  We consider the problem of learning Stochastic Differential Equations of the
form $dX_t = f(X_t)dt+\sigma(X_t)dW_t $ 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 $f$, $\sigma$, and stochastic process $dW_t$ representing the drift,
the diffusion, and the stochastic forcing terms, respectively. We propose a
method that combines Computational Graph Completion 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 $X_t \rightarrow X_{t+dt}$ as
a Computational Graph in which $f$, $\sigma$ and $dW_t$ 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.
