Abstract
In this article, we study random differential equations with discrete delay
with initial condition
The uncertainty in the problem is reflected via the outcome ω. The initial condition g(t) is a stochastic process. The term x(t) is a stochastic process that solves the random differential equation with delay in a probabilistic sense. In our case, we use the
random calculus approach. We extend the classical Picard theorem for deterministic ordinary differential equations to
calculus for random differential equations with delay, via Banach fixed-point theorem. We also relate
solutions with sample-path solutions. Finally, we utilize the theoretical ideas to solve the random autonomous linear differential equation with discrete delay.
Acknowledgements
The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.
Disclosure statement
The authors declare that there is no conflict of interests regarding the publication of this article.