In this paper, we derive an $L^p$-chaos expansion based on iterated Stratonovich integrals with respect to a given exponentially integrable continuous semimartingale. By omitting the orthogonality of the expansion, we show that every $p$-integrable functional, $p \in [1,\infty)$, can be approximated by a finite sum of iterated Stratonovich integrals. Using (possibly random) neural networks as integrands, we therefere obtain universal approximation results for $p$-integrable financial derivatives in the $L^p$-sense. Moreover, we can approximately solve the $L^p$-hedging problem (coinciding for $p = 2$ with the quadratic hedging problem), where the approximating hedging strategy can be computed in closed form within short runtime.

翻译：本文中，我们基于关于给定指数可积连续半鞅的迭代Stratonovich积分，推导出$L^p$-混沌展开。通过放弃展开的正交性，我们证明每一个$p$可积泛函（$p \in [1,\infty)$）都可以用迭代Stratonovich积分的有限和来逼近。利用（可能随机的）神经网络作为被积函数，我们因此在$L^p$意义下获得了对$p$可积金融衍生品的通用逼近结果。此外，我们能够近似求解$L^p$-对冲问题（当$p = 2$时与二次对冲问题一致），其中逼近对冲策略可以在短时间内以闭合形式计算得出。