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Lp-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term

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Lp-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term mathematics Article Lp-solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term Juan Carlos Cortés * and Marc Jornet Instituto Universitario de Matemática Multidisciplinar, Building 8G, access C, 2nd floor, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain; [email protected] * Correspondence: [email protected] 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 t > 0, by adding a random forcing term f (t) that varies with time: x0(t) = ax(t) + bx(t − t) + f (t), t ≥ 0, with initial condition x(t) = g(t), −t ≤ 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 Lp 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 Lp-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 t tends to 0, the random delay equation tends in Lp 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 Lp-calculus; uncertainty quantification 1. Introduction In this paper, we are concerned with random delay differential equations, defined as classical delay differential equations whose inputs (coefficients, forcing term, initial condition, ...) are considered as random variables or regular stochastic processes on an underlying complete probability space (W, F, P), which may take a wide variety of probability distributions, such as Binomial, Poisson, Gamma, Gaussian, etc. Equations of this kind should not be confused with stochastic differential equations of Itô type, forced by an irregular error term called White noise process (formal derivative of Brownian motion). In contrast to random differential equations, the solutions to stochastic differential equations exhibit nondifferentiable sample-paths. See [1] (pp. 96–98) for a detailed explanation of the difference between random and stochastic differential equations. See [1–6], for instance, for applications of random differential equations in engineering, physics, biology, etc. Thus, random differential equations require their own treatment and study: they model smooth random phenomena, with any type of input probability distributions. From a theoretical viewpoint, random differential equations may be studied in two senses: the sample-path sense or the Lp-sense. The former case considers the trajectories of the stochastic processes involved, so that the realizations of the random system correspond to deterministic versions of the p problem. The latter case works with the topology of the Lebesgue space (L , k · kp) of random p 1/p variables with finite absolute p-th moment, where the norm k · kp is defined as: kUkp = E[jUj ] Mathematics 2020, 8, 1013; doi:10.3390/math8061013 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1013 2 of 16 for 1 ≤ p < ¥ (E denotes the expectation operator), and kUk¥ = inffC ≥ 0 : jUj ≤ C almost surelyg p (essential supremum), U : W ! R being any random variable. The Lebesgue space (L , k · kp) has the structure of a Banach space. Continuity, differentiability, Riemann integrability, etc., can be considered in the aforementioned space Lp, which gives rise to the random Lp-calculus. In order to fix concepts, given a stochastic process x(t) ≡ x(t, w) on I × W, where I ⊆ R is an interval (notice that as usual the w-sample notation might be hidden), we say that x is p p L -continuous at t0 2 I if limh!0 kx(t0 + h) − x(t0)kp = 0. We say that x is L -differentiable at x(t0+h)−x(t0) 0 0 t0 2 I if limh!0 k h − x (t0)kp = 0, for certain random variable x (t0) (called the derivative p of x at t0). Finally, if I = [a, b], we say that x is L -Riemann integrable on [a, b] if there exists a sequence f g¥ = f = n < n < < n = g of partitions Pn n=1 with mesh tending to 0, Pn a t0 t1 ... trn b , such that, for n n n rn n n n p any choice of points si 2 [ti−1, ti ], i = 1, ... , rn, the limit limn!¥ ∑i=1 x(si )(ti − ti−1) exists in L . In this case, these Riemann sums have the same limit, which is a random variable and is denoted by R b a x(t) dt. This Lp-approach has been widely used in the context of random differential equations with no delay, especially the case p = 2 which corresponds to the Hilbert space L2 and yields the so-called mean square calculus; see [5,7–15]. Only recently, a theoretical probabilistic analysis of random differential equations with discrete constant delay has been addressed in [16–18]. In [16], general random delay differential equations in Lp were analyzed, with the goal of extending some of the existing results on random differential equations with no delay from the celebrated book [5]. In [17], we started our study on random delay differential equations with the basic autonomous-homogeneous linear equation, by proving the existence and uniqueness of Lp-solution under certain conditions. In [18], the authors studied the same autonomous-homogeneous random linear differential equation with discrete delay as [17], but considered the solution in the sample-path sense and computed its probability density function via the random variable transformation technique, for certain forms of the initial condition process. Other recent contributions for random delay differential equations, but focusing on numerical methods instead, are [19–21]. There is still a lack of theoretical analysis for important random delay differential equations. Motivated by this issue, the aim of this contribution is to advance further in the theoretical analysis of relevant random differential equations with discrete delay. In particular, in this paper we extend the recent study performed in [17] for the basic linear equation by adding a stochastic forcing term: ( x0(t, w) = a(w)x(t, w) + b(w)x(t − t, w) + f (t, w), t ≥ 0, w 2 W, (1) x(t, w) = g(t, w), −t ≤ t ≤ 0, w 2 W. The delay t > 0 is constant. The coefficients a and b are random variables. The forcing term f (t) and the initial condition g(t) are stochastic processes on [0, ¥) and [−t, 0] respectively, which depend on the outcome w 2 W of a random experiment which might be sometimes omitted in notation. The term x(t) represents the differentiable solution stochastic process in a certain probabilistic sense. Formally, according to the deterministic theory [22], we may express the solution process as a(w)(t+t) b1(w),t x(t, w) = e et g(−t, w) Z 0 a(w)(t−s) b1(w),t−t−s 0 + e et (g (s, w) − a(w)g(s, w)) ds −t Z t a(w)(t−s) b1(w),t−t−s + e et f (s, w) ds, (2) 0 Mathematics 2020, 8, 1013 3 of 16 where b = e−atb and 1 8 >0, −¥ < t < −t, > > −t ≤ t < >1, 0, > t > + ≤ < >1 c , 0 t t, <> 1! c,t t (t − t)2 et = 1 + c + c2 , t ≤ t < 2t, > 1! 2! > . > . > . > n > (t − (k − 1)t)k > ck , (n − 1)t ≤ t < nt, :> ∑ k=0 k! is the delayed exponential function [22] (Definition 1), c, t 2 R, and n = bt/tc + 1 (here b·c denotes the integer part defined by the so-called floor function). This formal solution is obtained with the method of steps and the method of variation of constants. The primary objective of this paper is to set probabilistic conditions under which x(t) is an Lp-solution to (1). We decompose the original problem (1) as ( y0(t, w) = a(w)y(t, w) + b(w)y(t − t, w), t ≥ 0, (3) y(t, w) = g(t, w), −t ≤ t ≤ 0, and ( z0(t, w) = a(w)z(t, w) + b(w)z(t − t, w) + f (t, w), t ≥ 0, (4) z(t, w) = 0, −t ≤ t ≤ 0. System (3) does not possess a stochastic forcing term, and it was deeply studied in the recent contribution [17]. Under certain assumptions, its Lp-solution is expressed as a(w)(t+t) b1(w),t y(t, w) = e et g(−t, w) Z 0 a(w)(t−s) b1(w),t−t−s 0 + e et (g (s, w) − a(w)g(s, w)) ds, (5) −t as a generalization of the deterministic solution to (3) obtained via the method of steps [22] (Theorem 1). Problem (4) is new and requires an analysis in the Lp-sense, in order to solve the initial problem (1). Its formal solution is given by Z t a(w)(t−s) b1(w),t−t−s z(t, w) = e et f (s, w) ds, (6) 0 see [22] (Theorem 2). In order to differentiate (6) in the Lp-sense, one requires the extension of the deterministic Leibniz’s integral rule for differentiation to the random scenario. This extension is an important piece of this paper.
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