  In this paper, we extend the Wiener-Ito chaos decomposition to the class of
continuous semimartingales that are exponentially integrable, which includes in
particular affine and some polynomial diffusion processes. By omitting the
orthogonality in the expansion, we are able to show that every $p$-integrable
functional of the semimartingale, for $p \in [1,\infty)$, can be represented as
a sum of iterated integrals thereof. Using finitely many terms of this
expansion and (possibly random) neural networks for the integrands, whose
parameters are learned in a machine learning setting, we show that every
financial derivative can be approximated arbitrarily well in the $L^p$-sense.
In particular, for $p = 2$, we recover the optimal hedging strategy in the
sense of quadratic hedging. Moreover, since the hedging strategy of the
approximating option can be computed in closed form, we obtain an efficient
algorithm to approximately replicate any sufficiently integrable financial
derivative within short runtime.
