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