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arXiv · 1908.03382

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

Christian Beck·Lukas Gonon·Martin Hutzenthaler·Arnulf Jentzen
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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.

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Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. 2019-08-09. On existence and uniqueness properties for solutions of stochastic fixed point equations. https://doi.org/10.3934/dcdsb.2020320

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