arXiv subjects

Xihao He

Publications and source records attributed to Xihao He.

11 recordsLinked to original sources

arXiv2026

Mean-field optimal stopping with endogenous quantile cutoffs

We study a mean-field optimal stopping problem with an endogenous population-level shutdown. All remaining agents stop when the survival mass falls below a prescribed threshold. We recast the discontinuous objective as the singular, nonconvex constraint that the survival mass lie in $\{0\}\cup[\alpha,1]$. We prove the equivalence of strong and weak values via an approximation and the existence of an optimal rule via compactness and penalization. We also prove a dynamic programming principle. The value is continuous away from the critical boundary but may be discontinuous at the boundary itself. Under strict initial feasibility, finite-population values converge to the mean-field value. In the same regime, the laws of near-optimal empirical measures are tight and every mean-field optimizer admits a recovery sequence. At the threshold, however, finite-population convergence may fail.

math.OC↗

arXiv2026

Sharp Wasserstein Convergence Rates for Empirical Path Laws of It\^o Processes

We establish the sharp logarithmic order $(\log N)^{-1/2}$ for the expected $p$-Wasserstein distance, induced by the supremum norm, between the empirical law of $N$ independent copies of a continuous It\^o process and their common path law. We only assume that the initial condition and the drift and diffusion integrands are controlled by a time-uniform random upper bound with a finite $\rho$-moment for some $\rho>p\geq1$. Under this assumption, we use an adaptive random time interval partition argument, which leads to a $(\log n)^{-1/2}$ functional quantization rate. A general transfer principle then converts the quantization estimate into a mean estimate and nonasymptotic deviation bounds for equal-weight empirical laws. Applications include empirical path-law estimates for path-dependent SDEs and a path-space propagation-of-chaos estimate for path-dependent McKean--Vlasov interacting particle systems.

math.PR↗

arXiv2026

Quantitative Particle Approximation for Controlled Nonlinear Filtering

We estimate convergence rates of value functions for particle approximations of a controlled nonlinear filtering problem. The state is a McKean--Vlasov diffusion on the flat torus, driven by hidden idiosyncratic noise and observed common noise. The filter---the conditional law of the state given the observations---serves as the state variable of the control problem, and the associated value function solves a second-order Hamilton--Jacobi--Bellman equation on the Wasserstein space. We approximate this problem by a centralized $N$-particle control problem with independent idiosyncratic noises and a common observation noise. The framework accommodates nonseparable rewards and controlled drifts. Since a single control is applied to the entire population, the Hamiltonian is defined by an optimization performed after integration over the population. Under smoothness of the data, uniform ellipticity, and regularity of this Hamiltonian, we establish uniform value-function error bounds of order $N^{-1/6}$ for $d=1$, $N^{-1/6}(\log N)^{1/3}$ for $d=2$, and $N^{-1/(3d)}$ for $d>2$. The proof combines a translation lift in the common-noise direction, Fourier--Wasserstein inf- and sup-convolutions, viscosity comparison, and particle derivative estimates uniform in $N$.

math.OC↗

arXiv2026

A comparison principle for Wasserstein PDEs with state- and law-dependent common noise

We prove a comparison principle for a class of second-order Hamilton--Jacobi--Bellman equations on the Wasserstein space whose second-order term is generated by a general common-noise Hessian. The main difficulty is that the relevant second-order direction is induced by a state- and measure-dependent coefficient, so the associated perturbation of the measure is no longer a translation or a fixed state-dependent transformation. We introduce a nonlinear flow of measures and use it to transform the Wasserstein-space equation into an augmented equation on $[0,T]\times \mathcal P_2(\mathbb R)\times\mathbb R$, where the general Hessian becomes an ordinary second derivative in the auxiliary variable. The construction may be viewed as a measure-dependent Lamperti transform: it removes the common-noise direction at the level of the equation, but unlike the classical one-dimensional Lamperti transform it permits degeneracy of the coefficient and dependence on the conditional law. We establish the spatial, measure-derivative, and negative-Sobolev estimates for this flow that are needed in the viscosity argument. Under structural assumptions on the transformed Hamiltonian, these estimates yield a Crandall--Ishii type comparison theorem for semicontinuous viscosity sub- and supersolutions. This gives, to the best of our knowledge, the first viscosity comparison framework of this kind for the filtering-driven equations considered here, and opens a new class of second-order PDEs on spaces of measures with state- and law-dependent common-noise directions. As an application, we identify the value function of a controlled stochastic filtering problem with state- and law-dependent common noise as the unique viscosity solution of its dynamic programming equation. We also explain how the same change-of-variable viewpoint applies to Zakai-type Kolmogorov equations on spaces of finite positive measures.

