**5. Conclusions**

In this paper, we have performed a comprehensive stochastic analysis of the random linear delay differential equation with stochastic forcing term. The equation considered has one discrete delay *τ* > 0, two random coefficients *a* and *b* (corresponding to the non-delay and the delay term, respectively) and two stochastic processes *f*(*t*) and *g*(*t*) (corresponding to the forcing term on [0, ∞) and the initial condition on [−*τ*, 0], respectively). Our setting supposes a step further than the previous contribution [17], in which no forcing term was considered (i.e., *f*(*t*) = 0). We have rigorously addressed the problem of extending the deterministic theory to the random scenario, by proving that the deterministic solution constructed via the method of steps and the method of variation of constants is an L*p*-solution, under certain assumptions on the random data. A new result, the random Leibniz's rule for L*p*-Riemann integration has been necessary to derive our conclusions. We have also studied the behavior in L*<sup>p</sup>* of the random delay equation when the delay tends to zero.

Our approach has been shown to be useful to approximate the statistical moments of the solution stochastic process, in particular its expectation and its variance. Thus, it is possible to perform uncertainty quantification. Our procedure is an alternative to the usual techniques for uncertainty quantification: Monte Carlo simulation, generalized polynomial chaos (gPC) expansions, etc.

Our approach could be extendable to other random differential equations with or without delay. As usual, one could prove that the deterministic solution also works in the random framework. To do

so, a rigorous and careful analysis of the probabilistic properties of the solution based on L*p*-calculus should be conducted.

Finally, we humbly think that advancing in theoretical aspects of random differential equations with delay will permit rigorously applying this class of equations to modeling phenomena involving memory and aftereffects together with uncertainties. In particular, they may be crucial to capture uncertainties inherent to some complex modeling problems, since input parameters of this type of equations may belong to a wider range of probability distributions than the ones considered in Itô differential equations.

**Author Contributions:** Investigation, M.J.; methodology, M.J.; software, M.J.; supervision, J.C.C.; validation, J.C.C.; visualization, J.C.C. and M.J.; writing—original draft, M.J.; writing—review and editing, J.C.C. and M.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work has been supported by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017–89664–P.

**Conflicts of Interest:** The authors declare that there is no conflict of interests regarding the publication of this article.
