Asymptotic behavior of a solution of the non-autonomous logistic stochastic differential equation

2021 ◽  
Vol 101 ◽  
pp. 39-50
Author(s):  
O. D. Borysenko ◽  
D. O. Borysenko
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Stochastic Differential Equation

scholarly journals A Stochastic Model with Jumps for Smoking

2019 ◽  
pp. 1-16
Author(s):  
Mohamed Coulibaly ◽  
Modeste N'Zi
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Diffusion Coefficient ◽  
Stochastic Differential Equation ◽  
Equilibrium State ◽  
Deterministic Model ◽  
Jump Diffusion ◽  
Epidemiological Models ◽  
Sirs Model ◽  
Theoretical Results

Some stochastic epidemiological models are less significant. They do not take into account some sudden events that could disrupt the behavior of the studied phenomenon. In this work, we introduce a white noise and jumps in a deterministic SIRS model for smoking to take into account of the effects of randomly fluctuation and such sudden factors respectively. First of all we prove that the solution of the stochastic differential equation with jumps of the new modelis positive. Then we study the asymptotic behavior around the smoking-free equilibrium state and the smoking-present equilibrium state of the original deterministic model. Under certain conditions, we show that the solution oscillate respectively around these equilibrium states. We prove that the intensity of these oscillations depends on the magnitude of noise and the jump diffusion coefficient of our stochastic differential equation with jumps. To support our theoretical results, we realise numerical simulations. The observations confirm our conclusions.

scholarly journals Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process

2018 ◽  
Vol 22 ◽  
pp. 58-95
Author(s):  
Emmanuelle Clément ◽  
Arnaud Gloter ◽  
Huong Nguyen
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Asymptotic Behavior ◽  
Stochastic Differential Equation ◽  
Malliavin Calculus ◽  
Stable Process ◽  
Small Time ◽  
Stable Lévy Process ◽  
Asymptotic Mixed Normality ◽  
Drift Coefficient

This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). We assume that the process depends on a parameter β = (θ, σ)T and we study the sensitivity of the density with respect to this parameter. This extends the results of [E. Clément and A. Gloter, Local asymptotic mixed normality property for discretely observed stochastic dierential equations driven by stable Lévy processes. Stochastic Process. Appl. 125 (2015) 2316–2352.] which was restricted to the index α ∈ (1, 2) and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density and its derivative as an expectation and a conditional expectation. This permits to analyze the asymptotic behavior in small time of the density, using the time rescaling property of the stable process.

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