This article examines neural network-based approximations for the superhedging price process of a contingent claim in a discrete time market model. First we prove that the α-quantile hedging price converges to the superhedging price at time 0 for α tending to 1, and show that the α-quantile hedging price can be approximated by a neural network-based price. This provides a neural network-based approximation for the superhedging price at time 0 and also the superhedging strategy up to maturity. To obtain the superhedging price process for t > 0, by using the Doob decomposition it is sufficient to determine the process of consumption. We show that it can be approximated by the essential supremum over a set of neural networks. Finally, we present numerical results.

This article examines neural network‐based approximations for the superhedging price process of a contingent claim in a discrete time market model. First we prove that the α‐quantile hedging price converges to the superhedging price at time 0 for α tending to 1, and show that the α‐quantile hedging price can be approximated by a neural network‐based price. This provides a neural network‐based approximation for the superhedging price at time 0 and also the superhedging strategy up to maturity. To obtain the superhedging price process for t>0$t>0$, by using the Doob decomposition, it is sufficient to determine the process of consumption. We show that it can be approximated by the essential supremum over a set of neural networks. Finally, we present numerical results.

<p style="text-indent:20px;">We study asset price bubbles in market models with proportional transaction costs <inline-formula><tex-math id="M1">\begin{document}$ \lambda\in (0, 1) $\end{document}</tex-math></inline-formula> and finite time horizon <inline-formula><tex-math id="M2">\begin{document}$ T $\end{document}</tex-math></inline-formula> in the setting of [<xref ref-type="bibr" rid="b49">49</xref>]. By following [<xref ref-type="bibr" rid="b29">29</xref>], we define the fundamental value <inline-formula><tex-math id="M3">\begin{document}$ F $\end{document}</tex-math></inline-formula> of a risky asset <inline-formula><tex-math id="M4">\begin{document}$ S $\end{document}</tex-math></inline-formula> as the price of a super-replicating portfolio for a position terminating in one unit of the asset and zero cash. We then obtain a dual representation for the fundamental value by using the super-replication theorem of [<xref ref-type="bibr" rid="b50">50</xref>]. We say that an asset price has a bubble if its fundamental value differs from the ask-price <inline-formula><tex-math id="M5">\begin{document}$ (1+\lambda)S $\end{document}</tex-math></inline-formula>. We investigate the impact of transaction costs on asset price bubbles and show that our model intrinsically includes the birth of a bubble.</p>

We extend the super-replication theorems of [27] in a dynamic setting, both in the numéraire-based as well as in the numéraire-free setting. For this purpose, we generalize the notion of admissible strategies. In particular, we obtain a well-defined super-replication price process, which is right-continuous under some regularity assumptions.

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