Stochastic HJB Equations and Regular Singular Points
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Authors
Krener, Arthur J.
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Date of Issue
2018
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ArXiv
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Abstract
horizon stochastic optimal control problems have a regular singular point at the origin. This makes them amenable to solution by power series techniques. This extends the work of Al’brecht who showed that the HJB equations of an infinite horizon deterministic optimal control problem can have a regular singular point at the origin, Al’brekht solved the HJB equations by power series, degree by degree. In particular, we show that the infinite horizon stochastic optimal control problem with linear dynamics, quadratic cost and bilinear noise leads to a new type of algebraic Riccati equation which we call the Stochastic Algebraic Riccati Equation (SARE). If SARE can be solved then one has a complete solution to this infinite horizon stochastic optimal control problem. We also show that a finite horizon stochastic optimal control problem with linear dynamics, quadratic cost and bilinear noise leads to a Stochastic Differential Riccati Equation (SDRE) that is well known. If these problems are the linear-quadratic-bilinear part of a nonlinear finite horizon stochastic optimal control problem then we show how the higher degree terms of the solutions can be computed degree by degree. To our knowledge this computation is new.
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Preprint
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Applied Mathematics
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Research supported in part by AFOSR
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16 p.
Citation
Krener, Arthur J. "Stochastic HJB Equations and Regular Singular Points." arXiv preprint arXiv:1806.04120 (2018).
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This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.