Advantages of using stochastic differential equations: Brownian bridge and geometric Brownian motion

Abstract

This article exposes several advantages of using stochastic differential equations (SDE) compared to classical differential equations. It addresses the difference between the exponential growth model represented as a classical differential equation and its stochastic counterpart, known as geometric Brownian motion. Additionally, it analyzes the Brownian bridge as opposed to a straight line; the latter model is used in ecological contexts to simulate migratory trajectories as seen in Kranstauber, B., et al. (2012). The article begins with an explanation of the most relevant results from stochastic analysis, which is essential for developing a model based on an SDE. Subsequently, it proceeds to explain the simulation of these stochastic differential equations using the Euler-Maruyama method and the stochastic Runge-Kutta method. These simulations are crucial for generating confidence curves associated with the average function.