Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

Abdulwahab Almutairi,I M Elbaz,M A Sohaly,H El-Metwally

doi:10.1186/s13662-021-03344-6

Abdulwahab Almutairi, I M Elbaz + Show 2 more

Open Access

https://doi.org/10.1186/s13662-021-03344-6

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Abstract

This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

Highlights

Many applications in many disciplines assume that the future state of the system is independent of the past states and is only determined by the current state
We propose the uncertainty through perturbing the system by the Brownian motion or through assuming that the coefficients, initial conditions, and forcing terms are random variables or stochastic terms
3 Stability theory of random/ stochastic DDE we show the proofs of stability theorems

Summary

Introduction

Many applications in many disciplines assume that the future state of the system is independent of the past states and is only determined by the current state. We propose the uncertainty through perturbing the system by the Brownian motion or through assuming that the coefficients, initial conditions, and forcing terms are random variables or stochastic terms. This is another way to introduce the uncertainty by assuming that the inputs (coefficients, initial conditions, forcing terms, etc.) are random variables and/or stochastic processes. In this case, we have a wider type of probability distributions such. We study the Black–Scholes delay market model driven by a Brownian motion with random variable coefficients This model has become the most known way to model pricing options in financial markets.

Preliminaries

Asymptotically stable in probability if

Results and discussions

Conclusion and further directions