Advances and applications of stochastic Ito-Taylor approximation and change of time method in the financial sector

Download
2013
Öz, Hacer
In this thesis, we discuss two different approaches for the solution of stochastic differential equations (SDEs): Ito-Taylor method (IT-M) and change of time method (CT-M). First approach is an approximation in space-domain and the second one is a probabilistic transformation in time-domain. Both approaches may be considered to substitute SDEs for more “practical” representations and solutions. IT-M was most studied for one-dimensional SDEs. The main aim of this work is to extend the theory of one-dimensional IT-M to the higher-dimensional SDEs. After covering IT-M for systems of SDEs with uncorrelated Brownian motions, we also consider the systems of SDEs with correlated Brownian motions. Then, discretization schemes are given and prepared to solve the systems of SDEs. As for the second approach, CT-M is discussed briefly. After this, applications of CT-M and IT-M are considered, especially, for most famous models, e.g., Cox-Ingersoll-Ross model and Ornstein-Uhlenbeck model. As an application of IT-M, stochastic control problems are also considered. In order to get an expression for the gradient of sensitivity, Malliavin calculus is used. Throughout the thesis we provide examples from the financial sector. This thesis ends with a conclusion and an outlook to future studies.