arXiv subjects

Zongxia Liang

Publications and source records attributed to Zongxia Liang.

At least 19 recordsLinked to original sources

arXiv2026

Equilibria for Time-inconsistent Regular-singular Control Problems

This paper studies a time-inconsistent mixed regular-singular control problem in continuous time with non-exponential discount. We seek time-consistent equilibria in an intrapersonal game framework and propose a novel definition of regular-singular equilibrium where both regular and singular controls are unified via control laws. We establish a verification theorem to provide a sufficient condition of the equilibrium. Furthermore, we propose a novel concept of mild equilibrium for which perturbations are applied separately to the regular and singular control and give the mild verification theorem. We illustrate the applicability of the theory to the effort and dividend problem and obtain an explicit equilibrium depending on the effort and dividend thresholds under a mixture of exponential discount functions and a pseudo-exponential discount function by solving the extended Hamilton--Jacobi--Bellman (HJB) system. The convexity of the value function is rigorously established, and the equilibrium conditions are verified. In addition, we establish the existence of the effort and dividend thresholds under a mixture of exponential discount functions. Numerical results reveal the impacts of exogenous parameters on the effort and dividend thresholds and the equilibrium value function, along with their economic implications.

math.OC↗

arXiv2026

Equilibrium for regular-singular control under mean-variance criterion: A unified approach via control laws

This paper studies a class of mixed regular-singular control problems under mean-variance criteria, where the drift and diffusion are allowed to depend on both the regular control and the level of singular control. We seek time-consistent equilibrium strategies in an intrapersonal game setting and propose a novel equilibrium notion where both regular and singular controls are unified via control laws, or equivalently, mappings on the augmented state space. Under which, we derive a verification theorem and necessary conditions providing a full mathematical characterization of the equilibrium. We apply the theory to a reinsurance problem. The equilibrium solution turns out to be nontrivially coupled, where the regular control depends on the level of singular control, and the free boundary of the singular control switches dynamically in accordance with the variation of the regular control expression. The free boundary is characterized in a piecewise semi-explicit manner and is proved to be $C^{1}$ across the switching point. In the degenerate case $\alpha_2=0$, the coupled solution reduces to the combination of two independent single-control equilibria and coincides with the limit as the parameter tends to zero.

math.OC↗

arXiv2026

Preference-fitting Framework: Elicited Utility Function and PHARA Approximation

The utility function plays a core role in portfolio selection, but its specific form is typically hard to elicit. We propose a definition of the elicited utility function and develop a preference-fitting method to obtain it. Basically, we use intuitive probability-wealth pairs to derive a fitted terminal wealth, a fitted portfolio and a fitted utility function, which converge to the optimal terminal wealth, the optimal portfolio and the elicited utility function of the investor, respectively. Specifically, we first establish a bijection between the utility functions and the terminal wealth functions, based on which we construct the fitted terminal wealth, and then obtain the fitted portfolio and the fitted utility function through the martingale-duality method. Next, we develop a piecewise hyperbolic absolute risk aversion (abbr. PHARA) utility approximation method, and verify the convergences in various senses: almost surely, $L^r$, uniform, etc. We demonstrate two applications of our method: obtaining asymptotically explicit portfolios and handling portfolio selection under Value-at-Risk (abbr. VaR) constraints, thereby illustrating its advantages including intuitiveness, analytical tractability, and ability to circumvent the Lagrange multiplier.

q-fin.PM↗

arXiv2026

Equilibrium singular dividend control under ambiguity aggregation of heterogeneous discount rates

This paper studies a singular dividend control problem for a firm with heterogeneous shareholders whose discount rates follow a given distribution. The central planner aggregates expected discounted payoffs using an ambiguity aggregation function $phi$, which captures shareholder heterogeneity and ambiguity attitudes but also leads to time inconsistency. To address this issue, we seek a time-homogeneous equilibrium dividend law characterized by a partition of the state space into waiting and dividend-paying regions. We provide a rigorous mathematical characterization by proving a verification theorem and deriving necessary conditions for the equilibrium law. We then analyze barrier-type equilibria, showing non-existence for a class of aggregation functions that includes power-type and logarithmic aggregation functions, and establishing existence and uniqueness under linear and exponential aggregation. In the linear case, the bounded-rate equilibrium is shown to converge to the singular barrier-type equilibrium as the dividend rate bound tends to infinity. Numerical examples illustrate the effects of discount-rate heterogeneity and ambiguity aversion on the equilibrium barrier.

