Subordinator

About: Subordinator is a research topic. Over the lifetime, 771 publications have been published within this topic receiving 15383 citations.

Gibbs Partitions (EPPF's) Derived From a Stable Subordinator are Fox H and Meijer G Transforms

Abstract: This paper derives explicit results for the infinite Gibbs partitions generated by the jumps of an $\alpha-$stable subordinator, derived in Pitman \cite{Pit02, Pit06}. We first show that for general $\alpha$ the conditional EPPF can be represented as ratios of Fox-$H$ functions, and in the case of rational $\alpha,$ Meijer-G functions. Furthermore the results show that the resulting unconditional EPPF's, can be expressed in terms of H and G transforms indexed by a function h. Hence when h is itself a H or G function the EPPF is also an H or G function. An implication, in the case of rational $\alpha,$ is that one can compute explicitly thousands of EPPF's derived from possibly exotic special functions. This would also apply to all $\alpha$ except that computations for general Fox functions are not yet available. However, moving away from special functions, we demonstrate how results from probability theory may be used to obtain calculations. We show that a forward recursion can be applied that only requires calculation of the simplest components. Additionally we identify general classes of EPPF's where explicit calculations can be carried out using distribution theory.

Coagulation–fragmentation duality, Poisson–Dirichlet distributions and random recursive trees

Abstract: In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the twoparameter family of Poisson–Dirichlet distributions PD(α, θ) that take values in this space. We introduce families of random fragmentation and coagulation operators Frag α and Coag α,θ , respectively, with the following property: if the input to Frag α has PD(α, θ) distribution, then the output has PD(α, θ + 1) distribution, while the reverse is true for Coag α,θ . This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between PD(α, θ) and PD(αβ, θ ). Repeated application of the Fragα operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation–fragmentation duality.

Evaluating Callable and Putable Bonds: An Eigenfunction Expansion Approach

Abstract: We propose an efficient method to evaluate callable and putable bonds under a wide class of interest rate models, including the popular short rate diffusion models, as well as their time changed versions with jumps. The method is based on the eigenfunction expansion of the pricing operator. Given the set of call and put dates, the callable and putable bond pricing function is the value function of a stochastic game with stopping times. Under some technical conditions, it is shown to have an eigenfunction expansion in eigenfunctions of the pricing operator with the expansion coefficients determined through a backward recursion. For popular short rate diffusion models, such as CIR, Vasicek, 3/2, the method is orders of magnitude faster than the alternative approaches in the literature. In contrast to the alternative approaches in the literature that have so far been limited to diffusions, the method is equally applicable to short rate jump-diffusion and pure jump models constructed from diffusion models by Bochner's subordination with a Levy subordinator.

Moments of convex distribution functions and completely alternating sequences

Abstract: We solve the moment problem for convex distribution functions on [0, 1] in terms of completely alternating sequences. This complements a recent solution of this problem by Diaconis and Freedman, and relates this work to the Levy-Khintchine formula for the Laplace transform of a subordinator, and to regenerative composition structures.

Modelling electricity prices: a time change approach

Abstract: To capture mean reversion and sharp seasonal spikes observed in electricity prices, this paper develops a new stochastic model for electricity spot prices by time changing the Jump Cox-Ingersoll-Ross (JCIR) process with a random clock that is a composite of a Gamma subordinator and a deterministic clock with seasonal activity rate. The time-changed JCIR process is a time-inhomogeneous Markov semimartingale which can be either a jump-diffusion or a pure-jump process, and it has a mean-reverting jump component that leads to mean reversion in the prices in addition to the smooth mean-reversion force. Furthermore, the characteristics of the time-changed JCIR process are seasonal, allowing spikes to occur in a seasonal pattern. The Laplace transform of the time-changed JCIR process can be efficiently computed by Gauss–Laguerre quadrature. This allows us to recover its transition density through efficient Laplace inversion and to calibrate our model using maximum likelihood estimation. To price electricity deri...