An explicit solution of a recurrence differential equation and its application in determining the conditional moments of quadratic variance diffusion processes

Abstract

This paper investigates solutions of a recurrence differential equation (RDE) of the form: ' 1 1 1 ' 2 2 2 2 1 ' 2 2 2 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), k k k k k k k A t a A t A t a A t b A t A t a A t b A t c A t + + + + + + = = + = + + ' 1 1 1 ' 2 2 2 2 1 ' 2 2 2 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), k k k k k k k A t a A t A t a A t b A t A t a A t b A t c A t + + + + + + = = + = + + ' 1 1 1 ' 2 2 2 2 1 ' 2 2 2 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), k k k k k k k A t a A t A t a A t b A t A t a A t b A t c A t + + + + + + = = + = + + for k K   1,2,..., 2 and any positive integer K ³ 3 subject to the initial conditions (0) i i A R = Î R for i K = 1,2,..., where , , i i i b c a Î Î C R and i j a a ¹ for i j ¹ . 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.

Keywords

• recurrence differential equation
• explicit solution
• conditional moments
• quadratic variance diffusion processes