Deep learning algorithms for solving high-dimensional nonlinear backward stochastic differential equations

Lorenc Kapllani; Long Teng

doi:10.3934/dcdsb.2023151

Article Contents

2024, Volume 29, Issue 4: 1695-1729. Doi: 10.3934/dcdsb.2023151

This issue Previous Article Feynman-Kac formula for tempered fractional general diffusion equations with nonautonomous external potential Next Article Optimal pointwise approximation of anticipating SDEs

Deep learning algorithms for solving high-dimensional nonlinear backward stochastic differential equations

Lorenc Kapllani^, and
Long Teng

Chair of Applied and Computational Mathematics, Faculty of Mathematics and Natural Sciences, University of Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany

^* Corresponding author: Lorenc Kapllani
^* Corresponding author: Lorenc Kapllani

Received: July 2022

Revised: August 2023

Early access: September 2023

Published: April 2024

Abstract / Introduction Full Text(HTML) Figure(24) / Table(11) Related Papers Cited by

Abstract

In this work, we propose a new deep learning-based scheme for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs). The idea is to reformulate the problem as a global optimization problem where local loss functions are included. Essentially, we approximate the unknown solution of a BSDE using a deep neural network and the gradient of the approximated solution with automatic differentiation. The approximations are performed by globally minimizing the quadratic local loss function defined at each time step, which always includes the terminal condition. This kind of loss functions are obtained by iterating the Euler-Maruyama discretization of the integrals with the terminal condition. Our formulation can prompt the stochastic gradient descent algorithm not only to take the accuracy at each time layer into account, but also converge to a good local minima, namely an estimated solution near to the exact solution. In order to demonstrate performances of our algorithm, several high-dimensional nonlinear BSDEs including pricing problems in finance are provided.

Keywords:

Backward stochastic differential equations,
High-dimensional problems,
deep neural network,
recurrent neural network,
automatic differentiation,
additive discretization,
Nonlinear option pricing.

Mathematics Subject Classification: Primary: 60H35, 65C30, 68T07; Secondary: 65C20.

Citation:

Full Text(HTML)

Figure 1. The architecture of the LaDBSDE scheme

Download: Full-size image PowerPoint slide

Figure 2. The empirical speed of convergence for $ (Y_0, Z_0) $ in Example 1 using $ d = 1 $. The average runtime of the algorithms is given in seconds

Download: Full-size image PowerPoint slide

Figure 3. Realizations of $ 5 $ independent paths for Example 1 using $ d = 1 $ and $ N = 256 $. $ (Y_t, Z_t) $ and $ (\mathcal{Y}_t^{\hat{\theta}},\mathcal{Z}_t^{\hat{\theta}}) $ are exact and learned solutions for $ t \in [0, T] $, respectively

Download: Full-size image PowerPoint slide

Figure 4. The mean regression errors $ (\bar{\epsilon}_{Y_i}, \bar{\epsilon}_{Z_i}) $ at time step $ t_i, i = 0, \ldots, N-1 $ for Example 1 using $ d = 1 $ and $ N = 256 $. The standard deviation is given in the shaded area

Download: Full-size image PowerPoint slide

Figure 5. The empirical speed of convergence for $ (Y_0, Z_0) $ in Example 1 using $ d = 100 $. The average runtime of the algorithms is given in seconds

Download: Full-size image PowerPoint slide

Figure 6. Realizations of $ 5 $ independent paths for Example 1 using $ d = 100 $ and $ N = 128 $. $ (Y_t, Z_t^1) $ and $ (\mathcal{Y}_t^{\hat{\theta}},\mathcal{Z}_t^{1,\hat{\theta}}) $ are exact and learned solutions for $ t \in [0, T] $, respectively

Download: Full-size image PowerPoint slide

Figure 7. The mean regression errors $ (\bar{\epsilon}_{Y_i}, \bar{\epsilon}_{Z_i}) $ at time step $ t_i, i = 0, \ldots, N-1 $ for Example 1 using $ d = 100 $ and $ N = 128 $. The standard deviation is given in the shaded area

Download: Full-size image PowerPoint slide

Figure 8. The empirical speed of convergence for $ (Y_0, Z_0) $ in Example 2 using $ d = 100 $. The average runtime of the algorithms is given in seconds

Download: Full-size image PowerPoint slide

Figure 9. Realizations of $ 5 $ independent paths for Example 2 using $ d = 100 $ and $ N = 128 $. $ (Y_t, Z_t^1) $ and $ (\mathcal{Y}_t^{\hat{\theta}},\mathcal{Z}_t^{1,\hat{\theta}}) $ are exact and learned solutions for $ t \in [0, T] $, respectively

