Massimiliano Tamborrino

The University of Warwick · Department of Statistics

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve i...

Sequential algorithms such as sequential importance sampling (SIS) and sequential Monte Carlo (SMC) have proven fundamental in Bayesian inference for models not admitting a readily available likelihood function. For approximate Bayesian computation (ABC), SMC-ABC is the state-of-art sampler. However, since the ABC paradigm is intrinsically wasteful...

Sequential numerical methods for integrating initial value problems (IVPs) can be prohibitively expensive when high numerical accuracy is required over the entire interval of integration. One remedy is to integrate in a parallel fashion, “predicting” the solution serially using a cheap (coarse) solver and “correcting” these values using an expensiv...

With the advent of supercomputers, multi-processor environments and parallel-in-time (PinT) algorithms offer ways to solve initial value problems for ordinary and partial differential equations (ODEs and PDEs) over long time intervals, a task often unfeasible with sequential solvers within realistic time frames. A recent approach, GParareal, combin...

The stochastic FitzHugh-Nagumo (FHN) model considered here is a two-dimensional nonlinear stochastic differential equation with additive degenerate noise, whose first component, the only one observed, describes the membrane voltage evolution of a single neuron. Due to its low dimensionality, its analytical and numerical tractability, and its neuron...

Parallel-in-time (PinT) techniques have been proposed to solve systems of time-dependent differential equations by parallelizing the temporal domain. Among them, Parareal computes the solution sequentially using an inaccurate (fast) solver, and then "corrects" it using an accurate (slow) integrator that runs in parallel across temporal subintervals...

In this article, we propose a 6N-dimensional stochastic differential equation (SDE), modelling the activity of N coupled populations of neurons in the brain. This equation extends the Jansen and Rit neural mass model, which has been introduced to describe human electroencephalography (EEG) rhythms, in particular signals with epileptic activity. Our...

Stochastic parareal (SParareal) is a probabilistic variant of the popular parallel-in-time algorithm known as parareal. Similarly to parareal, it combines fine- and coarse-grained solutions to an ordinary differential equation (ODE) using a predictor-corrector (PC) scheme. The key difference is that carefully chosen random perturbations are added t...

We are interested in the law of the first passage time of an Ornstein-Uhlenbeck process to time-varying thresholds. We show that this problem is connected to the laws of the first passage time of the process to members of a two-parameter family of functional transformations of a time-varying boundary. For specific values of the parameters, these tr...

Sequential numerical methods for integrating initial value problems (IVPs) can be prohibitively expensive when high numerical accuracy is required over the entire interval of integration. One remedy is to integrate in a parallel fashion, "predicting" the solution serially using a cheap (coarse) solver and "correcting" these values using an expensiv...

We provide a comparative analysis of qualitative features of different numerical methods for the inhomogeneous geometric Brownian motion (IGBM). The limit distribution of the IGBM exists, its conditional and asymptotic mean and variance are known and the process can be characterised according to Feller’s boundary classification. We compare the freq...

Parareal is a well-studied algorithm for numerically integrating systems of time-dependent differential equations by parallelising the temporal domain. Given approximate initial values at each temporal sub-interval, the algorithm locates a solution in a fixed number of iterations using a predictor-corrector, stopping once a tolerance is met. This i...

Efforts to suppress transmission of SARS-CoV-2 in the UK have seen non-pharmaceutical interventions being invoked. The most severe measures to date include all restaurants, pubs and cafes being ordered to close on 20th March, followed by a “stay at home” order on the 23rd March and the closure of all non-essential retail outlets for an indefinite p...

Shot noise processes have been extensively studied due to their mathematical properties and their relevance in several applications. Here, we consider nonnegative shot noise processes and prove their weak convergence to Lévy-driven Ornstein–Uhlenbeck (OU) process, whose features depend on the underlying jump distributions. Among others, we obtain t...

In this article, we construct and analyse explicit numerical splitting methods for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The methods are proved to be mean-square convergent of order 1 and to preserve i...

Shot noise processes have been extensively studied due to their mathematical properties and their relevance in several applications. Here, we consider nonnegative shot noise processes and prove their weak convergence to Lévy-driven Ornstein-Uhlenbeck (OU), whose features depend on the underlying jump distributions. Among others, we obtain the OU-Ga...

