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

Download

index.pdf

Date

2013

Author

Öz, Hacer

Metadata

Show full item record

Item Usage Stats

291
views

94
downloads

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.

Subject Keywords

Stochastic analysis., Stochastic differential equations., Time-domain analysis., Malliavin calculus.

URI

http://etd.lib.metu.edu.tr/upload/12616425/index.pdf
https://hdl.handle.net/11511/23094

Collections

Graduate School of Applied Mathematics, Thesis

Suggestions

OpenMETU
Core

Runge-Kutta scheme for stochastic optimal control problems Öz Bakan, Hacer; Weber, Gerhard Wilhelm; Yılmaz, Fikriye; Department of Financial Mathematics (2017) In this thesis, we analyze Runge-Kutta scheme for the numerical solutions of stochastic optimal control problems by using discretize-then-optimize approach. Firstly, we dis- cretize the cost functional and the state equation with the help of Runge-Kutta schemes. Then, we state the discrete Lagrangian and take the partial derivative of it with respect to its variables to get the discrete optimality system. By comparing the continuous and discrete optimality conditions, we ﬁnd a relationship between the Runge...
Backward stochastic differential equations and their applications to stochastic control problems Nalbant, Hanife Sevda; Yolcu Okur, Yeliz; Hayfavi, Azize; Department of Financial Mathematics (2013) Backward stochastic di fferential equations (BSDE) were firstly introduced by Bismut in 1973. Following decades, it has been great interest all over the world and appeared in numerious areas such as pricing and hedging claims, utility theory and optimal control theory. In 1997, El Karoui, Peng and Quenez brought together their brilliant studies in the article Backward Stochastic Di fferential Equations in Finance. They considered an adapted solution pair (Y,Z) of the following BSDE: -dY_t = f(t, Y_t,Z_t)dt ...
Periodic solutions and stability of differential equations with piecewise constant argument of generalized type Büyükadalı, Cemil; Akhmet, Marat; Department of Mathematics (2009) In this thesis, we study periodic solutions and stability of differential equations with piecewise constant argument of generalized type. These equations can be divided into three main classes: differential equations with retarded, alternately advanced-retarded, and state-dependent piecewise constant argument of generalized type. First, using the method of small parameter due to Poincaré, the existence and stability of periodic solutions of quasilinear differential equations with retarded piecewise constant...
Two studies on backward stochastic differential equations Tunç, Vildan; Sezer, Ali Devin; Department of Financial Mathematics (2012) Backward stochastic differential equations appear in many areas of research including mathematical finance, nonlinear partial differential equations, financial economics and stochastic control. The first existence and uniqueness result for nonlinear backward stochastic differential equations was given by Pardoux and Peng (Adapted solution of a backward stochastic differential equation. System and Control Letters, 1990). They looked for an adapted pair of processes {x(t); y(t)}; t is in [0; 1]} with values i...
Integral manifolds of differential equations with piecewise constant argument of generalized type Akhmet, Marat (Elsevier BV, 2007-01-15) In this paper we introduce a general type of differential equations with piecewise constant argument (EPCAG). The existence of global integral manifolds of the quasilinear EPCAG is established when the associated linear homogeneous system has an exponential dichotomy. The smoothness of the manifolds is investigated. The existence of bounded and periodic solutions is considered. A new technique of investigation of equations with piecewise argument, based on an integral representation formula, is proposed. Ap...

Citation Formats

H. Öz, “Advances and applications of stochastic Ito-Taylor approximation and change of time method in the financial sector,” M.S. - Master of Science, Middle East Technical University, 2013.