Theory of (non-linear) Stochastic Partial Differential Equations and its Applications to Interest Rates

Abstract

The aim of this Thesis is to study Stochastic Partial Differential Equations (SPDEs) by regarding the SPDE as a Stochastic Differential Equation taking values in an infinite-dimensional space, i.e. a space of functions. To do this, one must first build an integration theory for functions with values in an infinite-dimensional space. The weapon of choice in this Thesis is the Pettis integral in which the integral is build around studying one-dimensional projections. Stochastic integration is done with respect to cylindrical Brownian motion. After this is done, the Thesis studies existence and uniqueness of linear and non-linear SPDEs and a result on semi-linear backward SPDEs is also proved. Finally, the Thesis studies applications to interest rates. The starting point is the Heath-Jarrow-Morton model, but this is generalized by considering infinitely many sources of noise.