math.OC↗

arXiv2026

Robust Bayesian Portfolio Optimization with Discrepancy-based Posterior Ambiguity

We study a continuous-time robust Bayesian portfolio optimization problem under drift uncertainty of risky assets. The investor learns unknown asset drifts through Bayesian filtering while considering uncertainty around posterior estimates via discrepancy-based ambiguity sets, including Wasserstein and $L^p$ distances. To address the resulting time inconsistency, we introduce a feedback-type ambiguity framework that reformulates ambiguity conditionally on observable states. This leads to a modified Hamilton--Jacobi--Bellman--Isaacs (HJBI) equation characterizing the value function and the optimal strategy. For a semi-explicit solution example, we use the exponential utility to derive a reduced semilinear parabolic PDE and establish existence of classical solutions via a verification theorem.

math.OC↗

arXiv2026

Mean Field Competition of Optimal Switching: The Vanishing Entropy Regularization Approach

This paper studies a type of rank-based mean field game in which competing agents strategically switch among multiple effort regimes. We propose an entropy regularized auxiliary problem where the switching decisions are randomized to the control of transition probability for a continuous-time finite-state Markov chain. We first establish the existence of regularized equilibrium in this auxiliary problem. Assuming the convexity of reward scheme, we then prove that the equilibrium is unique and can be approximated by a fictitious play iteration scheme. Furthermore, as the entropy regularization vanishes, we establish the convergence analysis of the regularized equilibrium towards the relaxed equilibrium in the original MFG of optimal switching. The uniqueness of the population ranking distribution under the relaxed equilibrium is also obtained given a strictly convex reward scheme.

math.OC↗

arXiv2026

Mean-field game of mean-variance portfolio optimization with peer-based risk aversion

This paper investigates a class of mean-field game (MFG) for mean-variance (MV) portfolio optimization, highlighting a new type of relative performance encoded by the peer-based risk aversion. Specifically, the risk aversion is formulated as a piecewise form that depends on whether the individual's wealth is above or below the population average, leading to a time-inconsistent MFG. Our goal is to seek a mean-field equilibrium, characterized by a forward-backward stochastic differential equation (FBSDE) system and a mean-field consistency condition. The new challenge stems from the discontinuous coefficients induced by the piecewise risk aversion. In response, we first introduce a smooth regularization technique to establish the existence of a solution to the discontinuous multidimensional FBSDE; this solution then yields the existence of an intra-personal equilibrium for the representative agent. Finally, we conclude the existence of the mean-field equilibrium in the time-inconsistent MFG by invoking fixed-point arguments and convergence analysis as the smoothing regularization vanishes.

q-fin.MF↗

arXiv2026

Time-Inconsistent Singular Control Problems with a Running Minimum Process

This paper develops a time-inconsistent and path-dependent singular control framework incorporating a running minimum process. We derive a verification theorem that characterizes equilibria under substantially weaker regularity conditions than those imposed in the existing literature, and we obtain a stronger notion of equilibrium by enlarging the class of feasible perturbations. We first establish the mathematical foundations of the framework by proving the existence and uniqueness of strong solutions to a class of Skorokhod reflection problems involving the running minimum and by characterizing admissible singular control laws. We further demonstrate the existence of an equilibrium through a dividend problem, where the running minimum leads to a highly coupled and nonlinear differential-algebraic system. For this problem, we prove the monotonicity and local concavity of the dividend boundary, thereby providing a mathematical explanation for dividend smoothing and scarring effects. Numerical simulations confirm the robustness of the equilibrium across a wide range of parameter values.

math.OC↗

arXiv2026

Optimal Underreporting and Competitive Equilibrium

This paper develops a dynamic insurance market model comprising two competing insurance companies and a continuum of insureds, and examines the interaction between strategic underreporting by the insureds and competitive pricing between the insurance companies under a Bonus-Malus System (BMS) framework. For the first time in an oligopolistic setting, we establish the existence and uniqueness of the insureds' optimal reporting barrier, as well as its continuous dependence on the BMS premiums. For the 2-class BMS case, we prove the existence of Nash equilibrium premium strategies and conduct an extensive sensitivity analysis on the impact of the model parameters on the equilibrium premiums.

