Event Algebra Research Articles

Optimal Stopping in a Markov Process

Let $X = (X_t, \mathbf{F}_t, P^x)_{t\geqq0}$ be a Markov process where $(X_t; t \geqq 0)$ is the trajectory or sample path, $\mathbf{F}_t$ is the definitive $\sigma$-algebra of events generated by $(X_s; 0 \leqq s \leqq t)$, and $P^x$ is the probability distribution on sample paths corresponding to an initial state $x$. The state space is taken as the semi-compact $(E, \mathbf{C})$ where $E$ is a locally compact separable metric space with family of open sets $\mathbf{C}$. A non-negative extended real valued random variable $T$ such that for each $t \geqq 0, \{T \leqq t\} \varepsilon \mathbf{F}_t$ is called a Markov time or stopping time. This paper studies the problem of choosing a stopping time $T$ which, for a fixed $\lambda \geqq 0$, maximizes one of the following criteria:\begin{equation*}\tag{1} \Theta_T(x) = E^xe^{-\lambda T} g(X_T);\end{equation*}\begin{equation*}\tag{2} \lambda_T(x) = E^x\lbrack e^{-\lambda T} g(X_T) - \int^T_o e^{-\lambda s} c(X_s) ds\rbrack, \text{where} E^xT < \infty;\text{or}\end{equation*}\begin{equation*}\tag{3} \Phi_T(x) = E^x\lbrack g(X_T) - \int^T_0 c(X_s) ds\rbrack/E^xT, \text{where} 0 < E^xT < \infty;\end{equation*} where $g$ and $c$ are non-negative continuous functions defined on the state space of the process. Dynkin [9] studied criterion (1) where $\lambda = 0$ under the general assumption that $X$ is a standard process with a possibly random lifetime and under very weak continuity assumptions concerning the return function $g$. He showed that criterion (2) can often be transformed into criterion (1), and thus his approach is applicable in this case as well. This paper studies optimal stopping in a Markov process having a Feller transition function, a special case in Dynkin's development. We further specialize to exponentially distributed lifetimes which causes the appearance of a discount factor $e^{-\lambda t}$, with the natural interpretation that a dollar transaction $t$ time units hence has a present value of $e^{-\lambda t}$. Criterion (3) often has the meaning of a long-run time average return and a means of transforming this criterion into criterion (2) is given. Finally, some techniques for implementing Dynkin's approach in a variety of commonly occurring situations are given along with examples of their use.

The Local Limit Theorem for Markov Chains and Regularity Conditions

Previous article Next article The Local Limit Theorem for Markov Chains and Regularity ConditionsB. M. GurevichB. M. Gurevichhttps://doi.org/10.1137/1113019PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR (N.S.), 119 (1958), 861–864, (In Russian.) MR0103254 0083.10602 Google Scholar[2] I. A. Ibragimov and , Yu. V. Linnik, Independent and Stationarily Dependent Variables, “Nauka”, Moscow, 1965, 524–, (In Russian.) Google Scholar[3] B. M. Gurevich, On one class of special automorphisms and special flows, Dokl. Akad. Nauk SSSR, 153 (1963), 754–757, (In Russian.) Google Scholar[4] A. N. Kolmogorov, A local limit theorem for classical Markov chains, Izvestiya Akad. Nauk SSSR. Ser. Mat., 13 (1949), 281–300, (In Russian.) MR0031216 Google Scholar[5] I. Z. Rozenknop, On some properties of collections of closed paths in a system with n states and given connections among them, Izvestiya Akad. Nauk SSSR. Ser. Mat., 14 (1950), 95–100, (In Russian.) MR0033468 Google Scholar[6] I. V. Čulanovskii˘, On cycles in Markov chains, Doklady Akad. Nauk SSSR (N.S.), 69 (1949), 301–304, (In Russian.) MR0032120 Google Scholar[7] J. L. Doob, Stochastic processes, John Wiley & Sons Inc., New York, 1953viii+654 MR0058896 0053.26802 Google Scholar[8] Ya. G. Sinai, Dynamic systems and stationary Markov processes, Theory Prob. Applications, 4 (1960), 305–308 10.1137/1105029 LinkGoogle Scholar[9] V. A. RoKhlin, New progress in the theory of transformations with invariant measure, Uspekhi Mat. Nauk, 15 (1960), 3–26, (In Russian.) Google Scholar[10] B. M. Gurevich, Some conditions for the existence of K-partitions for special flows, Trudy Mosk. Matem. Ob-va, 17 (1967), 90–117, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails On the $\sigma $-Algebra of Symmetric Events for a Countable Markov ChainL. A. Grigorenko17 July 2006 | Theory of Probability & Its Applications, Vol. 24, No. 1AbstractPDF (723 KB) Volume 13, Issue 1| 1968Theory of Probability & Its Applications History Submitted:08 October 1966Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1113019Article page range:pp. 182-188ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

The Probabilistic Argument for a Non-Classical Logic of Quantum Mechanics

The aim of this paper is to state the single most powerful argument for use of a non-classical logic in quantum mechanics. In outline the argument is the following. The working logic of a science is the logic of the events and propositions to which probabilities are assigned. A probability should be assigned to every element of the algebra of events. In the case of quantum mechanics probabilities may be assigned to events but not, without restriction, to the conjunction of two events. The conclusion is that the working logic of quantum mechanics is not classical. The nature of the logic that is appropriate for quantum mechanics is examined.

Two Complete Axiom Systems for the Algebra of Regular Events

The theory of finite automata is closely linked with the theory of Kleene's regular expressions. In this paper, two formal systems for the algebraic transformation of regular expressions are developed. Both systems are consistent and complete; i.e., the set of equations derivable within the system equals the set of equations between two regular expressions denoting the same event. One of the systems is based upon the uniqueness of the solution of certain regular expression equations, whereas some facts concerning the representation theory of regular events are used in connection with the other.

A method of analysis of abstract automata using equations in the algebra of events

A method of analysis of abstract automata using equations in the algebra of events

Systems of equations in the algebra of events

Systems of equations in the algebra of events

Edmund C. Berkeley. The algebra of states and events. The scientific monthly, vol. 78 (1954), pp. 232–242.

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