We deal with the intervention options of the policy holder in life and pension insurance. Our approach is to let payments depend on and be driven by processes in which the policy holder is allowed to intervene. The main result is a quasi-variational inequality describing the market reserve on an insurance contract taking into account intervention options. The quasi-variational inequalty generalizes Thiele's differential equation used for calculation of reserves on a policy without intervention options. It also generalizes the classical variational inequality for the price of an American option.
Similar distributions appear in the theory of fluid queues.
The general theory is illustrated by detailed considerations of different cases for the integrated Gaussian process and the fractional Brownian motion.