Stochastic Gradient Descent in Continuous Time

TL;DR

This paper introduces SGDCT, a continuous-time stochastic gradient descent method for efficient online learning of continuous-time models, with proven convergence and applications to finance, including high-dimensional American option pricing.

Contribution

It develops a novel continuous-time SGD algorithm with convergence guarantees and demonstrates its effectiveness in complex financial modeling tasks.

Findings

Proves convergence of the continuous-time SGD algorithm.

Successfully applies SGDCT to high-dimensional American option pricing.

Shows advantages over traditional stochastic gradient descent in certain problems.

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 $\theta_t$ satisfying a stochastic differential equation. We prove that $\lim_{t \rightarrow \infty} \nabla \bar g(\theta_t) = 0$ 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).