arXiv subjects

Lijun Bo

Publications and source records attributed to Lijun Bo.

At least 19 recordsLinked to original sources

arXiv2026

Mean Field Stackelberg Game for Production and Carbon Emission Reduction with State Reflections

Global warming, driven by anthropogenic carbon emissions with transboundary pollution characteristics and irreversible damage, poses an existential threat to human society. This paper develops a novel two-level Stackelberg game with mean field interaction of controls and common noise which integrates hierarchical decision-making under a state-reflected emission dynamics. A central regulator (leader) adjusts product prices to guide $n$ heterogeneous competing regions (followers) while enforcing a hard emission cap via a reflection mechanism that models emergency reductions through a local time process. We establish the existence of an approximate Stackelberg equilibrium and perform sensitivity analysis via Monte Carlo simulations.

math.OC↗

arXiv2026

Dynamic Pricing for a Two-Sided Data Market Platform

We study a continuous-time dynamic pricing problem for a data platform that purchases raw data from privacy-sensitive providers and sells data products to consumers. The platform controls both the acquisition price offered to providers and the selling price charged to consumers. Provider and consumer arrivals are modeled by point processes whose intensities depend on the platform's current data stock, capturing feedback between data accumulation and market participation. We formulate the platform's problem as an infinite-horizon stochastic control problem with a jump-diffusion state process and derive the associated nonlinear integro-differential HJB equation. We prove that the value function is the unique viscosity solution, establish classical regularity under suitable conditions, and verify the optimal feedback pricing policy. Finally, we conduct numerical analyses to examine the influences of model parameters on the optimal pricing policies.

math.OC↗

arXiv2026

Optimal Consumption and Retirement Time under Shortfall Risk Measure

This paper studies the optimal portfolio, consumption, and endogenous early retirement problem within a benchmark tracking framework by incorporating a new relative performance evaluation. In this framework, the investor maximizes expected lifetime consumption utility while managing the maximum wealth shortfall relative to a benchmark, with shortfall-management costs that may differ before and after retirement. Mathematically, the problem is a hybrid stochastic control problem involving both regular controls and an optimal stopping time, in which the running maximum process records the investor's largest benchmark shortfall. We introduce an auxiliary reflected state process and establish an equivalent hybrid stochastic control problem. By proving the convex duality theorem, we technically transform the original problem into a two-dimensional pure optimal stopping problem with state reflection. This enables us to characterize the geometric structure of the stopping set and derive the feedback-form optimal retirement boundary, as well as optimal portfolio and consumption policies. Analytical examples and numerical simulations reveal a two-stage structure with more conservative investment and more aggressive consumption after retirement. Driven by the retirement option, the expected largest shortfall risk follows a pronounced U-shaped pattern with respect to wealth. Shortfall management costs, labor income, and leisure preference significantly influence retirement timing, investment, and consumption.

math.OC↗

arXiv2025

Dynamic Data Pricing: A Mean Field Stackelberg Game Approach

This paper studies the dynamic pricing mechanism for data products in demand-driven markets through a game-theoretic framework. We develop a three-tier Stackelberg game model to capture the hierarchical strategic interactions among key market entities: a single data buyer, an intermediary broker, and a competitive seller group. To characterize the temporal dynamics of data quality evolution, we establish a coupled system of stochastic differential equations (SDEs) where sellers' quality investments interact through mean field effects. Given exogenous pricing policies, we derive approximate Nash equilibrium adjustment strategies for competitive sellers using the mean field game (MFG) approach. The broker's optimal pricing strategy is subsequently established by solving a Stackelberg leadership problem, while the buyer's procurement policy is determined through an optimal control formulation involving conditional mean field forward-backward SDEs (FBSDEs). Under some regularity conditions, the proposed strategies are shown to collectively form an $(\epsilon_1,\epsilon_2,\epsilon_3)$-Stackelberg equilibrium.

