Deep backward stochastic differential equation method

Introduction

Deep BSDE (Deep Backward Stochastic Differential Equation) is a numerical method that combines deep learning with Backward stochastic differential equation (BSDE). This method is particularly useful for solving high-dimensional problems in financial derivatives pricing and risk management. By leveraging the powerful function approximation capabilities of deep neural networks, deep BSDE addresses the computational challenges faced by traditional numerical methods in high-dimensional settings^[1].

Example

Consider the high-dimensional Black-Scholes equation for European option pricing. Traditional numerical methods face significant challenges due to the curse of dimensionality. Deep BSDE uses neural networks to approximate the solution, significantly improving both accuracy and computational efficiency^[1].

Definitions

• **Backward Stochastic Differential Equations (BSDEs):** BSDEs are a class of stochastic differential equations where the terminal condition is specified, and the solution is sought backward in time^[2].
• **Curse of Dimensionality:** This refers to the exponential increase in computational resources needed to solve problems as the dimensionality of the data increases.

Further Examples

• **High-Dimensional Option Pricing:** Deep BSDE is used to price complex derivatives like basket options and Asian options, which involve multiple underlying assets^[1].
• **Risk Measurement:** The method is applied to calculate risk measures such as Conditional Value-at-Risk (CVaR) and Expected Shortfall (ES)^[3].
• **Dynamic Asset Allocation:** It helps in determining optimal strategies for asset allocation over time in a stochastic environment^[3].

History

BSDEs were first introduced by Pardoux and Peng in 1990^[2] and have since become essential tools in stochastic control and financial mathematics. The combination of deep learning with BSDEs, known as deep BSDE, was proposed by Han, Jentzen, and E in 2018 as a solution to the high-dimensional challenges faced by traditional numerical methods^[1].

Mathematical Construction

A standard BSDE can be expressed as: ${\displaystyle Y_{t}=\xi +\int _{t}^{T}f(s,Y_{s},Z_{s})ds-\int _{t}^{T}Z_{s}dW_{s}}$  where ${\displaystyle Y_{t}}$  is the target variable, ${\displaystyle \xi }$  is the terminal condition, ${\displaystyle f}$  is the driver function, and ${\displaystyle Z_{t}}$  is the process associated with the Brownian motion ${\displaystyle W_{t}}$ . The deep BSDE method constructs neural networks to approximate the solutions for ${\displaystyle Y}$  and ${\displaystyle Z}$ , and utilizes stochastic gradient descent and other optimization algorithms for training^[1].

Algorithm and Implementation

1. **Initialize the parameters of the neural network.** 2. **Generate Brownian motion paths using Monte Carlo simulation.** 3. **At each time step, calculate ${\displaystyle Y_{t}}$  and ${\displaystyle Z_{t}}$  using the neural network.** 4. **Compute the loss function based on the backward iterative formula of the BSDE.** 5. **Optimize the neural network parameters using stochastic gradient descent until convergence.**

The core of this method lies in designing an appropriate neural network structure (such as fully connected networks or recurrent neural networks) and selecting effective optimization algorithms^[1][3].

See Also

• Deep Learning
• Stochastic differential equation
• Financial Derivatives
• Monte Carlo Method
• Curse of Dimensionality

External Links

• [Deep Learning for High-Dimensional PDEs](https://arxiv.org/abs/1707.02568)
• [Backward Stochastic Differential Equations](https://www.math.ku.dk/english/research/conferences/2018/bsde/)
• [Curse of Dimensionality - Scholarpedia](http://www.scholarpedia.org/article/Curse_of_dimensionality)

References

1. Han, J.; Jentzen, A.; E, W. (2018). "Solving high-dimensional partial differential equations using deep learning". Proceedings of the National Academy of Sciences. 115 (34): 8505–8510.
2. Pardoux, E.; Peng, S. (1990). "Adapted solution of a backward stochastic differential equation". Systems & Control Letters. 14 (1): 55–61.
3. Beck, C.; E, W.; Jentzen, A. (2019). "Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations". Journal of Nonlinear Science. 29 (4): 1563–1619.