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16
The twoparameter PoissonDirichlet distribution derived from a stable subordinator.
, 1995
"... The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov ..."
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Cited by 234 (35 self)
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The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to VershikShmidtIgnatov, are generalized to the twoparameter case. The sizebiased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
On the Relative Lengths of Excursions Derived From a Stable Subordinator
, 1996
"... Results are obtained concerning the distribution of ranked relative lengths of excursions of a recurrent Markov process from a point in its state space whose inverse local time process is a stable subordinator. It is shown that for a large class of random times T the distribution of relative excursi ..."
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Cited by 15 (7 self)
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Results are obtained concerning the distribution of ranked relative lengths of excursions of a recurrent Markov process from a point in its state space whose inverse local time process is a stable subordinator. It is shown that for a large class of random times T the distribution of relative excursion lengths prior to T is the same as if T were a fixed time. It follows that the generalized arcsine laws of Lamperti extend to such random times T . For some other random times T , absolute continuity relations are obtained which relate the law of the relative lengths at time T to the law at a fixed time. 1 Introduction Following Lamperti [10], Wendel [24], Kingman [7], Knight [8], PermanPitman Yor [12, 13, 15], consider the sequence V 1 (T ) V 2 (T ) \Delta \Delta \Delta (1) of ranked lengths of component intervals of the set [0; T ]nZ, where T is a strictly positive random time, and Z is the zero set of a Markov process X started at zero, such as a Brownian motion or Bessel process,...
On the Constructions of the Skew Brownian Motion
, 2006
"... This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applicat ..."
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Cited by 5 (1 self)
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This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applications of the Skew Brownian motion.
Time change approach to generalized excursion measures, and its application to limit theorems
, 2006
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Hitting, Occupation, and Inverse Local Times of OneDimensional Diffusions: Martingale and Excursion Approaches
, 2001
"... Basic relations between the distributions of hitting, occupation, and inverse local times of a onedimensional diffusion process X , first discussed by ItoMcKean, are reviewed from the perspectives of martingale calculus and excursion theory. These relations, and the technique of conditioning o ..."
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Cited by 3 (1 self)
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Basic relations between the distributions of hitting, occupation, and inverse local times of a onedimensional diffusion process X , first discussed by ItoMcKean, are reviewed from the perspectives of martingale calculus and excursion theory. These relations, and the technique of conditioning on L y T , the local time of X at level y before a suitable random time T , yield formulae for the joint Laplace transform of L y T and the times spent by X above and below level y up to time T.
Simulation of a stochastic process in a discontinuous, layered media
, 2011
"... In this note, we provide a simulation algorithm for a diffusion process in a layered media. Our main tools are the properties of the Skew Brownian motion and a path decomposition technique for simulating occupation times. ..."
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Cited by 2 (1 self)
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In this note, we provide a simulation algorithm for a diffusion process in a layered media. Our main tools are the properties of the Skew Brownian motion and a path decomposition technique for simulating occupation times.
A conditional limit theorem for generalized diffusion processes
 J. Math. Kyoto Univ
, 2003
"... Let {X(t) : t ≥ 0} be a onedimensional generalized diffusion process with initial state X(0)> 0, hitting time τX(0) at the origin and speed measure m which is regularly varying at infinity with exponent 1/α − 1> 0. It is proved that, for a suitable function u(c), the probability law of {u(c)− ..."
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Cited by 2 (0 self)
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Let {X(t) : t ≥ 0} be a onedimensional generalized diffusion process with initial state X(0)> 0, hitting time τX(0) at the origin and speed measure m which is regularly varying at infinity with exponent 1/α − 1> 0. It is proved that, for a suitable function u(c), the probability law of {u(c)−1X(ct) : 0 < t ≤ 1} converges as c→ ∞ to the conditioned 2(1−α)dimensional Bessel excursion on natural scale and that the latter is equivalent to the 2(1 − α)dimensional Bessel meander up to a scale transformation. In particular, the distribution of u(c)−1X(c) converges to the Weibull distribution. From the conditional limit theorem we also derive a limit theorem for some of regenerative process associated with {X(t) : t ≥ 0}. Key words: generalized diffusion, hitting time, conditional limit theorem, Bessel diffusion, excursion, meander.
2011, Asymptotic Theory of Maximum Likelihood Estimator for Di¤usion Model. Working Paper
"... We derive the asymptotics of the maximum likelihood estimators for diffusion models. The models considered in the paper are very general, including both stationary and nonstationary diffusions. For such a broad class of diffusion models, we establish the consistency and derive the limit distribution ..."
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We derive the asymptotics of the maximum likelihood estimators for diffusion models. The models considered in the paper are very general, including both stationary and nonstationary diffusions. For such a broad class of diffusion models, we establish the consistency and derive the limit distributions of the exact maximum likelihood estimator, and also the quasi and approximate maximum likelihood estimators based on various versions of approximated transition densities. Our asymptotics are two dimensional, requiring the sampling interval to decrease as well as the time span of sample to increase. The two dimensional asymptotics provide a unifying framework for such a broad class of estimators for both stationary and nonstationary diffusion models. More importantly, they yield the primary asymptotics that are very useful to analyze the exact, quasi and approximate maximum likelihood estimators of the diffusion models, if the samplesarecollected at highfrequencyintervals over modestlengthsof sampling horizons as in the case of many practical applications.
Some Generalizations of Bessel Processes
, 1997
"... Contents Introduction 1 Main properties of Bessel processes : : : : : : : : : : : : : : : 2 1 First hitting times of radial OrnsteinUhlenbeck processes 7 2 BESQ processes with negative dimensions and extensions 14 3 Time reversal 22 3.1 Doob's htransform : : : : : : : : : : : : : : : : : ..."
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Contents Introduction 1 Main properties of Bessel processes : : : : : : : : : : : : : : : 2 1 First hitting times of radial OrnsteinUhlenbeck processes 7 2 BESQ processes with negative dimensions and extensions 14 3 Time reversal 22 3.1 Doob's htransform : : : : : : : : : : : : : : : : : : : : : : : : 25 3.2 Checking a time reversal theorem in ElworthyLiYor : : : : 28 3.3 The threeparametersfamily of processes with law P ffi;¯ : : : : 31 Bibliography 38 i Introduction Bessel processes are a oneparameter family of diffusion processes that appear in many financial problems and have remarkable properties. Following this introduction we recall the definition of Bessel processes and their properties in detail (see I.1  I.6). One important property of Bessel processes is, that the transition densities ar