Some Applications of Variational Inequalities in Mathematical Finance and Numerics

Abstract: This thesis contains two parts. The first part deals with a stochastic impulse control problem, subject to the restriction of a minimum time lapse in between interventions made by the controller. We prove existence of an optimal control and show that the value function of the control problem satisfies a system of quasi-variational inequalities. Furthermore, we apply the control method to price Swing options on the stock and commodity markets and to value a large position in a risky asset. In the second part we investigate a variational method for solving a class of linear parabolic partial differential equations. The method does not use time-stepping. The basic idea is to transform the non-coercive parabolic operators into equivalent coercive operators. We present one way to discretize the equations. We also give some numerical examples and results on convergence of the numerical scheme.