Analytic and numerical research on the first-hitting time applied to economic models

Abstract

The first-hitting time is a random variable that depends on the stochastic process that is researched. In this thesis the properties of the first-hitting time of the simple random walk, arithmetic Brownian motion, geometric Brownian motion and Ornstein-Uhlenbeck process get derived theoretically and get tested numerically. The numerical test are done by using Monte Carlo simulation in Python. The goal is to derive the probability density function, probability of absorption, expectation and variance. These properties are derived for the simple random walk, arithmetic Brownian motion and geometric Brownian motion. Only the Laplace transform and probability of absorption are derived for the Ornstein-Uhlenbeck process. In the comparison between the theoretical and numerical results we conclude if the time steps and number of runs increase, the numerically derived properties of the first-hitting times converge towards the theoretical derived properties for all stochastic processes that are researched.