Subordinator

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

Ruin probabilities for competing claim processes

Abstract: Let C1,C2,...,Cm be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x+ct C(t). The ruin probability is given by the well known Pollaczek-Hinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators whose sum constitutes C. Formulae for the probability that ruin is caused by Ci are derived. These formulae can be extended to perturbed risk processes of the

Parametric subordination in fractional diffusion processes 1

Abstract: We consider simulation of spatially one-dimensional space-time fractional diffusion. Whereas in an earlier paper of ours (Eur. Phys. J. Special Topics, Vol. 193, 119–132 (2011); E-print: http://arxiv.org/abs/1104.4041), we have developed the basic theory of what we call parametric subordination via three-fold splitting applied to continuous time random walk with subsequent passage to the diffusion limit, here we go the opposite way. Via Fourier-Laplace manipulations of the relevant fractional partial differential equation of evolution we obtain the subordination integral formula that teaches us how a particle path can be constructed by first generating the operational time from the physical time and then generating in operational time the spatial path. By inverting the generation of operational time from physical time we arrive at the method of parametric subordination. Due to the infinite divisibility of the stable subordinator we can simulate particle paths by discretization where the generated points of a path are precise snapshots of a true path. By refining the discretization more and more fine details of a path become visible.

Parametric estimation for subordinators and induced OU-processes

Abstract: Key words and Phrases: cumulant, empirical characteristic function, Levy process, self-decomposable distribution, stationary process. Consider a stationary sequence of random variables with infinitely divisible marginal law, characterized by its Levy density. We analyze the behavior of a so-called cumulant M-estimator, in case this Levy density is characterized by a Euclidean (finite-dimensional) parameter. Under mild conditions, we prove consistency and asymptotic normality of the estimator. The estimator is considered in the situation where the data are increments of a subordinator as well as the situation where the data consist of a discretely sampled Ornstein Uhlenbeck process induced by the subordinator. We illustrate our results for the Gamma-process and the Inverse-Gaussian-OU-process. For these processes we also explain how the estimator can be computed numerically.

Space--time duality for fractional diffusion

Abstract: Zolotarev (1961) proved a duality result that relates stable densities with different indices. In this paper we show how Zolotarev's duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivatives in place of the usual integer-order derivatives. They govern scaling limits of random walk models, with power-law jumps leading to fractional derivatives in space, and power-law waiting times between the jumps leading to fractional derivatives in time. The limit process is a stable Levy motion that models the jumps, subordinated to an inverse stable process that models the waiting times. Using duality, we relate the density of a spectrally negative stable process with index 1 < α < 2 to the density of the hitting time of a stable subordinator with index 1 /α, and thereby unify some recent results in the literature. These results provide a concrete interpretation of Zolotarev's duality in terms of the fractional diffusion model. They also illuminate a current controversy in hydrology, regarding the appropriate use of space- and time-fractional derivatives to model contaminant transport in river flows.

Anomalous diffusion schemes underlying the Cole–Cole relaxation: The role of the inverse-time α-stable subordinator

Abstract: The paper presents the random-variable formalism of the anomalous diffusion processes. The emphasis is on a rigorous presentation of asymptotic behaviour of random walk processes with infinite mean random time intervals between jumps. We elucidate the role of the so-called inverse-time stochastic process, the main mathematical tool that allows us to modify the dynamics of standard relaxation processes and give rise to the nonexponential decay of modes. In particular, we show that the Brownian motion in combination with an appropriate inverse-time process may lead not only to exponential but also to the nonexponential relaxation responses.