Deep backward stochastic differential equation method: Difference between revisions

Revision as of 09:41, 10 July 2024

Revision as of 10:03, 10 July 2024

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* <math> \sigma </math> is a known vector-valued function, <math> \sigma^T </math> denotes the transpose associated to <math> \sigma </math>, and <math> \text{Hess}_x u </math> denotes the Hessian of function <math> u </math> with respect to <math> x </math>.

* <math> \mu </math> is a known vector-valued function, and <math> f </math> is a known nonlinear function.

====2. Stochastic Processprocess Representationrepresentation====

Let <math> \{W_t\}_{t \geq 0} </math> be a <math> d </math>-dimensional Brownian motion and <math> \{X_t\}_{t \geq 0} </math> be a <math> d </math>-dimensional stochastic process which satisfies

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X_t = \xi + \int_0^t \mu(s, X_s) \, ds + \int_0^t \sigma(s, X_s) \, dW_s

</math>

====3. Backward stochastic differential equation(BSDE)====

Then the solution of the PDE satisfies the following BSDE: