Deep backward stochastic differential equation method: Difference between revisions

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=Deep BSDE=

==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<ref name="Han2018" />.

==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<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>.

==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<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>.

* **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<ref name="Han2018" />.

* **Risk Measurement:** The method is applied to calculate risk measures such as [[Conditional Value-at-Risk]] (CVaR) and [[Expected Shortfall]] (ES)<ref name="Beck2019">{{cite journal | last1=Beck | first1=C. | last2=E | first2=W. | last3=Jentzen | first3=A. | title=Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations | journal=Journal of Nonlinear Science | volume=29 | issue=4 | pages=1563-1619 | year=2019 }}</ref>.

* **Dynamic Asset Allocation:** It helps in determining optimal strategies for asset allocation over time in a stochastic environment<ref name="Beck2019" />.

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==References==

{{Reflist}}

* <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>

* <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>

* <ref name="Beck2019">{{cite journal | last1=Beck | first1=C. | last2=E | first2=W. | last3=Jentzen | first3=A. | title=Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations | journal=Journal of Nonlinear Science | volume=29 | issue=4 | pages=1563-1619 | year=2019 }}</ref>