  (Conditional) Generative Adversarial Networks (GANs) have found great success
in recent years, due to their ability to approximate (conditional)
distributions over extremely high dimensional spaces. However, they are highly
unstable and computationally expensive to train, especially in the time series
setting. Recently, it has been proposed the use of a key object in rough path
theory, called the signature of a path, which is able to convert the min-max
formulation given by the (conditional) GAN framework into a classical
minimization problem. However, this method is extremely expensive in terms of
memory cost, sometimes even becoming prohibitive. To overcome this, we propose
the use of \textit{Conditional Neural Stochastic Differential Equations}, which
have a constant memory cost as a function of depth, being more memory efficient
than traditional deep learning architectures. We empirically test that this
proposed model is more efficient than other classical approaches, both in terms
of memory cost and computational time, and that it usually outperforms them in
terms of performance.
