Applying the random variable transformation method to solve a class of random linear differential equation with discrete delay

We randomize the following class of linear differential equations with delay, xτ′(t)=axτ(t)+bxτ(t−τ),t > 0, and initial condition, xτ(t)=g(t),−τ≤t≤0, by assuming that coefficients a and b are random variables and the initial condition g(t) is a stochastic process. We consider two cases, depending on the functional form of the stochastic process g(t), and then we solve, from a probabilistic point of view, both random initial value problems by determining explicit expressions to the first probability density function, f(x, t; τ), of the corresponding solution stochastic processes. Afterwards, we establish sufficient conditions on the involved random input parameters in order to guarantee that f(x, t; τ) converges, as τ→0+, to the first probability density function, say f(x, t), of the corresponding associated random linear problem without delay (τ=0). The paper concludes with several numerical experiments illustrating our theoretical findings.