Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.24439

Dynamic Programming for Value-at-risk Markov Decision Processes in Continuous Spaces

Xian Chen·Erick Delage·Weikai Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Value-at-Risk (VaR) is a widely adopted quantile-based risk measure in practice, yet incorporating VaR objectives into Markov decision processes (VaR MDPs) remains a longstanding challenge. Recent work has derived dynamic programming equations for VaR MDPs in finite state and action spaces but left the case of more general spaces unresolved. This paper establishes, for the first time, the validity of these equations in continuous spaces, for both finite- and infinite-horizon settings. Such cases naturally arise in many real-world dynamic decision-making problems. As an application, we study an optimal execution problem with a VaR objective and demonstrate that our framework enables the characterization of optimal dynamic strategies, resolving an open question in the field.

Explore related subjects

math.OCq-fin.RM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xian Chen, Erick Delage, Weikai Wang. 2026-10-17. Dynamic Programming for Value-at-risk Markov Decision Processes in Continuous Spaces. https://arxiv.org/abs/2610.24439

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Achieving Linear Speedup and Near-Optimal Complexity for Decentralized Optimization over Row-stochastic Networks

A key challenge in decentralized optimization is determining the optimal convergence rate and designing algorithms to achieve it. While this problem has been extensively addressed for doubly-stochastic and column-stochastic mixing matrices, the row-stochastic scenario remains unexplored. This paper bridges this gap by introducing effective metrics to capture the influence of row-stochastic mixing matrices and establishing the first convergence lower bound for decentralized learning over row-stochastic networks. However, existing algorithms fail to attain this lower bound due to two key issues: deviation in the descent direction caused by the adapted gradient tracking (GT) and instability introduced by the Pull-Diag protocol. To address descent deviation, we propose a novel analysis framework demonstrating that Pull-Diag-GT achieves linear speedup, the first such result for row-stochastic decentralized optimization. Moreover, by incorporating a multi-step gossip (MG) protocol, we resolve the instability issue and attain the lower bound, achieving near-optimal complexity for decentralized optimization over row-stochastic networks.

math.OC↗
arXiv2026

Muon Dynamics as a Spectral Wasserstein Flow

Gradient normalization can stabilize deep-learning optimization, and spectral normalizations are especially natural for matrix-shaped parameter blocks; Muon is the motivating example. We study an idealized deterministic, continuous-time, vanishing-momentum version of this idea in the mean-field regime, where wide models are represented by probability measures on parameter space. Starting from normalized matrix flows, we introduce Spectral Wasserstein distances indexed by norms $γ$ on positive semidefinite matrices: the trace norm recovers the classical $W_2$ distance, the operator norm gives the Muon geometry, and Schatten norms interpolate between them. We develop a static Kantorovich formulation, a max--min robust-cost representation, and Gaussian reductions extending the Bures formula. For monotone norms, we prove equivalence with a Benamou--Brenier formulation and obtain a gradient-flow interpretation of normalized mean-field training dynamics. Numerical experiments illustrate the resulting flows for MMD objectives, Gaussian reductions, two-layer ReLU models, and shallow attention.

math.OC↗