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12. Risk-sensitive average optimality in Markov decision processes
- Creator:
- Sladký, Karel
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- controlled Markov processes, finite state space, asymptotic behavior, and risk-sensitive average optimality
- Language:
- English
- Description:
- In this note attention is focused on finding policies optimizing risk-sensitive optimality criteria in Markov decision chains. To this end we assume that the total reward generated by the Markov process is evaluated by an exponential utility function with a given risk-sensitive coefficient. The ratio of the first two moments depends on the value of the risk-sensitive coefficient; if the risk-sensitive coefficient is equal to zero we speak on risk-neutral models. Observe that the first moment of the generated reward corresponds to the expectation of the total reward and the second central moment of the reward variance. For communicating Markov processes and for some specific classes of unichain processes long run risk-sensitive average reward is independent of the starting state. In this note we present necessary and sufficient condition for existence of optimal policies independent of the starting state in unichain models and characterize the class of average risk-sensitive optimal policies.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
13. Second order linear $q$-difference equations: nonoscillation and asymptotics
- Creator:
- Řehák, Pavel
- Type:
- model:article and TEXT
- Subject:
- regularly varying functions, $q$-difference equations, asymptotic behavior, and oscillation
- Language:
- English
- Description:
- The paper can be understood as a completion of the $q$-Karamata theory along with a related discussion on the asymptotic behavior of solutions to the linear $q$-difference equations. The $q$-Karamata theory was recently introduced as the theory of regularly varying like functions on the lattice $q^{\mathbb {N}_0}:=\{q^k\colon k\in \mathbb {N}_0\}$ with $q>1$. In addition to recalling the existing concepts of $q$-regular variation and $q$-rapid variation we introduce $q$-regularly bounded functions and prove many related properties. The $q$-Karamata theory is then applied to describe (in an exhaustive way) the asymptotic behavior as $t\to \infty $ of solutions to the $q$-difference equation $D_q^2y(t)+p(t)y(qt)=0$, where $p\colon \smash {q^{\mathbb {N}_0}}\to \mathbb {R}$. We also present the existing and some new criteria of Kneser type which are related to our subject. A comparison of our results with their continuous counterparts is made. It reveals interesting differences between the continuous case and the $q$-case and validates the fact that $q$-calculus is a natural setting for the Karamata like theory and provides a powerful tool in qualitative theory of dynamic equations.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
14. Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations
- Creator:
- Poláčik, P.
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- parabolic equations, elliptic equations, hyperbolic equations, asymptotic behavior, and center manifold
- Language:
- English
- Description:
- We consider three types of semilinear second order PDEs on a cylindrical domain Ω × (0,∞), where Ω is a bounded domain in RN , N ≥ 2. Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of Ω × (0,∞) is reserved for time t, the third type is an elliptic equation with a singled out unbounded variable t. We discuss the asymptotic behavior, as t → ∞, of solutions which are defined and bounded on Ω × (0,∞).
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public