arXiv subjects

Zijiu Lyu

Publications and source records attributed to Zijiu Lyu.

4 recordsLinked to original sources

arXiv2026

The $L^q$-bounds for Derivatives of Unitary Developments of Random Continuous Geometric Rough Paths

In this paper we derive an explicit formula for derivatives of unitary developments of random continuous geometric rough paths of all orders and establish proper $L^q$-bounds for them, which allow us to study the analyticity of their characteristic functions in a quantitative manner and then prove that under some mild conditions the distributions of the signatures of random continuous geometric rough paths are determined by their expected signatures, provided the latter have positive radius of convergence. In particular, we give a partial affirmative answer to an open question posed by T. Lyons and H. Ni in ``Expected signature of Brownian motion up to the first exit time from a bounded domain'' (The Annals of Probability, 43(5), 2729-2762, 2015), concerning whether the expected signature of stopped Brownian motion determines the law of its signature.

math.PR↗

arXiv2025

Policy Transfer for Continuous-Time Reinforcement Learning: A (Rough) Differential Equation Approach

This paper studies policy transfer, one of the well-known transfer learning techniques adopted in large language models, for continuous-time reinforcement learning problems. In the case of continuous-time linear-quadratic systems with Shannon's entropy regularization, we fully exploit the Gaussian structure of their optimal policy and the stability of their associated Riccati equations. In the general case where the system has possibly non-linear and bounded dynamics, the key technical component is the stability of diffusion SDEs which is established by invoking the rough path theory. Our work provides the first theoretical proof of policy transfer for continuous-time RL: an optimal policy learned for one RL problem can be used to initialize to search for a near-optimal policy for another closely related RL problem, while achieving (at least) the same rate of convergence for the original algorithm. As a byproduct of our analysis, we derive the stability of a concrete class of continuous-time score-based diffusion models via their connection with LQRs. To illustrate the benefit of policy transfer for RL, we propose a novel policy learning algorithm for continuous-time LQRs, which achieves global linear convergence and local super-linear convergence.

cs.LG↗

arXiv2025

Mean-Field Games with Constraints

This paper introduces a framework of Constrained Mean-Field Games (CMFGs), where each agent solves a constrained Markov decision process (CMDP). This formulation captures scenarios in which agents' strategies are subject to feasibility, safety, or regulatory restrictions, thereby extending the scope of classical mean field game (MFG) models. We first establish the existence of CMFG equilibria under a strict feasibility assumption, and we further show uniqueness under a classical monotonicity condition. To compute equilibria, we develop Constrained Mean-Field Occupation Measure Optimization (CMFOMO), an optimization-based scheme that parameterizes occupation measures and shows that finding CMFG equilibria is equivalent to solving a single optimization problem with convex constraints and bounded variables. CMFOMO does not rely on uniqueness of the equilibria and can approximate all equilibria with arbitrary accuracy. We further prove that CMFG equilibria induce $O(1 / \sqrt{N})$-Nash equilibria in the associated constrained $N$-player games, thereby extending the classical justification of MFGs as approximations for large but finite systems. Numerical experiments on a modified Susceptible-Infected-Susceptible (SIS) epidemic model with various constraints illustrate the effectiveness and flexibility of the framework.

math.OC↗

arXiv2024

Restricted Path Characteristic Function Determines the Law of Stochastic Processes

A central question in rough path theory is characterising the law of stochastic processes on path spaces. It is established in [I. Chevyrev & T. Lyons, Characteristic functions of measures on geometric rough paths, Ann. Probab. 44 (2016), 4049--4082] that the characteristic function of a probability measure on group-like elements, which is a subspace of the extended tensor algebra $\mathcal{T}\left(\mathbb{R}^d\right) = \prod_{n=0}^\infty\left(\mathbb{R}^d\right)^{\otimes n}$, uniquely determines the measure. In this work, we show that the characteristic function restricted to special orthogonal Lie algebra $\mathfrak{so}(n)$ is sufficient to achieve this goal. The key to our arguments is an explicit algorithm -- as opposed to the non-constructive approach in [I. Chevyrev & T. Lyons, op. cit.] -- for determining a generic element $X \in \mathcal{T}\left(\mathbb{R}^d\right)$ from its generating function when restricting its domain to a sparse subspace of real tridiagonal skew-symmetric matrices. Our only assumption is that $X$ has a non-zero ROC, which relaxes the condition of having an infinite ROC in the literature. As an application, we propose the restricted path characteristic function distance (RPCFD), a novel distance function for probability measures on the path space that serves as the sparse counterpart of path characteristic function distance. It has enormous advantages in dimension reduction and potential in generative modeling for synthetic time series generation, validated in this paper via hypothesis testing on fractional Brownian motions.

math.PR↗