Portfolio control using the linear utility and differntiable trading strategies

Two approaches to portfolio selection using the linear utility of terminal wealth are proposed. The trading strategies are constrained to be differentiable and thus of finite variation. The linear utility is extended to include a quadratic penalty on the rate of change of the number of shares held in the risky asset. This removes the risk neutrality associated with the linear utility and portfolios with a certain risk aversion are obtained. The optimization tasks are formulated as control problems and solved explicitly using the method of dynamic programming