Deep backward stochastic differential equation method: Difference between revisions

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===Limitations of Traditional Numerical Methods===

TranditionalTraditional 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>

But as financial problems become more complex, traditional numerical methods for BSDEs (such as the [[Monte Carlo method]], [[finite difference method]], etc.) have shown limitations such as high computational complexity and the curse of dimensionality.<ref name="Han2018">{{cite journal | last1=Han | first1=J. | last2=Jentzen | first2=A. | last3=E | first3=W. | title=Solving high-dimensional partial differential equations using deep learning | journal=Proceedings of the National Academy of Sciences | volume=115 | issue=34 | pages=8505-8510 | year=2018 }}</ref>

#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=1.35| 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" />