arXiv subjects

Prakash Chakraborty

Publications and source records attributed to Prakash Chakraborty.

15 recordsLinked to original sources

arXiv2026

Quickest Detection with Rough Path Signatures

We develop a framework for quickest detection of distributional change in signals modeled as rough paths. By representing the pre-change and post-change dynamics as two independent rough paths and modeling the observed signal as their concatenation at an unknown random change-point, we formulate quickest detection as an optimal stopping problem using rough paths. We show that the optimal stopping rule takes the form of the first hitting time to a half-space by a linear functional of the rough path signature, and establish that the same structural form arises independently from the intrinsic geometry of the observed path. Statistical guarantees on detection delay and false alarm probability are derived, and the framework is extended to a distributionally robust formulation in which both the pre-change and post-change models are uncertain. The proposed rules are implemented via zeroth-order stochastic approximation over truncated-signature coefficients and evaluated numerically under Brownian and fractional Brownian dynamics, achieving performance comparable to optimal methods in the Brownian setting while outperforming them and remaining robust to adversarial path perturbations in the fractional Brownian setting.

math.ST↗

arXiv2026

Emergency Department Patient Flow Optimization with an Alternative Care Threshold Policy

Emergency department (ED) overcrowding and patient boarding represent critical systemic challenges that compromise care quality. We propose a threshold-based admission policy that redirects non-urgent patients to alternative care pathways, such as telemedicine, during peak congestion. The ED is modeled as a two-class $M/M/c$ preemptive-priority queuing system, where high-acuity patients are prioritized and low-acuity patients are subject to state-dependent redirection. Analyzed via a level-dependent Quasi-Birth-Death (QBD) process, the model determines the optimal threshold by maximizing a long-run time-averaged objective function comprising redirection-affected revenue and costs associated with patient balking and system occupancy. Structural analysis establishes monotone comparative statics relating the optimal threshold to all model parameters analytically. Numerical analysis using national healthcare data reveals that optimal policies are highly context-dependent. While rural EDs generally optimize at lower redirection thresholds, urban EDs exhibit performance peaks at moderate thresholds. Results indicate that our optimal policy yields significant performance gains of up to $4.84\%$ in rural settings and $5.90\%$ in urban environments. This research provides a mathematically rigorous framework for balancing clinical priority with operational efficiency across diverse ED settings.

cs.PF↗

arXiv2026

Complexity Guarantees for Zeroth-order Methods via Exponentially-shifted Gaussian Smoothing: Mitigating Dimension-dependence and Incorporating Decision-dependence

In this paper, we consider two distinct challenges in the resolution of nonsmooth stochastic optimization. Of these, the first pertains to the pronounced dependence of dimension in Gaussian smoothing-enabled zeroth-order schemes, impeding applications to large-scale settings. Second, no unified analysis {exists} for smoothing-enabled stochastic zeroth-order methods, allowing for capturing standard and decision-dependent stochastic optimization. To contend with the first challenge, we introduce a new exponentially-shifted Gaussian smoothing {\bf esGs} estimator whose moment bounds enjoy a linear dependence on dimension (rather than a quadratic dependence as in standard Gaussian smoothing estimators). Second, we show that such an estimator can be extended in two distinct ways to address decision-dependent regimes where the underlying densities are either available in closed form or not. Notably, the resulting gradient estimators continue to display a linear dependence in dimension. We then develop an {\bf esGs}-enabled stochastic zeroth-order method applicable to nonsmooth strongly convex, convex, and {non-}convex regimes. Importantly, our guarantees provide improved dimension-dependence in iteration complexities (by a factor of problem dimension $n$) while maintaing similar oracle complexities. In addition, we also provide novel high probability guarantees and almost sure sublinear rate guarantees in convex settings, while a new subsequential almost sure convergence guarantee is provided in nonconvex regimes. Preliminary numerics support our theoretical findings show that the proposed schemes display improved computational times and more refined empirical accuracies.

