Risk-Sensitive Control and an Abstract Collatz–Wielandt Formula

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• 作者：Ari Arapostathis ; Vivek S. Borkar ; K. Suresh Kumar
• 关键词：Risk ; sensitive control ; Collatz–Wielandt formula ; Nisio semigroup ; Variational formulation ; Principal eigenvalue ; Donsker–Varadhan functional
• 刊名：Journal of Theoretical Probability
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：29
• 期：4
• 页码：1458-1484
• 全文大小：562 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Statistics
  
• 出版者：Springer Netherlands
• ISSN：1572-9230
• 卷排序：29

文摘

The ‘value’ of infinite horizon risk-sensitive control is the principal eigenvalue of a certain positive operator. For the case of compact domain, Chang has built upon a nonlinear version of the Krein–Rutman theorem to give a ‘min–max’ characterization of this eigenvalue which may be viewed as a generalization of the classical Collatz–Wielandt formula for the Perron–Frobenius eigenvalue of a nonnegative irreducible matrix. We apply this formula to the Nisio semigroup associated with risk-sensitive control and derive a variational characterization of the optimal risk-sensitive cost. For the linear, i.e., uncontrolled case, this is seen to reduce to the celebrated Donsker–Varadhan formula for principal eigenvalue of a second-order elliptic operator.