Wiener–poisson chaos expansion and numerical solutions of the heath–jarrow–morton interest rate model

In this work we construct a chaotic representation of the solution of hyperbolic stochastic partial differential equations driven by Lévy noise. The driving Lévy process consists of a sequence of uncorrelated Wiener and Poisson random variables and the chaos expansion is based on the convolution of the Hermite and Charlier polynomials. We provide relations between the Hermite–Charlier basis polynomials, the Teugels time–space harmonic polynomials and the Kailath–Segall orthogonal polynomials, also used as the building blocks of the chaotic representation. In this setting, we provide certain existence, uniqueness and regularity results for the solution of the stochastic evolution equation under consideration. We apply the general construction to the estimation of Greek, Spanish, Portuguese and US government bond rates and prices during highly unstable periods of financial crises, under the Heath–Jarrow– Morton framework. The results show substantial speed-up and more accurate results compared to classical Wiener and Poisson chaos expansions and the Monte Carlo method. © Springer Science+Business Media New York 2015.