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

## TL;DR

This paper proves the existence and uniqueness of solutions for certain stochastic fixed point equations, especially those related to semilinear Kolmogorov PDEs, even when classical solutions are absent.

## Contribution

It establishes a general framework for the existence and uniqueness of SFPE solutions, extending applicability to PDEs lacking classical solutions.

## Key findings

- Proved existence of unique SFPE solutions in a broad setting.
- Applied results to semilinear Kolmogorov PDEs with Lipschitz nonlinearities.
- Extended solution concepts to cases without classical PDE solutions.

## 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.