math.OC↗

arXiv2026

The Pontryagin maximum principle and $Q$-functions in rough environments

We derive the Pontryagin maximum principle and $Q$-functions for the relaxed control of noisy rough differential equations. Our main tool is the development of a novel differentiation procedure along `spike variation' perturbations of the optimal state-control pair. We then exploit our development of the infinitesimal $Q$-function (also known as the $q$-function) to derive a policy improvement algorithm for settings with entropic cost constraints.

math.OC↗

arXiv2025

Fluid Limits for Time-Varying Many-Server Queues with Finite Capacity

This paper develops fluid limits for nonstationary many-server loss systems with general service-time distributions. For the zero-buffer $M_t/G/n/n$ queuing model, we prove a functional strong law of large numbers for the fraction of busy servers and characterize the limit by a nonlinear Volterra integral equation with discontinuous coefficients induced by instantaneous blocking. Well-posedness is established through an appropriate solution concept, yielding the time-varying acceptance probability without heuristic approximations. We then treat the finite-buffer $M_t/G/n/(n+b_n)$ regime, proving a functional strong law of large numbers for the triplet of fractions of busy servers, occupied buffers, and cumulative departures, whose limit satisfies a coupled system of three discontinuous Volterra equations capturing the interaction of service completions, buffer occupancy, and admission control at the capacity boundary. We establish well-posedness and convergence of the time-varying acceptance probability. Our theoretical results are supported by numerical simulations for both zero and finite-buffer regimes, illustrating the convergence of transient acceptance probabilities guaranteed by our theory. Finally, we use the fluid limits to derive optimal staffing and buffer-capacity for both time-varying loss systems.

math.PR↗

arXiv2025

$N$-Player Stochastic Differential Games with Regime Switching and Mean Field Convergence

In this study, we investigate $N$-player stochastic differential games with regime switching, where the player dynamics are modulated by a finite-state Markov chain. We analyze the associated Nash system, which consists of a system of coupled nonlinear partial differential equations, and establish the existence and uniqueness of solutions to this system, thereby proving the existence of a unique Nash equilibrium. Additionally, we examine the mean field game problem under the same regime-switching framework. We derive a connection between the Nash equilibrium of the MFG and forward-backward stochastic differential equation with jumps, and demonstrate the unique solvability of this equation. Finally, we explore the propagation of chaos and show that the optimal control obtained from the MFG serves as an approximate Nash equilibrium for the $N$-player problem.

math.PR↗

arXiv2024

Optimal Entry and Exit with Signature in Statistical Arbitrage

In this paper, we explore an optimal timing strategy for the trading of price spreads exhibiting mean-reverting characteristics. A sequential optimal stopping framework is formulated to analyze the optimal timings for both entering and subsequently liquidating positions, all while considering the impact of transaction costs. Then we leverages a refined signature optimal stopping method to resolve this sequential optimal stopping problem, thereby unveiling the precise entry and exit timings that maximize gains. Our framework operates without any predefined assumptions regarding the dynamics of the underlying mean-reverting spreads, offering adaptability to diverse scenarios. Numerical results are provided to demonstrate its superior performance when comparing with conventional mean reversion trading rules.

q-fin.CP↗

arXiv2024

Pathwise Relaxed Optimal Control of Rough Differential Equations

This note lays part of the theoretical ground for a definition of differential systems modeling reinforcement learning in continuous time non-Markovian rough environments. Specifically we focus on optimal relaxed control of rough equations (the term relaxed referring to the fact that controls have to be considered as measure valued objects). With reinforcement learning in view, our reward functions encompass forms that involve an entropy-type term favoring exploration. In this context, our contribution focuses on a careful definition of the corresponding relaxed Hamilton-Jacobi-Bellman (HJB)-type equation. A substantial part of our endeavor consists in a precise definition of the notion of test function and viscosity solution for the rough relaxed PDE obtained in this framework. Note that this task is often merely sketched in the rough viscosity literature, in spite of the fact that it gives a proper meaning to the differential system at stake. In the last part of the paper we prove that the natural value function in our context solves a relaxed rough HJB equation in the viscosity sense.

