator.

On a Monte Carlo scheme for some linear stochastic partial differential equations

The aim of this paper is to study the simulation of the expectation for the solution of linear stochastic partial differential equation driven by the space-time white noise with the bounded measurable coefficient and different boundary conditions. We first propose a Monte Carlo type method for the expectation of the solution of a linear stochastic partial differential equation and prove an upper bound for its weak rate error. In addition, we prove the central limit theorem for the proposed method in order to obtain confidence intervals for it. As an application, the Monte Carlo scheme applies to the stochastic heat equation with various boundary conditions, and we provide the result of numerical experiments which confirm the theoretical results in this paper.

Spatio-temporal prediction of missing temperature with stochastic Poisson equations

This paper presents our winning entry for the EVA 2019 data competition, the aim of which is to predict Red Sea surface temperature extremes over space and time. To achieve this, we used a stochastic partial differential equation (Poisson equation) based method, improved through a regularization to penalize large magnitudes of solutions. This approach is shown to be successful according to the competition’s evaluation criterion, i.e. a threshold-weighted continuous ranked probability score. Our stochastic Poisson equation and its boundary conditions resolve the data’s non-stationarity naturally and effectively. Meanwhile, our numerical method is computationally efficient at dealing with the data’s high dimensionality, without any parameter estimation. It demonstrates the usefulness of stochastic differential equations on spatio-temporal predictions, including the extremes of the process.