Random differential equations with discrete delay

Abstract

In this article, we study random differential equations with discrete delay $\tau >0:$ $x\prime (t,\omega )=f(x(t,\omega ),x(t-\tau ,\omega ),t,\omega ),t\ge t_0,$ with initial condition $x(t,\omega )=g(t,\omega ),t\in [t_0-\tau ,t_0].$ 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 ${\mathrm{L}}^p$ random calculus approach. We extend the classical Picard theorem for deterministic ordinary differential equations to ${\mathrm{L}}^p$ calculus for random differential equations with delay, via Banach fixed-point theorem. We also relate ${\mathrm{L}}^p$ solutions with sample-path solutions. Finally, we utilize the theoretical ideas to solve the random autonomous linear differential equation with discrete delay.

Keywords:

• Random differential equation with discrete delay
• stochastic process
• ${\mathrm{L}}^p$ random calculus
• Banach fixed-point theorem

AMS Classification 2010:

• 34F05
• 34K50
• 60H10
• 65C30

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.

Additional information

Funding

This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017–89664–P.