Existence and uniqueness of solutions to stochastic heat equations with Markovian switching and Feller property

In this paper, we focus on two-component Markov processes which consist of continuous dynamics and discrete events. Using the classical fixed point theorem for contractions to investigate the existence and uniqueness of solutions of stochastic heat equations with Markovian switching, then developing the corresponding the Feller property of the solution

Dynamics of a stochastic coronavirus (COVID-19) epidemic model with Markovian switching

In this paper, we analyze a stochastic coronavirus (COVID-19) epidemic model which is perturbed by both white noise and telegraph noise incorporating general incidence rate. Firstly, we investigate the existence and uniqueness of a global positive solution. Then, we establish the stochastic threshold for the extinction and the persistence of the disease. The data from Indian states, are used to confirm the results established along this paper

Analysis of a stochastic predator-prey system with foraging arena scheme

This paper focuses on a predator-prey system with foraging arena scheme incorporating stochastic noises. This SDE model is generated from a deterministic framework by the stochastic parameter perturbation. We then study how the correlations of the environmental noises affect the long-time behaviours of the SDE model. Later on the existence of a stationary distribution is pointed out under certain parametric restrictions. Numerical simulations are carried out to substantiate the analytical results

Explicit approximation of the invariant measure for SDDEs with the nonlinear diffusion term

To our knowledge, the existing measure approximation theory requires the diffusion term of the stochastic delay differential equations (SDDEs) to be globally Lipschitz continuous. Our work is to develop a new explicit numerical method for SDDEs with the nonlinear diffusion term and establish the measure approximation theory. Precisely, we construct a function-valued explicit truncated Euler-Maruyama segment process (TEMSP) and prove that it admits a unique ergodic numerical invariant measure. We also prove that the numerical invariant measure converges to the underlying one of SDDE in the Fortet-Mourier distance. Finally, we give an example and numerical simulations to support our theory.Comment: 31 pages, 2 figure

An improved two-step method in stochastic differential equation's structural parameter estimation

Non-parametric modelling is a method which relies heavily on data and motivated by the smoothness properties in estimating a function which involves spline and non-spline approaches. Spline approach consists of regression spline and smoothing spline. Regression spline characterised by the truncated power series basis with Bayesian approach is considered in the first step of a two-step method for estimating the structural parameters for stochastic differential equation (SDE). Previous methodology revealed the selection of knot and order of spline can be done heuristically based on a scatter plot. To overcome the subjective and tedious process of selecting the optimal knot and order of spline, an algorithm is proposed. A single optimal knot is selected out of all the points with exception of the first and the last data and the least value of Generalised Cross Validation is calculated for each order of spline. The spline model is later utilised in the second step to estimate the stochastic model parameters. In the second step, a non-parametric criterion is proposed for estimating the diffusion parameter of SDE. Linear and non-linear SDE consisting of Geometric Brownian Motion (GBM) for the former and logistic together with Lotka Volterra (LV) model for the later are tested using the two-step method for both simulated and real data. The results show high percentage of accuracy with 99.90% and 96.12% are obtained for GBM and LV model respectively for diffusion parameters of simulated data. This verifies the viability of the two-step method in the estimation of diffusion parameters of SDE with an improvement of a single knot selection