On existence and uniqueness properties for solutions of stochastic fixed point equations

Abstract

The Feynman-Kac formula implies that every suitable classical solution of a semilinear Kolmogorov partial differential equation (PDE) is also a solution of a certain stochastic fixed point equation (SFPE). In this article we study such and related SFPEs. In particular, the main result of this work proves existence of unique solutions of certain SFPEs in a general setting. As an application of this main result we establish the existence of unique solutions of SFPEs associated with semilinear Kolmogorov PDEs with Lipschitz continuous nonlinearities even in the case where the associated semilinear Kolmogorov PDE does not possess a classical solution.

33 pages

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Cited by in corpus (4)

• 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
• Generalised multilevel Picard approximations
• Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities