  A broad class of stochastic volatility models are defined by systems of
stochastic differential equations. While these models have seen widespread
success in domains such as finance and statistical climatology, they typically
lack an ability to condition on historical data to produce a true posterior
distribution. To address this fundamental limitation, we show how to re-cast a
class of stochastic volatility models as a hierarchical Gaussian process (GP)
model with specialized covariance functions. This GP model retains the
inductive biases of the stochastic volatility model while providing the
posterior predictive distribution given by GP inference. Within this framework,
we take inspiration from well studied domains to introduce a new class of
models, Volt and Magpie, that significantly outperform baselines in stock and
wind speed forecasting, and naturally extend to the multitask setting.
