Functional Ito Calculus
Specialeforsvar ved Jacob Daniel Vorstrup
Titel: Functional Ito Calculus
Resume: In this thesis, the stochastic calculus of Itō is extended to non-anticipative functionals of elements in Skorokhod space using ideas by Dupire. This is done in the full generality of càdlàg paths, based on the path-wise calculus of Föllmer and the recent developments of Fournié and Cont. An Itō decomposition formula is derived first in the analytic case and subsequently in the stochastic case for semimartingales and fractional Brownian motion, and it is shown that in the semimartingale setup the Itō integral coincides with the Föllmer integral almost surely. We also show that the functional derivative acts as a weak inverse to the stochastic Itō integral, allowing us to derive a integration by parts formula similar to that of the anticipative Malliavin calculus. As an application, sensitivities of path-dependent contingent claims are derived in the functional local volatility model using a classification scheme based on the Lie bracket, again deriving formulae that are equal the ones derived in the Malliavin calculus under Black-Scholes assumptions. Finally, a Tanaka-Meyer formula for convex functionals is proven.
Vejleder: Rofl Poulsen
Censor: David Skovmand