Abstract
The stochastic optimal control uses the differential equation of Bell-man and its solution—the Bellman function. We show how the homonym function in harmonic analysis is (and how it is not) the same stochastic optimal control Bellman function. Then we present several creatures from Bellman’s Zoo: a function that proves the inverse Hölder inequality, as well as several other harmonic analysis Bellman functions and their corresponding Bellman PDE’s. Finally we translate the approach of Burkholder to the language of “our” Bellman function.
The goal of this paper is almost entirely methodological: we relate the ideas between each other, rather than presenting the new ones.
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Nazarov, F., Treil, S., Volberg, A. (2001). Bellman function in stochastic control and harmonic analysis. In: Borichev, A.A., Nikolski, N.K. (eds) Systems, Approximation, Singular Integral Operators, and Related Topics. Operator Theory: Advances and Applications, vol 129. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8362-7_16
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DOI: https://doi.org/10.1007/978-3-0348-8362-7_16
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