Search Results (64)

This paper investigates a robust optimal reinsurance and investment problem for an insurance company operating in a Markov-modulated financial market. The insurer’s surplus process is modeled by a diffusion process with jumps, which is correlated with financial risky assets through a common shock structure. The economic regime switches according to a continuous-time Markov chain. To address model uncertainty concerning both diffusion and jump components, we formulate the problem within a robust optimal control framework. By applying the Girsanov theorem for semimartingales, we derive the dynamics of the wealth process under an equivalent martingale measure. We then establish the associated Hamilton–Jacobi–Bellman (HJB) equation, which constitutes a coupled system of nonlinear second-order integro-differential equations. An explicit form of the relative entropy penalty function is provided to quantify the cost of deviating from the reference model. The theoretical results furnish a foundation for numerical solutions using actor–critic reinforcement learning algorithms. Full article

The intelligent transformation of the wastewater treatment industry, as a core component of the modern environmental governance system, is of decisive significance for achieving sustainable development goals. This study focuses on the issue of multi-stakeholder collaborative governance in the intelligent transformation of the wastewater treatment industry, with differential game theory as the core framework. A tripartite game model involving the government, wastewater treatment enterprises, and digital twin platforms is developed to depict the dynamic interrelations and mutual influences of strategy choices, thereby capturing the coordination mechanisms among government regulation, enterprise technology adoption, and platform support in the transformation process. Based on the dynamic optimization properties of differential games, the Hamilton–Jacobi–Bellman (HJB) equation is employed to derive the long-term equilibrium strategies of the three parties, presenting the evolutionary paths under Nash non-cooperative games, Stackelberg games, and tripartite cooperative games. Furthermore, the Sobol global sensitivity analysis is applied to identify key parameters influencing system performance, while the response surface method (RSM) with central composite design (CCD) is used to quantify parameter interaction effects. The findings are as follows: (1) compared with Nash non-cooperative and Stackelberg games, the tripartite cooperative strategy based on the differential game model achieves global optimization of system performance, demonstrating the efficiency-enhancing effect of dynamic collaboration; (2) the most sensitive parameters are β, α, μ₃, and η₃, with β having the highest sensitivity index (ST_i = 0.459), indicating its dominant role in system performance; (3) significant synergistic enhancement effects are observed among α–β, α–μ₃, and β–μ₃, corresponding, respectively, to the “technology stability–benefit conversion” gain effect, the “technology decay–platform compensation” dynamic balance mechanism, and the “benefit conversion–platform empowerment” performance threshold rule. Full article

We study stochastic production planning in capacity-constrained manufacturing systems, where feasible operating states are restricted to a convex safe-operating region. The objective is to minimize the total cost that combines a quadratic production effort with an inventory holding cost, while automatically halting production when the state leaves the safe region. We derive the associated Hamilton–Jacobi–Bellman (HJB) equation, establish the existence and uniqueness of the value function under broad conditions, and prove a concavity property of the transformed value function that yields a robust gradient-based optimal feedback policy. From an operations perspective, the stopping mechanism encodes hard capacity and safety limits, ensuring bounded risk and finite expected costs. We complement the analysis with numerical methods based on finite differences and illustrate how the resulting policies inform real-time decisions through two application-inspired examples: a single-product case calibrated with typical process-industry parameters and a two-dimensional example motivated by semiconductor fabrication, where interacting production variables must satisfy joint safety constraints. The results bridge rigorous stochastic control with practical production planning and provide actionable guidance for operating under uncertainty and capacity limits. Full article

In this paper, we propose a novel image restoration framework that integrates optimal control techniques with the Hamilton–Jacobi–Bellman (HJB) equation. Motivated by models from production planning, our method restores degraded images by balancing an intervention cost against a state-dependent penalty that quantifies the loss of critical image information. Under the assumption of radial symmetry, the HJB equation is reduced to an ordinary differential equation and solved via a shooting method, from which the optimal feedback control is derived. Numerical experiments, supported by extensive parameter tuning and quality metrics such as PSNR and SSIM, demonstrate that the proposed framework achieves significant improvement in image quality. The results not only validate the theoretical model but also suggest promising directions for future research in adaptive and hybrid image restoration techniques. Full article

Investors face the challenge of how to incorporate economic and financial forecasts into their investment strategy, especially in times of financial crisis. To model this situation, we consider a financial market consisting of a risk-free asset with a constant interest rate as well as a risky asset whose drift and volatility is influenced by a stochastic process indicating the probability of potential market downturns. We use a dynamic portfolio optimization approach in continuous time to maximize the expected utility of terminal wealth and solve the corresponding HJB equations for the general class of HARA utility functions. The resulting optimal strategy can be obtained in closed form. It corresponds to a CPPI strategy with a stochastic multiplier that depends on the information from the crisis indicator. In addition to the theoretical results, a performance analysis of the derived strategy is implemented. The specified model is fitted using historic market data and the performance is compared to the optimal portfolio strategy obtained in a Black–Scholes framework without crisis information. The new strategy clearly dominates the BS-based CPPI strategy with respect to the Sharpe Ratio and Adjusted Sharpe Ratio. Full article