math.OC↗

arXiv2022

Mean field control and finite dimensional approximation for regime-switching jump diffusions

We consider a jump-diffusion mean field control problem with regime switching in the state dynamics. The corresponding value function is characterized as the unique viscosity solution of a HJB master equation on the space of probability measures. Using this characterization, we prove that the value function, which is not regular, is the limit of a finite agent centralized optimal control problem as the number of agents go to infinity, with an explicit convergence rate. Assuming in addition that the value function is smooth, we establish a quantitative propagation of chaos result for the optimal trajectory of agent states.

math.OC↗

arXiv2021

Optimal Dividends under Model Uncertainty

We consider a diffusive model for optimally distributing dividends, while allowing for Knightian model ambiguity concerning the drift of the surplus process. We show that the value function is the unique solution of a non-linear Hamilton-Jacobi-Bellman variational inequality. In addition, this value function embodies a unique optimal threshold strategy for the insurer's surplus, thereby making it the smooth pasting of a non-linear and linear part at the location of the threshold. Furthermore, we obtain continuity and monotonicity of the value function and the threshold strategy with respect to the parameter that measures ambiguity of our model.

math.OC↗

arXiv2019

A Many-Server Functional Strong Law For A Non-Stationary Loss Model

The purpose of this note is to show that it is possible to establish a many-server functional strong law of large numbers (FSLLN) for the fraction of occupied servers (i.e., the scaled number-in-system) without explicitly tracking either the age or the residual service times of the jobs in a non-Markovian, non-stationary loss model. This considerable analytical simplification is achieved by exploiting a semimartingale representation. The fluid limit is shown to be the unique solution of a Volterra integral equation.

math.PR↗

arXiv2019

Relativistic stable processes in quasi-ballistic heat conduction in thin film semiconductors

In this article, we show how relativistic alpha stable processes can be used to explain quasi-ballistic heat conduction in semiconductors. This is a method that can fit experimental results of ultrafast laser heating in alloys. It also provides a connection to a rich literature on Feynman-Kac formalism and random processes that transition from a stable Lévy process on short time and length scales to the Brownian motion at larger scales. This transition was captured by a heuristic truncated Lévy distribution in earlier papers. The rigorous Feynman-Kac approach is used to derive sharp bounds for the transition kernel. Future directions are briefly discussed.

cond-mat.mes-hall↗

arXiv2019

Strong Embeddings for Transitory Queueing Models

In this paper we establish strong embedding theorems, in the sense of the Komlos-Major-Tusnady framework, for the performance metrics of a general class of transitory queueing models of nonstationary queueing systems. The nonstationary and non-Markovian nature of these models makes the computation of performance metrics hard. The strong embeddings yield error bounds on sample path approximations by diffusion processes, in the form of functional strong approximation theorems.

math.PR↗

arXiv2018

Quenched asymptotics for a 1-d stochastic heat equation driven by a rough spatial noise

In this note we consider the parabolic Anderson model in one dimension with time-independent fractional noise $\dot{W}$ in space. We consider the case $H<\frac{1}{2}$ and get existence and uniqueness of solution. In order to find the quenched asymptotics for the solution we consider its Feynman-Kac representation and explore the asymptotics of the principal eigenvalue for a random operator of the form $\frac{1}{2} Δ+ \dot{W}$.

math.PR↗

arXiv2017

Rough differential equations with power type nonlinearities

In this note we consider differential equations driven by a signal $x$ which is $γ$-Hölder with $γ>1/3$, and is assumed to possess a lift as a rough path. Our main point is to obtain existence of solutions when the coefficients of the equation behave like power functions of the form $|ξ|^κ$ with $κ\in(0,1)$. Two different methods are used in order to construct solutions: (i) In a 1-d setting, we resort to a rough version of Lamperti's transform. (ii) For multidimensional situations, we quantify some improved regularity estimates when the solution approaches the origin.

math.PR↗