**Juan Carlos Cortés \* and Marc Jornet**

Instituto Universitario de Matemática Multidisciplinar, Building 8G, access C, *2*nd floor, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain; marjorsa@doctor.upv.es

**\*** Correspondence: jccortes@imm.upv.es

Received: 25 May 2020; Accepted: 18 June 2020; Published: 20 June 2020

**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*) = *ax*(*t*) + *bx*(*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*<sup>p</sup>* 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*<sup>p</sup>* 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.

**Keywords:** random linear delay differential equation; stochastic forcing term; random L*p*-calculus; uncertainty quantification
