The two-parameter Poisson-Dirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to Vershik-Shmidt-Ignatov, are generalized to the two-parameter case. The size-biased 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...

We investigate the relative importance of diffusion and jumps in a new jump diffusion model for asset returns. In contrast to the standard modelling of jumps for asset returns, the jump component of our process can display finite or infinite activity, and finite or infinite variation. Empirical investigations of time series indicate that index dynamics are essentially devoid of a diffusion component, while this component may be present in the dynamics of individual stocks. This result leads to the conjecture that the risk-neutral process should be free of a diffusion component for both indices and individual stocks. Empirical investigation of options data tends to confirm this conjecture. We conclude that the statistical and risk-neutral processes for indices and stocks tend to be pure jump processes of innite activity and finite variation.

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 size-weighted sum of order arrivals. The paper provides

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 Ray-Knight theorem describes the law of

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.

. In this paper we describe new fundamental properties of the law P \Gamma of the classical gamma process and related properties of the Poisson--Dirichlet measures PD(`). We prove the quasi-invariance of the measure P \Gamma with respect to an infinite-dimensional multiplicative group (the fact first discovered in [GGV83]) and the Markov--Krein identity as corollaries of the formula for the Laplace transform of P \Gamma . The quasi-invariance of the measure P \Gamma allows us to obtain new quasi-invariance properties of the measure PD(`). The corresponding invariance properties hold for oe-finite 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 ff-stable 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 Poisson--Dirichlet measures. Quasi-invariance du proces...

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The two-parameter Poisson-Dirichlet 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 two-parameter Poisson-Dirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we will be able to extend several results previously known for the one-parameter case, and the Markov-Krein identity for the generalized Dirichlet process is discussed from a point of view of functional analysis based on the twoparameter Poisson-Dirichlet distribution. 1

We investigate finite-time blow-up 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 non-global 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 α-stable generators. Systems of semilinear PDE’s with gamma generators are also considered.

We investigate nite-time blow-up and stability of semilinear partial dierential equations of the form @w t =@t = w t +t t , w 0 (x) = '(x) 0, x 2 R+ , where is the generator of the standard gamma process and > 0, 2 R, > 0 are constants. We show that any initial value satisfying c 1 x '(x), x > x 0 for some positive constants x 0 ; c 1 ; a 1 , yields a non-global solution if a 1 < 1 + , or if a 1 = 1 + and > 1. If '(x) c 2 x , x > x 0 ; where x 0 ; c 2 ; a 2 > 0, and a 2 > 1 + , then the solution w t is global and satis es 0 w t (x) Ct , x 0, for some constant C > 0. This extends the results previously obtained in the case of -stable generators. Systems of semilinear PDE's with gamma generators are also considered.