Research

Many questions in statistics and machine learning come down to drawing random samples from a complicated probability distribution. I study Langevin algorithms, which are Markov chain Monte Carlo methods that do this by simulating a random walk guided by the shape of the distribution. My work proves how quickly these samplers converge and designs versions that scale to networks of computers, respect constraints, and use higher-order dynamics to move faster.

  • I
    Decentralized sampling. Agents on a network learn a shared posterior without pooling their data.
  • II
    Constrained and singular dynamics. Samplers that stay inside a domain and avoid collisions.
  • III
    High-order Langevin Monte Carlo. Faster mixing from higher-order dynamics, with Wasserstein guarantees.
  • IV
    Financial engineering. Time-series prediction, derivative pricing, and risk.

At a glance

5papers
8talks
4mentees

How a Langevin sampler works

A short, hands-on introduction for anyone new to the topic.

Suppose we want random numbers whose histogram follows a given curve. In this example the target is a mixture of two familiar bell curves (normal distributions), one centred at \(-1\) that holds 60% of the probability and one centred at \(1.2\) that holds the other 40%:

\[\pi(x) \;=\; 0.6\cdot\frac{1}{0.7\sqrt{2\pi}}\,e^{-\frac{(x+1)^2}{2(0.7)^2}} \;+\; 0.4\cdot\frac{1}{0.6\sqrt{2\pi}}\,e^{-\frac{(x-1.2)^2}{2(0.6)^2}} .\]

We can write this formula down and evaluate it at any point, but there is no direct way to draw random numbers from it. A Langevin sampler gets around this with walkers that take small random steps guided by the shape of \(\pi\).

1

Turn probability into a landscape

Define the height \(U(x) = -\log \pi(x)\). Where \(\pi\) is large, \(U\) is low, so likely values become valleys and unlikely values become hills.

2

Roll downhill, with random kicks

At every step each walker takes a small step downhill and then gets a random push. The pull keeps it near likely values, and the kicks keep it from settling in one place.

3

Record where it has been

Every position a walker visits is one sample. With the right balance of pull and kicks, the histogram of all the samples settles onto \(\pi(x)\).

step 0 · 0 samples

Thirty walkers start high on the right-hand slope. Watch them roll into the valleys while the histogram fills in.

The update rule. With step size \(h = 0.03\) and a fresh standard normal draw \(\xi_k\) for each walker,

\[x_{k+1} = x_k \;\underbrace{-\,h\,U'(x_k)}_{\text{downhill pull}} \;+\; \underbrace{\sqrt{2h}\,\xi_k}_{\text{random kick}} .\]

How many samples? The demo runs 30 walkers side by side. Each step moves every walker once and records its new position, so each step adds 30 samples, and after \(k\) steps the histogram holds \(30k\).

Turn the kicks off and every walker stops at the bottom of its nearest valley. That is gradient descent: it finds a single best point but loses the spread of the distribution. With the kicks on, the walkers keep exploring and occasionally cross the hill between the valleys, so each valley ends up with its correct share: about 60% on the left and 40% on the right.

My research asks how quickly this happens, and how to keep it fast and accurate in high dimensions, across networks of computers, and inside constrained regions.

Current project

Simulation: particles following Langevin dynamics inside an elliptical constraint set while repelling one another
In progress · 2026

Constrained Langevin Dynamics with Singular Repulsion

With David P. Herzog, Iowa State University

Particles diffuse inside a bounded region while strongly repelling one another at short range. Standard numerical steps can push them toward collisions, and projecting back can create collisions of its own. We are developing a tamed-proximal Langevin scheme that handles the singular repulsion directly while respecting the boundary, with guarantees on stability, collision avoidance, and convergence.

  • Well-posedness
  • Invariant measures
  • Stable discretization
Project overview

Publications & preprints

Google Scholar

2026PreprintIllustration: a posterior distribution concentrated against the boundary of a circular constraint set

Decentralized Proximal Stochastic Gradient Langevin Dynamics

Mohammad Rafiqul Islam and Lingjiong Zhu
arXiv:2605.00723

A decentralized sampler for log-concave distributions on convex domains. A shared proximal regularization based on the Moreau–Yosida envelope lets agents respect the constraint without exchanging private data. Non-asymptotic 2-Wasserstein guarantees cover both individual agents and network averages and quantify the bias introduced by the proximal approximation.

PDFarXiv
BibTeX 
@article{islam2026depsgld,
  title={Decentralized Proximal Stochastic Gradient Langevin Dynamics},
  author={Islam, Mohammad Rafiqul and Zhu, Lingjiong},
  journal={arXiv preprint arXiv:2605.00723},
  year={2026}
}
2025PreprintIllustration: convergence curves of third- and fourth-order Langevin Monte Carlo

High-Order Langevin Monte Carlo Algorithms

Thanh L. Dang, Mert Gürbüzbalaban, Mohammad Rafiqul Islam, Nian Yao, and Lingjiong Zhu
arXiv:2508.17545

Langevin Monte Carlo algorithms built from third- and higher-order dynamics by combining splitting with accurate integration. The convergence guarantees improve their dependence on dimension and target accuracy as the order grows, and numerical experiments confirm the efficiency of the higher-order approach.

