  In this paper we study continuous-time stochastic control problems with both
monotone and classical controls motivated by the so-called public good
contribution problem. That is the problem of n economic agents aiming to
maximize their expected utility allocating initial wealth over a given time
period between private consumption and irreversible contributions to increase
the level of some public good. We investigate the corresponding social planner
problem and the case of strategic interaction between the agents, i.e. the
public good contribution game. We show existence and uniqueness of the social
planner's optimal policy, we characterize it by necessary and sufficient
stochastic Kuhn-Tucker conditions and we provide its expression in terms of the
unique optional solution of a stochastic backward equation. Similar stochastic
first order conditions prove to be very useful for studying any Nash equilibria
of the public good contribution game. In the symmetric case they allow us to
prove (qualitative) uniqueness of the Nash equilibrium, which we again
construct as the unique optional solution of a stochastic backward equation. We
finally also provide a detailed analysis of the so-called free rider effect.