Download: Full-size image PowerPoint slide

Figure 10. The mean regression errors $ (\bar{\epsilon}_{Y_i}, \bar{\epsilon}_{Z_i}) $ at time step $ t_i, i = 0, \ldots, N-1 $ for Example 2 using $ d = 100 $ and $ N = 128 $. The standard deviation is given in the shaded area

Download: Full-size image PowerPoint slide

Figure 11. Realizations of $ 5 $ independent paths for Example 3 using $ d = 2 $ and $ N = 128 $. $ (Y_t, Z_t^1) $ and $ (\mathcal{Y}_t^{\hat{\theta}},\mathcal{Z}_t^{1,\hat{\theta}}) $ are exact and learned solutions for $ t \in [0, T] $, respectively

Download: Full-size image PowerPoint slide

Figure 12. The mean regression errors $ (\bar{\epsilon}_{Y_i}, \bar{\epsilon}_{Z_i}) $ at time step $ t_i, i = 0, \ldots, N-1 $ for Examplek__ge 3 using $ d = 2 $ and $ N = 128 $. The standard deviation is given in the shaded area

Download: Full-size image PowerPoint slide

Figure 13. Realizations of $ 5 $ independent paths for Example 3 using $ d = 10 $ and $ N = 128 $. $ (Y_t, Z_t^1) $ and $ (\mathcal{Y}_t^{\hat{\theta}},\mathcal{Z}_t^{1,\hat{\theta}}) $ are exact and learned solutions for $ t \in [0, T] $, respectively

Download: Full-size image PowerPoint slide

Figure 14. Realizations of $ 5 $ independent paths for Example 3 using $ d = 10 $ and $ N = 128 $. $ (Z_t^4, Z_t^{10}) $ and $ (\mathcal{Z}_t^{4, \hat{\theta}},\mathcal{Z}_t^{10,\hat{\theta}}) $ are exact and learned solutions for $ t \in [0, T] $, respectively

Download: Full-size image PowerPoint slide

Figure 15. The mean regression errors $ (\bar{\epsilon}_{Y_i}, \bar{\epsilon}_{Z_i}) $ at time step $ t_i, i = 0, \ldots, N-1 $ for Example 3 using $ d = 10 $ and $ N = 128 $. The standard deviation is given in the shaded area

Download: Full-size image PowerPoint slide

Figure 16. The mean regression errors $ (\bar{\epsilon}_{Y_i}, \bar{\epsilon}_{Z_i}) $ at time step $ t_i, i = 0, \ldots, N-1 $ for Example 3 using $ d = 50 $ and $ N = 128 $. The standard deviation is given in the shaded area

Download: Full-size image PowerPoint slide

Figure 17. The mean loss values $ \bar{\mathbf{L}} $ for Example 3 using $ d = 50 $ and $ N = 128 $. The standard deviation is given in the shaded area

Download: Full-size image PowerPoint slide

Figure 18. The mean regression errors $ (\bar{\epsilon}_{Y_i}, \bar{\epsilon}_{Z_i}) $ at time step $ t_i, i = 0, \ldots, N-1 $ for Example 3 using $ d = 50 $ and $ N = 128 $. The standard deviation is given in the shaded area

Download: Full-size image PowerPoint slide

Figure 19. The empirical speed of convergence for $ (Y_0, Z_0) $ in Example 3 using $ d = 100 $. The average runtime of the algorithms is given in seconds

Download: Full-size image PowerPoint slide

Figure 20. The mean regression errors $ (\bar{\epsilon}_{Y_i}, \bar{\epsilon}_{Z_i}) $ at time step $ t_i, i = 0, \ldots, N-1 $ for Example 3 using $ d = 100 $ and $ N = 128 $. The standard deviation is given in the shaded area

Download: Full-size image PowerPoint slide

Figure 21. The empirical speed of convergence for $ Y_0 $ in Example 4 using $ d = 100 $. The average runtime of the algorithms is given in seconds

Download: Full-size image PowerPoint slide

Figure 22. The empirical speed of convergence for $ (Y_0, Z_0) $ in Example 5 using $ d = 100 $. The average runtime of the algorithms is given in seconds

Download: Full-size image PowerPoint slide

Figure 23. Realizations of $ 5 $ independent paths for Example 5 using $ d = 100 $ and $ N = 128 $. $ (Y_t, Z_t^1) $ and $ (\mathcal{Y}_t^{\hat{\theta}},\mathcal{Z}_t^{1,\hat{\theta}}) $ are exact and learned solutions for $ t \in [0, T] $, respectively

Download: Full-size image PowerPoint slide

Figure 24. The mean regression errors $ (\bar{\epsilon}_{Y_i}, \bar{\epsilon}_{Z_i}) $ at time step $ t_i, i = 0, \ldots, N-1 $ for Example 5 using $ d = 100 $ and $ N = 128 $. The standard deviation is given in the shaded area

