Segmented risk sharing in a continuous-time setting

The economy we study is comprised of a continuum of individuals. Each has a stochastic endowment that evolves continuously and independently of all other individuals' endowment processes. Individuals are risk averse and would therefore like to insure their endowment processes. The mutual independence of their endowment processes makes it feasible for them to obtain this insurance by pooling their endowments. We investigate whether such a scheme would survive as an equilibrium in a noncooperative setting.