2022/07/18 by Shivam Dhama, Dhama, Shivam Singh
Economics, Econometrics and Finance · Mathematics · #Applied mathematics #Brownian motion #Compound Poisson process #Differential equation #Dynamical Systems (math.DS) #FOS: Mathematics #Jump #Jump process #Mathematical Biology Tumor Growth #Mathematical analysis #Mathematics #Optimization and Control (math.OC) #Ordinary differential equation #Physics #Poisson distribution #Poisson process #Probability (math.PR) #Rate of convergence #Renewal theory #Statistical physics #Statistics #Stochastic differential equation #Stochastic process #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Wiener process
paper · pdf · doi:10.48550/arxiv.2207.08388
openalex publication_date 2022/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this article, we study the dynamics of a nonlinear system governed by an ordinary differential equation under the combined influence of fast periodic sampling with period δ and small jump noise of size ε, 0< ε,δ≪ 1. The noise is a combination of Brownian motion and Poisson random measure. The instantaneous rate of change of the state depends not only on its current value but on the most recent measurement of the state, as the state is measured at certain discrete-time instants. As ε,δ\searrow 0, the stochastic process of interest converges, in a suitable sense, to the dynamics of the deterministic equation. Next, the study of rescaled fluctuations of the stochastic process around its mean is found to vary depending on the relative rates of convergence of small parameters ε, δ in different asymptotic regimes. We show that the rescaled process converges, in a strong (path-wise) sense, to an effective process having an extra drift term capturing both the sampling and noise effect. Consequently, we obtain a first-order perturbation expansion of the stochastic process of interest, in terms of the effective process along with error bounds on the remainder.