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Infinite Anticipation Backward Stochastic Differential Equations

Guanwei Cheng, Shuzhen Yang

TL;DR

This work introduces infinite anticipation backward SDEs (IABSDEs) whose generators depend on the entire future trajectories of the solution over an unbounded horizon. It proves well-posedness under a weaker Lipschitz condition $(H1)$–$(H2)$, extends a 1D comparison theorem to the infinite-anticipation setting, and establishes a duality with SDEs with infinite delay (ISDDE) to tackle stochastic control problems. The duality yields a representation of the value function and an optimal control in terms of linear IABSDE and ISDDE dynamics, illustrating the framework’s applicability to long-memory phenomena such as climate-economy interactions. These results broaden the scope of anticipated BSDEs for infinite-horizon models and provide new tools for analysis and control of systems with enduring memory effects.

Abstract

In this paper, we introduce a new type of backward stochastic differential equations (BSDEs) with infinite anticipation, where the generator depends on the entire future values of the solution in infinite horizon. We show that the new BSDEs has a unique solution and admits a comparison result. In the end, we solve a stochastic control problem via a duality between BSDEs with infinite anticipation and stochastic differential equations (SDEs) with infinite delay.

Infinite Anticipation Backward Stochastic Differential Equations

TL;DR

This work introduces infinite anticipation backward SDEs (IABSDEs) whose generators depend on the entire future trajectories of the solution over an unbounded horizon. It proves well-posedness under a weaker Lipschitz condition –, extends a 1D comparison theorem to the infinite-anticipation setting, and establishes a duality with SDEs with infinite delay (ISDDE) to tackle stochastic control problems. The duality yields a representation of the value function and an optimal control in terms of linear IABSDE and ISDDE dynamics, illustrating the framework’s applicability to long-memory phenomena such as climate-economy interactions. These results broaden the scope of anticipated BSDEs for infinite-horizon models and provide new tools for analysis and control of systems with enduring memory effects.

Abstract

In this paper, we introduce a new type of backward stochastic differential equations (BSDEs) with infinite anticipation, where the generator depends on the entire future values of the solution in infinite horizon. We show that the new BSDEs has a unique solution and admits a comparison result. In the end, we solve a stochastic control problem via a duality between BSDEs with infinite anticipation and stochastic differential equations (SDEs) with infinite delay.

Paper Structure

This paper contains 12 sections, 10 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The Formulation and Priori Estimate
  4. The Formulation
  5. Comparative Analysis of Condition (H1)
  6. Priori Estimate
  7. The Existence and Uniqueness Theorem
  8. Comparison Theorems
  9. Stochastic Control Problem
  10. The Duality Property
  11. Optimal Control
  12. Conclusion

Key Result

Lemma 2.1

Let $g$ satisfy (a) and (b). Then for any given terminal condition $\xi \in L^2\left(\mathcal{F}_T ; \mathbb{R}^d\right)$, the BSDE has a unique solution, i.e., there exists a unique pair of $\mathcal{F}_{\text{t}}$-adapted processes $\left(Y., Z.\right)$$\in S_{\mathcal{F}}^2\left(0, T; \mathbb{R}^d\right) \times L_{\mathcal{F}}^2\left(0, T ; \mathbb{R}^{d \times m}\right)$ satisfying equation (

Theorems & Definitions (15)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 3.1
  • Example 3.2
  • Remark 3.3
  • Example 3.4
  • Lemma 3.5
  • Theorem 4.1
  • Theorem 5.1
  • ...and 5 more