الخوارزمية الجينية المسرعة لحل المعادلات التفاضلية التصادفية == Accelerated Genetic Algorithm Approach for Stochastic Differential Equations
Author name:
ياسين مرزة حمزة
Supervisor name:
ايمان علي حسين
General topic:
Mathematics
Specific topic:
Mathematics
Degree:
Doctorate
University:
Mustansiriyah University - College Of Science For Girls - Department Of Mathematics
Language:
English
University location:
Baghdad
First pages:
27T1014 - p.pdf
Abstract:
A novel method that considered herein to find numerical solutions of stochastic differential equations driven by Brownian motion are presented, using genetic algorithm method based on the Backus Naur Form (BNF). In this thesis the genetic algorithm was developed to be more efficient in solution of initial boundary value problems for these types of equations. The method was called accelerated genetic algorithm method (AGA). The following aspects of the research were regarded in this thesis as : Firstly, applying the genetic algorithm to find numerical solutions of ordinary and partial differential equations and verifying the results by comparing them with their exact solutions in order to confirm the accuracy of the method and the convergence. The implementation of the program revealed that it takes a long time as well as very large number of generations required to reach the exact solution or close to it, while these generations approach about 2000 generations or more. An attempt was made to speed up the algorithm required to get the required solutions .We found that the inserting vectors, which represent the initial and boundary conditions of a problem within the first generation of the algorithm, accelerated it so dramatically by shortest time of less than 10 generations. Each experiment in this thesis was performed 20 times. In the second aspect, the algorithm was applied to find numerical solutions of linear and nonlinear Black - Scholes models, which is the pricing models (European and American). We found that the results of solution are acceptable and convergent after comparing them with that of exact solutions and numerical solutions of other methods , however, the errors are very small with maximum 30 generations and tend to approach zero in most cases.IIformula to a system of partial differential equations and then find a numerical solution of this system, and then used the back substitutions to obtain the numerical solution of the original equations . The results of the application were compared with some numerical methods (such as Euler - Maruyama method and finite difference method) to verify the results and to investigate the efficiency of the algorithm performance in solution of these types of equations. It was found that the errors are very small with maximum 20 generations and tend to approach zero in most cases. In Fourth aspect, the algorithm also applied to find the numerical solution of stochastic partial differential equations which is transformed by using Doss - Sussman transformation into partial differential equations , and then find the numerical solutions of these equations which are ,consequently used to obtain the numerical solution to the original equation. The results of the application were compared with some numerical methods (such as Sual'yev method, finite difference method) to sustain the efficiency of the algorithm in solution of these types of equations, It was found that the errors are very small with maximum 30 generations and tend to approach zero in most cases. Fifthly, studying of the weak and strong convergences, and stability of these solutions obtained by the developed (accelerated) genetic algorithm was performed by comparing the results with convergence of some other methods . We found that the (AGA) method was excellent in convergence for all considered applications. Finally, Many programs are required for (AGA) method application and for other methods such as (Euler - Maruyama, Sual'yev, and finite difference method).These program were written using MATLAB programming language (MatlabR2010b), Because it is efficient and has the facilities do not exist in other languages.
