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The set of probability distribution solutions of a linear functional equation. (English) Zbl 1145.39305

Authors’ abstract: Let \(({\Omega}, {\mathcal A}, P)\) be a probability space and let \(\tau:{\mathbb R}\times{\Omega}\to{\mathbb R}\) be a function which is strictly increasing and continuous with respect to the first variable, measurable with respect to the second variable. Given the set of all continuous probability distribution solutions of the equation
\[ F(x)=\int_{{\Omega}}F(\tau(x,\omega))\,dP(\omega) \]
we determine the set of all its probability distribution solutions but fail to construct a single explicit one.

MSC:

39B12 Iteration theory, iterative and composite equations
39B22 Functional equations for real functions
45A05 Linear integral equations
45R05 Random integral equations
60H20 Stochastic integral equations
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