Download: Full-size image PowerPoint slide

Table 1. Hyperparameters for all the schemes

Scheme	Network parametrization				Learning rate decay
Scheme	# networks	L	n	$\varrho$	$\gamma_0$	$\gamma_{min}$
DBSDE	N-1	2	10+d	$\max(0, \cdot)$	$10^{-2}$	$10^{-4}$
LDBSDE	1	4	10+d	$\sin(\cdot)$	$10^{-3}$	$10^{-5}$
LaDBSDE	1	4	10+d	$\tanh(\cdot)$	$10^{-3}$	$10^{-5}$

| Show Table

DownLoad: CSV

Table 2. The mean absolute errors, convergence rates of $ (Y_0, Z_0) $ and the average runtime of the algorithms in seconds for Example 1 using $d = 1 $. The standard deviation of the absolute errors is given in parenthesis

Scheme	N = 4	N = 16	N = 64	N = 256
	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\beta_y$
	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\beta_z$
	$\bar{\tau}$	$\bar{\tau}$	$\bar{\tau}$	$\bar{\tau}$
DBSDE	NaN	NaN	NaN	NaN	NaN
	NaN	NaN	NaN	NaN	NaN
	-	-	-	-
LDBSDE	2.73e+0 (8.06e-3)	9.98e-1 (1.27e-2)	5.39e-1 (1.78e-1)	4.42e-1 (1.72e-1)	0.44
	1.22e+0 (6.25e-3)	5.58e-1 (7.58e-3)	3.45e-1 (1.30e-1)	1.99e-1 (1.48e-1)	0.43
	1.02e+2	3.52e+2	1.39e+3	5.46e+3
LDBSDE (RNN)	2.72e+0 (1.84e-2)	1.36e+0 (2.35e-1)	8.49e-1 (2.89e-1)	9.45e-1 (5.27e-1)	0.26
	1.23e+0 (1.50e-2)	7.46e-1 (1.32e-1)	5.49e-1 (1.91e-1)	6.60e-1 (4.06e-1)	0.16
	6.13e+1	2.02e+2	8.50e+2	3.33e+3
LDBSDE (LSTM)	2.73e+0 (6.51e-3)	1.69e+0 (2.22e-2)	1.11e+0 (9.51e-2)	9.44e-1 (1.38e-1)	0.26
	1.24e+0 (5.88e-3)	9.31e-1 (2.20e-2)	6.43e-1 (8.91e-2)	5.98e-1 (9.38e-2)	0.18
	1.29e+2	4.26e+2	1.76e+3	7.12e+3
LaDBSDE	2.37e+0 (6.22e-3)	2.90e-1 (2.19e-1)	1.67e-1 (5.33e-2)	7.52e-2 (2.77e-2)	0.79
	8.42e-1 (6.84e-3)	5.44e-2 (8.36e-2)	6.30e-2 (3.32e-2)	5.07e-2 (2.58e-2)	0.60
	8.52e+1	3.60e+2	1.39e+3	5.57e+3

| Show Table

DownLoad: CSV

Table 3. The mean absolute errors, convergence rates of $ (Y_0, Z_0) $ and the average runtime of the algorithms in seconds for Example 1 using $ d = 100 $. The standard deviation of the absolute errors is given in parenthesis

Scheme	N = 2	N = 8	N = 32	N = 128
	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\beta_y$
	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\beta_z$
	$\bar{\tau}$	$\bar{\tau}$	$\bar{\tau}$	$\bar{\tau}$
DBSDE	NaN	NaN	NaN	NaN	NaN
	NaN	NaN	NaN	NaN	NaN
	-	-	-	-
LDBSDE	1.66e-1 (1.64e-2)	8.93e-2 (1.15e-2)	7.74e-2 (7.59e-3)	9.29e-2 (1.29e-2)	0.14
	5.04e-2 (1.39e-3)	2.54e-2 (6.77e-4)	8.13e-3 (6.89e-4)	6.82e-3 (1.87e-3)	0.52
	1.02e+3	3.31e+3	1.24e+4	4.82e+4
LaDBSDE	9.28e-2 (1.79e-2)	1.22e-2 (9.15e-3)	8.96e-3 (6.89e-3)	7.91e-3 (5.59e-3)	0.56
	3.46e-2 (1.51e-3)	9.95e-3 (2.02e-3)	4.05e-3 (1.31e-3)	3.64e-3 (1.52e-3)	0.55
	8.61e+2	3.09e+3	1.24e+4	4.74e+4

| Show Table

DownLoad: CSV

Table 4. The mean absolute errors, convergence rates of $ (Y_0, Z_0) $ and the average runtime of the algorithms in seconds for Example 2 using $ d = 100 $. The standard deviation of the absolute errors is given in parenthesis

