Deep backward stochastic differential equation method: Difference between revisions

Revision as of 12:48, 9 July 2024

Revision as of 12:49, 9 July 2024

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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<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>.

Fix a terminal time <math>T>0</math> and a [[probability space]] <math>(\Omega,\mathcal{F},\mathbb{P})</math>. Let <math>(B_t)_{t\in [0,T]}</math> be a [[Brownian motion]] with natural filtration <math>(\mathcal{F}_t)_{t\in [0,T]}</math>. A backward stochastic differential equation is an integral equation of the type