# Stochastic Gradient Descent in Continuous Time

arXiv: https://arxiv.org/abs/1611.05545 · HTML: https://mlsanity.com/paper/1611.05545/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2016.

## Built on by (most cited first, 14 of 14)

- [Stochastic gradient descent with noise of machine learning type. Part II: Continuous time analysis](https://mlsanity.com/paper/2106.02588/) (2021)
- [A Continuous-time Stochastic Gradient Descent Method for Continuous Data](https://mlsanity.com/paper/2112.03754/) (2021)
- [Modeling AdaGrad, RMSProp, and Adam with Integro-Differential Equations](https://mlsanity.com/paper/2411.09734/) (2024)
- [Continuous-time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations](https://mlsanity.com/paper/2202.06637/) (2022)
- [A Forward Propagation Algorithm for Online Optimization of Nonlinear Stochastic Differential Equations](https://mlsanity.com/paper/2207.04496/) (2022)
- [Two-Timescale Stochastic Approximation for Bilevel Optimisation Problems in Continuous-Time Models](https://mlsanity.com/paper/2206.06995/) (2022)
- [Safe Gradient Flow for Bilevel Optimization](https://mlsanity.com/paper/2501.16520/) (2025)
- [From Adam to Adam-Like Lagrangians: Second-Order Nonlocal Dynamics](https://mlsanity.com/paper/2602.09101/) (2026)
- [Trading Signals In VIX Futures](https://mlsanity.com/paper/2103.02016/) (2021)
- [OGF: An Online Gradient Flow Method for Optimizing the Statistical Steady-State Time Averages of Unsteady Turbulent Flows](https://mlsanity.com/paper/2507.05149/) (2025)
- [Convergence of the generalization error for deep gradient flow methods for PDEs](https://mlsanity.com/paper/2512.25017/) (2025)
- [Recursive Maximum Likelihood Estimation for Interacting Particle Systems using Virtual Particles](https://mlsanity.com/paper/2605.00786/) (2026)
- [Policy Gradient for Continuous-Time Robust Markov Decision Processes](https://mlsanity.com/paper/2606.04335/) (2026)
- [Asymptotically-informed neural networks for Black-Scholes implied volatility computation](https://mlsanity.com/paper/2609.05491/) (2026)

## Abstract (arXiv)

Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in science, engineering, and finance. The SGDCT algorithm follows a (noisy) descent direction along a continuous stream of data. SGDCT performs an online parameter update in continuous time, with the parameter updates θt satisfying a stochastic differential equation. We prove that limt rightarrow ∞ nabla bar g(θt) = 0 where bar g is a natural objective function for the estimation of the continuous-time dynamics. The convergence proof leverages ergodicity by using an appropriate Poisson equation to help describe the evolution of the parameters for large times. SGDCT can also be used to solve continuous-time optimization problems, such as American options. For certain continuous-time problems, SGDCT has some promising advantages compared to a traditional stochastic gradient descent algorithm. As an example application, SGDCT is combined with a deep neural network to price high-dimensional American options (up to 100 dimensions).