Second-Order Weak Approximations of CKLS and CEV Processes by Discrete Random Variables

In this paper, we construct second-order weak split-step approximations of the CKLS and CEV processes that use generation of a three−valued random variable at each discretization step without switching to another scheme near zero, unlike other known schemes (Alfonsi, 2010; Mackevičius, 2011). To the best of our knowledge, no second-order weak approximations for the CKLS processes were constructed before. The accuracy of constructed approximations is illustrated by several simulation examples with comparison with schemes of Alfonsi in the particular case of the CIR process and our first-order approximations of the CKLS processes (Lileika– Mackevičius, 2020).

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IMPRECISE SECOND-ORDER MODEL FOR A SYSTEM OF INDEPENDENT RANDOM VARIABLES

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488505003382 ◽

2005 ◽

Vol 13 (02) ◽

pp. 177-193 ◽

Cited By ~ 2

Author(s):

LEV V. UTKIN

Keyword(s):

Random Variables ◽

Random Variable ◽

Second Order ◽

Independent Random Variables ◽

Order Model ◽

First Order ◽

Probability Bounds ◽

Uncertainty Model ◽

Lower And Upper Probabilities ◽

Interval Valued

A new hierarchical uncertainty model for combining different evidence about a system of statistically independent random variable is studied in the paper. It is assumed that the first-order level of the model is represented by sets of lower and upper previsions (expectations) of random variables and the second-order level is represented by sets of lower and upper probabilities which can be viewed as confidence weights for interval-valued expectations of the first-order level. The model is rather general and allows us to compute probability bounds and "average" bounds for previsions of a function of random variables. A numerical example illustrates this model.

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Weak Approximations of the Wright–Fisher Process

Mathematics ◽

10.3390/math10010125 ◽

2022 ◽

Vol 10 (1) ◽

pp. 125

Author(s):

Vigirdas Mackevičius ◽

Gabrielė Mongirdaitė

Keyword(s):

Random Variables ◽

Second Order ◽

Fisher Equation ◽

Discretization Schemes ◽

Weak Approximations ◽

Discretization Step ◽

Fisher Process

In this paper, we construct first- and second-order weak split-step approximations for the solutions of the Wright–Fisher equation. The discretization schemes use the generation of, respectively, two- and three-valued random variables at each discretization step. The accuracy of constructed approximations is illustrated by several simulation examples.

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Reliability Methods for Bimodal Distribution With First-Order Approximation1

ASCE-ASME J Risk and Uncert in Engrg Sys Part B Mech Engrg ◽

10.1115/1.4040000 ◽

2018 ◽

Vol 5 (1) ◽

Cited By ~ 1

Author(s):

Zhangli Hu ◽

Xiaoping Du

Keyword(s):

Probability Density ◽

Order Approximation ◽

Limit State ◽

Random Variables ◽

Random Variable ◽

State Function ◽

First Order ◽

Reliability Method ◽

Reliability Methods ◽

Basic Random Variable

In traditional reliability problems, the distribution of a basic random variable is usually unimodal; in other words, the probability density of the basic random variable has only one peak. In real applications, some basic random variables may follow bimodal distributions with two peaks in their probability density. When binomial variables are involved, traditional reliability methods, such as the first-order second moment (FOSM) method and the first-order reliability method (FORM), will not be accurate. This study investigates the accuracy of using the saddlepoint approximation (SPA) for bimodal variables and then employs SPA-based reliability methods with first-order approximation to predict the reliability. A limit-state function is at first approximated with the first-order Taylor expansion so that it becomes a linear combination of the basic random variables, some of which are bimodally distributed. The SPA is then applied to estimate the reliability. Examples show that the SPA-based reliability methods are more accurate than FOSM and FORM.

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A storage process by semi-Markov input

Advances in Applied Probability ◽

10.2307/1426402 ◽

1975 ◽

Vol 7 (4) ◽

pp. 830-844 ◽

Cited By ~ 5

Author(s):

Lajos Takács

Keyword(s):

Recurrence Formula ◽

Random Variables ◽

Random Variable ◽

Discrete Random Variable ◽

Storage Process ◽

Markov Sequence ◽

Discrete Random Variables ◽

Matrix Factorisation

A sequence of random variables η0, η1, …, ηn, … is defined by the recurrence formula ηn = max (ηn–1 + ξn, 0) where η0 is a discrete random variable taking on non-negative integers only and ξ1, ξ2, … ξn, … is a semi-Markov sequence of discrete random variables taking on integers only. Define Δ as the smallest n = 1, 2, … for which ηn = 0. The random variable ηn can be interpreted as the content of a dam at time t = n(n = 0, 1, 2, …) and Δ as the time of first emptiness. This paper deals with the determination of the distributions of ηn and Δ by using the method of matrix factorisation.

