The main goal of this thesis is to develop Malliavin Calculus for Lévy processes. This will be achieved by using the representation property of square integrable functions; every Lévy process can be decomposed into a Wiener and Poisson random measure part. In the first part of the thesis we prove a chaos expansion for Lévy spaces. We can then define directional derivatives in the Wiener and Poisson Random measure directions, and reach an extended Clark-Ocone-Haussmann formula. Following that we define and study the properties of the adjoint operators of the directional derivatives - the Skorohod integrals in both directions. The theoretical part is concluded by studying under which conditions a solution of a stochastic differential equation belongs to the domain of the two directional derivatives. The last part of the thesis is devoted to the applications of the developed theory to finance. This will include the explicit calculation of minimal variance hedging strategies in incomplete markets and the computation of the Greeks.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:484722 |
| Date | January 2007 |
| Creators | Petrou, Evangelia |
| Contributors | Gibbons, John |
| Publisher | Imperial College London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/10044/1/1263 |
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