Kiattisak Prathom and Sanae Rujivan

DOI: 10.14456/sjst-psu.2021.123

pp. 936 - 947

Abstract

This paper investigates solutions of a recurrence differential equation (RDE) of the form

A'₁(t) = a₁A₁(t)

A'₂(t) = a₂A₂(t) + b₂A₁(t)

A'_k+2(t) = a_k+2A_k+2(t) + b_k+2A_k+1(t) + c_k+2A_k(t)

for k = 1,2,..., K - 2 and any positive integer K ≥ 3 subject to the initial conditions A_i(0) = R_i ∈ R for i = 1,2,...,K where b_i,c_i ∈ C, a_i ∈ R and a_i ≠ a_j for i ≠ j. Firstly, we apply Laplace transform to the RDE to obtain a difference equation in Laplace space. Our success in performing Laplace inverse transform leads to an explicit solution of the RDE. Finally, we present
an application of our results by deriving closed-form formulas for the conditional moment, variance, covariance, and correlation
of quadratic variance diffusion processes which are commonly used for studying model variance or interest rate processes in
financial engineering.