Representation of Forward Performance Criteria with Random Endowment via FBSDE and Its Application to Forward Optimized Certainty Equivalent

We extend the notion of forward performance criteria to settings with random endowment in incomplete markets. Building on these results, we introduce and develop the novel concept of forward optimized certainty equivalent (forward OCE) , which offers a genuinely dynamic valuation mechanism that accommodates progressively adaptive market model updates, stochastic risk preferences, and incoming claims with arbitrary maturities. In parallel, we develop a new methodology to analyze the emerging stochastic optimization problems by directly studying the candidate optimal control processes for both the primal and dual problems. Specifically, we derive two new systems of forward‐backward stochastic differential equations (FBSDEs) and establish necessary and sufficient conditions for optimality, and various equivalences between the two problems. We provide representative examples for forward performance criteria with random endowment and forward OCE. For the case of exponential criteria, we investigate the connection between forward OCE and forward entropic risk measure.