Optimal Stabilization Policies for Stochastic Linear Systems: The Case of Correlated Multiplicative and Additive Disturbances

In their comment on my paper [4], Chang and Stekler [2] make two points. First, they show that had I begun by specifying the stochastic version of the model in level form, in which the additive and multiplicative disturbances are uncorrelated, the corresponding disturbances in the derived deviation equation would necessarily have been correlated. This would have led to an adjustment of the control law derived in [4]. Secondly, they indicate how, using a discrete time approximation to the continuous time stochastic processes, they can treat the case where both the multiplier coefficient and the accelerator coefficient are stochastic simultaneously. In my paper I assumed, largely for convenience (but without considering the full implications), that the multiplicative and additive disturbances of the equation specified in deviation form are uncorrelated. The optimal stabilization policy was then derived by applying some results due to Wonham [6]. While one is quite at liberty to begin directly with a stochastic specification in deviation form, in most economic applications it is more natural to postulate the underlying relation (or set of relations) in levels and to transform the system to deviations in the process of deriving the solution. In view of the ChangStekler observation that such a transformation will almost inevitably induce correlation between the additive and multiplicative disturbances, it becomes important to extend the theory of optimal linear stabilization policy to include this case. This is the purpose of the present note. We shall develop the policy using discrete time. As well as being probably more relevant for short-run stabilization problems, this has the important advantage that the stochastic disturbances need not be white noise. Thus the stochastic terms appearing in the dynamics of the system could be composite expressions, such as products, enabling us to treat any nuimber of stochastic disturbances simultaneously, in a similar fashion to Chang and Stekler. Unlike them, however, we are not viewing the discrete formulation as being an approximation to any continuous process. The nature of the problem can be seen by considering the following first-order linear stochastic system, specified in level form