Deep learning (DL) is becoming indispensable to contemporary stochastic analysis and finance; nevertheless, it is still unclear how to design a principled DL framework for approximating infinite-dimensional causal operators. This paper proposes a "geometry-aware" solution to this open problem by introducing a DL model-design framework that takes a suitable infinite-dimensional linear metric spaces as inputs and returns a universal sequential DL models adapted to these linear geometries: we call these models Causal Neural Operators (CNO). Our main result states that the models produced by our framework can uniformly approximate on compact sets and across arbitrarily finite-time horizons H\"older or smooth trace class operators which causally map sequences between given linear metric spaces. Consequentially, we deduce that a single CNO can efficiently approximate the solution operator to a broad range of SDEs, thus allowing us to simultaneously approximate predictions from families of SDE models, which is vital to computational robust finance. We deduce that the CNO can approximate the solution operator to most stochastic filtering problems, implying that a single CNO can simultaneously filter a family of partially observed stochastic volatility models.
翻译:深度学习( DL) 正在成为当代随机分析和融资所不可或缺的; 然而, 仍然不清楚如何设计一个原则性 DL 框架, 以近似于无限因果操作者。 本文提出一个“ 地球测量- 识” 解决方案, 通过引入一个 DL 模型设计框架, 将合适的无限线性线性光学空间作为投入, 并返回一个适合这些线性地形的通用连续DL 模型: 我们称这些模型为 Causal神经操作员( CNO ) 。 我们的主要结果显示, 我们框架生成的模型可以统一地接近集束和横跨任意的有限时间地平线 H\\" older 或顺带级操作者, 从而在给给给特定线性测量空间之间绘制序列。 因此, 我们推论, 一个单一的 CNO 能够有效地将解决方案操作员与广泛的 SDE 模型组合相近, 从而让我们同时将 SDE 模型的预测相近, 这对计算稳健的融资至关重要 。 我们推论, CNO 可以将解决方案操作员与大多数随机过滤问题的解决方案操作员相近,, 意味着一个单一的 CNO 可以同时筛选模型 。