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.

中文翻译：

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随机不动点方程解的存在性和唯一性

Feynman-Kac公式意味着，半线性Kolmogorov偏微分方程（PDE）的每个合适的经典解也是某个随机不动点方程（SFPE）的解。在本文中，我们将研究此类和相关的SFPE。特别是，这项工作的主要结果证明了在一般情况下某些SFPE的独特解决方案的存在。作为此主要结果的应用，即使在关联的半线性Kolmogorov PDE不具有经典解的情况下，我们也确定了具有Lipschitz连续非线性的，与半线性Kolmogorov PDE相关的SFPE的唯一解的存在。