Seminarium i matematik: Bernt Øksendal, del 3

Hela serien

• Onsdag 24 april 2019 kl 10:15-12:00 i sal D0073, hus D, Växjö
• Torsdag 25 april 2019 kl 8:15-10:00 i sal B2034, hus B, Växjö
• Fredag 26 april 2019 kl 9:00-10:30 i sal B2034, hus B, Växjö

Abstract

The stochastic calculus of variations, now also know as Malliavin calculus, was introduced by P. Malliavin (1978) as a tool for studying the smoothness of densities of solutions of stochastic differential equations. Subsequently other applications of this theory was found. Later Ocone (1984) used Malliavin calculus to prove an explicit representation theorem for Brownian motion functionals and in a subsequent paper Karatzas and Ocone applied this to study portfolio problems in finance.

The original presentation of Malliavin was quite complicated, but subsequently simpler constructions of this theory have been found. See e.g. the book by Di Nunno et al (2009). In particular, we think that the use of white noise theory makes the theory of Malliavin calculus (in this context also known as Hida-Malliavin calculus) quite natural within the context of directional derivatives and Fréchet derivatives on the space of tempered distributions, both in the Brownian motion case and in the case of Poisson random measure.

As observed by K. Aase et al (2000) a major advantage of presenting the Malliavin calculus in the context of white noise theory is that the corresponding Hida-Malliavin derivative can be extended from the subspace D_{1,2} to all of L^2(P). This enables us to prove stronger results compared with what would be possible in the classical setting. In particular, we will prove

1. A general integration by parts formula and duality theorem for Skorohod integrals (and hence  in particular for Itô integrals),
2. a generalised fundamental theorem of stochastic calculus,
3. a general Clark-Ocone theorem, valid for all F in L^2(P),
4. a general representation for solutions of backward stochastic differential equations (BSDEs) with jumps, in terms of Hida-Malliavin derivatives, and
5. applications to e.g. mathematical finance.

The presentation is based on joint works with Nacira Agram, Linneaus University.