Scheme	N = 2	N = 8	N = 32	N = 128
	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\beta_y$
	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\beta_z$
	$\bar{\tau}$	$\bar{\tau}$	$\bar{\tau}$	$\bar{\tau}$
DBSDE	1.75e-3 (3.37e-4)	9.63e-4 (4.54e-4)	7.68e-4 (2.60e-4)	1.04e-3 (2.42e-4)	0.13
	3.30e-4 (1.42e-5)	4.56e-4 (3.24e-5)	6.51e-4 (5.72e-5)	9.37e-4 (5.55e-5)	-0.25
	2.73e+2	1.21e+3	5.09e+3	2.25e+4
LDBSDE	1.98e-3 (8.47e-4)	9.23e-4 (5.99e-4)	1.09e-3 (6.75e-4)	3.85e-3 (5.98e-3)	-0.16
	8.29e-5 (2.87e-5)	1.37e-4 (5.41e-5)	2.29e-4 (3.28e-5)	6.67e-4 (2.21e-4)	-0.49
	7.15e+2	2.25e+3	8.70e+3	3.30e+4
LaDBSDE	1.33e-3 (1.03e-3)	7.97e-4 (5.75e-4)	8.11e-4 (1.25e-3)	6.40e-4 (3.50e-4)	0.16
	7.52e-5 (6.21e-6)	1.48e-4 (3.35e-5)	1.65e-4 (4.15e-5)	1.63e-4 (3.92e-5)	-0.18
	5.75e+2	2.08e+3	8.47e+3	3.32e+4

| Show Table

DownLoad: CSV

Table 5. The mean absolute errors of $ (Y_0, Z_0) $ and the average runtime of the algorithms in seconds for Example 3 using $ d = 2 $ and $ N = 128 $. The standard deviation of the absolute errors is given in parenthesis

Scheme	$ \bar{\epsilon}_{Y_0} $ (Std. Dev.)	$ \bar{\epsilon}_{Z_0} $ (Std. Dev.)	$ \bar{\tau} $
DBSDE	7.86e-4 (3.38e-4)	3.35e-3 (2.98e-3)	4.63e+3
LDBSDE	1.26e-3 (9.19e-4)	1.44e-2 (9.77e-3)	3.94e+3
LaDBSDE	2.49e-3 (1.88e-3)	4.44e-3 (2.31e-3)	3.94e+3

| Show Table

DownLoad: CSV

Table 6. The mean absolute errors of $ (Y_0, Z_0) $ and the average runtime of the algorithms in seconds for Example 3 using $ d = 10 $ and $ N = 128 $. The standard deviation of the absolute errors is given in parenthesis

Scheme	$ \bar{\epsilon}_{Y_0} $ (Std. Dev.)	$ \bar{\epsilon}_{Z_0} $ (Std. Dev.)	$ \bar{\tau} $
DBSDE	1.12e-2 (1.96e-3)	1.70e-2 (1.32e-3)	5.79e+3
LDBSDE	1.68e-2 (1.17e-2)	5.32e-2 (1.87e-2)	5.51e+3
LaDBSDE	5.45e-3 (4.93e-3)	7.85e-3 (3.42e-3)	5.23e+3

| Show Table

DownLoad: CSV

Table 7. The mean absolute errors of $ (Y_0, Z_0) $ and the average runtime of the algorithms in seconds for Example 3 using $ d = 50 $ and $ N = 128 $. The standard deviation of the absolute errors is given in parenthesis

Scheme	$ \bar{\epsilon}_{Y_0} $ (Std. Dev.)	$ \bar{\epsilon}_{Z_0} $ (Std. Dev.)	$ \bar{\tau} $
DBSDE	2.14e+0 (2.83e-1)	1.10e-1 (6.95e-3)	1.76e+4
LDBSDE	1.58e-1 (6.24e-2)	6.81e-2 (9.80e-3)	2.39e+4
LaDBSDE	1.33e-1 (4.66e-2)	3.40e-2 (3.73e-3)	2.28e+4

| Show Table

DownLoad: CSV

Table 8. The mean absolute errors of $ (Y_0, Z_0) $ and the average runtime of the algorithms in seconds for Example 3 using $ d = 50 $ and $ N = 128 $. The standard deviation of the absolute errors is given in parenthesis

Scheme	$ \bar{\epsilon}_{Y_0} $ (Std. Dev.)	$ \bar{\epsilon}_{Z_0} $ (Std. Dev.)	$ \bar{\tau} $
DBSDE	5.17e-1 (4.10e-2)	7.73e-2 (2.55e-3)	1.72e+4
LDBSDE	2.04e-1 (2.31e-2)	7.18e-2 (7.07e-3)	2.33e+4
LaDBSDE	3.07e-2 (1.62e-2)	8.26e-3 (8.87e-4)	2.23e+4

| Show Table

DownLoad: CSV

Table 9. The mean absolute errors, convergence rates of $ (Y_0, Z_0) $ and the average runtime of the algorithms in seconds for Example 3 using $ d = 100 $. The standard deviation of the absolute errors is given in parenthesis

