Towards a Characterization of Markov Processes Enjoying the Time-Inversion Property

We give a necessary and sufficient condition for a homogeneous Markov process taking values in ℝ ⁿ to enjoy the time-inversion property of degree α. The condition sets the shape for the semigroup densities of the process and allows to further extend the class of known processes satisfying the time-inversion property. As an application we recover the result of Watanabe (Z. Wahrscheinlichkeitstheor. Verwandte Geb. 31:115–124, 1975) for continuous and conservative Markov processes on ℝ₊. As new examples we generalize Dunkl processes and construct a matrix-valued process with jumps related to the Wishart process by a skew-product representation.      

Modelling a Multitype Branching Brownian Motion: Filtering of a Measure-Valued Process

In the framework of marked trees, a multitype branching brownian motion, described by measure-valued processes, is studied. By applying the strong branching property, the Markov property and the expression of the generator are derived for the process whose components are the measure-valued processes associated to each type particles. The conditional law of the measure-valued process describing the whole population observing the cardinality of the subpopulation of a given type particles is characterized as the unique weak solution of the Kushner‐Stratonovich equation. An explicit representation of the filter is obtained by Feyman–Kac formula using the linearized filtering equation.     

The Heckman–Opdam Markov processes

We introduce and study the natural counterpart of the Dunkl Markov processes in a negatively curved setting. We give a semimartingale decomposition of the radial part, and some properties of the jumps. We prove also a law of large numbers, a central limit theorem, and the convergence of the normalized process to the Dunkl process. Eventually we describe the asymptotic behavior of the infinite loop as it was done by Anker, Bougerol and Jeulin in the symmetric spaces setting in (Iberoamericana 18: 41–97, 2002). Partially supported by the European Commission (IHP Network HARP 2002–2006).     

The Upper Envelope of Positive Self-Similar Markov Processes

We establish integral tests in connection with laws of the iterated logarithm at 0 and at +∞, for the upper envelope of positive self-similar Markov processes. Our arguments are based on the Lamperti representation and on the study of the upper envelope of the future infimum due to the author (see Pardo in Stoch. Stoch. Rep. 78:123–155, [2006]). These results extend laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erdős (Proceedings of the Second Berkeley Symposium, [1951]) and stable Lévy processes with no positive jumps conditioned to stay positive due to Bertoin (Stoch. Process. Appl. 55:91–100, [1995]). Research supported by a grant from CONACYT (Mexico).     

Absolute continuity of the invariant measure in piecewise deterministic Markov Processes having degenerate jumps

We consider piecewise deterministic Markov processes with degenerate transition kernels of the house-of-cards- type. We use a splitting scheme based on jump times to prove the absolute continuity, as well as some regularity, of the invariant measure of the process. Finally, we obtain finer results on the regularity of the one-dimensional marginals of the invariant measure, using integration by parts with respect to the jump times.     

Bayesian estimation of stochastic volatility models based on OU processes with marginal Gamma law

This paper discusses practical Bayesian estimation of stochastic volatility models based on OU processes with marginal Gamma laws. Estimation is based on a parameterization which is derived from the Rosiński representation, and has the advantage of being a non-centered parameterization. The parameterization is based on a marked point process, living on the positive real line, with uniformly distributed marks. We define a Markov chain Monte Carlo (MCMC) scheme which enables multiple updates of the latent point process, and generalizes single updating algorithm used earlier. At each MCMC draw more than one point is added or deleted from the latent point process. This is particularly useful for high intensity processes. Furthermore, the article deals with superposition models, where it discuss how the identifiability problem inherent in the superposition model may be avoided by the use of a Markov prior. Finally, applications to simulated data as well as exchange rate data are discussed.     

Markov selections for the 3D stochastic Navier–Stokes equations

We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier–Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection principle, that there are Markov solutions to the Navier–Stokes equations. Due to the lack of continuity of solutions in the space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise, we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences of these facts, together with a new sufficient condition for well-posedness.      

On <Emphasis Type="Italic">g</Emphasis>-measures in symbolic dynamics

Finitary Markov processes are described in G. Morvai and B. Weiss, Prediction for discrete time series, Probability Theory and Related Fields 132 (2005), 1–12. The transition functions of finitary Markov processes are residually locally constant g-functions that can be extended by continuity to their maximal domain of definition. The study of their associated symbolic dynamics leads one to the D-shifts as introduced in W. Krieger, On g-functions for subshifts, Institute of Mathematical Statistics Lecture Notes-Monograph Series, Vol. 48, Dynamics & Stochastics, arXiv:math.DS/0608259, (2006), 306–316, We study the phenomena that can arise in residually locally constant and locally constant maximally defined g-functions on D-shifts, Markov shifts and synchronizing systems with respect to future measures and g-measures     

α-Transience and α-Recurrence of Right Processes

In this article, we mainly discuss some potential theory in the framework of right Markov processes. We introduce the concept of α-excessive function, α-recurrence and α-transience for right processes with α ≤ 0, and give a thorough investigation.     