PDFarXiv
BibTeX 
@article{dang2025highorder,
  title={High-Order Langevin Monte Carlo Algorithms},
  author={Dang, Thanh L. and Gurbuzbalaban, Mert and Islam, Mohammad Rafiqul and Yao, Nian and Zhu, Lingjiong},
  journal={arXiv preprint arXiv:2508.17545},
  year={2025}
}
2024PreprintIllustration: EXTRA SGLD removing the network bias of DE-SGLD, with a ring network inset

Generalized EXTRA Stochastic Gradient Langevin Dynamics

Mert Gürbüzbalaban, Mohammad Rafiqul Islam, Xiaoyu Wang, and Lingjiong Zhu
arXiv:2412.01993

Decentralized stochastic-gradient Langevin methods carry a persistent network bias. Inspired by EXTRA-type optimization, the proposed sampler removes that bias in the full-batch setting and gives sharper mini-batch bounds, so agents can learn a Bayesian model together without centralizing their data.

PDFarXiv
BibTeX 
@article{gurbuzbalaban2024extra,
  title={Generalized EXTRA Stochastic Gradient Langevin Dynamics},
  author={Gurbuzbalaban, Mert and Islam, Mohammad Rafiqul and Wang, Xiaoyu and Zhu, Lingjiong},
  journal={arXiv preprint arXiv:2412.01993},
  year={2024}
}
2021JournalIllustration: Monte Carlo simulated cryptocurrency price paths with a 95% band

GJR-GARCH Volatility Modeling under NIG and ANN for Predicting Top Cryptocurrencies

Fahad Mostafa, Pritam Saha, Mohammad Rafiqul Islam, and Nguyet Nguyen
Journal of Risk and Financial Management 14(9), 421

Combines GJR-GARCH volatility modeling, Monte Carlo simulation, value-at-risk backtesting, and neural networks to study ten major cryptocurrencies. The models capture asymmetric volatility and tail risk, and the neural-network forecasts track market prices closely.

PDFDOI
BibTeX 
@article{mostafa2021gjr,
  title={GJR-GARCH Volatility Modeling under NIG and ANN for Predicting Top Cryptocurrencies},
  author={Mostafa, Fahad and Saha, Pritam and Islam, Mohammad Rafiqul and Nguyen, Nguyet},
  journal={Journal of Risk and Financial Management},
  volume={14}, number={9}, pages={421}, year={2021},
  doi={10.3390/jrfm14090421}
}
2020JournalIllustration: actual index level against ARIMA, GBM, and neural-network predictions

Comparison of Financial Models for Stock Price Prediction

Mohammad Rafiqul Islam and Nguyet Nguyen
Journal of Risk and Financial Management 13(8), 181

Compares ARIMA, artificial neural networks, and geometric Brownian motion on historical daily stock prices. For next-day prediction, the classical statistical and stochastic models approximate observed prices more closely than the neural network does.

PDFDOI
BibTeX 
@article{islam2020comparison,
  title={Comparison of Financial Models for Stock Price Prediction},
  author={Islam, Mohammad Rafiqul and Nguyen, Nguyet},
  journal={Journal of Risk and Financial Management},
  volume={13}, number={8}, pages={181}, year={2020},
  doi={10.3390/jrfm13080181}
}

Talks & presentations

Undergraduate research

I have mentored undergraduates at Florida State through two research programs.

UROP · 2025 – 2026

Machine learning and Florida traffic fatalities

Sarah Thapa: predictive models for factors associated with Florida road crashes and the economic impact of crash-related casualties.

UROP · 2025 – 2026

Machine learning vs. Black–Scholes–Merton

Aiden Marin: comparing classical Black–Scholes–Merton pricing with modern machine-learning models for options on different underlying stocks.

Directed Reading Program · 2024 – 2025

Identifying spam and non-spam emails

Lily Kinney and Kiera Agic: studying and implementing machine-learning methods for an NLP classification project.

Course & thesis projects

Florida State · Advisor: Giray Ökten

Option pricing: randomized quasi-Monte Carlo vs. the Fourier-cosine method

Prices European options with random-start Halton RQMC and the COS method, benchmarked against Black–Scholes–Merton, and identifies several reasons to prefer RQMC.

University of Dhaka · Advisor: Md. Abdus Samad

Higher-order Runge–Kutta methods and their applications

Fourth- and sixth-order Runge–Kutta schemes in FORTRAN 90/95 for initial- and boundary-value problems, including the shooting method and a step-size error study.