Stochastic Gradient Descent in Continuous Time

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. For certain continuous-time problems, SGDCT has some promising advantages compared to a traditional stochastic gradient descent algorithm. This paper mainly focuses on applications in finance, such as model estimation for stocks, bonds, interest rates, and financial derivatives. SGDCT can also be used for the optimization of high-dimensional continuous-time models, such as American options. As an example application, SGDCT is combined with a deep neural network to price high-dimensional American options (up to 100 dimensions).