Fröbenius Expansions for Second-Order Random Differential Equations: Stochastic Analysis and Applications to Lindley-Type Damping Models

Abstract

This paper develops a Frobenius series framework for the stochastic analysis of second-order random differential equations of the form Y(t) + A(t)Y(t) = 0, where the damping coefficient A(t) is a positive stochastic process and the initial conditions are square-integrable random variables. Assuming mean-square analyticity of A(t) in a neighborhood of the initial time, we establish existence and uniqueness of the solution in L2(Omega) and derive exponentially convergent truncation error bounds for the associated Frobenius expansion. The resulting series representation enables the numerical approximation of the probability density function of Y(t) via Monte Carlo simulation. To improve computational efficiency, a control variates strategy is incorporated for variance reduction. A comprehensive numerical study is conducted for a broad family of positive, right-skewed damping distributions, including the Lindley, XLindley, New XLindley (NXLD), Gamma-Lindley, Inverse-Lindley, Truncated-Lindley, Log-Lindley, and a newly proposed Mixed Lindley-Uniform model. The simulations illustrate how different tail behaviors and boundedness properties of the damping coefficient influence the stochastic dynamics and the accuracy of density estimation. Finally, stylized applications to option pricing and Value-at-Risk estimation are presented to illustrate how the Frobenius-based framework and control variates methodology can be embedded within standard uncertainty quantification workflows. Overall, the proposed approach provides a flexible and computationally efficient tool for the analysis of randomly damped dynamical systems.