Deep learning scheme for forward utilities using ergodic BSDEs

Guillaume Broux-Quemerais; Sarah Kaakaï; Anis Matoussi; Wissal Sabbagh

doi:10.3934/puqr.2024009

Article Contents

2024, Volume 9, Issue 2: 149-180. Doi: 10.3934/puqr.2024009

This issue Previous Article Mean field and n-player games in Ito-diffusion markets under forward performance criteria Next Article Rank-dependent predictable forward performance processes

Deep learning scheme for forward utilities using ergodic BSDEs

1.
Laboratoire Manceau de Mathématiques & FR CNRS No 2962, Institut du Risque et de l’Assurance, Le Mans University, France
2.
Centre de Mathématiques Appliquées (CMAP), CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France

Corresponding author: Anis Matoussi
Corresponding author: Anis Matoussi

Received: April 04, 2024

Accepted: May 20, 2024

Early access: June 2024

Published: June 2024

The authors research is part of the ANR project DREAMeS (ANR-21-CE46-0002) and benefited from the support of respectively the “Chair Risques Emergents en Assurance” and “Chair Impact de la Transition Climatique en Assurance” under the aegis of Fondation du Risque, a joint initiative by Risk and Insurance Institute of Le Mans, and MMA-Covéa and Groupama respectively. The authors thank Z. Bensaid (LMM-Le Mans University) for helpful discussions on deep learning methods for the simulation of BSDEs. The authors thank the anonymous referees and the associate editor for their helpful comments..

Abstract / Introduction Full Text(HTML) Figure(19) / Table(4) Related Papers Cited by

Abstract

In this paper, we present a probabilistic numerical method for a class of forward utilities in a stochastic factor model. For this purpose, we use the representation of forward utilities using the ergodic Backward Stochastic Differential Equations (eBSDEs) introduced by Liang and Zariphopoulou in [27]. We establish a connection between the solution of the ergodic BSDE and the solution of an associated BSDE with random terminal time $ \tau $, defined as the hitting time of the positive recurrent stochastic factor. The viewpoint based on BSDEs with random horizon yields a new characterization of the ergodic cost $ \lambda $ which is a part of the solution of the eBSDEs. In particular, for a certain class of eBSDEs with quadratic generator, the Cole-Hopf transformation leads to a semi-explicit representation of the solution as well as a new expression of the ergodic cost $ \lambda $. The latter can be estimated with Monte Carlo methods. We also propose two new deep learning numerical schemes for eBSDEs. Finally, we present numerical results for different examples of eBSDEs and forward utilities together with the associated investment strategies.

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Mathematics Subject Classification: 60H30, 60H10, 91B16.

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Figure 1. Empirical loss function $ L^{B_{\epsilon}} $

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Figure 2. Convergence of $ \lambda $

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Figure 3. Mean relative error (5.12) on Y for different time steps

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Figure 6. Example of trajectory of Y over $ {\left[{0, \tau}\right]} $ with $ h=0.01 $

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Figure 4. Integral error (5.13) on $ Y $ over $ [0, T] $

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Figure 5. Integral error (5.13) on $ Z $ over $ [0, T] $

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Figure 7. Empirical loss function $ L^{B_{\epsilon}} $

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Figure 8. Convergence of $ \bar{\lambda} $

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Figure 11. Integral error (5.13) on $ Z $ over $ [0, T] $

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Figure 9. Mean relative error (5.12) on $ Y $ for different time steps

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Figure 10. Integral error (5.13) on $ Y $ over $ [0, T] $

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Figure 19. Rescaled optimal strategy $ \pi_{t}^{1} $

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Figure 12. Empirical loss function $ L^{B^{\epsilon}} $

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Figure 13. Convergence of $ \bar{\lambda} $

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Figure 14. Empirical loss function $ L^{B^{\epsilon}} $

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Figure 15. Convergence of $ \bar{\lambda} $

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Figure 16. Dynamics of approximated utility $ U(t, x) $

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Figure 17. Monotonicity and concavity of approximated utility $ U(t,x) $

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Figure 18. Random field $ U(t, x) $

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Table 1. Mean absolute error (variance) on $ \hat{\lambda} $ for Example 5.1. The exact value of $ \lambda $ is $ 0 $

$ h $	M=1000	M=10000	M=100000
0.05	0.009743 (4.87e-05)	0.007224 (7.48e-06)	0.005691 (1.37e-06)
0.02	0.009107 (4.66e-05)	0.006597 (4.63e-06)	0.004895 (1.69e-06)
0.01	0.008833 (3.73e-05)	0.005778 (7.85e-06)	0.004374 (9.73e-07)

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Table 2. Mean absolute error (variance) on $ \hat{\lambda} $ for Example 5.2. The exact value of $ \lambda $ given in (5.14) is $ 0.398942 $

$ h $	M=1000	M=10000	M=100000
0.05	0.002988 (2.86e-06)	0.002955 (2.50e-07)	0.002904 (1.99e-08)
0.02	0.002136 (2.30e-06)	0.001617 (4.98e-07)	0.001528 (1.67e-08)
0.01	0.001637 (1.29e-06)	0.000780 (2.62e-07)	0.000939 (3.55e-08)

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Table 3. Comparison of $ \lambda $ approximations for parameters $ v_{0} = 0.5 $, $ T= 1, \, h=0.01, \, \kappa= 2, \, \mu=2, \, C_{v}=1 $

Example	Exact	MC	GeBSDE	LAeBSDE
Example 5.1	0	−0.004374 (9.73e-07)	−0.003782 (4.28e-05)	−0.004280 (3.07e-05)
Example 5.2	0.398942	0.399882 (3.55e-08)	0.400130 (1.53e-05)	0.397600 (4.63e-05)

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Table 4. Mean (variance) on $ \bar{\lambda} $ on $ 10 $ independent runs

$ h $	M=1000	M=10000	M=100000
0.10	0.192683 (3.48e-04)	0.201704 (5.5e-05)	0.206999 (3.9e-05)
0.05	0.187675 (8.03e-04)	0.183987 (2.17e-04)	0.182469 (2.6e-05)
0.02	0.159276 (1.55e-03)	0.173734 (4.17e-04)	0.173557 (1.31e-04)
0.01	0.159119 (1.16e-03)	0.174493 (1.61e-03)	0.169749 (1.8e-04)

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