A mathematical model and the optimal strategy in the transactions between one bank and the central bank

We present a new mathematical model, based on stochastic control, for an economy formed by one bank and the Central Bank. In comparison to [6], the model allows for transactions to be discounted at different rates. We formulate and solve the bank problem of finding the optimal strategy when the underlying process is modeled by a Brownian motion with drift. We extend the model to involve the bank's asset size. In comparison to [1] and [3], we obtain that the optimal upper barrier for selling is a linear function of the asset size. As a consequence, using the double Skorokhod formula, the net purchase amount turns to be linear in the asset size.