Optimal Stopping

Joint Stochastic Optimal Control and Stopping in Aquaculture: Finite-Difference and PINN-Based Approaches ArXiv ID: 2510.02910 “View on arXiv” Authors: Kevin Kamm Abstract This paper studies a joint stochastic optimal control and stopping (JCtrlOS) problem motivated by aquaculture operations, where the objective is to maximize farm profit through an optimal feeding strategy and harvesting time under stochastic price dynamics. We introduce a simplified aquaculture model capturing essential biological and economic features, distinguishing between biologically optimal and economically optimal feeding strategies. The problem is formulated as a Hamilton-Jacobi-Bellman variational inequality and corresponding free boundary problem. We develop two numerical solution approaches: First, a finite difference scheme that serves as a benchmark, and second, a Physics-Informed Neural Network (PINN)-based method, combined with a deep optimal stopping (DeepOS) algorithm to improve stopping time accuracy. Numerical experiments demonstrate that while finite differences perform well in medium-dimensional settings, the PINN approach achieves comparable accuracy and is more scalable to higher dimensions where grid-based methods become infeasible. The results confirm that jointly optimizing feeding and harvesting decisions outperforms strategies that neglect either control or stopping. ...

Target search optimization by threshold resetting ArXiv ID: 2504.13501 “View on arXiv” Authors: Unknown Abstract We introduce a new class of first passage time optimization driven by threshold resetting, inspired by many natural processes where crossing a critical limit triggers failure, degradation or transition. In here, search agents are collectively reset when a threshold is reached, creating event-driven, system-coupled simultaneous resets that induce long-range interactions. We develop a unified framework to compute search times for these correlated stochastic processes, with ballistic- and diffusive- searchers as key examples uncovering diverse optimization behaviors. A cost function, akin to breakdown penalties, reveals that optimal resetting can forestall larger losses. This formalism generalizes to broader stochastic systems with multiple degrees of freedom. ...

Stochastic Optimal Control of Iron Condor Portfolios for Profitability and Risk Management ArXiv ID: 2501.12397 “View on arXiv” Authors: Unknown Abstract Previous research on option strategies has primarily focused on their behavior near expiration, with limited attention to the transient value process of the portfolio. In this paper, we formulate Iron Condor portfolio optimization as a stochastic optimal control problem, examining the impact of the control process ( u(k_i, τ) ) on the portfolio’s potential profitability and risk. By assuming the underlying price process as a bounded martingale within $[“K_1, K_2”]$, we prove that the portfolio with a strike structure of $k_1 < k_2 = K_2 < S_t < k_3 = K_3 < k_4$ has a submartingale value process, which results in the optimal stopping time aligning with the expiration date $τ= T$. Moreover, we construct a data generator based on the Rough Heston model to investigate general scenarios through simulation. The results show that asymmetric, left-biased Iron Condor portfolios with $τ= T$ are optimal in SPX markets, balancing profitability and risk management. Deep out-of-the-money strategies improve profitability and success rates at the cost of introducing extreme losses, which can be alleviated by using an optimal stopping strategy. Except for the left-biased portfolios $τ$ generally falls within the range of [“50%,75%”] of total duration. In addition, we validate these findings through case studies on the actual SPX market, covering bullish, sideways, and bearish market conditions. ...

A deep primal-dual BSDE method for optimal stopping problems ArXiv ID: 2409.06937 “View on arXiv” Authors: Unknown Abstract We present a new deep primal-dual backward stochastic differential equation framework based on stopping time iteration to solve optimal stopping problems. A novel loss function is proposed to learn the conditional expectation, which consists of subnetwork parameterization of a continuation value and spatial gradients from present up to the stopping time. Notable features of the method include: (i) The martingale part in the loss function reduces the variance of stochastic gradients, which facilitates the training of the neural networks as well as alleviates the error propagation of value function approximation; (ii) this martingale approximates the martingale in the Doob-Meyer decomposition, and thus leads to a true upper bound for the optimal value in a non-nested Monte Carlo way. We test the proposed method in American option pricing problems, where the spatial gradient network yields the hedging ratio directly. ...