Scheme	N = 2	N = 8	N = 32	N = 128
	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\beta_y$
	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\beta_z$
	$\bar{\tau}$	$\bar{\tau}$	$\bar{\tau}$	$\bar{\tau}$
DBSDE	1.16e+0 (5.14e-3)	4.29e-1 (2.20e-2)	2.73e+0 (1.32e-1)	4.76e+0 (4.10e-2)	-0.44
	6.89e-2 (6.22e-4)	3.98e-2 (2.39e-3)	1.83e-1 (3.46e-3)	1.89e-1 (5.72e-3)	-0.33
	4.10e+2	1.86e+3	7.97e+3	3.37e+4
LDBSDE	1.09e+0 (3.25e-2)	2.12e-1 (3.24e-2)	1.90e-1 (5.40e-2)	2.64e-1 (6.19e-2)	0.32
	7.55e-2 (1.41e-3)	3.13e-2 (2.12e-3)	4.56e-2 (6.93e-3)	6.86e-2 (4.13e-3)	-0.01
	1.04e+3	3.27e+3	1.28e+4	4.85e+4
LaDBSDE	1.07e+0 (5.37e-2)	1.88e-1 (6.04e-2)	5.46e-2 (3.63e-2)	7.52e-2 (5.89e-2)	0.66
	7.17e-2 (2.89e-3)	2.15e-2 (2.63e-3)	1.19e-2 (2.35e-3)	1.22e-2 (2.79e-3)	0.43
	8.19e+2	3.17e+3	1.27e+4	4.86e+4

| Show Table

DownLoad: CSV

Table 10. The mean absolute errors, convergence rates of $ Y_0 $ and the average runtime of the algorithms in seconds for Example 4 using $ d = 100 $. The standard deviation of the absolute errors is given in parenthesis

Scheme	N = 2	N = 8	N = 32	N = 128
	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\beta_y$
	$\bar{\tau}$	$\bar{\tau}$	$\bar{\tau}$	$\bar{\tau}$
	DBSDE	1.44e+0 (4.36e-3)	3.04e-1 (4.31e-2)	9.67e-2 (3.01e-2)	3.72e-2 (1.38e-2)	0.88
2.09e+2	DBSDE	9.31e+2	4.07e+3	1.66e+4
LDBSDE	8.64e-1 (2.16e-2)	2.68e-1 (1.59e-2)	4.11e-1 (1.56e-2)	4.24e-1 (1.85e-2)	0.12
LDBSDE	2.86e+3	4.03e+3	8.97e+3	2.91e+4
LaDBSDE	1.16e+0 (2.16e-2)	2.38e-2 (1.43e-2)	1.81e-1 (2.31e-2)	2.00e-1 (1.90e-2)	0.23
LaDBSDE	2.76e+3	3.81e+3	8.33e+3	2.54e+4

| Show Table

DownLoad: CSV

Table 11. The mean absolute errors, convergence rates of $ (Y_0, Z_0) $ and the average runtime of the algorithms in seconds for Example 5 using $ d = 100 $. The standard deviation of the absolute errors is given in parenthesis

Scheme	N = 2	N = 8	N = 32	N = 128
	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\bar{\epsilon}_{Y_0}$ (Std. Dev.)	$\beta_y$
	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\bar{\epsilon}_{Z_0}$ (Std. Dev.)	$\beta_z$
	$\bar{\tau}$	$\bar{\tau}$	$\bar{\tau}$	$\bar{\tau}$
DBSDE	1.06e-1 (3.67e-4)	1.93e-2 (3.47e-4)	2.23e-3 (1.37e-4)	1.30e-3 (2.73e-4)	1.11
	8.29e-4 (6.59e-6)	4.35e-4 (2.33e-5)	2.91e-4 (1.24e-5)	3.69e-4 (1.60e-5)	0.20
	5.22e+2	2.33e+3	9.87e+3	4.24e+4
LDBSDE	9.97e-2 (5.13e-4)	1.59e-2 (1.21e-3)	2.65e-3 (1.42e-3)	1.25e-3 (8.03e-4)	1.08
	9.49e-4 (4.30e-6)	7.06e-4 (2.72e-5)	4.09e-4 (4.77e-5)	2.85e-4 (1.06e-4)	0.30
	1.17e+3	3.84e+3	1.50e+4	5.64e+4
LaDBSDE	1.05e-1 (4.64e-4)	2.13e-2 (7.80e-4)	4.56e-3 (1.28e-3)	1.12e-3 (9.03e-4)	1.09
	8.68e-4 (4.42e-6)	5.01e-4 (1.53e-5)	1.74e-4 (2.53e-5)	1.58e-4 (2.84e-5)	0.45
	9.94e+2	3.61e+3	1.43e+4	5.63e+4

