The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function

Abstract

This paper deals with the damped pendulum random differential equation:Ẍ(t) + 2ω 0 ξẊ(t) + ω 2 0 X(t) = Y (t), t ∈ [0, T ], with initial conditions X(0) = X 0 andẊ(0) = X 1 . The forcing term Y (t) is a stochastic process and X 0 and X 1 are random variables in a common underlying complete probability space (Ω, F, P). The term X(t) is a stochastic process that solves the random differential equation in both the sample path and in the L p senses. To understand the probabilistic behaviour of X(t), we need its joint finite-dimensional distributions. We establish mild conditions under which X(t) is an absolutely continuous random variable, for each t, and we find its probability density function f X(t) (x). Thus, we obtain the first finite-dimensional distributions. In practice, we deal with two types of forcing term: Y (t) is a Gaussian process, which occurs with the damped pendulum stochastic differential equation of Itô type; and Y (t) can be approximated by a sequence {Y N (t)} ∞ N =1 in L 2 ([0, T ] × Ω), which occurs with Karhunen-Loève expansions and some random power series. Finally, we provide numerical examples in which we choose specific random variables X 0 and X 1 and a specific stochastic process Y (t), and then, we find the probability density function of X(t).

Computational methods for random differential equations: probability density function and estimation of the parameters

“…then it suffices that one of the ξ j 's or one of the Y j 's is absolutely continuous and independent of the rest of random coefficients, by [17…”

“…We have fully extended the results of the recent contribution [42], which is the first one in dealing with the nonautonomous case of this important equation for growth modeling. The idea of our approach, which could be used for other random differential equations as we did in [17], could be summarized as follows: (i) by assuming absolute continuity only for the initial condition P 0 , we have proved that P (t) is absolutely continuous with density function f 1 (p, t) expressed in terms of an expectation, via a generalization of the random variable transformation technique to our particular setting; (ii) when A(t) is expanded or approximated in terms of a sequence of stochastic processes {A…”

“…or discretizations from numerical methods are usually required. By truncating these infinite expansions or from the discretizations, one constructs a sequence of approximating processes, whose exact density functions (computed via the RVT technique) form a sequence of approximating density functions that converge rapidly [17,19,45,57]. In terms of rapidity and accuracy, this sort of approximations are preferable to Monte Carlo simulation and Kernel density estimations.…”

“…), one usually constructs a sequence of approximating processes, whose exact density functions (computed via the RVT technique) form a sequence of approximating density functions that converge rapidly. This methodology has been carried out via numerical methods [57], Karhunen-Loève expansions [17,19] (see [89] for the theory on these expansions), polynomial expansions [14], or random power series [17,45], and it has supposed an improvement compared with Monte Carlo simulation and Kernel density estimations in terms of rate of convergence and accuracy.…”

“…In this chapter, we tackle the approximation to f 1 (p, t) analogously to our recent contribution [17]. We consider a general sequence {A N (t)} ∞ N =1 that approximates A(t) in the Hilbert space L 2 ([t 0 , T ] × Ω) (mean square approximation/expansion of A(t)).…”