math.OC↗

arXiv2025

Privacy Data Pricing: A Stackelberg Game Approach

Data markets are emerging as key mechanisms for trading personal and organizational data. Traditional data pricing studies -- such as query-based or arbitrage-free pricing models -- mainly emphasize price consistency and profit maximization but often neglect privacy constraints and strategic interactions. The widespread adoption of differential privacy (DP) introduces a fundamental privacy-utility trade-off: noise protects individuals' privacy but reduces data accuracy and market value. This paper develops a Stackelberg game framework for pricing DP data, where the market maker (leader) sets the price function and the data buyer (follower) selects the optimal query precision under DP constraints. We derive the equilibrium strategies for both parties under a balanced pricing function where the pricing decision variable enters linearly into the original pricing model. We obtain closed-form solutions for the optimal variance and pricing level, and determine the boundary conditions for market participation. Furthermore, we extend the analysis to Stackelberg games involving nonlinear power pricing functions. The model bridges DP and economic mechanism design, offering a unified foundation for incentive-compatible and privacy-conscious data pricing in data markets.

cs.GT↗

arXiv2025

EV Charging in Smart Grids: Mean Field Equilibrium and Approximate Non-Cooperative and Cooperative Strategies

We study the optimal charging strategies for large-scale electric vehicles in smart grids within a finite-horizon mean field game framework. We first establish the existence and uniqueness of the solution to the consistency condition equation, which characterizes the optimal charging behavior in the mean field limit. Building on this result, we construct approximate optimal charging strategies for a finite population of vehicles in both non-cooperative and cooperative settings. Finally, we provide numerical analyses that illustrate and compare the approximate strategies in the non-cooperative and cooperative games.

math.OC↗

arXiv2025

Mean Field Control with Poissonian Common Noise: A Pathwise Compactification Approach

This paper contributes to the compactification approach to study mean-field control problems with Poissonian common noise. To overcome the lack of compactness and continuity issues caused by common noise, we exploit the point process representation of the Poisson random measure with finite intensity and propose a pathwise formulation in a two-step procedure by freezing a sample path of the common noise. In the first step, we establish the existence of optimal relaxed controls in the pathwise formulation as if common noise is absent, but with finite deterministic jumping times. The second step plays the key role in our approach, which is to aggregate the optimal solutions in the pathwise formulation over all sample paths of common noise and show that it yields an optimal solution in the original model. To this end, with the help of concatenation techniques, we first develop a pathwise superposition principle in the model with deterministic jumping times, drawing a relationship between the pathwise relaxed control problem and the pathwise measure-valued control problem. We then further bridge the equivalence among different problem formulations and verify that the constructed solution under aggregation is indeed optimal in the original problem.

math.OC↗

arXiv2025

Mean Field Game of Optimal Tracking Portfolio

This paper studies the mean field game (MFG) problem arising from a large population competition in fund management, featuring a new type of relative performance via the benchmark tracking. In the $n$-player model, each agent aims to minimize the expected largest shortfall of the wealth with reference to the benchmark process, which is modeled by a linear combination of the population's average wealth process and a market index process. With a continuum of agents, we formulate the MFG problem with a reflected state process. We establish the existence of the mean field equilibrium (MFE) using the partial differential equation (PDE) approach. Firstly, by applying the dual transform, the best response control of the representative agent can be characterized in analytical form in terms of a dual reflected diffusion process. As a novel contribution, we verify the consistency condition of the MFE in separated domains with the help of the duality relationship and properties of the dual process. Moreover, based on the MFE, we construct an approximate Nash equilibrium for the $n$-player game when the number $n$ is sufficiently large.

math.OC↗

arXiv2025

Mean Field Game of Controls with State Reflections: Existence and Limit Theory

This paper studies mean field game (MFG) of controls by featuring the state-control joint distribution and the reflected state process at an exogenous stochastic reflection boundary. We contribute to the literature with a customized relaxed formulation and some new compactification arguments to establish the existence of a Markovian mean field equilibrium (MFE) in the weak sense. We consider an enlarged canonical space, utilizing the dynamic Skorokhod mapping, to accommodate the stochastic reflection boundary process. A fixed-point argument on the extended space using an extension transformation technique is developed to tackle challenges from the joint measure flow of the state and the relaxed control that may not be continuous. Furthermore, the bidirectional connections between the MFG and the $N$-player game are established. We first show that any weak limit of empirical measures induced by $\boldsymbol{\epsilon}$-Nash equilibria in $N$-player games must be supported exclusively on the set of relaxed mean field equilibria, analogous to the propagation of chaos in mean field control problems. We then prove the convergence result that a Markovian MFE in the weak sense can be approximated by a sequence of constructed $\boldsymbol{\epsilon}$-Nash equilibria in the weak sense in $N$-player games when $N$ tends to infinity.

