Predicting the conditional evolution of Volterra processes with stochastic
volatility is a crucial challenge in mathematical finance. While deep neural
network models offer promise in approximating the conditional law of such
processes, their effectiveness is hindered by the curse of dimensionality
caused by the infinite dimensionality and non-smooth nature of these problems.
To address this, we propose a two-step solution. Firstly, we develop a stable
dimension reduction technique, projecting the law of a reasonably broad class
of Volterra process onto a low-dimensional statistical manifold of non-positive
sectional curvature. Next, we introduce a sequentially deep learning model
tailored to the manifold's geometry, which we show can approximate the
projected conditional law of the Volterra process. Our model leverages an
auxiliary hypernetwork to dynamically update its internal parameters, allowing
it to encode non-stationary dynamics of the Volterra process, and it can be
interpreted as a gating mechanism in a mixture of expert models where each
expert is specialized at a specific point in time. Our hypernetwork further
allows us to achieve approximation rates that would seemingly only be possible
with very large networks.

预测具有随机波动率的 Volterra 过程的条件演变是数学金融中的关键挑战，我们提出了一种两步解决方案，通过稳定的降维技术和适合流形几何的逐步深度学习模型来近似投影 Volterra 过程的条件律。