q-fin.MF↗

arXiv2026

Mean-Field Games Under Model Uncertainty

We study discrete-time, finite-state mean-field games (MFGs) under model uncertainty, where agents face ambiguity about the state transition probabilities. Each agent maximizes its expected payoff against the worst-case transitions within an uncertainty set. Unlike in classical MFGs, model uncertainty renders the population distribution flow stochastic. This leads us to consider strategies that depend on both individual states and the realized distribution of the population. Our main results establish the asymptotic relationship between $N$-agent games and MFGs: every MFG equilibrium constitutes an $\varepsilon$-Nash equilibrium for sufficiently large populations, and conversely, limits of $N$-agent equilibria are MFG equilibria. We also prove the existence of equilibria for finite-agent games and construct a solvable mean-field example with closed-form solutions.

math.OC↗

arXiv2025

Reinforcement learning for irreversible reinsurance problems: the randomized singular control approach

This paper studies the continuous-time reinforcement learning for stochastic singular control with the application to an infinite-horizon irreversible reinsurance problem. The singular control is equivalently characterized as a pair of regions of time and the augmented states, called the singular control law. To encourage the exploration in the learning procedure, we propose a randomization method by considering an auxiliary singular control and entropy regularization. The exploratory singular control problem is formulated as a two-stage optimal control problem, in which the time-inconsistency issue arises in the outer problem. Existence of equilibrium singular control law for the time-inconsistent outer problem is rigorously established. Taking advantage of the solution structure, we utilize a proper parameterization and neural networks to devise the actor-critic reinforcement learning algorithm. In the numerical experiment, we show the superior convergence of parameter iterations based on the randomized equilibrium policy and illustrate how the exploration may advance the learning performance.

math.OC↗

arXiv2025

Equilibrium Investment with Random Risk Aversion: (Non-)uniqueness, Optimality, and Comparative Statics

This paper studies a continuous-time portfolio selection problem under a general distribution of random risk aversion (RRA). We provide a complete characterization of all deterministic equilibrium strategies in closed form. Our results show that the structure of the solution depends crucially on the distribution of RRA: the equilibrium is unique (if exits) when the expectation of RRA is finite, whereas an infinite expectation leads either to infinitely many equilibria or to a unique trivial one (i.e. risk-free investment). To resolve this multiplicity of equilibria, we select, among all deterministic equilibria, the one that maximizes the objective functional at the initial time. We establish a necessary and sufficient condition for the existence of such an optimal equilibrium, which is then shown to be unique and uniformly optimal. Finally, we conduct a comparative statics. Using counterexamples based on two-point distributed RRA, we demonstrate that a larger risk aversion in the sense of first-order stochastic dominance does not necessarily lead to less risky investment. Within the two-point distribution framework, we further examine the single-crossing property of equilibrium strategies and the monotonicity of the crossing time. We show that a larger risk aversion under a stronger stochastic order -- the reverse hazard rate order -- always leads to less risky investment. In addition, we analyze how the convex combination of independent and identically distributed RRAs influences investment.

q-fin.MF↗

arXiv2025

Mean-Field Game of Relative Performance Portfolio for Two Populations with Poisson Common Noise

This paper studies the mean field game (MFG) and N-player game on relative performance portfolio management with two heterogeneous populations. In addition to the Brownian idiosyncratic and common noise, the first population invests in assets driven by idiosyncratic Poisson jump risk, while the second population invests in assets subject to Poisson common noise. We establish the characterization of the mean-field equilibrium (MFE) in MFG with two populations as well as the Nash equilibrium in the $N_1+N_2$-player game. Furthermore, we prove the convergence of the Nash equilibrium in the $N_1+N_2$-player game to the MFE as the number of players in two populations tends to infinity. We also discuss some impacts on MFE by the Poisson idiosyncratic risk and Poisson common noise in the context of relative performance, compensated by some numerical examples and financial implications.

math.OC↗

arXiv2025

Equilibrium Portfolio Selection under Utility-Variance Analysis of Log Returns in Incomplete Markets

