Stochastic Calculations with Applications to Finance

This thesis presents a variety of probabilistic and stochastic calculations related to the

Ornstein-Uhlenbeck process, the weighted self-normalized sum of exchangeable variables,

various operators defined on the Wiener space and Greeks in mathematical finance.

First, we discuss some properties of the weighted self-normalized sum of exchangeable

variables. Then we show two methods to compute the different order moments of the

Brownian motion via the definition of expactation and the so-called Malliavin calculus,

repectively. We also show how to compute the different order moments of the Ornstein-

Uhlenbeck process by using Itô calculus and generlize it to the Itô processes of the Ornstein-

Uhlenbeck type.

Finally we show how to apply the Malliavin calculus to compute different operators

defined on the Wiener space such as the derivative opertor, the divergence opertor, the infinitesimal

generator of the Ornstein-Uhlenbeck semigroup and the associated characteristics.

We also apply Malliavin calculus to compute Greeks for European options as well as exotic

options, where the integration by parts formula provides a powerful tool. In addition,

we demonstrate the computation of Greeks for the models where we treat share price Itô

martingale models such as Wt and Wt2−t.