Recursive Optimal Stopping with Poisson Stopping Constraints

2024-07-25

AI Summary

This paper investigates a recursive optimal stopping problem under Poisson process constraints: stopping is permitted only at Poisson jump times, and each jump simultaneously affects the stopping rule, the structure of the objective functional, and the model coefficients. Motivated by modeling stochastic intervention decisions—such as constrained American options—in nonlinear markets, we first integrate the Jacod–Pham decomposition with jump-driven recursive backward stochastic differential equations (BSDEs) to derive an explicit representation of the value function over Poisson inter-arrival intervals. Furthermore, leveraging the comparison theorem for jump-type BSDEs, we uniformly handle both concave and convex generator cases. Theoretical contributions include establishing a complete structural framework for this class of recursive optimal stopping problems, achieving an explicit characterization of the value function, and providing a rigorous pricing and hedging methodology for Poisson-constrained American options in nonlinear markets.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Search and Retrieval-Augmented AI: Assisted, interactive, and conversational search

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.

Problem

• Solves recursive optimal stopping with Poisson constraints
• Decomposes problem between Poisson stopping times
• Applies solution to American option pricing

Innovation

• PBSDE with jumps for recursive stopping
• Decomposition method based on Jacod-Pham
• BSDE comparison theorem for value representation