This paper investigates a time-inconsistent portfolio selection problem in the incomplete mar ket model, integrating expected utility maximization with risk control. The objective functional balances the expected utility and variance on log returns, giving rise to time inconsistency and motivating the search of a time-consistent equilibrium strategy. We characterize the equilibrium via a coupled quadratic backward stochastic differential equation (BSDE) system and establish the existence theory in two special cases: (i)the two Brownian motions driven the price dynamics and the factor process are independent with $\rho = 0$; (ii) the trading strategy is constrained to be bounded. For the general case with correlation coefficient $\rho \neq 0$, we introduce the notion of an approximate time-consistent equilibrium. Employing the solution structure from the equilibrium in the case $\rho = 0$, we can construct an approximate time-consistent equilibrium in the general case with an error of order $O(\rho^2)$. Numerical examples and financial insights are also presented based on deep learning algorithms.

q-fin.PM↗

arXiv2025

Mean Field Game with Reflected Jump Diffusion Dynamics: A Linear Programming Approach

This paper develops a linear programming approach for mean field games with reflected jump-diffusion dynamics. We first prove the equivalence between the mean field equilibria in the linear programming formulation and those in the weak relaxed control formulation under some measurability and growth conditions on model coefficients. Building upon the characterization of the occupation measure in the equivalence result, we further establish the existence of linear programming mean field equilibria under fairly general conditions on model coefficients. Finally, a numerical example is presented to illustrate the computation of a mean field equilibrium using the linear programming formulation.

math.OC↗

arXiv2025

Portfolio Selection with Costly Information Acquisition

We investigate joint optimization on information acquisition and portfolio selection within a Bayesian adaptive framework. The investor dynamically controls the precision of a private signal and incurs costs while updating her belief about the unobservable asset drift. Controllable information acquisition fails the classical separation principle of stochastic filtering. We adopt functional modeling of control to address the consequential endogeneity issues, then solve our optimization problem through dynamic programming. When the unknown drift follows a Gaussian prior, the HJB equation is often explicitly solvable via the method of characteristics, yielding sufficiently smooth classical solution to establish a verification theorem and confirm the optimality of feedback controls. In such settings, we find that the investor's information acquisition strategy is deterministic and could be decoupled from her trading strategy, indicating a weaker separation property. In some degenerate cases where classical solutions may fail, semi-explicit optimal controls remain attainable by regularizing the information cost.

math.OC↗

arXiv2025

Robust Utility Maximization with Intractable Claims under Distributional Ambiguity: A Random Distributionally Robust Optimization Approach

This paper studies a robust utility maximization problem for intractable claims under distributional ambiguity, where the distribution of the claim cannot be inferred from market information and its dependence with tradable assets is largely unknown. We extend the existing framework for intractable claims in two directions. First, we allow the marginal distribution of the claim to vary within a $\varphi$-divergence ambiguity set, capturing statistical uncertainty in its estimation. Second, we consider a general (possibly non-additive) bivariate utility function, which enables more flexible interactions between the decision and the claim beyond the classical additive specification. To analyze this problem, we adopt a random distributionally robust optimization (RDRO) formulation, which lifts the optimization to the space of joint distributions and provides a convenient representation of the coupling between the decision and the uncertain claim. We establish the existence of optimal decisions using tools from optimal transport and develop a Legendre-Fenchel duality framework that links the constrained and penalized formulations, leading to uniqueness results and tractable reformulations. Finally, we propose a numerical algorithm based on unbalanced optimal transport scaling combined with projected gradient methods, and illustrate the relationship between the parameters in the constrained and penalized formulations.

math.OC↗

arXiv2025

A Reinforcement Learning Framework for Some Singular Stochastic Control Problems

We develop a continuous-time reinforcement learning framework for a class of singular stochastic control problems without entropy regularization. The optimal singular control is characterized as the optimal singular control law, which is a pair of regions of time and the augmented states. The goal of learning is to identify such an optimal region via the trial-and-error procedure. In this context, we generalize the existing policy evaluation theories with regular controls to learn our optimal singular control law and develop a policy improvement theorem via the region iteration. To facilitate the model-free policy iteration procedure, we further introduce the zero-order and first-order q-functions arising from singular control problems and establish the martingale characterization for the pair of q-functions together with the value function. Based on our theoretical findings, some q-learning algorithms are devised accordingly and a numerical example based on simulation experiment is presented.

math.OC↗