Optimal investment strategies in financial asset portfolios

By the scheme offered by Merton the problem of the financial asset portfolio management in two versions is discussed: 1) except risky assets the investor in the financial market has access to some riskless asset, 2) one trades in the market only risky assets. It is considered that the dynamics of the prices of risky assets follows the Ito processes and the portfolio diversification can change continuously in time. The optimal strategies are determined in explicit form in the sense of the maximal utility of portfolio diversification taking into account that during time the investor consumes a part of this wealth.