This paper deals with the optimal control of grid-connected Battery Energy Storage Systems (BESSs) operating for energy arbitrage. An important issue is that BESSs degrade over time, according to their use, and thus they are usable only for a limited number of cycles. Therefore, the time horizon of the optimization depends on the actual operation of the BESS. We focus on Li-ion batteries and use an empirical model to describe battery degradation. The BESS model includes an equivalent circuit for the battery and a simplified model for the power converter. In order to model the energy price variations, we use a linear stochastic model that includes the effect of the time-of-the-day. The problem of maximizing the revenues obtained over the BESS lifetime is formulated as a stochastic optimal control problem with a long, operation-dependent time horizon. First, we divide this problem into a finite set of subproblems, such that for each one of them, the State of Health (SoH) of the battery is approximately constant. Next, we reformulate approximately every subproblem into the minimization of the ratio of two long-time average-cost criteria and use a value-iteration-type algorithm to derive the optimal policy. Finally, we present some numerical results and investigate the effects of the energy loss parameters, degradation parameters, and price dynamics on the optimal policy.