Multi-level Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities

Abstract

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.

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Cited by in corpus (7)

• Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
• Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
• Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations
• On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations
• Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations
• Monte Carlo for high-dimensional degenerated Semi Linear and Full Non Linear PDEs
• A Numerical Scheme For High-dimensional Backward Stochastic Differential Equation Based On Modified Multi-level Picard Iteration