We consider a portfolio optimization problem for financial markets described by semi-martingales with independent increments and jumps defined through Lévy processes. For this problem we show the corresponding verification theorem and construct the optimal consumption/investment strategies. For the power utility functions we find the optimal strategies in the explicit form and then we apply these strategies to markets with transaction costs. Based on the Leland-Lepinette approach we develop asymptotic optimal investment and consumption method when the number of portfolio revision tends to infinity. Finally, we cared out Monte Carlo simulations to illustrate numerically the obtained theoretical results in practice.