Stochastic control problems where small intervention costs have big effects

We study an impulse control problem where the cost of interfering in a stoc hastic system with an impulse of size zeta is an element of R is given by c + lambda\zeta\, where c and lambda are positive constants. We call lambda the proportional cost coefficient and c the intervention cost. We find the value/cost functi on V-c for this problem for each c > 0 and we show that lim(c-->0+) V-c = W , where W is the value function far the corresponding singular stochastic c ontrol problem. Our main result is that dV(c)/dc = infinity at c = 0. This illustrates that the introduction of an intervention cost c > 0, howev er small, into a system can have a big effect on the value function: the in crease in the value function is in no proportion to the increase in c (from c = 0).