In this paper we introduce methods based upon Lie symmetry analysis for the construction of explicit fundamental solutions of multidimensional parabolic PDEs. We give applications to the problem of finding transition probability densities for multidimensional diffusions and to representation theory.

This paper uses Lie symmetry group methods to obtain transition probability densities for scalar diffusions, where the diffusion coefficient is given by a power law. We will show that if the drift of the diffusion satisfies a certain family of Riccati equations, then it is possible to compute a...

This paper makes use of an integrated benchmark modelling framework that allows us to model credit risk. We demonstrate how to price contingent claims by taking expectations under the real world probability measure in a benchmarked world. Furthermore, put and call options on an index are studied...

We consider different approaches to the problem of numerically inverting Laplace transforms in finance. In particular, we discuss numerical inversion techniques in the context of Asian option pricing.

We analyse the procedure for determining volatility presented by Lagnado and Osher, and explain in some detail where the scheme comes from. We present an alternative scheme which avoids some of the technical complications arising in Lagnado and Osher's approach. An algorithm for solving the...

This paper uses Lie symmetry group methods to analyse a class of partial differential equations of he form It is shown that when the drift function f is a solution of a family of Ricatti equations, then symmetry techniques can be used to find the characteristic functions and transition densities...

We obtain fundamental solutions for PDEs of the form ut = x uxx +f(x)ux ??xru by showing that if the symmetry group of the PDE is nontrivial, it contains a standard integral transform of the fundamental solution. We show that in this case, the problem of finding a fundamental solution can be...