Deep backward stochastic differential equation method: Difference between revisions

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==Model==

===Mathematical Method===

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>.

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>.

A backward stochastic differential equation (BSDE) can be formulated as:

<math> Y_t = \xi + \int_t^T f(s, Y_s, Z_s) \, ds - \int_t^T Z_s \, dW_s, \quad t \in [0, T] </math>

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

In this equation:

* <math> Y_t </math> represents the solution process.

{{NumBlk|:|<math> Y_t = \xi + \int_t^T f(s, Y_s, Z_s) \, dsmathrm{d}s - \int_t^T Z_s \mathrm{d}B_s, dW_s, \quad t \in [0, T] ,</math>|{{EquationRef|1}}}}

* <math> \xi </math> is the terminal condition specified at time <math> T </math>.

* <math> f </math> is the driver function, which can depend on time <math> s </math>, the solution <math> Y_s </math>, and the control process <math> Z_s </math>.

In this equation:

* <math> Z_s </math> is the control process, which is adapted to the [[Brownian motion|Brownian motion]] <math> W_s </math>.

* <math>f:[0,T]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}</math> is called the generator of the BSDE,

* <math> W_s </math> is a standard Brownian motion.

* <math> \xi </math> is an <math>\mathcal{F}_T</math>-measurable random variable and the terminal condition specified at time <math> T </math>.

* <math>(Y_t,Z_t)_{t\in[0,T]}</math> is the solution process, which consists of stochastic processes <math>(Y_t)_{t\in[0,T]}</math> and <math>(Z_t)_{t\in[0,T]}</math>

* <math>(Y_t)_{t\in[0,T]}</math> and <math>(Z_t)_{t\in[0,T]}</math> which are adapted to the filtration <math>(\mathcal{F}_t)_{t\in [0,T]}</math>.

* <math> Z_sB_s </math> is thea control process, which is adapted to thestandard [[Brownian motion|Brownian motion]] <math> W_s </math>.

The goal is to find adapted processes <math> Y_t </math> and <math> Z_t </math> that satisfy this equation. Traditional numerical methods struggle with BSDEs due to the curse of dimensionality, which makes computations in high-dimensional spaces extremely challenging.

===Neural Network Architecture===

[[File:Deep BSDE Method.png|thumb|Neural Network Framework of Deep BSDE Method]]