math.AP↗

arXiv2026

Mean-field games with rough common noise: the compactification approach

We study mean-field game (MFG) problems with rough common noise, in which the representative state dynamics are governed by a controlled rough stochastic differential equation driven by an idiosyncratic Brownian motion and a deterministic rough-path signal that affects the whole population. Within this new framework, we introduce a canonical weak formulation based on relaxed controls and rough martingale problems. We prove the existence of a pathwise mean-field equilibrium by developing new compactification tools that accommodate rough integration and differ substantially from classical compactification arguments in the literature. Finally, we discuss the relationship between the pathwise problem and the classical MFG problem with randomized Brownian common noise. Using the notion of a pathwise admissible set, we recast mean-field game problems with common noise as optimization problems over an extended space of probability measures. We establish an equivalent characterization of Carmona-Delarue-Lacker's weak equilibrium and, as an application, give an alternative proof of strong equilibrium without first establishing pathwise uniqueness.

math.PR↗

arXiv2026

Comparison of viscosity solutions for a class of non-linear PDEs on the space of finite nonnegative measures

We establish a comparison principle for viscosity solutions of a class of nonlinear partial differential equations posed on the space of nonnegative finite measures, thereby extending recent results for PDEs defined on the Wasserstein space of probability measures. As an application, we study a controlled branching McKean-Vlasov diffusion and characterize the associated value function as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. This yields a PDE-based approach to the optimal control of branching processes.

math.PR↗

arXiv2025

Graphon particle systems with common noise

We study a nonlinear graphon particle system driven by both idiosyncratic and common noise, where interactions are governed by a graphon and represented as positive finite measures. Each particle evolves via a McKean-Vlasov-type SDE with graphon-weighted conditional laws. We prove a law of large numbers for the empirical and interaction measures, using generalized Wasserstein metrics and weak convergence techniques suited for the non-Markovian structure induced by common noise.

math.PR↗

arXiv2025

Comparison for semi-continuous viscosity solutions for second order PDEs on the Wasserstein space

In this paper, we prove a comparison result for semi-continuous viscosity solutions of a class of second-order PDEs in the Wasserstein space. This allows us to remove the Lipschitz continuity assumption with respect to the Fourier-Wasserstein distance in AriX: 2309.05040 and obtain uniqueness by directly working in the Wasserstein space. In terms of its application, we characterize the value function of a stochastic control problem with partial observation as the unique viscosity solution to its corresponding HJB equation. Additionally, we present an application to a prediction problem under partial monitoring, where we establish an upper bound on the limit of regret using our comparison principle for degenerate dynamics.

math.AP↗

arXiv2023

On the limit theory of mean field optimal stopping with non-Markov dynamics and common noise

This paper focuses on a mean-field optimal stopping problem with non-Markov dynamics and common noise, inspired by Talbi, Touzi, and Zhang \cite{TalbiTouziZhang1,TalbiTouziZhang3}. The goal is to establish the limit theory and demonstrate the equivalence of the value functions between weak and strong formulations. The difference between the strong and weak formulations lies in the source of randomness determining the stopping time on a canonical space. In the strong formulation, the randomness of the stopping time originates from Brownian motions. In contrast, this may not necessarily be the case in the weak formulation. Additionally, a $(H)$-Hypothesis-type condition is introduced to guarantee the equivalence of the value functions. The limit theory encompasses the convergence of the value functions and solutions of the large population optimal stopping problem towards those of the mean-field limit, and it shows that every solution of the mean field optimal stopping problem can be approximated by solutions of the large population optimal stopping problem.

math.PR↗

arXiv2023

A mean-field version of Bank-El Karoui's representation of stochastic processes

We study a mean-field version of Bank-El Karoui's representation theorem of stochastic processes. Under different technical conditions, we establish some existence and uniqueness results. As motivation and first applications, our mean-field representation results provide a unified approach to study different Mean-Field Games (MFGs) in the setting with common noise and multiple populations, including the MFG of timing, the MFG with singular control, etc. As a crucial technical step, we provide a stability result on the classical Bank-El Karoui's representation theorem, which has its own interests and other applications, such as in deriving stability results of the optimizers (in the strong sense) for a class of optimal stopping problems and singular control problems.

math.PR↗

arXiv2021

An exit contract optimization problem

We study an exit contract design problem, where one provides a universal exit contract to multiple heterogeneous agents, with which each agent chooses an optimal (exit) stopping time. The problem consists in optimizing the universal exit contract w.r.t. some criterion depending on the contract as well as the agents' exit times. Under a technical monotonicity condition, and by using Bank-El Karoui's representation of stochastic processes, we are able to transform the initial contract optimization problem into an optimal control problem. The latter is also equivalent to an optimal multiple stopping problem and the existence of the optimal contract is proved. We next show that the problem in the continuous-time setting can be approximated by a sequence of discrete-time ones, which would induce a natural numerical approximation method. We finally discuss the optimaization problem over the class of all Markovian and/or continuous exit contracts.

math.PR↗