Background: Efforts to suppress transmission of SARS-CoV-2 in the UK have seen non-pharmaceutical interventions being invoked. The most severe measures to date include all restaurants, pubs and cafes being ordered to close on 20th March, followed by a "stay at home" order on the 23rd March and the closure of all non-essential retail outlets for an...

Approximate Bayesian computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed to an established tool for modelling time-dependent, real-world phenomena with underlying random effects. When applying ABC to stoch...

We introduce the inhomogeneous geometric Brownian motion (IGBM) as a test equation for analysing qualitative features of numerical methods applied to multiplicative noise stochastic ordinary differential equations of Ito type with an inhomogeneous drift. The usual linear stability analysis of a constant equilibrium (in the mean-square or almost-sur...

The special issue is available from: https://www.aimspress.com/newsinfo/1269.html.

The output signal is examined for the Jacobi neuronal model which is characterized by input-dependent multiplicative noise. The dependence of the noise on the rate of inhibition turns out to be of primary importance to observe maxima both in the output firing rate and in the diffusion coefficient of the spike count and, simultaneously, a minimum in...

Approximate Bayesian Computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed to an established tool for modelling time dependent, real world phenomena with underlying random effects. When applying ABC to stoch...

The Jacobi process is a stochastic diffusion characterized by a linear drift and a special form of multiplicative noise which keeps the process confined between two boundaries. One example of such a process can be obtained as the diffusion limit of the Stein’s model of membrane depolarization which includes both excitatory and inhibitory reversal p...

When identifying confusable visual stimuli, accumulation of information over time is an obvious strategy of the observer. However, the nature of the accumulation process is unresolved: for example it may be discrete or continuous in terms of the information encoded. Another unanswered question is whether or not stimulus sampling continues after the...

It is widely accepted that neuronal firing rates contain a significant amount of information about the stimulus intensity. Nevertheless, theoretical studies on the coding accuracy inferred from the exact spike counting distributions are rare. We present an analysis based on the number of observed spikes assuming the stochastic perfect integrate-and...

The time to the first spike after stimulus onset typically varies with the stimulation intensity. Experimental evidence suggests that neural systems use such response latency to encode information about the stimulus. We investigate the decoding accuracy of the latency code in relation to the level of noise in the form of presynaptic spontaneous act...

The first passage time density of a diffusion process to a time varying threshold is of primary interest in different fields. Here we consider a Brownian motion in presence of an exponentially decaying threshold to model the neuronal spiking activity. Since analytical expressions of the first passage time density are not available, we propose to ap...

Given a two-dimensional correlated diffusion process, we determine the joint density of the first passage times of the process to some constant boundaries. This quantity depends on the joint density of the first passage time of the first crossing component and of the position of the second crossing component before its crossing time. First we show...

A latent internal process describes the state of some system, e.g. the social tension in a political conflict, the strength of an industrial component or the health status of a person. When this process reaches a predefined threshold, the process terminates and an observable event occurs, e.g. the political conflict finishes, the industrial compone...

Choosing the best model for analysis of count data in randomised clinical trials is complicated. In this paper, we review count data analysis with different parametric and non-parametric methods used in randomised clinical trials, and we define procedures for choosing between the two methods and their subtypes. We focus on analysis of simple count...

We consider the multivariate point process determined by the crossing times of the components of a multivariate jump process through a multivariate boundary, assuming to reset each component to an initial value after its boundary crossing. We prove that this point process converges weakly to the point process determined by the crossing times of the...

Online material for the paper PARAMETER INFERENCE FROM HITTING TIMES FOR PERTURBED BROWNIAN MOTION

In many physical systems there is a time delay before an applied input (stimulation) has an impact on the output (response), and the quantification of this delay is of paramount interest. If the response can only be observed on top of an indistinguishable background signal, the estimation can be highly unreliable, unless the background signal is ac...

The dynamics of a neuron are influenced by the connections with the network where it lies. Recorded spike trains exhibit patterns due to the interactions between neurons. However, the structure of the network is not known. A challenging task is to investigate it from the analysis of simultaneously recorded spike trains. We develop a non-parametric...

We propose a model able to describe the Interspike Intervals of two or more neurons subject to common inputs from the network. The single neuron dynamic is described through a classical Leaky Integrate and Fire model, but the model also catches the joint behavior of two neurons resorting to the use of copulas. Copulas are mathematical objects large...