Hyperbolic SPDEs: analytical and numerical study using Wiener Chaos approach

Abstract

In the first part of this dissertation we propose a constructive approach forgeneralized weighted Wiener Chaos solutions of linear hyperbolic SPDEsdriven by a cylindrical Brownian Motion. Explicit conditions for the existence,uniqueness and regularity of generalized (Wiener Chaos) solutions areestablished in Sobolev spaces. An equivalence relation between the WienerChaos solution and the traditional one is established.In the second part we propose a novel numerical scheme based on theWiener Chaos expansion for solving hyperbolic stochastic PDEs. Throughthe Wiener Chaos expansion the stochastic PDE is reduced to an infinite hierarchy of deterministic PDEs which is then truncated to a finite system ofPDEs, that can be addressed by standard techniques. A priori and a posterioriconvergence results for the method are provided. The proposed methodis applied to solve the stochastic forward rate Heath-Jarrow-Morton modelwith the Musiela parametrization and the results are compared to those derivedby the Monte Carlo method. The main advantage of the proposedscheme is that it is significantly faster than the Monte Carlo (MC) simulationmethod, for the same order of accuracy. It also provides a convenientway to compute not only the solution but also the statistical moments of thesolution numerically.