Stochastic Partial Differential Equations : Approximations and Applications

Abstract

For many people the behaviour of stock prices may appear to be unpredictable. The price dynamics seem to exhibit no regularity. Although it might be hard to believe, mathematicians and physisists have managed to explain this behaviour via functions whose characteristics match those of the observed phenomena. In mathematics we model such curves with stochastic equations (driven by stochastic processes). They describe chaotic behaviour and can be used to produce computer simulations. The (standard) theory is quite well known and established. However, when one studies more complex financial markets and products, the complexity of the stochastic equations increases considerably. As an extension to the text-book theory, one could devise models in more than one dimension. Eventually this would lead to the notion of stochastic equations taking values in some function space (stochastic partial differential equations) or random fields. The simulation of stochastic partial differential equations is the main contribution of this work. We show convergence of discretizations as the simulation becomes more precise. We introduce as well possible applications like forward pricing in energy markets, or hedging against weather risk due to temperature uncertainty. A Finite Element Method is used for the discretization. This is a well established numerical method for deterministic problems. When we deal with stochastic equations, however, the world is not smooth and thus the problems become more daunting. In this work we introduce Finite Element Methods for stochastic partial differntial equations driven by different noise processes.