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

by Ch. Beck and L. Gonon and M. Hutzenthaler and A. Jentzen

(Report number 2019-47)

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

Keywords:

@Techreport{BGHJ19_851,
  author = {Ch. Beck and L. Gonon and M. Hutzenthaler and A. Jentzen},
  title = {On existence and uniqueness properties for solutions of stochastic fixed point equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-47},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-47.pdf },
  year = {2019}
}

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