Deep Penalty Methods: A Class of Deep Learning Algorithms for Solving High Dimensional Optimal Stopping Problems ArXiv ID: 2405.11392 “View on arXiv” Authors: Unknown Abstract We propose a deep learning algorithm for high dimensional optimal stopping problems. Our method is inspired by the penalty method for solving free boundary PDEs. Within our approach, the penalized PDE is approximated using the Deep BSDE framework proposed by \cite{“weinan2017deep”}, which leads us to coin the term “Deep Penalty Method (DPM)” to refer to our algorithm. We show that the error of the DPM can be bounded by the loss function and $O(\frac{“1”}λ)+O(λh) +O(\sqrt{“h”})$, where $h$ is the step size in time and $λ$ is the penalty parameter. This finding emphasizes the need for careful consideration when selecting the penalization parameter and suggests that the discretization error converges at a rate of order $\frac{“1”}{“2”}$. We validate the efficacy of the DPM through numerical tests conducted on a high-dimensional optimal stopping model in the area of American option pricing. The numerical tests confirm both the accuracy and the computational efficiency of our proposed algorithm. ...

Nash Equilibria in Greenhouse Gas Offset Credit Markets ArXiv ID: 2401.01427 “View on arXiv” Authors: Unknown Abstract One approach to reducing greenhouse gas (GHG) emissions is to incentivize carbon capturing and carbon reducing projects while simultaneously penalising excess GHG output. In this work, we present a novel market framework and characterise the optimal behaviour of GHG offset credit (OC) market participants in both single-player and two-player settings. The single player setting is posed as an optimal stopping and control problem, while the two-player setting is posed as optimal stopping and mixed-Nash equilibria problem. We demonstrate the importance of acting optimally using numerical solutions and Monte Carlo simulations and explore the differences between the homogeneous and heterogeneous players. In both settings, we find that market participants benefit from optimal OC trading and OC generation. ...

Optimal Retirement Choice under Age-dependent Force of Mortality ArXiv ID: 2311.12169 “View on arXiv” Authors: Unknown Abstract This paper examines the retirement decision, optimal investment, and consumption strategies under an age-dependent force of mortality. We formulate the optimization problem as a combined stochastic control and optimal stopping problem with a random time horizon, featuring three state variables: wealth, labor income, and force of mortality. To address this problem, we transform it into its dual form, which is a finite time horizon, three-dimensional degenerate optimal stopping problem with interconnected dynamics. We establish the existence of an optimal retirement boundary that splits the state space into continuation and stopping regions. Regularity of the optimal stopping value function is derived and the boundary is proved to be Lipschitz continuous, and it is characterized as the unique solution to a nonlinear integral equation, which we compute numerically. In the original coordinates, the agent thus retires whenever her wealth exceeds an age-, labor income- and mortality-dependent transformed version of the optimal stopping boundary. We also provide numerical illustrations of the optimal strategies, including the sensitivities of the optimal retirement boundary concerning the relevant model’s parameters. ...

On an Optimal Stopping Problem with a Discontinuous Reward ArXiv ID: 2311.03538 “View on arXiv” Authors: Unknown Abstract We study an optimal stopping problem with an unbounded, time-dependent and discontinuous reward function. This problem is motivated by the pricing of a variable annuity contract with guaranteed minimum maturity benefit, under the assumption that the policyholder’s surrender behaviour maximizes the risk-neutral value of the contract. We consider a general fee and surrender charge function, and give a condition under which optimal stopping always occurs at maturity. Using an alternative representation for the value function of the optimization problem, we study its analytical properties and the resulting surrender (or exercise) region. In particular, we show that the non-emptiness and the shape of the surrender region are fully characterized by the fee and the surrender charge functions, which provides a powerful tool to understand their interrelation and how it affects early surrenders and the optimal surrender boundary. Under certain conditions on these two functions, we develop three representations for the value function; two are analogous to their American option counterpart, and one is new to the actuarial and American option pricing literature. ...

Optimal Entry and Exit with Signature in Statistical Arbitrage ArXiv ID: 2309.16008 “View on arXiv” Authors: Unknown Abstract In this paper, we explore an optimal timing strategy for the trading of price spreads exhibiting mean-reverting characteristics. A sequential optimal stopping framework is formulated to analyze the optimal timings for both entering and subsequently liquidating positions, all while considering the impact of transaction costs. Then we leverages a refined signature optimal stopping method to resolve this sequential optimal stopping problem, thereby unveiling the precise entry and exit timings that maximize gains. Our framework operates without any predefined assumptions regarding the dynamics of the underlying mean-reverting spreads, offering adaptability to diverse scenarios. Numerical results are provided to demonstrate its superior performance when comparing with conventional mean reversion trading rules. ...