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	K. Andersson, A. Andersson and C. W. Oosterlee, Convergence of a robust deep FBSDE method for stochastic control, SIAM J. Sci. Comput., 45 (2023), A226-A255. doi: 10.1137/22M1478057.
[2]	S. Ankirchner, C. Blanchet-Scalliet and A. Eyraud-Loisel, Credit risk premia and quadratic BSDEs with a single jump, Int. J. Theor. Appl. Finance, 13 (2010), 1103-1129. doi: 10.1142/S0219024910006133.
[3]	C. Beck, S. Becker, P. Cheridito, A. Jentzen and A. Neufeld, Deep splitting method for parabolic PDEs, SIAM J. Sci. Comput., 43 (2021), A3135-A3154. doi: 10.1137/19M1297919.
[4]	S. Becker, R. Braunwarth, M. Hutzenthaler, A. Jentzen and P. von Wurstemberger, Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations, Commun. Comput. Phys., 28 (2020), 2109-2138. doi: 10.4208/cicp.OA-2020-0130.
[5]	C. Bender and J. Zhang, Time discretization and Markovian iteration for coupled FBSDEs, Ann. Appl. Probab., 18 (2008), 143-177. doi: 10.1214/07-aap448.
[6]	Y. Z. Bergman, Option pricing with differential interest rates, Rev. Financ. Stud., 8 (1995), 475-500. doi: 10.1093/rfs/8.2.475.
[7]	B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stoch. Process. Their Appl., 111 (2004), 175-206. doi: 10.1016/j.spa.2004.01.001.
[8]	Q. Chan-Wai-Nam, J. Mikael and X. Warin, Machine learning for semi linear PDEs, J. Sci. Comput., 79 (2019), 1667-1712. doi: 10.1007/s10915-019-00908-3.
[9]	J.-F. Chassagneux, J. Chen, N. Frikha and C. Zhou, A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs, preprint, arXiv:2102.12051.
[10]	Y. Chen and J. W. L. Wan, Deep neural network framework based on backward stochastic differential equations for pricing and hedging American options in high dimensions, Quant. Finance, 21 (2021), 45-67. doi: 10.1080/14697688.2020.1788219.
[11]	D. Crisan and K. Manolarakis, Solving backward stochastic differential equations using the cubature method: application to nonlinear pricing, SIAM J. Financial Math., 3 (2012), 534-571. doi: 10.1137/090765766.
[12]	W. E, J. Han and A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Commun. Math. Stat., 5 (2017), 349-380. doi: 10.1007/s40304-017-0117-6.
[13]	W. E, M. Hutzenthaler, A. Jentzen and T. Kruse, On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations, J. Sci. Comput., 79 (2019), 1534-1571. doi: 10.1007/s10915-018-00903-0.
[14]	W. E, M. Hutzenthaler, A. Jentzen and T. Kruse, Multilevel Picard iterations for solving smooth semilinear parabolic heat equations, Partial Differ. Equ. Appl., 2 (2021), 1-31. doi: 10.1007/s42985-021-00089-5.
[15]	A. Eyraud-Loisel, Backward stochastic differential equations with enlarged filtration: Option hedging of an insider trader in a financial market with jumps, Stoch. Process. Their Appl., 115 (2005), 1745-1763. doi: 10.1016/j.spa.2005.05.006.
[16]	A. Fahim, N. Touzi and X. Warin, A probabilistic numerical method for fully nonlinear parabolic PDEs, Ann. Appl. Probab., 21 (2011), 1322-1364. doi: 10.1214/10-aap723.
[17]	Y. Fu, W. Zhao and T. Zhou, Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs, Discrete Continuous Dynam. Systems - B, 22 (2017), 3439-3458. doi: 10.3934/dcdsb.2017174.
[18]	M. Fujii, A. Takahashi and M. Takahashi, Asymptotic expansion as prior knowledge in deep learning method for high dimensional BSDEs, Asia-Pac. Financ. Markets, 26 (2019), 391-408. doi: 10.1007/s10690-019-09271-7.
[19]	M. Germain, H. Pham and X. Warin, Approximation error analysis of some deep backward schemes for nonlinear PDEs, SIAM J. Sci. Comput., 44 (2022), A28-A56. doi: 10.1137/20M1355355.
[20]	X. Glorot and Y. Bengio, Understanding the difficulty of training deep feedforward neural networks, Proc. Mach. Learn. Res., 9 (2010), 249-256. Available at PMLR: https://proceedings.mlr.press/v9/glorot10a.html.
[21]	A. Gnoatto, A. Picarelli and C. Reisinger, Deep xVA solver-A neural network based counterparty credit risk management framework,, SIAM J. Financial Math., 14 (2023), 314-352. doi: 10.1137/21M1457606.
[22]	E. Gobet and C. Labart, Solving BSDE with adaptive control variate, SIAM J. Numer. Anal., 48 (2010), 257-277. doi: 10.1137/090755060.
