O Movimento Browniano: aspectos teóricos e abordagem computacional

Abstract

The main objective of this work is the resolution of a Stochastic Differential Equation(EDE) represented by a super-damped physical system, under the effect of a multiplicative noise in the system. As is well known, it is immensely difficult to solve an EDE analytically because the equation now contains a term directly related to noise.Thus, in order to develop its solution, properties present in the stochastic calculus are employed, and no longer those present in the usual calculus previously seen. As a result, iterative methods that were able to integrate systems whose modeling used such equations were investigated. Different numerical methods such as Euler-Maruyama, Euler-Heun, Milstein, Milstein without derivative and Stochastic Runge-Kutta were then programmed in the Mat lab programming language in order to computation ally solve an EDE in the particular case of an over-damped harmonic oscillator. For the results of the different numerical methods, a statistical analysis was performed to identify which method has the tendency to go faster to the equilibrium region.