École Nationale Supérieure de
Techniques Avancées

Abstract

In this report, we will discuss the application of stochastic calculus and probability tools to determine the best tax policy to be applied by the government towards companies. We will be interested in the case of two ompanies and try to deal with two different optimization problems: maximizing the probability that the two firms will occur eternally on one hand, and maximizing the estimated number of firms that will occur eternally on the other. The aim of working with different criterions is to nderstand how the choice of the problem to be optimized affects the choice of the policy to be applied: republican or democratic in our case. First of all, we will start with some basic probability calculations using Itô’s formula and the stopped processes. Then, we introduce much more developed methods such as the stochastic optimization problems and HJB equations. Finally, we try to interpret results from the numerical observations obtained with the finite difference method to sum up all the work done and mention the limitations of the problem.