[23]	E. Gobet, J.-P. Lemor and X. Warin, A regression-based Monte Carlo method to solve backward stochastic differential equations, Ann. Appl. Probab., 15 (2005), 2172-2202. doi: 10.1214/105051605000000412.
[24]	E. Gobet, J. G. López-Salas, P. Turkedjiev and C. Vázquez, Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs, SIAM J. Sci. Comput., 38 (2016), C652-C677. doi: 10.1137/16M106371X.
[25]	E. Gobet and P. Turkedjiev, Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions, Math. Comp., 85 (2016), 1359-1391. doi: 10.1090/mcom/3013.
[26]	J. Han, A. Jentzen and W. E, Solving high-dimensional partial differential equations using deep learning, Proc. Natl. Acad. Sci. U.S.A., 115 (2018), 8505-8510. doi: 10.1073/pnas.1718942115.
[27]	J. Han and J. Long, Convergence of the deep BSDE method for coupled FBSDEs, Probab. Uncertain. Quant. Risk, 5 (2020), 1-33. doi: 10.1186/s41546-020-00047-w.
[28]	K. Hornik, M. Stinchcombe and H. White, Multilayer feedforward networks are universal approximators, Neural Netw., 2 (1989), 359-366. doi: 10.1016/0893-6080(89)90020-8.
[29]	C. Huré, H. Pham and X. Warin, Deep backward schemes for high-dimensional nonlinear PDEs, Math. Comp., 89 (2020), 1547-1579. doi: 10.1090/mcom/3514.
[30]	M. Hutzenthaler, A. Jentzen and T. Kruse, Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities, Found. Comput. Math., 22 (2022), 905-966. doi: 10.1007/s10208-021-09514-y.
[31]	M. Hutzenthaler, A. Jentzen, T. Kruse and T. A. Nguyen, A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations, SN Partial Differ. Equ. Appl., 1 (2020), 1-34. doi: 10.1007/s42985-019-0006-9.
[32]	M. Hutzenthaler, A. Jentzen, T. Kruse and T. A. Nguyen, Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities, preprint, arXiv:2009.02484.
[33]	M. Hutzenthaler, A. Jentzen, T. Kruse and T. A. Nguyen, Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations, J. Numer. Math., 31 (2023), 1-28. doi: 10.1515/jnma-2021-0111.
[34]	M. Hutzenthaler, A. Jentzen, T. Kruse, T. A. Nguyen and P. von Wurstemberger, Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations, Proc. A., 476 (2020), 20190630, 25 pp. doi: 10.1098/rspa.2019.0630.
[35]	M. Hutzenthaler, A. Jentzen, B. Kuckuck and J. L. Padgett, Strong ${L}^{p} $-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations, preprint, arXiv:2110.08297.
[36]	M. Hutzenthaler and T. Kruse, Multilevel Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities, SIAM J. Numer. Anal., 58 (2020), 929-961. doi: 10.1137/17M1157015.
[37]	M. Hutzenthaler and T. A. Nguyen, Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions, Appl. Numer. Math., 181 (2022), 151-175. doi: 10.1016/j.apnum.2022.05.009.
[38]	M. Hutzenthaler and T. A. Nguyen, Multilevel Picard approximations for high-dimensional decoupled forward-backward stochastic differential equations, preprint, arXiv:2204.08511.
[39]	S. Ioffe and C. Szegedy, Batch normalization: Accelerating deep network training by reducing internal covariate shift, Proc. Mach. Learn. Res., 37 (2015), 448-456. Available at PMLR: https://proceedings.mlr.press/v37/ioffe15.html.
[40]	S. Ji, S. Peng, Y. Peng and X. Zhang, Three algorithms for solving high-dimensional fully coupled FBSDEs through deep learning, IEEE Intell. Syst., 35 (2020), 71-84. doi: 10.1109/MIS.2020.2971597.
[41]	S. Ji, S. Peng, Y. Peng and X. Zhang, A control method for solving high-dimensional Hamiltonian systems through deep neural networks, preprint, arXiv:2111.02636.
[42]	S. Ji, S. Peng, Y. Peng and X. Zhang, A deep learning method for solving stochastic optimal control problems driven by fully-coupled FBSDEs, preprint, arXiv:2204.05796.
[43]	Y. Jiang and J. Li, Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficients, Probab. Uncertain. Quant. Risk, 6 (2021), 391-408. doi: 10.3934/puqr.2021019.
[44]	L. Kapllani and L. Teng, Multistep schemes for solving backward stochastic differential equations on GPU, J. Math. Industry, 12 (2022), 1-22. doi: 10.1186/s13362-021-00118-3.
[45]	N. E. Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Financ., 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.
[46]	D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, preprint, arXiv:1412.6980.
