Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.01585

Distribution-constrained maximum stopping of maximum type

Shuoqing Deng·Xin Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the distribution-constrained optimal stopping problem $\sup_{τ\sim μ} \mathbb E[B^*_τ]$, where $μ$ is a probability distribution on $\mathbb R_+$, and $(B^*_t)$ denotes the running maximum of a standard Brownian motion. This problem was introduced in Beiglbock et al. (PTRF, 2018), where a monotonicity principle is used to establish the optimal stopping time as the hitting time of a specific boundary. In this paper, we characterize this boundary by a variational inequality. In the spirit of Cox et al. (PTRF, 2019), we provide a novel probabilistic representation for the variational inequality as a time-reversed optimal stopping problem. A key ingredient for proving the viscosity solution property and comparison principle is a quantitative estimate near the singular corner of the time-space domain, where the initial and boundary conditions are incompatible. We then prove the optimality of the resulting hitting time through a discrete-time Snell envelope construction and a stability argument for the associated stopping times.

Explore related subjects

math.PRq-fin.MF

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shuoqing Deng, Xin Zhang. 2026-10-01. Distribution-constrained maximum stopping of maximum type. https://arxiv.org/abs/2610.01585

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

KEEP EXPLORING

Related papers

arXiv2026

Averaging principle for slow-fast systems of rough differential equations via controlled paths

In this paper we prove the strong averaging principle for a slow-fast system of rough differential equations. The slow and the fast component of the system are driven by a rather general random rough path and Brownian rough path, respectively. These two driving noises are assumed to be independent. A prominent example of the driver of the slow component is fractional Brownian rough path with Hurst parameter between 1/3 and 1/2. We work in the framework of controlled path theory, which is one of the most widely-used frameworks in rough path theory. To prove our main theorem, we carry out Khas'minskii's time-discretizing method in this framework.

math.PR↗
arXiv2026

Statistical inference for a multiscale stochastic model of enzyme kinetics via propagation of chaos

We study a class of Stochastic Differential Equations (SDEs) with jumps modeling multistage Michaelis--Menten enzyme kinetics, in which a substrate is sequentially transformed into a product via a cascade of intermediate complexes. These networks are typically high-dimensional and exhibit multiscale behavior with a strong coupling between different components, posing substantial analytical and computational challenges. In particular, the problem of statistical inference of reaction rates is significantly difficult and becomes even more intricate when direct observations of system states are unavailable and only a random sample of product formation times is observed. We address this problem in two stages. First, in a suitable scaling regime consistent with the Quasi-Steady State Approximation (QSSA), we rigorously establish a stochastic averaging principle yielding a reduced model for the product-substrate dynamics. Guided by the reduced-order dynamics, we next construct a novel Interacting Particle System (IPS) that approximates the product-substrate process at the particle level. This IPS plays a pivotal role in the inference methodology, and we prove several non-asymptotic bounds and limiting results for this system. These results facilitate the construction of an estimator based on a product-form approximate likelihood requiring only a random sample of product formation times. This approach does not need access to the system states, and we rigorously prove consistency of the estimator.

math.PR↗
arXiv2026

Large deviations for stochastic evolution equations beyond the coercive case

Using the weak convergence approach, we prove the small-noise large deviation principle (LDP) for stochastic evolution equations in an $L^2$-setting. As the coefficients are allowed to be non-coercive, our framework encompasses a much broader scope than variational settings. In place of coercivity, we require only well-posedness of the stochastic evolution equation and two concrete, verifiable a priori estimates. Furthermore, we accommodate drift nonlinearities satisfying a modified criticality condition and we allow for vanishing drift perturbations. The latter permits the inclusion of Itô--Stratonovich correction terms, enabling the treatment of both noise interpretations. We apply our results to a reaction-diffusion system that lacks coercivity. In separate joint work, the present framework has been used to establish an LDP for the 3D primitive equations with full transport noise, further demonstrating its versatility. Finally, we show that even in the coercive case, our framework yields new LDP results for equations with critical nonlinearities that rely on our modified criticality condition, including the stochastic 2D Allen--Cahn equation in the weak setting.

math.PR↗