Optimal investment with taxes : an optimal control problem with endogeneous delay

Here, we study the case where the portfolio rebalancement involves the payment of taxes on benefits. Then, the purchasing time of the asset to be sold has to be recorded in order to compute the amount of tax to be paid. In addition to the no-short-selling constraint, our model assumes that sales are subject to the first-in-first-out priority rule on sales. A precise description of the model is given in Section 2. The agent problem turns out to be a nonclassical optimal control problem with endogeneous delay and with complex nonnegativity constraint on consumption. Section 3 is devoted to the proof of the following economic appealing result. An optimal strategy can always be chosen such that the agent never sells out of his portfolio and buy new Financial assets simultaneously. Using this property, the nonnegativity constraint on consumption is simplified and reduced to a classical constraint on the controls and the state variables. Namely, the nonnegativity constraint on consumption can be expressed simply in terms of the investment and the disinvestment functions. In Section 4, we assume some additional smoothness conditions on the optimal strategy in order to derive the first-order conditions associated to the control problem of interest. The usual variational methods are adapted to handle the endogeneous delay function.