Numerical methods for simulation of stochastic differential equations

Numerical Methods for Simulation of Stochastic Differential Equations Stochastic differential equations are used to describe various processes, which are exposed to many small disturbances. These equation are widely used in to describe phenomenons from various areas, such as: physics, economics, biology, mathematics, etc. Often times finding exact solutions in stochastic differential equations is too complicated or imposible. Methods for finding numerical approximate solutions are used to solve this problem. The aim of the bachelor’s thesis - finding possible numerical solutions of stochastic differential equations using several weak and strong approximation methods and develop Python module, which would help to do that. To achieve this goal, the strong Euler-Marujama and Milstein and weak Euler, first-order and second-order Milstein approximations are examined. Using the Euler-Marujama strong approximation several possible trajectories of the geometric Brownian motion and the Ornstein-Ulenbek process have been realized. Investigating the geometric Brownian motion with different disturbances and step sizes Milstein strong approximation is compared with Euler strong approximation. Observed, that Milstein approximation error is smaller than Euler with different step sizes and the difference between errors becomes bigger when processes are exposed to bigger disturbances. Several possible trajectories of the Cox-Ingersoll-Ross and Ornstein-Ulenbek processes were also realized using strong approximations. In further studies geometric Brownian motion were used to compare weak approximations. While using Euler’s, first-order and second-order Milstein approximations, it was investigated how the error differs not only due to the chosen approximation, but also due to the selected number of trajectory realizations and step size. It has been observed that the error decreases strongly with decreasing the step, but when the step is chosen to be sufficiently “good”, the chosen number of realizations has a greater influence. It is also confirmed that the errors of Euler and Milstein’s first-order approximations are almost the same, and the errors of Milstein’s second-order approximations are smaller. All the above calculations were realized using the developed Python module, which is publicly available in Google Colab system