Stochastic Calculations with Applications to Finance

<div>This thesis presents a variety of probabilistic and stochastic calculations related to the</div><div>Ornstein-Uhlenbeck process, the weighted self-normalized sum of exchangeable variables,</div><div>various operators defined on the Wiener space and Greeks in mathematical finance.</div><div>First, we discuss some properties of the weighted self-normalized sum of exchangeable</div><div>variables. Then we show two methods to compute the different order moments of the</div><div>Brownian motion via the definition of expactation and the so-called Malliavin calculus,</div><div>repectively. We also show how to compute the different order moments of the Ornstein-</div><div>Uhlenbeck process by using Itô calculus and generlize it to the Itô processes of the Ornstein-</div><div>Uhlenbeck type.</div><div>Finally we show how to apply the Malliavin calculus to compute different operators</div><div>defined on the Wiener space such as the derivative opertor, the divergence opertor, the infinitesimal</div><div>generator of the Ornstein-Uhlenbeck semigroup and the associated characteristics.</div><div>We also apply Malliavin calculus to compute Greeks for European options as well as exotic</div><div>options, where the integration by parts formula provides a powerful tool. In addition,</div><div>we demonstrate the computation of Greeks for the models where we treat share price Itô</div><div>martingale models such as Wt and Wt2−t.</div>