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The combination of the It\^o formula and the Bismut-Elworthy-Li formulaimplies that suitable smooth solutions of semilinear Kolmogorov partialdifferential equations (PDEs) are also solutions to certain stochastic fixedpoint equations (SFPEs). In this paper we generalize known results on existenceand uniqueness of solutions of SFPEs associated with PDEs with Lipschitzcontinuous, gradient-independent nonlinearities to the case ofgradient-dependent nonlinearities. The main challenge arises from the fact thatin the case of a non-differentiable terminal condition and a gradient-dependentnonlinearity the Bismut-Elworthy-Li formula leads to a singularity of thesolution of the SFPE in the last time point.

On existence and uniqueness properties for solutions of stochastic fixed point equations with gradient-dependent nonlinearities