Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Gechun Liang, Yifan Sun, Thaleia Zariphopoulou

TL;DR

The paper addresses time-consistent valuation under random endowment in incomplete markets by developing a forward FBSDE framework for primal and dual problems, and introducing the forward optimized certainty equivalent (forward OCE) as a horizon-invariant, dynamically adjustable criterion. It shows that the primal and dual FBSDEs form a convex dual pair with explicit optimal controls and state-price densities, and uses the decoupling-field method to handle general stochastic-factor models. The exponential forward performance case yields a direct link between forward OCE and the forward entropic risk measure, while the complete-market and Markovian settings provide explicit solutions and intuition. Overall, the work offers a comprehensive, horizon-independent approach to forward-type valuations with random endowment and establishes a solid connection to existing backward SPDE methods, enriching the toolkit for forward-looking risk and indifference pricing.

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.

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

TL;DR

The paper addresses time-consistent valuation under random endowment in incomplete markets by developing a forward FBSDE framework for primal and dual problems, and introducing the forward optimized certainty equivalent (forward OCE) as a horizon-invariant, dynamically adjustable criterion. It shows that the primal and dual FBSDEs form a convex dual pair with explicit optimal controls and state-price densities, and uses the decoupling-field method to handle general stochastic-factor models. The exponential forward performance case yields a direct link between forward OCE and the forward entropic risk measure, while the complete-market and Markovian settings provide explicit solutions and intuition. Overall, the work offers a comprehensive, horizon-independent approach to forward-type valuations with random endowment and establishes a solid connection to existing backward SPDE methods, enriching the toolkit for forward-looking risk and indifference pricing.

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.
Paper Structure (36 sections, 20 theorems, 248 equations, 1 figure)

This paper contains 36 sections, 20 theorems, 248 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Background results
  4. The incomplete market
  5. Forward performance criteria and their convex dual representation
  6. Forward performance criteria with random endowment: an FBSDE approach
  7. Optimal policy characterization using the primal FBSDE
  8. Primal FBSDE and the dual problem
  9. Optimal state price density characterization using the dual FBSDE
  10. Dual FBSDE and the primal problem
  11. Relations between the primal and dual FBSDEs
  12. Maturity independence of the value functions in forward performance criteria with random endowment
  13. Solving the primal and dual FBSDEs with random endowment
  14. Complete market
  15. Stochastic factor model and exponential forward criteria
  16. ...and 21 more sections

Key Result

Lemma 2.1

For each $T>0$, $0\leq t\leq T$, $\pi \in \mathcal{A}_{[t,T]}$ and $q\in \mathcal{Q}_{[t,T]}$, the process is a true $\mathbb{F}$-martingale on $[t,T]$.

Figures (1)

  • Figure 1: The FBSDE approach road map of main results.

Theorems & Definitions (39)

  • Lemma 2.1
  • Definition 2.2
  • Remark 2.4
  • Remark 2.5
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • Remark 3.4
  • Theorem 3.5
  • Remark 3.6
  • ...and 29 more