Some Applications of Variational Inequalities in Mathematical Finance and Numerics

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.