Multi-self-similar Markov processes on ℝ<Subscript>+</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> and their Lamperti representations

 A classical result, due to Lamperti, establishes a one-to-one correspondence between a class of strictly positive Markov processes that are self-similar, and the class of one-dimensional Lévy processes. This correspondence is obtained by suitably time-changing the exponential of the Lévy process. In this paper we generalise Lamperti's result to processes in n dimensions. For the representation we obtain, it is essential that the same time-change be applied to all coordinates of the processes involved. Also for the statement of the main result we need the proper concept of self-similarity in higher dimensions, referred to as multi-self-similarity in the paper. The special case where the Lévy process ξ is standard Brownian motion in n dimensions is studied in detail. There are also specific comments on the case where ξ is an n-dimensional compound Poisson process with drift. Finally, we present some results concerning moment sequences, obtained by studying the multi-self-similar processes that correspond to n-dimensional subordinators. Received: 22 August 2002 / Revised version: 10 February 2003 Published online: 15 April 2003 RID="*" ID="*" MaPhySto – Centre for Mathematical Physics and Stochastics, funded by a grant from the Danish National Research Foundation Mathematics Subject Classification (2000): 60G18, 60G51, 60J25, 60J60, 60J75 Key words or phrases: Lévy process – Self-similarity – Time-change – Exponential functional – Brownian motion – Bessel process – Piecewise deterministic Markov process – Moment sequence     

Moment Solutions for the State Exiting Counting Processes of a Markov Renewal Process

Important performance measures for many Markov renewal processes are the counts of the exits from each state. We present solutions for the conditional first, second, and covariance moments of the state exiting counting processes for a Markov renewal process, and solutions for the unconditional equilibrium versions of the moments. We demonstrate the relationship between the conditional first moments for the state exiting and the state entering counting processes. For analytical and illustrative purposes, we concentrate on the two state case. Two asymptotic expansions for the moment functions are proposed and evaluated both analytically and empirically. The two approximations are shown to be competitive in terms of absolute relative error, but the second approximation has a simpler analytical form which is useful in analyzing more complex stochastic processes having an underlying MRP structure.     

On statistical manifolds of solutions of martingale problems

We consider geometrically regular statistical models defined by local densities of probability measures corresponding to discrete or continuous time Markov processes and smoothly depending on a finite dimensional parameter. Evolution equations are derived in terms of the generators of the underlying Markov additive processes for the elements of the related Fisher information matrix and the skewness tensor defining the Riemannian metric and the Amari-Chentsov's affine α-connections as functions of time and starting points of Markov processes. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikes Rinkinys, Vol. 35, No. 4, pp. 456–468, October–December, 1995.     

Finite absolute continuity of Gaussian measures on infinite-dimensional spaces

We study the notion of finite absolute continuity for measures on infinite-dimensional spaces. For Gaussian product measures on and Gaussian measures on a Hilbert space, we establish criteria for finite absolute continuity. We consider cases where the condition of finite absolute continuity of Gaussian measures is equivalent to the condition of their equivalence. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1367–1377, October, 2008.     

Some examples and counterexamples of convergence of σ-algebras and filtrations

We study some properties of the weak convergence of filtrations, in particular, its behavior under elementary set operations. We also derive relations between the convergence of filtrations generated by point processes with a single jump and the convergence of their compensators or distributions of their jump moments. Finally, we apply a lemma on the intersection of σ-algebras to filtrations generated by different discretizations of a single process. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 295–306, July–September, 2000. Translated by V. Mackevičius     

Limiting Distributions for Multitype Branching Processes

In this article, the asymptotic behavior of multitype Markov branching processes with discrete or continuous time is investigated in the positive regular and nonsingular case when both the initial number of ancestors and the time tend to infinity. Some limiting distributions are obtained as well as multivariate asymptotic normality is proved. The article also considers the relative frequencies of distinct types of individuals motivated by applications in the field of cell biology. We obtained non-random limits for the frequencies and multivariate asymptotic normality when the initial number of ancestors is large and the time of observation increases to infinity. In fact this paper continues the investigations of Yakovlev and Yanev [32 Yakovlev , A.Y. , and Yanev , N.M. 2009 . Relative frequencies in multitype branching processes . Annals of Applied Probability 19 ( 1 ): 1 – 14 . [Google Scholar]] where the time was fixed. The new obtained limiting results are of special interest for cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement.     

On the Properties of Some Class of Measures in a Hilbert Space

We find conditions for absolute continuity of transition probabilities of Markov processes in a Hilbert space.     

A structured pattern matrix algorithm for multichain Markov decision processes

In this paper, we are concerned with a new algorithm for multichain finite state Markov decision processes which finds an average optimal policy through the decomposition of the state space into some communicating classes and a transient class. For each communicating class, a relatively optimal policy is found, which is used to find an optimal policy by applying the value iteration algorithm. Using a pattern matrix determining the behaviour pattern of the decision process, the decomposition of the state space is effectively done, so that the proposed algorithm simplifies the structured one given by the excellent Leizarowitz’s paper (Math Oper Res 28:553–586, 2003). Also, a numerical example is given to comprehend the algorithm.     

Diffusions on path and loop spaces: existence, finite dimensional approximation and Hölder continuity

Summary. We construct Ornstein–Uhlenbeck processes and more general diffusion processes on path and loop spaces of Riemannian manifolds by finite dimensional approximation. We also show H?lder continuity of the sample paths w.r.t. the supremum norm. The proofs are based on the Lyons–Zheng decomposition. Received: 6 September 1996 / In revised form: 1 April 1997     

Filtration and prediction of random solutions of a system of linear differential equations with coefficients depending on a finite-valued Markov process

We obtain an equation of optimal filtration for processes of Markov random evolution, which is a solution of systems of linear differential equations with Markov switchings. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 997–1000, July, 1998.