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An Integrated Transportation Distance Between Kernels and Approximate Dynamic Risk Evaluation in Markov Systems

Zhengqi Lin, Andrzej Ruszczynski

TL;DR

A distance between kernels based on the Wasserstein distances between their values is introduced, its properties are studied, and it is proved that it is a metric on an appropriately defined space of kernels.

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 to recursively approximate 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.

An Integrated Transportation Distance Between Kernels and Approximate Dynamic Risk Evaluation in Markov Systems

TL;DR

A distance between kernels based on the Wasserstein distances between their values is introduced, its properties are studied, and it is proved that it is a metric on an appropriately defined space of kernels.

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 to recursively approximate 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.
Paper Structure (10 sections, 10 theorems, 81 equations, 6 figures, 3 tables)

This paper contains 10 sections, 10 theorems, 81 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Markov risk measures
  4. The Wasserstein distance
  5. The Integrated Transportation Distance Between Kernels
  6. Approximate Risk Evaluation in Markov Systems
  7. Kernel Approximation by Particles
  8. Numerical Illustration
  9. Proofs of the statements in § \ref{['s:kernel-distance']}
  10. Comparison of Kernel Distances on Gaussian Mixture Models

Key Result

Theorem 2.3

For any $\mu, \nu$ in $\mathcal{P}_{1}(\mathcal{X})$, where $\|\psi\|_{\text{\rm Lip}}$ denotes the minimal Lipschitz constant of the function $\psi:\mathcal{X}\to \mathbbm{R}$.

Figures (6)

  • Figure 1: The approximate value of the American basket put option as a function of the number of time steps
  • Figure 2: All sample points (blue, N = 1000) vs. selected grid points (black, M= 46).
  • Figure 3: Monte Carlo simulation vs. the grid selection method.
  • Figure 4: Gaussian Mixture model with 5 centers and samples of 400 drawn from each center; $\text{dim}(\beta) = 1000000$, $\text{dim}(\gamma) = 500$, and 100 selected representative points.
  • Figure 5: Gaussian Mixture model with 10 centers and samples of 200 drawn from each center; $\text{dim}(\beta) = 1000000$, $\text{dim}(\gamma) = 500$, and 100 selected representative points.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • ...and 18 more