Impulse control and its application in portfolio and hedging with both fixed and proportional transaction costs

In this paper, we will state impulse control and its application in portfolio selection. For this purpose, we first introduce Quasi Variational Inequalities. Introducing impulse control, in stochastic stopping times we have jumps with stochastic size. Then by using approximative Markov chain, the optimal impulse control is obtained in portfolio, including European option. Finally, considering default parameters, the numerical results in optimal impulse control is obtained.