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.

翻译：预测具有随机波动率的Volterra过程的条件演化是数学金融中的一个关键挑战。尽管深度神经网络模型在逼近此类过程的条件分布方面展现出潜力，但由于这些问题的无限维性和非光滑特性所导致的维数灾难，其有效性受到限制。为解决这一问题，我们提出了一种两步解决方案。首先，我们开发了一种稳定的降维技术，将一类相当广泛的Volterra过程的分布投影到一个具有非正截面曲率的低维统计流形上。其次，我们引入了一种针对该流形几何结构设计的序列深度学习模型，并证明该模型能够逼近Volterra过程的投影条件分布。我们的模型利用一个辅助超网络动态更新其内部参数，使其能够编码Volterra过程的非平稳动态，并且可被解释为混合专家模型中的门控机制，其中每个专家专精于特定时间点。我们的超网络进一步使我们能够达到看似只有使用极大网络才可能实现的逼近速率。