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The Law of the Iterated Logarithm for Linear Processes Generated by a Sequence of Stationary Independent Random Variables under the Sub-Linear Expectation

Entropy ◽

10.3390/e23101313 ◽

2021 ◽

Vol 23 (10) ◽

pp. 1313

Author(s):

Wei Liu ◽

Yong Zhang

Keyword(s):

Independent Random Variable ◽

Random Variables ◽

Random Variable ◽

Second Order ◽

Law Of Iterated Logarithm ◽

Independent Random Variables ◽

Linear Processes ◽

Iterated Logarithm ◽

The Law

In this paper, we obtain the law of iterated logarithm for linear processes in sub-linear expectation space. It is established for strictly stationary independent random variable sequences with finite second-order moments in the sense of non-additive capacity.

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A second-order weak approximation of Heston model by discrete random variables

Lithuanian Mathematical Journal ◽

10.1007/s10986-015-9298-4 ◽

2015 ◽

Vol 55 (4) ◽

pp. 555-572 ◽

Cited By ~ 1

Author(s):

Antanas Lenkšas ◽

Vigirdas Mackevičius

Keyword(s):

Random Variables ◽

Second Order ◽

Heston Model ◽

Weak Approximation ◽

Discrete Random Variables

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Probability Mass Functions

Bayesian Statistics for Beginners ◽

10.1093/oso/9780198841296.003.0008 ◽

2019 ◽

pp. 87-107

Author(s):

Therese M. Donovan ◽

Ruth M. Mickey

Keyword(s):

Bayesian Inference ◽

Binomial Distribution ◽

Probability Distributions ◽

Random Variables ◽

Random Variable ◽

Posterior Distributions ◽

Bernoulli Distribution ◽

Discrete Random Variables ◽

Probability Mass ◽

Mass Functions

This chapter focuses on probability mass functions. One of the primary uses of Bayesian inference is to estimate parameters. To do so, it is necessary to first build a good understanding of probability distributions. This chapter introduces the idea of a random variable and presents general concepts associated with probability distributions for discrete random variables. It starts off by discussing the concept of a function and goes on to describe how a random variable is a type of function. The binomial distribution and the Bernoulli distribution are then used as examples of the probability mass functions (pmf’s). The pmfs can be used to specify prior distributions, likelihoods, likelihood profiles and/or posterior distributions in Bayesian inference.

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ASYMPTOTIC EXPANSIONS OF GENERALIZED QUANTILES AND EXPECTILES FOR EXTREME RISKS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964815000017 ◽

2015 ◽

Vol 29 (3) ◽

pp. 309-327 ◽

Cited By ~ 10

Author(s):

Tiantian Mao ◽

Kai Wang Ng ◽

Taizhong Hu

Keyword(s):

Value At Risk ◽

Risk Measures ◽

Domain Of Attraction ◽

Random Variable ◽

Second Order ◽

Underlying Distribution ◽

First Order ◽

Extreme Risk ◽

Special Cases ◽

Extreme Risks

Generalized quantiles of a random variable were defined as the minimizers of a general asymmetric loss function, which include quantiles, expectiles and M-quantiles as their special cases. Expectiles have been suggested as potentially better alternatives to both Value-at-Risk and expected shortfall risk measures. In this paper, we first establish the first-order expansions of generalized quantiles for extreme risks as the confidence level α↑ 1, and then investigate the first-order and/or second-order expansions of expectiles of an extreme risk as α↑ 1 according to the underlying distribution belonging to the max-domain of attraction of the Fréchet, Weibull, and Gumbel distributions, respectively. Examples are also presented to examine whether and how much the first-order expansions have been improved by the second-order expansions.

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Random Variables and Probability

10.1093/oso/9780192843470.003.0007 ◽

2021 ◽

pp. 109-124

Author(s):

Timothy E. Essington

Keyword(s):

Parameter Estimation ◽

Model Selection ◽

Beta Distribution ◽

Random Variables ◽

Random Variable ◽

Density Functions ◽

The Core ◽

Discrete Random Variables ◽

Probability Mass ◽

T Distribution

The chapter “Random Variables and Probability” serves as both a review and a reference on probability. The random variable is the core concept in understanding probability, parameter estimation, and model selection. This chapter reviews the basic idea of a random variable and discusses the two main kinds of random variables: discrete random variables and continuous random variables. It covers the distinction between discrete and continuous random variables and outlines the most common probability mass or density functions used in ecology. Advanced sections cover distributions such as the gamma distribution, Student’s t-distribution, the beta distribution, the beta-binomial distribution, and zero-inflated models.

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Peluang, Peubah Acak Diskrit, dan Sebaran Peluang Peubah Acak Diskrit

10.31219/osf.io/sq8t3 ◽

2020 ◽

Author(s):

Ahmad Sudi Pratikno

Keyword(s):

Probability Distribution ◽

Random Variables ◽

Random Variable ◽

Discrete Random Variable ◽

Discrete Random Variables

Probability to learn someone's chance in getting or winning an event. In the discrete random variable is more identical to repeated experiments, to form a pattern. Discrete random variables can be calculated as the probability distribution by calculating each value that might get a certain probability value.

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