Numerical schemes based on geometric brownian motion for the stochastic differential equations

Stochastic Differential Equations (SDEs) are effectively utilized in modeling financial, biological, and physical systems. Due to the noise term they contain, these equations are classified differently from deterministic differential equations. Since obtaining exact solutions for Stochastic Differential Equations is generally not feasible, the development and analysis of numerical approaches are of significant importance. In this thesis, after presenting the fundamental concepts related to Stochastic Differential Equations, classical numerical methods used for these equations are introduced, followed by numerical tests of these methods. In the case where the drift and diffusion coefficient functions in SDEs are semi-linear, methods based on Geometric Brownian Motion have been examined under global Lipschitz and local Lipschitz conditions. The performance comparisons of the examined methods are conducted through numerical tests. Convergence analysis has been conducted for the case where the diffusion coefficient function is linear.