Stochastic Control Problems where Small Intervention Costs Have Big Effects

We study an impulse control problem where the cost of interfering in a stochastic system with an impulse of size ζ∈ R is given by c+λ|ζ|, where c and λ are positive constants. We call λ the proportional cost coefficient and c the intervention cost . We find the value/cost function 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 for the corresponding singular stochastic control problem. Our main result is that This illustrates that the introduction of an intervention cost c>0 , however small, into a system can have a big effect on the value function: the increase in the value function is in no proportion to the increase in c (from c=0 ). Accepted 23 April 1998