With the increasing deployment of Unmanned Aerial Vehicle (UAV) swarms in uncertain and dynamically changing environments, optimal cooperative control has become essential for ensuring robust and efficient system coordination. To this end, this paper designs a data-driven optimal bipartite containment tracking control scheme for multi-UAV systems under compound uncertainties. A novel Dynamic Iteration Regulation Strategy (DIRS) is proposed, which enables real-time adjustment of the learning iteration step according to the task-specific demands. Compared with conventional fixed-step data-driven algorithms, the DIRS provides greater flexibility and computational efficiency, allowing for better trade-offs between the performance and cost. First, the optimal bipartite containment tracking control problem is formulated, and the associated coupled Hamilton–Jacobi–Bellman (HJB) equations are established. Then, a data-driven iterative policy learning algorithm equipped with the DIRS is developed to solve the optimal control law online. The stability and convergence of the proposed control scheme are rigorously analyzed. Furthermore, the control law is approximated via the neural network framework without requiring full knowledge of the model. Finally, numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed DIRS-based optimal containment tracking scheme for multi-UAV systems, which can reduce the number of iterations by 88.27% compared to that for the conventional algorithm. Full article

In this paper, the problem of pitching attitude finite-horizon optimization for aircraft is posed with system uncertainties, external disturbances, and input constraints. First, a neural network (NN) and a nonlinear disturbance observer (NDO) are employed to estimate the value of system uncertainties and external disturbances. Taking input constraints into account, an auxiliary system is designed to compensate for the constrained input. Subsequently, the backstepping control containing NN and NDO is used to ensure the stability of systems and suppress the adverse effects caused by the system uncertainties and external disturbances. In order to avoid the derivation operation in the process of backstepping, a dynamic surface control (DSC) technique is utilized. Simultaneously, the estimations of the NN and NDO are applied to derive the backstepping control law. For the purpose of achieving finite-horizon optimization for pitching attitude control, an adaptive method termed adaptive dynamic programming (ADP) with a single NN-termed critic is applied to obtain the optimal control. Time-varying feature functions are applied to construct the critic NN in order to approximate the value function in the Hamilton–Jacobi–Bellman (HJB) equation. Furthermore, a supplementary term is added to the weight update law to minimize the terminal constraint. Lyapunov stability theory is used to prove that the signals in the control system are uniformly ultimately bounded (UUB). Finally, simulation results illustrate the effectiveness of the proposed finite-horizon optimal attitude control method. Full article

This paper investigates a multiple unmanned aerial vehicle (UAV) enabled network for supporting emergency communication services, where each drone acts as a base station (also called the drone small cell (DSC)). The novelty of this paper is that a mean field game (MFG)-based strategy is conceived for jointly controlling the three-dimensional (3D) locations of these drones to guarantee the distortion requirement of lossy communications, while considering the inter-cell interference and the flight energy consumption of drones. More explicitly, we derive the Hamilton–Jacobi–Bellman (HJB) and Fokker–Planck–Kolmogorov (FPK) equations, and propose an algorithm where both the Lax–Friedrichs scheme and the Lagrange relaxation are invoked for solving the HJB and FPK equations with 3D control vectors and state vectors. The numerical results show that the proposed algorithm can achieve a higher access rate with a similar flight energy consumption. Full article

This paper is devoted to the study of stochastic optimal control of averaged stochastic differential delay equations (SDDEs) with semi-Markov switchings and their applications in economics. By using the Dynkin formula and solution of the Dirichlet–Poisson problem, the Hamilton–Jacobi–Bellman (HJB) equation and the inverse HJB equation are derived. Applications are given to a new Ramsey stochastic models in economics, namely the averaged Ramsey diffusion model with semi-Markov switchings. A numerical example is presented as well. Full article

In recent years, high-frequency trading has become increasingly popular in financial markets, making the dynamic modeling of the limit book and the optimization of market maker strategies become key topics. However, existing studies often lacked detailed descriptions of order books and failed to fully characterize the optimal decisions of market makers in complex market environments, especially in China’s A-share market. Based on Markov queue theory, this paper proposes the dynamic model of the limit order and the optimal strategy of the market maker. The model uses a state transition probability matrix to refine the market diffusion state, order generation, and trading process and incorporates indicators such as optimal quote deviation and restricted order trading probability. Then, the optimal control model is constructed and the reference strategy is derived using the Hamilton–Jacobi–Bellman (HJB) equation. Then, the key parameters are estimated using the high-frequency data of Ping An Bank for a single trading day. In the empirical aspect, the six-month high-frequency trading data of 114 representative stocks in different market states such as the bull market and bear market in China’s A-share market were selected for strategy verification. The results showed that the proposed strategy had robust returns and stable profits in the bull market and that frequent capture of market fluctuations in the bear market can earn relatively high returns while maintaining 50% of the order coverage rate and 66% of the stable order winning rate. Our study used Markov queuing theory to describe the state and price dynamics of the limit order book in detail and used optimization methods to construct and solve the optimal market maker strategy. The empirical aspect broadens the empirical scope of market maker strategies in the Chinese market and studies the stability and effectiveness of market makers in different market states. Full article

