Optimal impulse control for a multidimensional cash management system with generalized cost functions

We consider the optimal control of a multidimensional cash management system where the cash balances fluctuate as a homogeneous diffusion process in Rⁿ${\mathbb{R}}^n$. We formulate the model as an impulse control problem on an unbounded domain with unbounded cost functions. Under general assumptions we characterize the value function as a weak solution of a quasi-variational inequality in a weighted Sobolev space and we show the existence of an optimal policy. Moreover we prove the local uniform convergence of a finite element scheme to compute numerically the value function and the optimal cost. We compute the solution of the model in two-dimensions with linear and distance cost functions, showing what are the shapes of the optimal policies in these two simple cases. Finally our third numerical experiment computes the solution in the realistic case of the cash concentration of two bank accounts made by a centralized treasury.