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Stochastic Calculations with Applications to Finance

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thesis
posted on 14.11.2019, 11:38 by Kuo Wang
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.

History

Supervisor(s)

Sergey Utev

Date of award

20/08/2019

Author affiliation

Department of Mathematics

Awarding institution

University of Leicester

Qualification level

Doctoral

Qualification name

PhD

Language

en

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