算法交易与随机积分

中文摘要 AI 辅助

我们研究了系数由神经网络表示的简单可预测过程。在有限测度空间上，我们建立了神经网络在Orlicz空间中的稠密性结果。对于由随机过程生成的滤子，包括停时处的可测随机变量，可以通过依赖于有限观测的神经网络来逼近。关于半鞅的每个随机积分，都可以在半鞅拓扑下，通过此类简单可预测过程的积分来逼近。我们证明了将交易策略限制在这一类别中，在部分信息下，最小均方误差对冲误差保持不变，并获得了关于等价鞅测度的无免费午餐特征。最后，Bichteler-Dellacherie关于半鞅的特征描述，即使在将可预测被积函数限制为系数由神经网络表示的那些函数时，仍然有效。

英文摘要

We study simple predictable processes whose coefficients are represented by neural networks. On finite measure spaces, we establish density results for neural networks in Orlicz spaces. For filtrations generated by a stochastic process, measurable random variables, including at stopping times, can be approximated by neural networks depending on finitely many observations. Every stochastic integral with respect to a semimartingale can then be approximated, in the semimartingale topology, by integrals of such simple predictable processes. We show that restricting trading strategies to this class leaves the minimal mean-variance hedging error under partial information unchanged and obtain a no-free-lunch characterization in terms of equivalent martingale measures. Finally, the Bichteler-Dellacherie characterization of semimartingales remains valid even upon restricting the predictable integrands to those whose coefficients are represented by neural networks.

发表机构

• ETH Zurich（苏黎世联邦理工学院）
• WU Vienna University of Economics and Business（维也纳经济与商业大学）
• TU Wien（维也纳工业大学）

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