[47]	P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.
[48]	S. Kremsner, A. Steinicker and M. Szölgyenyi, A deep neural network algorithm for semilinear elliptic PDEs with applications in insurance mathematics, Risks, 8 (2020), 136. doi: 10.3390/risks8040136.
[49]	C. Labart and J. Lelong, A parallel algorithm for solving BSDEs: Application to the pricing and hedging of American options, Monte Carlo Methods Appl., 19 (2013), 11-39. doi: 10.1515/mcma-2013-0001.
[50]	J.-P. Lemor, E. Gobet and X. Warin, Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations, Bernoulli, 12 (2006), 889-916. doi: 10.3150/bj/1161614951.
[51]	J. Liang, Z. Xu and P. Li, Deep learning-based least squares forward-backward stochastic differential equation solver for high-dimensional derivative pricing, Quant. Finance, 21 (2021), 1309-1323. doi: 10.1080/14697688.2021.1881149.
[52]	J. Ma, J. Shen and Y. Zhao, On numerical approximations of forward-backward stochastic differential equations, SIAM J. Numer. Anal., 46 (2008), 2636-2661. doi: 10.1137/06067393X.
[53]	B. Negyesi, K. Andersson and C. W. Oosterlee, The one step Malliavin scheme: New discretization of BSDEs implemented with deep learning regressions, preprint, arXiv:2110.05421.
[54]	B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.
[55]	E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Syst. Control. Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.
[56]	M. A. Pereira, Z. Wang, I. Exarchos and E. A. Theodorou, Learning deep stochastic optimal control policies using forward-backward SDEs, preprint, arXiv:1902.03986.
[57]	H. Pham, X. Warin and M. Germain, Neural networks-based backward scheme for fully nonlinear PDEs, SN Partial Differ. Equ. Appl., 2 (2021), 1-24. doi: 10.1007/s42985-020-00062-8.
[58]	M. Raissi, Forward-backward stochastic neural networks: Deep learning of high-dimensional partial differential equations, preprint, arXiv:1804.07010.
[59]	M. J. Ruijter and C. W. Oosterlee, Numerical Fourier method and second-order Taylor scheme for backward SDEs in finance, Appl. Numer. Math., 103 (2016), 1-26. doi: 10.1016/j.apnum.2015.12.003.
[60]	M. J. Ruijter and C. W. Oosterlee, A Fourier cosine method for an efficient computation of solutions to BSDEs, SIAM J. Sci. Comput., 37 (2015), A859-A889. doi: 10.1137/130913183.
[61]	D. E. Rumelhart, G. E. Hinton and R. J. Williams, Learning representations by back-propagating errors, Nature, 323 (1986), 533-536. doi: 10.1038/323533a0.
[62]	A. M. Schäfer and H. G. Zimmermann, Recurrent neural networks are universal approximators, Lecture Notes in Computer Science, 4131 (2006), 632-640. doi: 10.1007/11840817_66.
[63]	A. Takahashi, Y. Tsuchida and T. Yamada, A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver, J. Comput. Phys., 454 (2022), 110956. doi: 10.1016/j.jcp.2022.110956.
[64]	L. Teng, A review of tree-based approaches to solving forward-backward stochastic differential equations, J. Comput. Finance, 25 (2021), 125-159. doi: 10.21314/JCF.2021.010.
[65]	L. Teng, Gradient boosting-based numerical methods for high-dimensional backward stochastic differential equations, Appl. Math. Comput., 426 (2022), 127119. doi: 10.1016/j.amc.2022.127119.
[66]	H. Wang, H. Chen, A. Sudjianto, R. Liu and Q. Shen, Deep learning-based BSDE solver for LIBOR market model with application to Bermudan swaption pricing and hedging, preprint, arXiv:1807.06622.
[67]	G. Zhang, M. Gunzburger and W. Zhao, A sparse-grid method for multi-dimensional backward stochastic differential equations, J. Comp. Math., 31 (2013), 221-248. doi: 10.4208/jcm.1212-m4014.
[68]	J. Zhang, A numerical scheme for BSDEs, Ann. Appl. Probab., 14 (2004), 459-488. doi: 10.1214/aoap/1075828058.
[69]	W. Zhao, L. Chen and S. Peng, A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), 1563-1581. doi: 10.1137/05063341X.
[70]	W. Zhao, Y. Fu and T. Zhou, New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations, SIAM J. Sci. Comput., 36 (2014), A1731-A1751. doi: 10.1137/130941274.
[71]	W. Zhao, G. Zhang and L. Ju, A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), 1369-1394. doi: 10.1137/09076979X.