arXiv subjects

Hongjiang Qian

Publications and source records attributed to Hongjiang Qian.

6 recordsLinked to original sources

arXiv2026

Smoluchowski-Kramers Approximation Meets Khasminskii Averaging Principles in Nonequilibrium Random Environments I

This work establishes a simultaneous Smoluchowski-Kramers approximation and Khasminskii averaging principle for a class of second-order stochastic differential equations (SDEs) in nonequilibrium random environments. The system describes the motion of a particle of mass $m>0$ subject to external forces, friction, and noise, all of which depend on a fluctuating environment such as a stochastic heat bath. The environment is modeled by a fast-varying first-order SDE, where a parameter $0<\epsilon \ll 1$ encodes the time-scale separation. Under the scaling $m=\epsilon^2$, the slow process converges in probability to an effective dynamics with averaged drift and noise-induced coefficients. Our analysis utilizes a pathwise integration-by-parts formula and Poisson equations associated with the fast dynamics. Finally, numerical experiments are provided for demonstration.

math.PR↗

arXiv2026

Maximum Principles for Partially Observed Controls of Forward SPDEs and Backward SDEs with Jumps

This work establishes two versions of the Pontryagin-type maximum principles for partially observed optimal control of coupled forward stochastic partial differential equations (FSPDEs) and backward stochastic differential equations (BSDEs) with jumps in convex control domains. The FSPDE-BSDE system is driven by cylindrical Wiener processes, finite-dimensional Brownian motions, and compensated Poisson random measures. For systems with deterministic coefficients, a direct method is employed and particular attention is focused on establishing the well-posedness of a singular backward SPDE with jumps. For systems with random coefficients, a Malliavin calculus approach is developed. The main novelty here is the establishment of the well-posedness of an operator-valued SPDE with jumps, which provides a new stochastic flow representation for linear SPDEs with jumps.

math.OC↗

arXiv2025

Moderate Deviation Principles for Stochastic Differential Equations in Fast-Varying Markovian Environment

In this paper, we proved moderate deviation principles for a fully coupled two-time-scale stochastic systems, where the slow process is given by stochastic differential equations with small noise, while the fast process is a rapidly changing purely jump process on finite state space. The system is fully coupled in that the drift and diffusion coefficients of the slow process, as well as the jump distribution of the fast process, depend on states of both processes. Moreover, the diffusion component in the slow process can be degenerate. Our approach is based on the combination of the weak convergence method from [A. Budhiraja, P. Dupuis, and A. Ganguly, Electron. J. Probab. 23 (2018), pp. 1-33; Ann. Probab. 44 (2016), pp. 1723-1775] with Poisson equation for the fast-varying purely jump process.

math.PR↗

arXiv2025

Optimal Control of Stochastic Partial Differential Equations with Partial Observations: Stochastic Maximum Principles and Numerical Approximation

This work establishes a general stochastic maximum principle for partially observed optimal control of semi-linear stochastic partial differential equations in a nonconvex control domain. The state evolves in a Hilbert space driven by a cylindrical Wiener process and finitely many Brownian motions, while observations are in an Euclidean space having correlated noise. For convex control domain and diffusion coefficients in the state being control-independent, numerical algorithms are developed to solve the partially observed optimal control problems using stochastic gradient descent algorithm combined with finite element approximations and the branching filtering algorithm. Numerical experiments are conducted for demonstration.

math.OC↗

arXiv2025

Numerical approximations for partially observed optimal control of stochastic partial differential equations

In this paper, we study numerical approximations for optimal control of a class of stochastic partial differential equations with partial observations. The system state evolves in a Hilbert space, whereas observations are given in finite-dimensional space $\rr^d$. We begin by establishing stochastic maximum principles (SMPs) for such problems, where the system state is driven by a cylindrical Wiener process. The corresponding adjoint equations are characterized by backward stochastic partial differential equations. We then develop numerical algorithms to solve the partially observed optimal control. Our approach combines the stochastic gradient descent method, guided by the SMP, with a particle filtering algorithm to estimate the conditional distributions of the state of the system. Finally, we demonstrate the effectiveness of our proposed algorithm through numerical experiments.

math.OC↗

arXiv2024

Large Deviations Analysis For Regret Minimizing Stochastic Approximation Algorithms

Motivated by learning of correlated equilibria in non-cooperative games, we perform a large deviations analysis of a regret minimizing stochastic approximation algorithm. The regret minimization algorithm we consider comprises multiple agents that communicate over a graph to coordinate their decisions. We derive an exponential decay rate towards the algorithm's stable point using large deviations theory. Our analysis leverages the variational representation of the Laplace functionals and weak convergence methods to characterize the exponential decay rate.

math.OC↗