Stochastic control problems where small intervention costs have big effects
This paper deals with impulse control problems with the cost given by \(c+\lambda|\xi|\), where \(\xi\) is the size of the impulse and \(c\) and \(\lambda\) are positive constants. The author investigates how the value function \(V_c(y)\) depends on \(c\), where \(y\) is the starting point of the system, and he shows that (1) as \(c\) tends to \(0\), \(V_c\) converges to the value function for the corresponding singular stochastic control problem, (2) \(V_c\) is not differentiable at \(c=0\), precisely \({dV_c\over dc} (y)= \infty\) for all \(y\). This means that going from \(c=0\) to a small positive \(c\) can have a big effect on the value function.
scientific article; zbMATH DE number 1369642