Searcharxiv⌕ Search

arXiv subjects

Michael Grabchak

Publications and source records attributed to Michael Grabchak.

At least 19 recordsLinked to original sources
arXiv2026

The Test and Find Model

We introduce the Test and Find (TF) problem, where a decision maker (DM) faces the following situation: $k$ identical objects are randomly allocated to $n$ distinct boxes (sites) according to some distribution $\pi$, with no more than one object to a box. The DM tests all of the boxes. However, the tests are imperfect: they can give false positive or false negative results. DM has $m$ tags, $1\leq m\leq n$, and, after testing all boxes, she can place a tag on any box that she thinks has a hidden object. She is rewarded $c_i$ for a correct guess and penalized $d_i$ for a wrong guess in box $i$. DM knows all of the parameters of the model and her goal is to maximize the expected reward. We give an explicit solution to this problem. We then turn to the symmetric case, for which we derive more computationally efficient results. We also consider several extensions of the TF model and give detailed solutions. One of these extensions is to the realistic case where $k$, the number of objects, is unknown and random.

math.PR↗
arXiv2025

Risk-Neutral Pricing of Random-Expiry Options Using Trinomial Trees

Random-expiry options are nontraditional derivative contracts that may expire early based on a random event. We develop a methodology for pricing these options using a trinomial tree, where the middle path is interpreted as early expiry. We establish that this approach is free of arbitrage, derive its continuous-time limit, and show how it may be implemented numerically in an efficient manner.

q-fin.PR↗
arXiv2025

Confidence Intervals Using Turing's Estimator: Simulations and Applications

Turing's estimator allows one to estimate the probabilities of outcomes that either do not appear or only rarely appear in a given random sample. We perform a simulation study to understand the finite sample performance of several related confidence intervals (CIs) and introduce an approach for selecting the appropriate CI for a given sample. We give an application to the problem of authorship attribution and apply it to a dataset comprised of tweets from users on X (Twitter). Further, we derive several theoretical results about asymptotic normality and asymptotic Poissonity of Turing's estimator for two important discrete distributions.

math.ST↗
arXiv2024

On Approximations of Subordinators in $L^p$ and the Simulation of Tempered Stable Distributions

Subordinators are infinitely divisible distributions on the positive half-line. They are often used as mixing distributions in Poisson mixtures. We show that appropriately scaled Poisson mixtures can approximate the mixing subordinator and we derive a rate of convergence in $L^p$ for each $p\in[1,\infty]$. This includes the Kolmogorov and Wasserstein metrics as special cases. As an application, we develop an approach for approximate simulation of the underlying subordinator. In the interest of generality, we present our results in the context of more general mixtures, specifically those that can be represented as differences of randomly stopped L\'evy processes. Particular focus is given to the case where the subordinator belongs to the class of tempered stable distributions.

math.PR↗
arXiv2023

Representation and Simulation of Multivariate Dickman Distributions and Vervaat Perpetuities

A multivariate extension of the Dickman distribution was recently introduced, but very few properties have been studied. We discuss several properties with an emphasis on simulation. Further, we introduce and study a multivariate extension of the more general class of Vervaat perpetuities and derive a number of properties and representations. Most of our results are presented in the even more general context of so-called $\alpha$-times self-decomposable distributions.

math.PR↗
arXiv2022

Efficient Simulation of $p$-Tempered $α$-Stable OU Processes

We develop efficient methods for simulating processes of Ornstein-Uhlenbeck type related to the class of $p$-tempered $α$-stable ($\ts$) distributions. Our results hold for both the univariate and multivariate cases and we consider both the case where the $\ts$ distribution is the stationary law and where it is the distribution of the background driving Lévy process (BDLP). In the latter case, we also derive an explicit representation for the transition law as this was previous known only in certain special cases and only for $p=1$ and $α\in[0,1)$. Simulation results suggest that our methods work well in practice.

math.PR↗
arXiv2020

A Zero-One Law for Markov Chains

We prove an analog of the classical Zero-One Law for both homogeneous and nonhomogeneous Markov chains (MC). Its almost precise formulation is simple: given any event $A$ from the tail $σ$-algebra of MC $(Z_n)$, for large $n$, with probability near one, the trajectories of the MC are in states $i$, where $P(A|Z_n=i)$ is either near $0$ or near $1$. A similar statement holds for the entrance $σ$-algebra, when $n$ tends to $-\infty$. To formulate this second result, we give detailed results on the existence of nonhomogeneous Markov chains indexed by $\mathbb Z_-$ or $\mathbb Z$ in both the finite and countable cases. This extends a well-known result due to Kolmogorov. Further, in our discussion, we note an interesting dichotomy between two commonly used definitions of MCs.

