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Stochastic differential equations and stochastic linear quadratic optimal control problem with Lévy processes

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Abstract

In this paper, the authors first study two kinds of stochastic differential equations (SDEs) with Lévy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations (BSDEs) driven by Lévy processes, the authors proceed to study a stochastic linear quadratic (LQ) optimal control problem with a Lévy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results.

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Correspondence to Huaibin Tang.

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This work was supported by the National Basic Research Program of China (973 Program) under Grant No. 2007CB814904, the Natural Science Foundation of China under Grant No. 10671112 and Shandong Province under Grant No. Z2006A01, and Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20060422018.

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Tang, H., Wu, Z. Stochastic differential equations and stochastic linear quadratic optimal control problem with Lévy processes. J Syst Sci Complex 22, 122–136 (2009). https://doi.org/10.1007/s11424-009-9151-0

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  • DOI: https://doi.org/10.1007/s11424-009-9151-0

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