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13
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 221 (37 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...
Asset Prices are Brownian motion: only in Business Time
, 1998
"... This paper argues that asset price processes arising from market clearing conditions should be modeled as pure jump processes, with no continuous martingale component. However, we show that continuity and normality can always be obtained after a time change. We study various examples of time changes ..."
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Cited by 19 (2 self)
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This paper argues that asset price processes arising from market clearing conditions should be modeled as pure jump processes, with no continuous martingale component. However, we show that continuity and normality can always be obtained after a time change. We study various examples of time changes and show that in all cases they are related to measures of economic activity. For the most general class of processes, the time change is a sizeweighted sum of order arrivals. The paper provides
The Brownian Burglar: conditioning Brownian motion by its local time process.
, 1998
"... this paper our normalisation of local time is such that it is an occupation density with respect to Lebesgue measure. Let T 1 be the first time t such that W t = 1. The celebrated RayKnight theorem describes the law of ..."
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Cited by 15 (4 self)
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this paper our normalisation of local time is such that it is an occupation density with respect to Lebesgue measure. Let T 1 be the first time t such that W t = 1. The celebrated RayKnight theorem describes the law of
Stochastic Volatility, Jumps and Hidden Time Changes
, 2001
"... Stochastic volatility and jumps are viewed as arising from Brownian subordination given here by an independent purely discontinuous process and we inquire into the relation between the realized variance or quadratic variation of the process and the time change. The class of models considered encompa ..."
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Cited by 10 (2 self)
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Stochastic volatility and jumps are viewed as arising from Brownian subordination given here by an independent purely discontinuous process and we inquire into the relation between the realized variance or quadratic variation of the process and the time change. The class of models considered encompasses a wide range of models employed in practical financial modeling. It is shown that in general the time change cannot be recovered from the composite process and we obtain its conditional distribution in a variety of cases. The implications of our results for working with stochastic volatility models in general is also described. We solve the recovery problem, i.e. the identification the conditional law for a variety of cases, the simplest solution being for the gamma time change when this conditional law is that of the first hitting time process of Brownian motion with drift attaining the level of the variation of the time changed process. We also introduce and solve in certain cases the problem of stochastic scaling. A stochastic scalar is a subordinator that recovers the law of a given subordinator when evaluated at an independent and time scaled copy of the given subordinator. These results are of importance in comparing price quality delivered by alternate exchanges.
QuasiInvariance of the gamma Process and Multiplicative Properties of the PoissonDirichlet Measures
, 1999
"... . In this paper we describe new fundamental properties of the law P \Gamma of the classical gamma process and related properties of the PoissonDirichlet measures PD(`). We prove the quasiinvariance of the measure P \Gamma with respect to an infinitedimensional multiplicative group (the fact ..."
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Cited by 7 (3 self)
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. In this paper we describe new fundamental properties of the law P \Gamma of the classical gamma process and related properties of the PoissonDirichlet measures PD(`). We prove the quasiinvariance of the measure P \Gamma with respect to an infinitedimensional multiplicative group (the fact first discovered in [GGV83]) and the MarkovKrein identity as corollaries of the formula for the Laplace transform of P \Gamma . The quasiinvariance of the measure P \Gamma allows us to obtain new quasiinvariance properties of the measure PD(`). The corresponding invariance properties hold for oefinite analogues of P \Gamma and PD(`). We also show that the measure P \Gamma can be considered as a limit of measures corresponding to the ffstable L'evy processes when parameter ff tends to zero. Our approach is based on simultaneous considering the gamma process (especially its Laplace transform) and its simplicial part  the PoissonDirichlet measures. Quasiinvariance du proces...
The twoparameter PoissonDirichlet point process
, 2007
"... The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtai ..."
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Cited by 3 (0 self)
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The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Relying on this, we apply the theory of point processes to reveal mathematical structure of the twoparameter PoissonDirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we will be able to extend several results previously known for the oneparameter case, and the MarkovKrein identity for the generalized Dirichlet process is discussed from a point of view of functional analysis based on the twoparameter PoissonDirichlet distribution. 1
Invariant measures for the continual Cartan subgroup
 J. Funct. Anal
"... To Professor Malliavin with deep respect We construct and study the oneparameter semigroup of σfinite measures L θ, θ> 0, on the space of Schwartz distributions that have an infinitedimensional abelian group of linear symmetries; this group is a continual analog of the classical Cartan subgroup o ..."
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Cited by 2 (0 self)
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To Professor Malliavin with deep respect We construct and study the oneparameter semigroup of σfinite measures L θ, θ> 0, on the space of Schwartz distributions that have an infinitedimensional abelian group of linear symmetries; this group is a continual analog of the classical Cartan subgroup of diagonal positive matrices of the group SL(n, R). The parameter θ is the degree of homogeneity with respect to homotheties of the space, we prove uniqueness theorem for measures with given degree of homogeneity, and call the measure with degree of homogeneity equal to one the infinitedimensional Lebesgue measure L. The structure of these measures is very closely related to the socalled Poisson–Dirichlet measures PD(θ), and to the wellknown gamma process. The nontrivial properties of the Lebesgue measure are related to the superstructure of the measure PD(1), which is called the conic Poisson–Dirichlet measure – CPD. This is the most interesting σfinite measure on the set of positive convergent monotonic real series.
Blowup and stability of semilinear PDEs with Gamma generators
 J. Math. Anal. Appl
"... We investigate finitetime blowup and stability of semilinear partial differential equations of the form ∂wt/∂t = Γ wt +νt σw 1+β, w0(x) = ϕ(x) ≥ 0, x ∈ R+, t where Γ is the generator of the standard gamma process and ν> 0, σ ∈ R, β> 0 are constants. We show that any initial value satisfying c1x− ..."
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Cited by 1 (0 self)
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We investigate finitetime blowup and stability of semilinear partial differential equations of the form ∂wt/∂t = Γ wt +νt σw 1+β, w0(x) = ϕ(x) ≥ 0, x ∈ R+, t where Γ is the generator of the standard gamma process and ν> 0, σ ∈ R, β> 0 are constants. We show that any initial value satisfying c1x−a1 ≤ ϕ(x), x> x0 for some positive constants x0, c1, a1, yields a nonglobal solution if a1β < 1 + σ. If ϕ(x) ≤ c2x−a2, x> x0, where x0, c2, a2> 0, and a2β> 1 + σ, then the solution wt is global and satisfies 0 ≤ wt(x) ≤ Ct−a2, x ≥ 0, for some constant C> 0. This complements the results previously obtained in [3, 10, 22] for symmetric αstable generators. Systems of semilinear PDE’s with gamma generators are also considered.