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Subordinator
About: Subordinator is a research topic. Over the lifetime, 771 publications have been published within this topic receiving 15383 citations.
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03 Dec 2012TL;DR: It is shown that the nonconvex logarithmic and exponential penalty functions are the Laplace exponents of Gamma and compound Poisson subordinators, respectively, and the relationship between these two penalties is due to asymmetricity of the KL distance.
Abstract: In this paper we study sparsity-inducing nonconvex penalty functions using Levy processes. We define such a penalty as the Laplace exponent of a subordinator. Accordingly, we propose a novel approach for the construction of sparsity-inducing nonconvex penalties. Particularly, we show that the nonconvex logarithmic (LOG) and exponential (EXP) penalty functions are the Laplace exponents of Gamma and compound Poisson subordinators, respectively. Additionally, we explore the concave conjugate of nonconvex penalties. We find that the LOG and EXP penalties are the concave conjugates of negative Kullback-Leiber (KL) distance functions. Furthermore, the relationship between these two penalties is due to asymmetricity of the KL distance.
17 citations
TL;DR: In this article, it was shown that the probability of extinction of a Galton-Watson process is less than one if the mean number of offspring is larger than one in the proofs.
Abstract: Appropriate constant multiples of the function $t^{1/\alpha}$ are "the" "maximal local lower envelope" and "the" "minimal local upper envelope" for the sample functions of a strictly stable subordinator of index $\alpha$. The fact that the probability of extinction of a Galton-Watson process is less than one if the mean number of offspring is larger than one is used in the proofs.
17 citations
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TL;DR: In this paper, the marginal distributions of a quantile clock, at each fixed time, equate with the marginal distribution of a single subordinator, and the authors show that there are many quantile clocks where one can specify $L$, such that their marginal distributions have a desired law in the class of generalized $s$-self decomposable distributions.
Abstract: Quantile clocks are defined as convolutions of subordinators $L$, with quantile functions of positive random variables We show that quantile clocks can be chosen to be strictly increasing and continuous and discuss their practical modeling advantages as business activity times in models for asset prices We show that the marginal distributions of a quantile clock, at each fixed time, equate with the marginal distribution of a single subordinator Moreover, we show that there are many quantile clocks where one can specify $L$, such that their marginal distributions have a desired law in the class of generalized $s$-self decomposable distributions, and in particular the class of self-decomposable distributions The development of these results involves elements of distribution theory for specific classes of infinitely divisible random variables and also decompositions of a gamma subordinator, that is of independent interest As applications, we construct many price models that have continuous trajectories, exhibit volatility clustering and have marginal distributions that are equivalent to those of quite general exponential L\'{e}vy price models In particular, we provide explicit details for continuous processes whose marginals equate with the popular VG, CGMY and NIG price models We also show how to perfectly sample the marginal distributions of more general classes of convoluted subordinators when $L$ is in a sub-class of generalized gamma convolutions, which is relevant for pricing of European style options
17 citations
TL;DR: In this article, the authors investigated the potential densities of subordinators which are constant to the right of a positive number in AMS 2000 Mathematics Subject Classification: Primary 60G51, Secondary 60J45, 60J75.
Abstract: A subordinator is called special if the restriction of its potential measure to (0,∞) has a decreasing density with respect to Lebesgue measure. In this note we investigate what type of measures μ on (0,∞) can arise as Levy measures of special subordinators and what type of functions u : (0,∞)→ [0,∞) can arise as potential densities of special subordinators. As an application of the main result, we give examples of potential densities of subordinators which are constant to the right of a positive number. AMS 2000 Mathematics Subject Classification: Primary 60G51, Secondary 60J45, 60J75.
17 citations
19 Aug 2014
TL;DR: In this paper, a discretization scheme for a large class of stochastic differential equations driven by a time-changed Brownian motion with drift is established, where the time change is given by a general inverse subordinator.
Abstract: This paper establishes a discretization scheme for a large class of stochastic differential equations driven by a time-changed Brownian motion with drift, where the time change is given by a general inverse subordinator. The scheme involves two types of errors: one generated by application of the Euler–Maruyama scheme and the other ascribed to simulation of the inverse subordinator. With the two errors carefully examined, the orders of strong and weak convergence are established. In particular, an improved error estimate for the Euler–Maruyama scheme is derived, which is required to guarantee the strong convergence. Numerical examples are attached to support the convergence results.
17 citations