Strong Markov Process Research Articles

Fictitious states, coupled laws and local time

The work of Ray and Neveu has established that, for any transition function P on a countable set E, (i) there exists a best possible entrance boundary E + supporting a right continuous, strong Markov process X with transition function P and that (ii) the points y of E + are in one-one correspondence with the extremal entrance laws g y of P. Here, it is shown that, if a point y of E + is regular for itself, then the derived characteristic f y of the local time at y is a regular extremal entrance law “coupled” with g y in the sense of Neveu. Further, coupled laws arise only in this fashion. By using excursion theory, a simple explicit formula for f y in terms of g y may be obtained. The paper contains a conjecture about the intrinsic character of the Ray-Neveu topology and an example which shows emphatically that, in general, local time is not a derivative of occupation time.

Markov process with creation and annihilation

A simple way is given to construct probabilistically a strong Markov process (yi, Px, η ζ of an extended sense such that the expectation u = Ex[f(yt)] provides the solution of (1.1) for a Borel measurable function c (i.e. with both terms of creation and annihilation of mass), where η and ζ stand for the dates of birth and death of a particle, respectively. Actually the (non-probability) measure Px is given as the sum of induced measures of probability measure (of Brownian motion with age) by a set of mappings. The probability measure is obtained by making killed processes of Brownian motion and piecing them together in a specific way.

Converse Theorems for Stochastic Liapunov Functions

If a continuous strong Markov process has certain stochastic stability properties, it is proved that a stochastic Liapunov function (see Kushner [1], [2], [3]) exists.' This is a stochastic counterpart to a converse theorem of Massera [4] for the deterministic case, where the "process" is the solution to an ordinary differential equation.

The exit characteristics of Markov processes with applications to continuous martingales in Rⁿ

Introduction. The main results of this paper were announced in [4]. A weaker form of the results in ??1 and 2 appeared in the author's thesis(1) [5], the research for which was supported by the National Science Foundation. Paul Levy showed in [11] that Brownian motion on the line can be characterized in terms of two martingale functions. Recently Arbib [1] extended this result to include all diffusions on the line with natural boundaries at infinity. His method was to characterize a diffusion by its exit characteristics, but his proof was only of use in the specific cases he was considering. In ?1 we obtain, by probabilistic methods, a similar characterization of minimal (no return from the boundary) right continuous strong Markov processes on a general state space. In ?2 we use this to generalize Levy's theorem to Rn, and in ?4 indicate some extension of Arbib's results to Rn. In ?3 we make use of our general Levy-type theorem to obtain a characterization of all continuous martingales in Rn, extending a result of Dambis [3] on the line. It was recently brought to my attention that Kunita and Watanabe [10] have used quite different methods to obtain somewhat weaker results which were also, in essence, reported by Dubins and Schwarz [7], who used still another method of proof.

The hitting characteristics of a strong Markov process, with applications to continuous martingales in 𝑅ⁿ

The hitting characteristics of a strong Markov process, with applications to continuous martingales in 𝑅ⁿ

Hitting and martingale characterizations of one-dimensional diffusions

The main theorem of the paper is that, for a large class of one-dimensional diffusions (i. e. strong Markov processes with continuous sample paths): if x(t) is a continuous stochastic process possessing the hitting probabilities and mean exit times of the given diffusion, then x(t) is Markovian, with the transition probabilities of the diffusion.

On Markov Random Sets

A Markov random set is a time-homogeneous random closed set on the half-line $t \geqq 0$, satisfying the Markov property of independence between the future and the past when the present is known. Such sets are introduced as a special class of Markov processes. They may be described by a non-increasing right-continuous positive function $g(x)$, $x > 0$, integrable near 0 and a non-negative number $\alpha $, determined up to an arbitrary positive constant factor. If $y(t)$ is a continuous strong Markov process, the t-set $\{ {y(t) = {\text{const}}} \}$ is a Markov random set. The most interesting Markov sets are obtained by simple transformations from the Brownien motion process.

Invariability of the Strong Markov Property in the Transformations of Dynkin

E. B. Dynkin in [2] has defined a large class of transformations of Markov processes. He calls processes obtained by these transformations $\mathop X\limits^ \sim $ and $(\alpha ,\xi )$-subprocesses of the process X. In this paper we prove that if the process X is a strong Markov process, then, with some conditions imposed on $\alpha $ and $\xi $, the $(\alpha ,\xi )$-subprocess will also be a strong Markov process. One special case of a $(\alpha ,\xi )$ subprocess has been examined earlier by E. B. Dynkin in his book [1]. He proved in particular that the strong Markov property does not change in the formation of subprocesses. We show that the method he used for the special case can be carried over to the general case.

