Deep backward stochastic differential equation method: Difference between revisions

Revision as of 22:05, 23 August 2024

Revision as of 12:23, 11 September 2024

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[[Deep Learning]] is a [[machine learning]] method based on multilayer [[neural networks]]. 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. In 2006, the [[Deep Belief Networks]] proposed by [[Geoffrey Hinton]] and others rekindled interest in deep learning. Since then, deep learning has made groundbreaking advancements in [[image processing]], [[speech recognition]], [[natural language processing]], and other fields.<ref name="NatureBengio">{{cite journal |last1=LeCun |first1= Yann|last2=Bengio |first2=Yoshua | last3=Hinton | first3= Geoffrey|s2cid=3074096 |year=2015 |title=Deep Learning |journal=Nature |volume=521 |issue=7553 |pages=436–444 |doi=10.1038/nature14539 |pmid=26017442|bibcode=2015Natur.521..436L |url= https://hal.science/hal-04206682/file/Lecun2015.pdf}}</ref>

===Limitations of Traditionaltraditional Numericalnumerical Methodsmethods===

Traditional numerical methods for solving stochastic differential equations<ref name="kloeden">Kloeden, P.E., Platen E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin, Heidelberg. DOI: https://doi.org/10.1007/978-3-662-12616-5</ref> include the [[Euler–Maruyama method]], [[Milstein method]], [[Runge–Kutta method (SDE)]] and methods based on different representations of iterated stochastic integrals.<ref name="Kuznetsov">Kuznetsov, D.F. (2023). Strong approximation of iterated Itô and Stratonovich stochastic integrals: Method of generalized multiple Fourier series. Application to numerical integration of Itô SDEs and semilinear SPDEs. Differ. Uravn. Protsesy Upr., no. 1. DOI: https://doi.org/10.21638/11701/spbu35.2023.110</ref><ref name="Rybakov">Rybakov, K.A. (2023). Spectral representations of iterated stochastic integrals and their application for modeling nonlinear stochastic dynamics. Mathematics, vol. 11, 4047. DOI: https://doi.org/10.3390/math11194047</ref>

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</math>

====3. Backward stochastic differential equation (BSDE)====

Then the solution of the PDE satisfies the following BSDE:

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This function implements the Adam<ref name="Adam2014">{{cite arXiv |eprint=1412.6980 |class=cs.LG |first1=Diederik |last1=Kingma |first2=Jimmy |last2=Ba |title=Adam: A Method for Stochastic Optimization |year=2014}}</ref> algorithm for minimizing the target function <math>\mathcal{G}(\theta)</math>.

'''Function:''' ADAM(<math>\alpha</math>, <math>\beta_1</math>, <math>\beta_2</math>, <math>\epsilon</math>, <math>\mathcal{G}(\theta)</math>, <math>\theta_0</math>) '''is'''

<math>m_0 := 0</math> ''// Initialize the first moment vector''

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This function implements the backpropagation algorithm for training a multi-layer feedforward neural network.

'''Function:''' BackPropagation(''set'' <math>D=\left\{(\mathbf{x}_k,\mathbf{y}_k)\right\}_{k=1}^{m}</math>) '''is'''

''// Step 1: Random initialization''

''// Step 2: Optimization loop''

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This function calculates the optimal investment portfolio using the specified parameters and stochastic processes.

'''function''' OptimalInvestment(<math>W_{t_{i+1}} - W_{t_i}</math>, <math>x</math>, <math>\theta=(X_{0}, H_{0}, \theta_{1}, \theta_{2}, \dots, \theta_{N-1})</math>) '''is'''

''// Step 1: Initialization''

'''for''' <math>k := 0</math> '''to''' maxstep '''do'''

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===Advantages===

Sources:<ref name="Han2018" /><ref name="Beck2019" />

# High-Dimensionaldimensional Capabilitycapability: Compared to traditional numerical methods, deep BSDE performs exceptionally well in high-dimensional problems.

# Flexibility: The incorporation of deep neural networks allows this method to adapt to various types of BSDEs and financial models.

# Parallel Computingcomputing: Deep learning frameworks support GPU acceleration, significantly improving computational efficiency.

===Disadvantages===

Sources:<ref name="Han2018" /><ref name="Beck2019" />

# Training Timetime: Training deep neural networks typically requires substantial data and computational resources.

# Parameter Sensitivitysensitivity: The choice of neural network architecture and hyperparameters greatly impacts the results, often requiring experience and trial-and-error.

==See also==

{{Div col|colwidth=22em}}

* [[Bellman equation]]

* [[Dynamic programming]]

* [[Applications of artificial intelligence]]

* [[List of artificial intelligence projects]]

* [[Backward stochastic differential equation]]

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* {{cite journal|last1=Higham.|first1=Desmond J.|title=An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations|journal=SIAM Review|date=January 2001|volume=43|issue=3|pages=525–546|doi=10.1137/S0036144500378302|bibcode=2001SIAMR..43..525H|citeseerx=10.1.1.137.6375}}

* Desmond Higham and Peter Kloeden: "An Introduction to the Numerical Simulation of Stochastic Differential Equations", SIAM, {{ISBN|978-1-611976-42-7}} (2021).

{{Numerical PDE}}

{{Industrial and applied mathematics}}

{{DEFAULTSORT:Numerical Partial Differential Equations}}

[[Category:Numerical differential equations| ]]