math.OC↗

arXiv2025

Constrained mean-field control with singular controls: Existence, stochastic maximum principle and constrained FBSDE

This paper studies a class of mean-field control (MFC) problems with singular controls under general dynamic state-control-law constraints. We first propose a customized relaxed control formulation to cope with the dynamic mixed constraints and establish the existence of an optimal control using compactification argument in the proper canonical spaces to accommodate singular controls. To further characterize the optimal pair of regular and singular controls, we treat the controlled McKean-Vlasov process as an infinite-dimensional equality constraint and recast the MFC problem as an optimization problem on canonical spaces with constraints on Banach space, allowing us to derive the stochastic maximum principle (SMP) and a class of constrained BSDE using a new Lagrange multipliers method. Additionally, we investigate the uniqueness and the stability result of the solution to the constrained FBSDE associated with the constrained MFC with singular controls.

math.OC↗

arXiv2024

Mean field game problem for the optimal control of neuronal spiking activity

We study the mean field game problem for a nervous system consisting of a large number of neurons with mean-field interaction. In this system, each neuron can modulate its spiking activity by controlling its membrane potential to synchronize with others, thereby giving rise to a finite-player game problem. To address this, we first examine the corresponding mean field game problem and characterize the mean field equilibrium by solving a fixed point problem. Subsequently, leveraging the obtained mean field equilibrium, we construct an approximate Nash equilibrium for the finite-player game as the number of neurons is large.

math.OC↗

arXiv2024

Optimal consumption under relaxed benchmark tracking and consumption drawdown constraint

This paper studies an optimal consumption problem with both relaxed benchmark tracking and consumption drawdown constraint, leading to a stochastic control problem with dynamic state-control constraints. In our relaxed tracking formulation, it is assumed that the fund manager can strategically inject capital to the fund account such that the total capital process always outperforms the benchmark process, which is described by a geometric Brownian motion. We first transform the original regular-singular control problem with state-control constraints into an equivalent regular control problem with a reflected state process and consumption drawdown constraint. By utilizing the dual transform and the optimal consumption behavior, we then turn to study the linear dual PDE with both Neumann boundary condition and free boundary condition in a piecewise manner across different regions. Using the smoothfit principle and the super-contact condition, we derive the closed-form solution of the dual PDE, and obtain the optimal investment and consumption in feedback form. We then prove the verification theorem on optimality by some novel arguments with the aid of an auxiliary reflected dual process and some technical estimations. Some numerical examples and financial insights are also presented.

math.OC↗

arXiv2024

Extended mean-field control under constraints: The generalized Fritz-John conditions and Lagrangian method

This paper studies mean-field control with joint law dependence under dynamic expectation constraints and/or dynamic state-control-law constraints. We pioneer the establishment of the stochastic maximum principle (SMP) and the derivation of the backward SDE (BSDE) from the perspective of constrained optimization using the method of Lagrangian multipliers. We first propose to embed the constrained mean-field control (C-MFC) with joint-law dependence into some abstract optimization problems with constraints on Banach spaces, for which we develop the generalized Fritz-John (FJ) optimality conditions. We then prove the stochastic maximum principle (SMP) for C-MFC by transforming the FJ conditions into an equivalent stochastic first-order condition associated with a general type of constrained forward-backward SDEs (FBSDEs). Contrary to the existing literature, we treat the McKean-Vlasov SDE as an infinite-dimensional equality constraint such that the BSDE induced by the FJ first-order optimality conditions can be interpreted as the generalized Lagrange multiplier. We also employ the methodology to stochastic control and mean field game problems under dynamic constraints.

math.OC↗

arXiv2024

Extended mean-field control problems with Poissonian common noise: Stochastic maximum principle and Hamiltonian-Jacobi-Bellman equation