On the Definition of a Strong Markov Process

Previous article Next article On the Definition of a Strong Markov ProcessA. A. YushkevichA. A. Yushkevichhttps://doi.org/10.1137/1105021PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] E. B. Dynkin, Inhomogeneous strong Markov processes, Dokl. Akad. Nauk SSSR (N.S.), 113 (1957), 261–263, (In Russian.) MR0100923 (20:7348) 0084.13302 Google Scholar[2] E. B. Dynkin, Foundations of the Theory of Markov Processes, Moscow, 1959, (In Russian.) Google Scholar[3] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950xi+304, (Russian translation, 1953.) MR0033869 (11,504d) 0040.16802 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 5, Issue 2| 1960Theory of Probability & Its Applications History Submitted:26 March 1959Published online:28 July 2006 InformationCopyright © 1960 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1105021Article page range:pp. 216-220ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

One-Dimensional Continuous Strong Markov Processes

One-dimensional temporally homogeneous strong Markov processes with continuous path functions are considered. No regularity conditions are assumed. Infinitesimal operators for all these processes are calculated. These calculations are based on a preliminary analysis of the local behaviour of path functions. The general results of sections 1—3 are used in sections 4—5 for conducting a more detailed investigation of the process in intervals of its regularity. Some results contained in this paper have been published in [3] without proofs.

Random Substitution of Time in Strong Markov Processes

The terminology and symbols are as in [7] and [1].Let $x(t,\omega )$ be a homogeneous strong Markov process, and $\tau _t (\omega )$ be a random function not decreasing for increasing t. The process $y_t = x(\tau _t (\omega ),\omega )$ is called a process obtained from $x_t (\omega )$ by means of a random substitution of time $\tau _t $.The conditions sufficient for the process $y_t $ to be a Markov or a strong Markov process are formulated (Theorems 1 and 2).In [1] it is shown that the infinitesimal operator A of a Feller strong Markov process continuous on the right is a contraction of a certain operator $\mathfrak{A}$, which is called the extended operator. It is shown that if $x_t $ and $x(\tau _t )$ are Feller processes continuous on the right and $\tau _t $ is determined by equation (2), where $\varphi (x) > 0$, and continuous, then their extended operator is $\mathfrak{A}$, where $\mathfrak{A}$ satisfies the equation $t = \varphi (x)\mathfrak{A}$ (Theorem 3).In Theorem 4 and in its corollary it is ...

On Strong Markov Processes

The problem of constructing a strong Markov process with a given measurable Markov transition function $p(s,x,t,\Gamma )$ is considered. The space X of possible states is supposed to be given together with the function $p(s,x,t,\Gamma )$. If it is required that the sample functions $x(t,\omega )$ be defined for each $\omega $ at all $t \in [0,\infty )$, then as shown by the examples described, the three following types of measurable transition functions exist: 1) every Markov process with a given transition function is a strong Markov process; 2) there are both strong Markov processes, and measurable not strong Markov processes with a given transition function; 3) there are no strong Markov processes with a given transition function. To avoid the third case, it is important to know, whether there exists a process with a given transition function whose sample functions are continuous from the right. Under some general assumptions about the space X, a process with such sample functions exists if for each $s \geqq 0$ and $\varepsilon \geqq 0$ uniformly in $x \in X$\[ {\text{(I)}}\qquad \mathop {\lim }\limits_{t \downarrow s} p(u,x,t,V_\varepsilon (x)) = 1,\quad s \leqq u \leqq t, \] where $V_\varepsilon (x)$ is the $\varepsilon $-neighborhood of the point x. For the discrete topology, every Markov process with sample functions continuous from the right is a strong Markov process. If for each $s \geqq 0$ uniformly in $x \in X$\[ {\text{(II)}} \qquad \mathop {\lim }\limits_{t \downarrow s} p(s,x,t,x) = 1, \] then the sample functions of the process can be assumed to be continuous from the right in the discrete topology. For the case of a finite number of states, condition (II) is also necessary for continuity of the sample functions from the right. If it is not required that the sample functions $x(t,\omega )$ be defined for each $\omega $ at all t, then in all known examples of measurable transition functions of type (3), one can obtain the strong Markov property, which is now defined in accordance with the new character of the sample functions. This can be done for every transition function of a time-homogeneous process with a denumerable number of states for which the condition (II) is true (without uniformity in x).