We consider the Cauchy problem for a second order parabolic PDE in half spaces, arising from the stochastic modelling of a multidimensional European financial option. To improve generality, the asset price drift and volatility in the underlying stochastic model are taken time and space-dependent and the payoff function is not specified. The numerical methods and possible approximation results are strongly linked to the theory on the solvability of the PDE. We make use of two theories: the theory of linear PDE in Hölder spaces and the theory of linear PDE in Sobolev spaces. First, instead of the problem in half spaces, we consider the correspondent problem in domains. This localised PDE problem is solvable in Hölder spaces. The solution is numerically approximated, using finite differences (with both the explicit and implicit schemes) and the rate of convergence of the time-space finite differences scheme is estimated. Finally, we estimate the localization error. Then, using the L<sup>2</sup> theory of solvability in Sobolev spaces and in weighted Sobolev spaces, the solution of the PDE problem is approximated in space, also using finite differences. The approximation in time is considered in abstract spaces for evolution equations (making use of both the explicit and implicit schemes) and then specified to the second-order parabolic PDE problem. The rates of convergence are estimated for the approximation in space and in time.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:651579 |
| Date | January 2007 |
| Creators | Gonçalves, Fernando |
| Publisher | University of Edinburgh |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/1842/10926 |
Page generated in 0.0023 seconds