Suppose that in the absence of intervention the cash on hand

Suppose that in the absence of intervention, the cash on hand for a certain corporation fluctuates according to a standard Brownian motion {B(t), t Greaterthanorequalto 0}. The company manages its cash using an (s, S) policy: If the cash level ever drops to zero, it is instantaneously replenished up to level s; If the cash level ever rises up to S, sufficient cash is invested in long term securities to bring the cash-on-hand down to level s. In the long run, what fraction of cash interventions are investments of excess cash?

Solution

the fraction for cash interventions that are investments of excess cash is 3/4