An insurance network: Nash equilibrium

We consider d insurance companies whose surplus processes are r.c.l.l. functions (like the sample paths of perturbed Levy processes). Suppose they have a treaty to diversify risk; accordingly, if one company needs a certain amount to prevent ruin, the other companies pitch in previously – agreed – upon fractions of the amount, and any shortfall is got from external sources. With each company trying to minimise its repayment liability, the situation is viewed upon as a d-person dynamic game with state space constraints and a Nash equilibrium is sought. Under certain natural conditions, it is shown that the Skorokhod problem of probability theory provides a (unique) Nash equilibrium. The thrust of the paper is entirely deterministic.