Quantile Mechanics II: Changes of Variables in Monte Carlo Methods and GPU-Optimized Normal Quantiles

@article{Shaw2009QuantileMI,
  title={Quantile Mechanics II: Changes of Variables in Monte Carlo Methods and GPU-Optimized Normal Quantiles},
  author={William T. Shaw and Thomas C. Luu and Nick Brickman},
  journal={ERN: Estimation (Topic)},
  year={2009}
}
This article presents differential equations and solution methods for the functions of the form $Q(x) = F^{-1}(G(x))$, where $F$ and $G$ are cumulative distribution functions. Such functions allow the direct recycling of Monte Carlo samples from one distribution into samples from another. The method may be developed analytically for certain special cases, and illuminate the idea that it is a more precise form of the traditional Cornish-Fisher expansion. In this manner the model risk of… 
Monte Carlo sampling given a Characteristic Function: Quantile Mechanics in Momentum Space
In mathematical finance and other applications of stochastic processes, it is frequently the case that the characteristic function may be known but explicit forms for density functions are not
Series representations and approximation of some quantile functions appearing in finance
TLDR
The main focus of this thesis will be to develop Taylor and asymptotic series representations for quantile functions of the following probability distributions; Variance Gamma, Generalized Inverse Gaussian, Hyperbolic, -Stable and Snedecor’s F distributions.
Closed Form Expressions for the Quantile Function of the Erlang Distribution Used in Engineering Models
TLDR
The closed form expression for the quantile function obtained here is very useful in modeling physical and engineering systems that are completely described by or fitted with the Erlang distribution.
Quantile Diffusions for Risk Analysis
This paper focuses on the development of a new class of diffusion processes that allows for direct and dynamic modelling of quantile diffusions. We constructed quantile diffusion processes by
Quantile mechanics: Issues arising from critical review
  • Okagbue et al.
  • Mathematics
    International Journal of ADVANCED AND APPLIED SCIENCES
  • 2019
Article history: Received 26 June 2018 Received in revised form 27 October 2018 Accepted 27 October 2018 Approximations are the alternative way of obtaining the Quantile function when the inversion
Efficient estimation of financial risk by regressing the quantiles of parametric distributions: An application to CARR models
Abstract Risk measures such as value-at-risk (VaR) and expected shortfall (ES) may require the calculation of quantile functions from quantile regression models. In a parametric set-up, we propose to
Efficient and Accurate Parallel Inversion of the Gamma Distribution
  • T. Luu
  • Computer Science, Mathematics
    SIAM J. Sci. Comput.
  • 2015
TLDR
A method for parallel inversion of the gamma distribution is described, which has accuracy close to a choice of single- or double-precision machine epsilon for random number generation in Monte Carlo simulations where gamma variates are required.
Classes of Ordinary Differential Equations Obtained for the Probability Functions of Gumbel Distribution
In this paper, the differential calculus was used to obtain some classes of ordinary differential equations (ODEs) for the probability density function, quantile function, survival function,
Classes of Ordinary Differential Equations Obtained for the Probability Functions of Kumaraswamy Distribution
In this paper, differential calculus was used to obtain the ordinary differential equations (ODE) of the probability density function (PDF), Quantile function (QF), survival function (SF), inverse
...
...

References

SHOWING 1-10 OF 24 REFERENCES
Quantile mechanics
In both modern stochastic analysis and more traditional probability and statistics, one way of characterizing a static or dynamic probability distribution is through its quantile function. This paper
Sampling Student’s T distribution – use of the inverse cumulative distribution function
With the current interest in copula methods, and fat-tailed or other non-normal distributions, it is appropriate to investigate technologies for managing marginal distributions of interest. We
The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map
Motivated by the need for parametric families of rich and yet tractable distributions in financial mathematics, both in pricing and risk management settings, but also considering wider statistical
The Variance Gamma (V.G.) Model for Share Market Returns
A new stochastic process, termed the variance gamma process, is proposed as a model for the uncertainty underlying security prices. The unit period distribution is normal conditional on a variance
On the Distributional Characterization of Daily Log‐Returns of a World Stock Index
In this paper distributions are identified which suitably fit log‐returns of the world stock index when these are expressed in units of different currencies. By searching for a best fit in the class
Exponentially decreasing distributions for the logarithm of particle size
  • O. Barndorff-Nielsen
  • Mathematics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1977
The family of continuous type distributions such that the logarithm of the probability (density) function is a hyperbola (or, in several dimensions, a hyperboloid) is introduced and investigated. It
Monte Carlo Methods in Financial Engineering
TLDR
This paper presents a meta-modelling procedure that automates the very labor-intensive and therefore time-heavy and therefore expensive and expensive process of manually computing random numbers and random Variables.
The Physics of Blown Sand and Desert Dunes
LIEUT.–COLONEL BAGNOLD is well kno for his scientific publications on desert sands and dunes, and for a book describing his travels in the Egyptian and Libyan Deserts. The present book combines these
The full Monte
TLDR
This paper deals with computing based on various forms of random sampling, or Monte Carlo methods, with applications to evaluate integrals and simple simulation.
Statistical Modelling with Quantile Functions
INTRODUCTION An Overview Describing the Sample Describing the Population Statistical Foundations QUANTILE MODELS AND THEIR CONSTRUCTION Foundation Distributions Distributional Model Building THE
...
...