The Almost-Sure Asymptotic Behavior Of The Solution To The Stochastic Heat Equation With Levy Noise

We examine the almost-sure asymptotics of the solution to the stochastic heat equation driven by a Levy space-time white noise. When a spatial point is fixed and time tends to infinity, we show that the solution develops unusually high peaks over short time intervals, even in the case of additive noise, which leads to a breakdown of an intuitively expected strong law of large numbers. More precisely, if we normalize the solution by an increasing nonnegative function, we either obtain convergence to 0, or the limit superior and/or inferior will be infinite. A detailed analysis of the jumps further reveals that the strong law of large numbers can be recovered on discrete sequences of time points increasing to infinity. This leads to a necessary and sufficient condition that depends on the Levy measure of the noise and the growth and concentration properties of the sequence at the same time. Finally, we show that our results generalize to the stochastic heat equation with a multiplicative nonlinearity that is bounded away from zero and infinity.

Path properties of the solution to the stochastic heat equation with Levy noise

Carsten Hao Ye Chong, Robert Dalang, Thomas Marie Jean-Baptiste Humeau

We consider sample path properties of the solution to the stochastic heat equation, in Rd or bounded domains of Rd, driven by a Levy space-time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a cadlag modification in fractional Sobolev spaces of index less than - Concerning the partial regularity of the solution in time or space when the other variable is fixed, we determine critical values for the Blumenthal-Getoor index of the Levy noise such that noises with a smaller index entail continuous sample paths, while Levy noises with a larger index entail sample paths that are unbounded on any non-empty open subset. Our results apply to additive as well as multiplicative Levy noises, and to light- as well as heavy-tailed jumps.

SPRINGER2019

Statistical Inference for Partially Observed Stochastic Epidemics

Andrea Kraus

This work is concerned with the estimation of the spreading potential of the disease in the initial stages of an epidemic. A speedy and accurate estimation is important for determining whether or not interventions are necessary to prevent a major outbreak. At the same time, the information available in the early stages is scarce and data collection imperfect. We consider an epidemic in a large susceptible population, and address the estimation based on temporally aggregated counts of new cases that are subject to unknown random under-reporting. We allow for an influence of the detection process on the evolution of the epidemic. While the proportion of infectious individuals in the population is small, the role of chance in the spread of the disease may be substantial. Therefore, stochastic epidemic models are applied. As these are difficult to analyse, the time evolution of the number of infectious individuals is approximated by branching processes. We study the estimation in a partially observed Galton–Watson process and in a partially observed linear birth and death process; and in each case focus on the parameter characterising the growth of the process. We aim at estimators that perform well in the asymptotic sense where a single trajectory is observed over a long period of time, and study the asymptotics conditionally on the eventual explosion of the process. The partially observed Galton–Watson process has been recently proposed in the literature as a model for the initial stages of an epidemic. Its probabilistic structure has been explored and estimation has been partially addressed, in that consistent estimators have been constructed. However, the estimation-related uncertainty has not been evaluated. We address this issue here by constructing estimators that are motivated from the asymptotic dependence structure of the process. We show that they are consistent and asymptotically normal, consistently estimate their asymptotic variances, and construct asymptotic confidence intervals. In addition, we evaluate their finite-sample performance in a simulation study and their practical performance on real data. The observation mechanism in the partially observed Galton–Watson process is inherently discrete. To allow for continuous-time dynamics, we incorporate partial observation in the linear birth and death process. In particular, we propose a model where the birth process is completely unobserved, while a random proportion of the death process is observed at discrete time points. We study the estimation in this model. Motivated by its counting process structure, we arrive at consistent and asymptotically normal estimators, consistently estimate their asymptotic variances, and construct asymptotic confidence intervals. We also evaluate the finite-sample and practical performance of the estimators in a simulation study and on real data.

EPFL2013

Stochastic partial differential equations driven by Lévy white noises

Thomas Marie Jean-Baptiste Humeau

We study various aspects of stochastic partial differential equations driven by Lévy white noise. This driving noise, which is a generalization of Gaussian white noise, can be viewed either as a generalized random process or as an independently scattered random measure. After unifying these approaches and establishing appropriate stochastic integral representations, we show that a necessary and sufficient condition for a Lévy white noise to have values in the space of tempered Schwartz distributions, is that the underlying Lévy measure have a positive absolute moment. In the case of a linear stochastic partial differential equation with a general differential operator and driven by a symmetric pure jump Lévy white noise, we show that when the mild solution is locally Lebesgue integrable, then it is equal to the generalized solution, and that a random field representation exists for the generalized solution if and only if the fundamental solution of the operator has certain integrability properties. In that case, we show that the random field representation is equal to the mild solution. For this purpose, a new stochastic Fubini theorem is proved. These results are applied to the linear stochastic heat and wave equations driven by a symmetric alpha-stable noise. We then study the non-linear stochastic heat equation driven by a general type of Lévy white noise, possibly with heavy tails and non-summable small jumps. Our framework includes in particular the alpha-stable noise. In the case of the equation on the whole space, we show that the law of the solution that we construct does not depend on the space variable. Then we show in various domains D that the solution u to the stochastic heat equation is such that t -> u(t,·) has a càdlàg version in a fractional Sobolev space of order r < -d /2. Finally, we show that the partial functions have a continuous version under some optimal moment conditions. In the alpha-stable case, we show that for the choices of alpha for which this moment condition is not satisfied, the sample paths of the partial functions are unbounded on any non-empty open subset.