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==History==
===Backwards stochastic differential equations===
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<ref name="Pardoux1990">{{cite journal | last1=Pardoux | first1=E. | last2=Peng | first2=S. | title=Adapted solution of a backward stochastic differential equation | journal=Systems & Control Letters | volume=14 | issue=1 | pages=55-61 | year=1990 }}</ref>.
===Deep learning===
[[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 Traditional Numerical Methods===
Tranditional 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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#In high-dimensional scenarios, the Monte Carlo method requires numerous simulation paths to ensure accuracy, resulting in lengthy computation times. In particular, for nonlinear BSDEs, the convergence rate is slow, making it challenging to handle complex financial derivative pricing problems<ref name="puc">{{cite web | title = Real Options with Monte Carlo Simulation | url = http://www.puc-rio.br/marco.ind/monte-carlo.html | access-date = 2010-09-24 | archive-url = https://web.archive.org/web/20100318060412/http://www.puc-rio.br/marco.ind/monte-carlo.html | archive-date = 2010-03-18 | url-status = dead }}</ref><ref>{{cite web | title = Monte Carlo Simulation | url = http://www.palisade.com/risk/monte_carlo_simulation.asp | publisher = Palisade Corporation | year = 2010 | access-date = 2010-09-24 }}</ref>. [[File:Pi monte carlo all.gif|thumb|upright=1.3| Monte Carlo method applied to approximating the value of {{pi}}]]
#The finite difference method, on the other hand, experiences exponential growth in the number of computation grids with increasing dimensions, leading to significant computational and storage demands. This method is generally suitable for simple boundary conditions and low-dimensional BSDEs, but it is less effective in complex situations<ref name="GrossmannRoos2007">{{cite book|author1=Christian Grossmann|author2=Hans-G. Roos| author3=Martin Stynes|title=Numerical Treatment of Partial Differential Equations| url=https://archive.org/details/numericaltreatme00gros_820|url-access=limited| year=2007| publisher=Springer Science & Business Media| isbn=978-3-540-71584-9|page=[https://archive.org/details/numericaltreatme00gros_820/page/n34 23]}}</ref>.
===Deep BSDE method===
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. The specific idea is to represent the solution of a BSDE as the output of a neural network and train the network to approximate the solution<ref name="Han2018" />.
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