Deep backward stochastic differential equation method
Deep backward stochastic differential equation method 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] BSDEs were first introduced by Pardoux and Peng in 1990 and have since become essential tools in stochastic control and financial mathematics.
In the 1990s, Étienne Pardoux and Shige Peng established the existence and uniqueness theory for BSDE solutions, applying BSDEs to financial mathematics and control theory.
For instance, BSDEs have been widely used in option pricing, risk measurement, and dynamic hedging.
Its core concept can be traced back to the neural computing models of the 1940s.
In the 1980s, the proposal of the backpropagation algorithm made the training of multilayer neural networks possible.
Since then, deep learning has made groundbreaking advancements in image processing, speech recognition, natural language processing, and other fields.
have shown limitations such as high computational complexity and the curse of dimensionality.
[1] 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.
The Deep BSDE approach leverages the powerful nonlinear fitting capabilities of deep learning, approximating the solution of BSDEs by constructing neural networks.
[1] Backward Stochastic Differential Equations (BSDEs) represent a powerful mathematical tool extensively applied in fields such as stochastic control, financial mathematics, and beyond.
Unlike traditional Stochastic differential equations (SDEs), which are solved forward in time, BSDEs are solved backward, starting from a future time and moving backwards to the present.
This unique characteristic makes BSDEs particularly suitable for problems involving terminal conditions and uncertainties.
Traditional numerical methods struggle with BSDEs due to the curse of dimensionality, which makes computations in high-dimensional spaces extremely challenging.
Stack all sub-networks in the approximation step to form a deep neural network.
Source:[1] Deep learning encompass a class of machine learning techniques that have transformed numerous fields by enabling the modeling and interpretation of intricate data structures.
These methods, often referred to as deep learning, are distinguished by their hierarchical architecture comprising multiple layers of interconnected nodes, or neurons.
This architecture allows deep neural networks to autonomously learn abstract representations of data, making them particularly effective in tasks such as image recognition, natural language processing, and financial modeling.
[3] The choice of deep BSDE network architecture, the number of layers, and the number of neurons per layer are crucial hyperparameters that significantly impact the performance of the deep BSDE method.
The deep BSDE method constructs neural networks to approximate the solutions for
, and utilizes stochastic gradient descent and other optimization algorithms for training.
[1] The fig illustrates the network architecture for the deep BSDE method.
is the multilayer feedforward neural network approximating the spatial gradients at time
is the forward iteration providing the final output of the network as an approximation of
is the shortcut connecting blocks at different times, characterized by Eqs.
This function implements the backpropagation algorithm for training a multi-layer feedforward neural network.
Source:[1] This function calculates the optimal investment portfolio using the specified parameters and stochastic processes.
Deep BSDE is widely used in the fields of financial derivatives pricing, risk management, and asset allocation.
