Simulating stochastic differential equations using Ito-Taylor schemes

Download

index.pdf

Date

2014

Author

Baylan, Ekin

Metadata

Show full item record

Item Usage Stats

119
views

93
downloads

The aim of this work is to understand the stochastic Taylor schemes and to measure the accuracy of them by comparing their closeness to the exact solutions at the discretization points. Our assumption is that when we use the stochastic Taylor schemes with higher orders, we obtain better approximation processes to exact solutions of the stochastic differential equations. We give the stochastic Taylor schemes with different orders by regarding the convergence criteria for the stochastic differential equations. While Euler-Maruyama and Milstein schemes are derived by using the derivatives of stochastic Taylor expansion, stochastic Runge-Kutta schemes do not need these derivatives in their constructions. Therefore, we have the chance to get higher order stochastic Taylor schemes with less computational difficulties in Runge-Kutta schemes. Moreover, in the application part of the thesis, we observe that the stochastic Runge-Kutta schemes supply the best approximate processes to the exact solutions, for instance, in simulating Orsntein-Uhlenbeck process and in Monte Carlo method for option pricing models.

Subject Keywords

Stochastic differential equations., Euler characteristic., Monte Carlo method.

URI

http://etd.lib.metu.edu.tr/upload/12618003/index.pdf
https://hdl.handle.net/11511/24126

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...
Probabilistic Slope Stability Analyses Using Limit Equilibrium and Finite Element Methods Akbas, Burak; Huvaj Sarıhan, Nejan (2015-10-16) This paper compares the results of different probabilistic approaches and emphasizes the necessity of probabilistic analyses in slope stability studies. To do that, Limit Equilibrium Method (LEM) and Finite Element Method (FEM) are utilized and their outputs are compared in terms of probability of failure (PF), reliability index (RI), factor of safety (FS) and the failure surface. Lastly, concept of Random Finite Element Method (RFEM) is studied and effects of spatial correlation distance are investigated.
Asymptotic integration of impulsive differential equations Doğru Akgöl, Sibel; Ağacık, Zafer; Özbekler, Abdullah; Department of Mathematics (2017) The main objective of this thesis is to investigate asymptotic properties of the solutions of differential equations under impulse effect, and in this way to fulfill the gap in the literature about asymptotic integration of impulsive differential equations. In this process our main instruments are fixed point theorems; lemmas on compactness; principal and nonprincipal solutions of impulsive differential equations and Cauchy function for impulsive differential equations. The thesis consists of five chapters....
Efficient Analysis of Reflectarrays through the Use of Characteristic Modes Ercil, Erdinc; Alatan, Lale; Aydın Çivi, Hatice Özlem (2015-05-17) The aim of this study is to establish an efficient analysis method for reflectarrays with varying element sizes. The efficiency of the method relies on utilization of characteristic modes as macro basis functions, and thereby reducing the MoM matrix equation size dramatically. Although the reflectarrays on the focus of this study are comprised of differently sized elements, it was observed that, when same numerical characteristic mode is used to represent the current on all patches, the resultant approximat...
Advances and applications of stochastic Ito-Taylor approximation and change of time method in the financial sector Öz, Hacer; Weber, Gerhard Wilhelm; Department of Financial Mathematics (2013) 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-dimension...

Citation Formats

E. Baylan, “Simulating stochastic differential equations using Ito-Taylor schemes,” M.S. - Master of Science, Middle East Technical University, 2014.