We consider the problem of learning Stochastic Differential Equations of the form
dXt = f(Xt)dt + σ(Xt)dWt from one sample trajectory. This problem is more challenging
than learning deterministic dynamical systems because one sample trajectory only provides
indirect information on the unknown functions f, σ, and stochastic process dWt representing
the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a method
that combines Computational Graph Completion [46] and data adapted kernels learned via a
new variant of cross validation. Our approach can be decomposed as follows: (1) Represent the
time-increment map Xt → Xt+dt as a Computational Graph in which f, σ and dWt appear
as unknown functions and random variables. (2) Complete the graph (approximate unknown
functions and random variables) via Maximum a Posteriori Estimation (given the data) with
Gaussian Process (GP) priors on the unknown functions. (3) Learn the covariance functions
(kernels) of the GP priors from data with randomized cross-validation. Numerical experiments
illustrate the efficacy, robustness, and scope of our method.