Subordinator

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

Sharp bounds on the density, Green function and jumping function of subordinate killed BM

Abstract: Subordination of a killed Brownian motion in a domain D ⊂ R d via an α/2-sta- ble subordinator gives rise to a process Zt whose infinitesimal generator is −(−� |D) α/2 , the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green function and jumping function of Zt when D is either a bounded C 1,1 domain or an exterior C 1,1 domain. Our estimates are sharp in the sense that the upper and lower estimates differ only by a multiplicative constant.

Universality of the REM for dynamics of mean-field spin glasses

Abstract: We consider a version of a Glauber dynamics for a p-spin Sherrington--Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for any p>2 and any inverse temperature \beta>0, there exist constants g>0, such that for all exponential time scales, $\exp(\gamma N)$, with $\gamma< g$, the properly rescaled clock process (time-change process), converges to an \alpha-stable subordinator where \alpha=\gamma/\beta^2<1. Moreover, the dynamics exhibits aging at these time scales with time-time correlation function converging to the arcsine law of this \alpha-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system), the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud's REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p=2) seems to belong to a different universality class.

Stochastic volatility, jumps and hidden time changes

Abstract: 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.

Particle tracking for fractional diffusion with two time scales

Abstract: Previous work [Y. Zhang, M.M. Meerschaert, B. Baeumer, Particle tracking for time-fractional diffusion, Phys. Rev. E 78 (2008) 036705] showed how to solve time-fractional diffusion equations by particle tracking. This paper extends the method to the case where the order of the fractional time derivative is greater than one. A subordination approach treats the fractional time derivative as a random time change of the corresponding Cauchy problem, with a first derivative in time. One novel feature of the time-fractional case of order greater than one is the appearance of clustering in the operational time subordinator, which is non-Markovian. Solutions to the time-fractional equation are probability densities of the underlying stochastic process. The process models movement of individual particles. The evolution of an individual particle in both space and time is captured in a pair of stochastic differential equations, or Langevin equations. Monte Carlo simulation yields particle location, and the ensemble density approximates the solution to the variable coefficient time-fractional diffusion equation in one or several spatial dimensions. The particle tracking code is validated against inverse transform solutions in the simplest cases. Further applications solve model equations for fracture flow, and upscaling flow in complex heterogeneous porous media. These variable coefficient time-fractional partial differential equations in several dimensions are not amenable to solution by any alternative method, so that the grid-free particle tracking approach presented here is uniquely appropriate.