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Recursive Optimal Stopping with Poisson Stopping Constraints

Gechun Liang, Wei Wei, Zhen Wu, Zhenda Xu

TL;DR

This work addresses a recursive optimal stopping problem where stopping is constrained to Poisson intervention times, formulated via penalized backward SDEs with jumps. By employing the Jacod-Pham decomposition, the authors decompose the problem into interarrival Poisson intervals and derive a representation of the value function as the solution to a PBSDE, valid under convexity of the generator in $(y,z,c)$ and compactness of the dual domain. They develop representations in cases where the generator is independent of $(Z,C)$, linear in $(Z,C)$, and convex/concave in $(Z,C)$, using adjoint processes and duality arguments, respectively. The results are then applied to pricing American options in a nonlinear market with Poisson stopping constraints, including a staircase-payoff example and accompanying PDE characterizations. The work suggests natural extensions to infinite horizon, recursive switching, and risk-sensitive formulations, demonstrating the broad applicability of PBSDEs with jumps for constrained optimal stopping problems.

Abstract

This paper solves a recursive optimal stopping problem with Poisson stopping constraints using the penalized backward stochastic differential equation (PBSDE) with jumps. Stopping in this problem is only allowed at Poisson random intervention times, and jumps play a significant role not only through the stopping times but also in the recursive objective functional and model coefficients. To solve the problem, we propose a decomposition method based on Jacod-Pham that allows us to separate the problem into a series of sub-problems between each pair of consecutive Poisson stopping times. To represent the value function of the recursive optimal stopping problem when the initial time falls between two consecutive Poisson stopping times and the generator is concave/convex, we leverage the comparison theorem of BSDEs with jumps. We then apply the representation result to American option pricing in a nonlinear market with Poisson stopping constraints.

Recursive Optimal Stopping with Poisson Stopping Constraints

TL;DR

This work addresses a recursive optimal stopping problem where stopping is constrained to Poisson intervention times, formulated via penalized backward SDEs with jumps. By employing the Jacod-Pham decomposition, the authors decompose the problem into interarrival Poisson intervals and derive a representation of the value function as the solution to a PBSDE, valid under convexity of the generator in and compactness of the dual domain. They develop representations in cases where the generator is independent of , linear in , and convex/concave in , using adjoint processes and duality arguments, respectively. The results are then applied to pricing American options in a nonlinear market with Poisson stopping constraints, including a staircase-payoff example and accompanying PDE characterizations. The work suggests natural extensions to infinite horizon, recursive switching, and risk-sensitive formulations, demonstrating the broad applicability of PBSDEs with jumps for constrained optimal stopping problems.

Abstract

This paper solves a recursive optimal stopping problem with Poisson stopping constraints using the penalized backward stochastic differential equation (PBSDE) with jumps. Stopping in this problem is only allowed at Poisson random intervention times, and jumps play a significant role not only through the stopping times but also in the recursive objective functional and model coefficients. To solve the problem, we propose a decomposition method based on Jacod-Pham that allows us to separate the problem into a series of sub-problems between each pair of consecutive Poisson stopping times. To represent the value function of the recursive optimal stopping problem when the initial time falls between two consecutive Poisson stopping times and the generator is concave/convex, we leverage the comparison theorem of BSDEs with jumps. We then apply the representation result to American option pricing in a nonlinear market with Poisson stopping constraints.
Paper Structure (12 sections, 6 theorems, 128 equations, 2 figures)

This paper contains 12 sections, 6 theorems, 128 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary and main result
  3. Recursive optimal stopping problem with Poisson stopping constraints
  4. PBSDE with jumps
  5. Decomposition for PBSDE with jumps
  6. Main result
  7. Proof of Theorem \ref{['result-zc-nonlinear']}
  8. Representation for the case with generator independent of Z and C
  9. Representation for the case with generator linear with respect to Z and C
  10. Representation for the case with convex/concave generator
  11. Application to American option pricing in a nonlinear market with Poisson stopping constraints
  12. Conclusion and extensions

Key Result

Lemma 2.3

\newlabelBSDE-RandomDuration0 Suppose that (H1)(H2) hold. The BSDE with jumps FBSDE-Stopping admits a unique adapted solution $({X}, \tilde{Y}^\tau, \tilde{Z}^\tau, \tilde{C}^\tau) \in S_\mathbb{G}^2(0, T;\mathbb{R}^n) \times S_\mathbb{G}^2(0, T;\mathbb{R}) \times L_\mathbb{G}^{2,p}(0, T;\mathbb{R

Figures (2)

  • Figure 1: The asymptotic behaviours of the value functions and the free boundaries with respect to $\lambda$. Parameters: $\sigma = 0.2, \underline{r} = 0.02, \overline{r} = 0.05, \eta= 0.1, T = 5, k=4.$
  • Figure 2: The value functions and the free boundaries at Poisson random times. Parameters: $\sigma = 0.2, \underline{r} = 0.02, \overline{r} = 0.05, \eta= 0.1, T = 5, k=4, \lambda=0.2.$

Theorems & Definitions (18)

  • Example 1.1: American option pricing in a nonlinear market with Poisson stopping constraints
  • Lemma 2.3
  • Lemma 2.5
  • Theorem 2.6
  • Proof 1
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.10
  • Remark 2.11
  • Remark 2.12
  • ...and 8 more