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Solving high dimensional FBSDE with deep signature techniques with application to nonlinear options pricing

Hui Sun, Feng Bao

TL;DR

This work reports two methods for solving FBSDEs of path dependent types of high dimensions using path signatures as underlying features and proposes a deep learning framework for solving such problems using path signatures as underlying features.

Abstract

We report two methods for solving FBSDEs of path dependent types of high dimensions. Specifically, we propose a deep learning framework for solving such problems using path signatures as underlying features. Our two methods (forward/backward) demonstrate comparable/better accuracy and efficiency compared to the state of the art techniques. More importantly, we are able to solve the problem of high dimension which is a limitation in the conventional methods. We also provide convergence proof for both methods with the proof of the backward methods in the Markovian case.

Solving high dimensional FBSDE with deep signature techniques with application to nonlinear options pricing

TL;DR

This work reports two methods for solving FBSDEs of path dependent types of high dimensions using path signatures as underlying features and proposes a deep learning framework for solving such problems using path signatures as underlying features.

Abstract

We report two methods for solving FBSDEs of path dependent types of high dimensions. Specifically, we propose a deep learning framework for solving such problems using path signatures as underlying features. Our two methods (forward/backward) demonstrate comparable/better accuracy and efficiency compared to the state of the art techniques. More importantly, we are able to solve the problem of high dimension which is a limitation in the conventional methods. We also provide convergence proof for both methods with the proof of the backward methods in the Markovian case.
Paper Structure (14 sections, 10 theorems, 67 equations, 2 figures, 3 tables, 2 algorithms)

This paper contains 14 sections, 10 theorems, 67 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Algorithm and Notations
  3. A quick introduction to Signature methods
  4. Algorithms
  5. Proof of convergence
  6. Assumptions
  7. Markovian Case
  8. Forward Algorithm
  9. Backward Algorithm
  10. Non-Markovian Generalization
  11. Numerical Examples
  12. Lookback Options
  13. A Higher dimension example
  14. An example of non-linear type: Amerasian Option

Key Result

Theorem 2.1

(Chen's Identity Theorem 2 lyons4). Let $X: [a,b] \rightarrow \mathbb{R}^d$ and $Y: [b,c] \rightarrow \mathbb{R}^d$ be two paths. Then

Figures (2)

  • Figure 1: Predicted $Y_0$ value versus the number of interations trained.
  • Figure 2: Predicted $Y_0$ value versus the number of interations trained.

Theorems & Definitions (24)

  • Definition 2.1
  • Example 2.1
  • Example 2.2
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.1
  • Definition 2.4
  • Proposition 2.2
  • Theorem 3.1
  • Lemma 3.2
  • ...and 14 more