SIAM Journal on Control and Optimization, Volume 61, Issue 6, Page 3559-3583, December 2023.
Abstract. We introduce a distance between kernels based on the Wasserstein distances between their values, study its properties, and prove that it is a metric on an appropriately defined space of kernels. We also relate it to various modes of convergence in the space of kernels. Then we consider the problem of approximating solutions to forward–backward systems, where the forward part is a Markov system described by a sequence of kernels, and the backward part calculates the values of a risk measure by operators that may be nonlinear with respect to the system’s kernels. We propose recursively approximating the forward system with the use of the integrated transportation distance between kernels and we estimate the error of the risk evaluation by the errors of individual kernel approximations. We illustrate the results on stopping problems and several well-known risk measures. Then we develop a particle-based numerical procedure, in which the approximate kernels have finite support sets. Finally, we illustrate the efficacy of the approach on the financial problem of pricing an American basket option.

中文翻译：

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马尔可夫系统中核间综合运输距离与近似动态风险评估

SIAM 控制与优化杂志，第 61 卷，第 6 期，第 3559-3583 页，2023 年 12 月。
摘要。我们基于核值之间的 Wasserstein 距离引入核之间的距离，研究其属性，并证明它是适当定义的核空间上的度量。我们还将它与核空间中的各种收敛模式联系起来。然后我们考虑前向-后向系统的近似解问题，其中前向部分是由一系列核描述的马尔可夫系统，后向部分通过算子计算风险度量的值，该值可能与系统的内核。我们建议使用核之间的综合运输距离来递归地逼近前向系统，并通过各个核近似的误差来估计风险评估的误差。我们说明了阻止问题的结果和几种众所周知的风险措施。然后我们开发了一个基于粒子的数值程序，其中近似核具有有限的支持集。最后，我们说明了该方法对美国一篮子期权定价的财务问题的有效性。