Some Control Problems With Random Intervention Times

The singular stochastic control problems have been extensively studied in the last three decades and have found applications in many areas such as engineering, economics, finance and biology, etc[1],[3],[5],[18]. One major attraction of this formulation lies in the possibility of obtaining explicit solutions, especially when the time horizon is infinite. However, the optimal strategies in such occasions are often instantaneously and continuously, which made them very hard to implement in practice. An alternative type of control problems allow the control policies to be adjusted only at times which are multiples of some fixed positive number[19],[20]. while this discrete time formulation seems much more realistic it is usually very difficult to obtain explicit solutions even when the time horizon is infinite. Paper[21] study a kind of control problems governed by Poisson process, one feature of this model is that the control policy is essentially discrete, in the sense that the control can only be exerted at the times when the Poisson process has a jump. Firstly, we generalize the controlled state process and the cost function of the model in paper[21], it benefit for the extensively application.Secondly, we discuss the optimal investment model , We introduce the mathematical model of multiple risky investment with transaction costs and make a expectation for the next progress.