Stochastic Gradient Descent in Continuous Time

24 Pages Posted: 20 Apr 2017

See all articles by Justin Sirignano

Justin Sirignano

Imperial College London - Department of Mathematics; University of Illinois at Urbana-Champaign

Konstantinos Spiliopoulos

Brown University - Division of Applied Mathematics

Date Written: April 17, 2017

Abstract

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→∞ ∇g(θt) = 0 where 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).

Keywords: Statistical Learning, Machine Learning, Finance, American Options, Financial Engineering, Financial Math

Suggested Citation

Sirignano, Justin and Spiliopoulos, Konstantinos, Stochastic Gradient Descent in Continuous Time (April 17, 2017). Available at SSRN: https://ssrn.com/abstract=2954149 or http://dx.doi.org/10.2139/ssrn.2954149

Justin Sirignano (Contact Author)

Imperial College London - Department of Mathematics ( email )

South Kensington Campus
Imperial College
LONDON, SW7 2AZ
United Kingdom

HOME PAGE: http://jasirign.github.io

University of Illinois at Urbana-Champaign ( email )

601 E John St
Champaign, IL 61820
United States

Konstantinos Spiliopoulos

Brown University - Division of Applied Mathematics ( email )

Providence, RI 02912
United States

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