This paper is dedicated to studying the optimal investment proportions of three types of assets with symmetry, namely, risky assets, risk-free assets, and wealth management products, when the stochastic expenditure process follows a jump-diffusion model. The stochastic expenditure process is treated as an exogenous cash flow and is assumed to follow a stochastic differential process with jumps. Under the Cox–Ingersoll–Ross interest rate term structure, it is presumed that the prices of multiple risky assets evolve according to a multi-dimensional geometric Brownian motion. By employing stochastic control theory, the Hamilton–Jacobi–Bellman (HJB) equation for the household portfolio problem is formulated. Considering various risk-preference functions, particularly the Hyperbolic Absolute Risk Aversion (HARA) function, and given the algebraic form of the objective function through the terminal-value maximization condition, an explicit solution for the optimal investment strategy is derived. The findings indicate that when household investment behavior is characterized by random expenditures and symmetry, as the risk-free interest rate rises, the optimal proportion of investment in wealth-management products also increases, whereas the proportion of investment in risky assets continually declines. As the expected future expenditure increases, households will decrease their acquisition of risky assets, and the proportion of risky-asset purchases is sensitive to changes in the expectation of unexpected expenditures. Full article

In the digital economy era, information symmetry, transparency, and traceability in food supply chains have increasingly garnered consumer attention. To motivate supply chain members to engage in product traceability, this paper examines the competitive and cooperative dynamics among participants in the food supply chain over continuous time. By developing a differential game model involving manufacturers and retailers with three decision-making modes, we solve the model using the Hamilton–Jacobi–Bellman (HJB) equation and perform a simulation analysis to assess the impact of different modes on overall supply chain profits. Additionally, we analyze how various parameters affect the profits of manufacturers and retailers. The key findings of this study indicate that centralized decision-making enhances the overall benefits of the food supply chain. Among the three decision-making models, the cost-sharing model proves to be the optimal approach, as it leads to a Pareto improvement in the profits of both manufacturers and retailers. These conclusions provide valuable insights for supply chain members seeking to optimize product traceability and enhance supply chain efficiency, as well as for government authorities involved in traceable supply chain governance. Full article

In this paper, we address the utility maximization problem of an infinitely lived agent who has the option to increase their income. The agent can increase their income at any time, but doing so incurs a wealth cost proportional to the amount of the increase. To prevent the agent from infinitely increasing their income and borrowing against future income, we additionally consider a non-negative wealth constraint that prohibits borrowing based on future income. This utility maximization problem is a mixture of stochastic control, where the agent chooses consumption and investment, and singular control, where the agent chooses a non-decreasing income process. To solve this non-trivial and challenging problem, we derive the Hamilton–Jacobi–Bellman (HJB) equation with a gradient constraint using the dynamic programming principle (DPP). Then, using the guess-and-verify method and a linearization technique, we obtain a closed-form solution to the HJB equation and, based on this, find the optimal strategy. Full article

To better simulate the prices of underlying assets and improve the accuracy of pricing financial derivatives, an increasing number of new models are being proposed. Among them, the Lévy process with jumps has received increasing attention because of its capacity to model sudden movements in asset prices. This paper explores the Hamilton–Jacobi–Bellman (HJB) equation with a fractional derivative and an integro-differential operator, which arise in the valuation of American options and stock loans based on the Lévy-$\alpha$-stable process with jumps model. We design a fast solution strategy that includes the policy iteration method, Krylov subspace method, and banded preconditioner, aiming to solve this equation rapidly. To solve the resulting HJB equation, a finite difference method including an upwind scheme, shifted Grünwald approximation, and trapezoidal method is developed with stability and convergence analysis. Then, an algorithmic framework involving the policy iteration method and the Krylov subspace method is employed. To improve the performance of the above solver, a banded preconditioner is proposed with condition number analysis. Finally, two examples, sugar option pricing and stock loan valuation, are provided to illustrate the effectiveness of the considered model and the efficiency of the proposed preconditioned policy–Krylov subspace method. Full article

This paper proposes an optimal tracking control scheme through adaptive dynamic programming (ADP) for a class of partially unknown discrete-time (DT) nonlinear systems based on a radial basis function neural network (RBF-NN). In order to acquire the unknown system dynamics, we use two RBF-NNs; the first one is used to construct the identifier, and the other one is used to directly approximate the steady-state control input, where a novel adaptive law is proposed to update neural network weights. The optimal feedback control and the cost function are derived via feedforward neural network approximation, and a means of regulating the tracking error is proposed. The critic network and the actor network were trained online to obtain the solution of the associated Hamilton–Jacobi–Bellman (HJB) equation within the ADP framework. Simulations were carried out to verify the effectiveness of the optimal tracking control technique using the neural networks. Full article