For semilinear Kolmogorov PDEs with Lipschitz nonlinearities, a unique continuous at-most-polynomially-growing solution to the associated stochastic fixed point equation exists, even without a classical PDE solution.
Multi-level Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities
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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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On existence and uniqueness properties for solutions of stochastic fixed point equations
For semilinear Kolmogorov PDEs with Lipschitz nonlinearities, a unique continuous at-most-polynomially-growing solution to the associated stochastic fixed point equation exists, even without a classical PDE solution.