Multi-level Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities
Provides a theoretical guarantee for polynomial-time approximation of high-dimensional PDEs with gradient-dependent nonlinearities, which are common in financial derivative pricing and hedging.
The authors prove that semilinear heat equations with gradient-dependent nonlinearities can be approximated with computational complexity growing polynomially in dimension and reciprocal accuracy, enabling efficient high-dimensional PDE solutions for financial applications.
Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) have a wide range of applications. In particular, high-dimensional PDEs with gradient-dependent nonlinearities appear often in the state-of-the-art pricing and hedging of financial derivatives. In this article we prove that semilinear heat equations with gradient-dependent nonlinearities can be approximated under suitable assumptions with computational complexity that grows polynomially both in the dimension and the reciprocal of the accuracy.