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Deep learning algorithms for FBSDEs with jumps: Applications to option pricing and a MFG model for smart grids

Clémence Alasseur, Zakaria Bensaid, Roxana Dumitrescu, Xavier Warin

TL;DR

The efficiency of the deep-learning algorithms to solve a coupled multi-dimensional FBSDE system driven by a time-inhomogeneous jump process with stochastic intensity is shown and the Nash equilibria for a specific mean-field game (MFG) problem is developed.

Abstract

In this paper, we introduce various machine learning solvers for (coupled) forward-backward systems of stochastic differential equations (FBSDEs) driven by a Brownian motion and a Poisson random measure. We provide a rigorous comparison of the different algorithms and demonstrate their effectiveness in various applications, such as cases derived from pricing with jumps and mean-field games. In particular, we show the efficiency of the deep-learning algorithms to solve a coupled multi-dimensional FBSDE system driven by a time-inhomogeneous jump process with stochastic intensity, which describes the Nash equilibria for a specific mean-field game (MFG) problem for which we also provide the complete theoretical resolution. More precisely, we develop an extension of the MFG model for smart grids introduced in Alasseur, Campi, Dumitrescu and Zeng (Annals of Operations Research, 2023) to the case when the random jump times correspond to the jump times of a doubly Poisson process. We first provide an existence result of an equilibria and derive its semi-explicit characterization in terms of a system of FBSDEs in the linear-quadratic setting. We then compare the MFG solution to the optimal strategy of a central planner and provide several numerical illustrations using the deep-learning solvers presented in the first part of the paper.

Deep learning algorithms for FBSDEs with jumps: Applications to option pricing and a MFG model for smart grids

TL;DR

The efficiency of the deep-learning algorithms to solve a coupled multi-dimensional FBSDE system driven by a time-inhomogeneous jump process with stochastic intensity is shown and the Nash equilibria for a specific mean-field game (MFG) problem is developed.

Abstract

In this paper, we introduce various machine learning solvers for (coupled) forward-backward systems of stochastic differential equations (FBSDEs) driven by a Brownian motion and a Poisson random measure. We provide a rigorous comparison of the different algorithms and demonstrate their effectiveness in various applications, such as cases derived from pricing with jumps and mean-field games. In particular, we show the efficiency of the deep-learning algorithms to solve a coupled multi-dimensional FBSDE system driven by a time-inhomogeneous jump process with stochastic intensity, which describes the Nash equilibria for a specific mean-field game (MFG) problem for which we also provide the complete theoretical resolution. More precisely, we develop an extension of the MFG model for smart grids introduced in Alasseur, Campi, Dumitrescu and Zeng (Annals of Operations Research, 2023) to the case when the random jump times correspond to the jump times of a doubly Poisson process. We first provide an existence result of an equilibria and derive its semi-explicit characterization in terms of a system of FBSDEs in the linear-quadratic setting. We then compare the MFG solution to the optimal strategy of a central planner and provide several numerical illustrations using the deep-learning solvers presented in the first part of the paper.
Paper Structure (25 sections, 6 theorems, 69 equations, 10 figures, 7 tables, 5 algorithms)

This paper contains 25 sections, 6 theorems, 69 equations, 10 figures, 7 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. General Deep Learning algorithms for coupled FBSDEs with jumps
  3. Preliminaries
  4. Deep Learning algorithms
  5. Discrete-time approximation.
  6. From the discrete-time FBSDE to neural networks.
  7. First class of deep-learning algorithms.
  8. Second class of deep-learning algorithms.
  9. Numerical tests for option pricing
  10. The models
  11. Results
  12. Conclusion
  13. Application to an MFG model with jumps for smart grids
  14. Extended MFG model for Demand Side Management with Cox process
  15. Model with $n$-consumers.
  16. ...and 10 more sections

Key Result

Theorem 2.1

Under the Assumptions assume:Regularity and assume:Monotonicity, for any $\xi \in L^2(\mathcal{F}_0, \mathbb{R}^d)$, FBSDE (eqn:FBSDEDL) has a unique solution $(X, Y, Z, U) \in \mathcal{S}^2 \times \mathcal{S}^2 \times \mathcal{H}^2 \times \mathcal{H}^2_{\nu}$.

Figures (10)

  • Figure 1: Convergence of the 7 algorithms in the BS model
  • Figure 2: Convergence of the 7 algorithms in the Merton model
  • Figure 3: Convergence of the 7 algorithms in the Variance Gamma model
  • Figure 4: Convergence of the 5 algorithms in the MFG model
  • Figure 5: Trajectories over 48 hours of the consumption for 2 different consumers in kW (upper figure left) and the common intensity of jumps for the divergence costs (upper figure right).
  • ...and 5 more figures

Theorems & Definitions (15)

  • Theorem 2.1: Well-posedness for arbitrary time horizon
  • Theorem 2.2: Well-posedness in small time
  • Theorem 2.3: Universal Approximation Theorem
  • Theorem 2.4: Universal Approximation Theorem
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 3.1: Mean-field Nash Equilibrium
  • Remark 5
  • ...and 5 more