Representation of forward performance criteria with random endowment via FBSDE and its application to forward optimized certainty equivalent

TL;DR

This paper extends forward performance criteria to incomplete markets with random endowment, introducing the novel forward OCE concept and developing new FBSDE-based methods for dynamic valuation and optimal control analysis.

Contribution

It introduces the forward OCE, a new dynamic valuation tool, and develops FBSDE systems for analyzing stochastic optimization problems in this context.

Findings

Derived necessary and sufficient conditions for optimality.

Established equivalences between primal and dual problems.

Explored connections between forward OCE and entropic risk measures.

Abstract

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 \textit{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. This new approach is general and complements the existing one for forward performance criteria with random endowment based on backward stochastic partial differential equations (backward SPDEs) for the related value functions. We, also, consider representative examples for both forward performance criteria with random endowment and for forward OCE. Furthermore, for the case of exponential criteria, we investigate the connection between forward OCE and forward entropic risk measures.