Before revision:  We study the corresponding social planner problem and the case of strategic interaction between the agents . These problems are set up as stochastic control problems with both monotone and classical controls representing the cumulative contribution into the public good and the consumption of the private good, respectively. We characterize the optimal investment policies by a set of necessary and sufficient stochastic Kuhn-Tucker conditions , which in turn allow to identify a universal signal process that triggers the public good investments.
After revision:  We study 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 , which in turn allow to identify a universal signal process that triggers the public good investments.