Lp-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term

Juan Carlos Cortés,Marc Jornet

doi:10.3390/math8061013

Abstract

This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay τ > 0 , by adding a random forcing term f ( t ) that varies with time: x ′ ( t ) = a x ( t ) + b x ( t − τ ) + f ( t ) , t ≥ 0 , with initial condition x ( t ) = g ( t ) , − τ ≤ t ≤ 0 . The coefficients a and b are assumed to be random variables, while the forcing term f ( t ) and the initial condition g ( t ) are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space L p of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an L p -solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz’s integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay τ tends to 0, the random delay equation tends in L p to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification.

Highlights

We are concerned with random delay differential equations, defined as classical delay differential equations whose inputs are considered as random variables or regular stochastic processes on an underlying complete probability space (Ω, F, P), which may take a wide variety of probability distributions, such as Binomial, Poisson, Gamma, Gaussian, etc
We state the preliminary results on L p -calculus needed for the following sections
If a stochastic process depending on two variables is L p -continuous, the two iterated L p -Riemann integrals can be interchanged

Summary

Introduction

We are concerned with random delay differential equations, defined as classical delay differential equations whose inputs (coefficients, forcing term, initial condition, . . .) are considered as random variables or regular stochastic processes on an underlying complete probability space (Ω, F , P), which may take a wide variety of probability distributions, such as Binomial, Poisson, Gamma, Gaussian, etc.Equations of this kind should not be confused with stochastic differential equations of Itô type, forced by an irregular error term called White noise process (formal derivative of Brownian motion).In contrast to random differential equations, the solutions to stochastic differential equations exhibit nondifferentiable sample-paths. . .) are considered as random variables or regular stochastic processes on an underlying complete probability space (Ω, F , P), which may take a wide variety of probability distributions, such as Binomial, Poisson, Gamma, Gaussian, etc. We are concerned with random delay differential equations, defined as classical delay differential equations whose inputs Equations of this kind should not be confused with stochastic differential equations of Itô type, forced by an irregular error term called White noise process (formal derivative of Brownian motion). Random differential equations require their own treatment and study: they model smooth random phenomena, with any type of input probability distributions

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Conclusion