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

2023-12-29

AI Summary

This paper addresses the dynamic valuation problem under stochastic returns and stochastic endowments in incomplete markets. Methodologically, it introduces the novel concept of forward optimization certainty equivalent (forward OCE) and develops a dynamic utility modeling framework that adapts in real time to market evolution, drifting risk preferences, and arbitrary maturity claims. It proposes two original forward–backward stochastic differential equation (FBSDE) systems, yielding necessary and sufficient conditions for optimality—and their dual characterization—thereby overcoming key limitations of conventional backward stochastic PDE approaches in non-Markovian settings and under non-exponential preferences. Theoretical contributions include: (i) a rigorous characterization of forward performance criteria under stochastic endowments; (ii) an analytical connection between forward OCE and forward entropy risk measures under exponential preferences; and (iii) a new paradigm for dynamic valuation and risk management that is both theoretically rigorous and numerically implementable.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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 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 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 forward OCE, and for the case of exponential criteria, we investigate the connection between forward OCE and forward entropic risk measures.

Problem

• Extending forward performance criteria to incomplete markets with random endowment
• Developing forward optimized certainty equivalent for dynamic valuation mechanisms
• Establishing optimality conditions via novel forward-backward stochastic differential equations

Innovation

• Forward OCE enables dynamic valuation with adaptive updates
• FBSDE systems establish optimality conditions for control processes
• Methodology complements backward SPDE approaches for performance criteria