This paper studies mean-field control problems with state-control joint law dependence and Poissonian common noise. We develop the stochastic maximum principle (SMP) and establish its connection to the Hamiltonian-Jacobi-Bellman (HJB) equation on the Wasserstein space. The presence of the conditional joint law and its discontinuity under Poissonian common noise bring new technical challenges. To develop the SMP when the control domain is not necessarily convex, we first consider a strong relaxed control formulation that allows us to perform the first-order variation. We propose the technique of extension transformation to overcome the compatibility issues arising from the joint law in the relaxed control formulation. By further establishing the equivalence between the relaxed control and the strict control formulations, we obtain the SMP for the original problem with strict controls. In the part to investigate the HJB equation, we formulate an equivalent controlled Fokker-Planck problem subjecting to a controlled measure-valued dynamics with Poisson jumps, which allows us to derive the HJB equation of the original problem under open-loop strict controls. We also establish the connection between the SMP and the HJB equation.

math.OC↗

arXiv2024

Continuous-time q-Learning for Jump-Diffusion Models under Tsallis Entropy

This paper studies the continuous-time reinforcement learning in jump-diffusion models by featuring the q-learning (the continuous-time counterpart of Q-learning) under Tsallis entropy regularization. Contrary to the Shannon entropy, the general form of Tsallis entropy renders the optimal policy not necessarily a Gibbs measure. Herein, the Lagrange multiplier and KKT condition are needed to ensure that the learned policy is a probability density function. As a consequence, the characterization of the optimal policy using the q-function also involves a Lagrange multiplier. In response, we establish the martingale characterization of the q-function and devise two q-learning algorithms depending on whether the Lagrange multiplier can be derived explicitly or not. In the latter case, we consider different parameterizations of the optimal q-function and the optimal policy, and update them alternatively in an Actor-Critic manner. We also study two numerical examples, namely, an optimal liquidation problem in dark pools and a non-LQ control problem. It is interesting to see therein that the optimal policies under the Tsallis entropy regularization can be characterized explicitly, which are distributions concentrated on some compact support. The satisfactory performance of our q-learning algorithms is illustrated in each example.

math.OC↗

arXiv2023

On optimal tracking portfolio in incomplete markets: The reinforcement learning approach

This paper studies an infinite horizon optimal tracking portfolio problem using capital injection in incomplete market models. The benchmark process is modelled by a geometric Brownian motion with zero drift driven by some unhedgeable risk. The relaxed tracking formulation is adopted where the fund account compensated by the injected capital needs to outperform the benchmark process at any time, and the goal is to minimize the cost of the discounted total capital injection. When model parameters are known, we formulate the equivalent auxiliary control problem with reflected state dynamics, for which the classical solution of the HJB equation with Neumann boundary condition is obtained explicitly. When model parameters are unknown, we introduce the exploratory formulation for the auxiliary control problem with entropy regularization and develop the continuous-time q-learning algorithm in models of reflected diffusion processes. In some illustrative numerical example, we show the satisfactory performance of the q-learning algorithm.

q-fin.PM↗

arXiv2023

De Finetti's Poissonian Dividend Control Problem under Spectrally Positive Markov Additive Process

We study a De Finetti's optimal dividend and capital injection problem under a Markov additive model. The surplus process without dividend and capital injection is assumed to follow a spectrally positive Markov additive process (MAP). Dividend payments are made at the jump times of an independent Poisson process and capitals are injected to avoid bankruptcy. The aim of the paper is to characterize an optimal dividend and capital injection strategy that maximizes the expected total discounted dividends subtracted by the total discounted costs of capital injection. Applying the fluctuation and excursion theory for Levy processes and the stochastic control theory, we first address an auxiliary dividend and capital injection control problem with a terminal payoff under the spectrally positive Levy model. Using results obtained for this auxiliary problem and a fixed point argument for iterations induced by dynamic program, we characterize the optimal strategy of our prime control problem as a regime-modulated double-barrier Poissonian-continuous-reflection dividend and capital injection strategy. Besides, a numerical example is provided to illustrate the features of the optimal strategies. The impacts of model parameters are also studied.

math.OC↗

arXiv2023

A decomposition-homogenization method for Robin boundary problems on the nonnegative orthant

This paper studies the existence and uniqueness of a classical solution to a type of Robin boundary problems on the nonnegative orthant. We propose a new decomposition-homogenization method for the Robin boundary problem based on probabilistic representations, which leads to two auxiliary Robin boundary problems admitting some simplified probabilistic representations. The auxiliary probabilistic representations allow us to establish the existence of a unique classical solution to the original Robin boundary problem using some stochastic flow analysis.

math.PR↗