Existence of an equilibrium with limited participation

A limited participation economy models the real-world phenomenon that some economic agents have access to more of the financial market than others. We prove the global existence of a Radner equilibrium with limited participation, where the agents have exponential preferences and derive utility from both running consumption and terminal wealth. Our analysis centers around a coupled quadratic backward stochastic differential equation (BSDE) system whose equations describe the economic agents’ stochastic control solutions and equilibrium prices. We define a candidate equilibrium in terms of the BSDE system solution and prove through a verification argument that the candidate is a Radner equilibrium with limited participation. Finally, we prove that the BSDE system has a unique solution in \({\mathcal{S}}^{\infty }\times \text{bmo}\). This work generalises the model of Basak and Cuoco (Rev. Financ. Stud. 11:309–341, 1998) to allow a stock with a general dividend stream and agents with stochastic income streams and exponential preferences. We also provide an explicit example.