math.PR↗
arXiv2020

An Exact Method For Simulating Rapidly Decreasing Tempered Stable Distributions

Rapidly decreasing tempered stable distributions are useful models for financial applications. However, there has been no exact method for simulation available in the literature. We remedy this by introducing an exact simulation method in the finite variation case. Our methodology works for the wider class of $p$-RDTS distributions.

math.PR↗
arXiv2020

On the Transition Laws of $p$-Tempered $α$-Stable OU-Processes

We derive an explicit representation for the transition law of a $p$-tempered $α$-stable process of Ornstein-Uhlenbeck-type and use it to develop a methodology for simulation. Our results apply in both the univariate and multivariate cases. Special attention is given to the case where $p\leα$, which is more complicated and requires additional care.

math.PR↗
arXiv2020

On the Simulation of General Tempered Stable Ornstein-Uhlenbeck Processes

We give an explicit representation for the transition law of a tempered stable Ornstein-Uhlenbeck process and use it to develop a rejection sampling algorithm for exact simulation of increments from this process. Our results apply to general classes of both univariate and multivariate tempered stable distributions and contain a number of previously studied results as special cases.

math.PR↗
arXiv2018

On the occupancy problem for a regime switching model

This article studies the expected occupancy probabilities on an alphabet. Unlike the standard situation, where observations are assumed to be independent and identically distributed (iid), we assume that they follow a regime switching Markov chain. For this model, we 1) give finite sample bounds on the occupancy probabilities, and 2) provide detailed asymptotics in the case where the underlying distribution is regularly varying. We find that, in the regularly varying case, the finite sample bounds are rate optimal and have, up to a constant, the same rate of decay as the asymptotic result.

math.PR↗
arXiv2018

Rejection Sampling for Tempered Levy Processes

We extend the idea of tempering stable Levy processes to tempering more general classes of Levy processes. We show that the original process can be decomposed into the sum of the tempered process and an independent point process of large jumps. We then use this to set up a rejection sampling algorithm for sampling from the tempered process. A small scale simulation study is given to help understand the performance of this algorithm.

math.PR↗
arXiv2017

Domains of Attraction for Positive and Discrete Tempered Stable Distributions

We introduce a large and flexible class of discrete tempered stable distributions, and analyze the domains of attraction for both this class and the related class of positive tempered stable distributions. Our results suggest that these are natural models for sums of independent and identically distributed random variables with tempered heavy tails tails, i.e. tails that appear to be heavy up to a point, but ultimately decay faster.

math.PR↗
arXiv2016

Finite sample properties of the mean occupancy counts and probabilities

For a probability distribution $P$ on an at most countable alphabet $\mathcal A$, this article gives finite sample bounds for the expected occupancy counts $\mathbb E K_{n,r}$ and probabilities $\mathbb E M_{n,r}$. Both upper and lower bounds are given in terms of the counting function $ν$ of $P$. Special attention is given to the case where $ν$ is bounded by a regularly varying function. In this case, it is shown that our general results lead to an optimal-rate control of the expected occupancy counts and probabilities with explicit constants. Our results are also put in perspective with Turing's formula and recent concentration bounds to deduce bounds in probability. At the end of the paper, we discuss an extension of the occupancy problem to arbitrary distributions in a metric space.

math.ST↗
arXiv2015

Three Upsilon Transforms Related to Tempered Stable Distributions

We discuss the properties of three upsilon transforms, which are related to the class of $p$-tempered $α$-stable ($TS^p_α$) distributions. In particular, we characterize their domains and show how they can be represented as compositions of each other. Further, we show that if $-\infty<β<α<2$ and $0<q<p<\infty$ then they can be used to transform the Lévy measures of $TS^p_β$ distributions into those of $TS^q_α$.

math.PR↗
arXiv2014

Entropic Representation and Estimation of Diversity Indices

This paper serves a twofold purpose. First, a unified perspective on diversity indices is introduced based on an entropic basis. It is shown that the class of all linear combinations of the entropic basis, referred to as the class of linear diversity indices, covers a wide range of diversity indices used in the literature. Second, a class of estimators for linear diversity indices is proposed and it is shown that these estimators have rapidly decaying biases and asymptotic normality.

math.ST↗