Weak approximations of CKLS model by discrete random variables

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

Mathematics ◽

10.3390/math9121337 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1337

Author(s):

Gytenis Lileika ◽

Vigirdas Mackevičius

Keyword(s):

Random Variables ◽

Random Variable ◽

Second Order ◽

First Order ◽

Discrete Random Variables ◽

Weak Approximations ◽

Discretization Step ◽

Cir Process

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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Normal approximations to discrete unimodal distributions

Journal of Applied Probability ◽

10.1017/s0021900200115931 ◽

1986 ◽

Vol 23 (04) ◽

pp. 1013-1018

Author(s):

B. G. Quinn ◽

H. L. MacGillivray

Keyword(s):

Stationary Distribution ◽

Sufficient Conditions ◽

Random Variables ◽

Hypergeometric Distribution ◽

Death Process ◽

Normal Approximations ◽

Discrete Random Variables ◽

Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

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The number of failed components in a coherent working system when the lifetimes are discretely distributed

Metrika ◽

10.1007/s00184-021-00817-2 ◽

2021 ◽

Author(s):

Krzysztof Jasiński

Keyword(s):

Random Variables ◽

Continuous Case ◽

Coherent System ◽

Coherent Systems ◽

Discrete Random Variables ◽

Independent And Identically Distributed

AbstractIn this paper, we study the number of failed components of a coherent system. We consider the case when the component lifetimes are discrete random variables that may be dependent and non-identically distributed. Firstly, we compute the probability that there are exactly i, $$i=0,\ldots ,n-k,$$ i = 0 , … , n - k , failures in a k-out-of-n system under the condition that it is operating at time t. Next, we extend this result to other coherent systems. In addition, we show that, in the most popular model of independent and identically distributed component lifetimes, the obtained probability corresponds to the respective one derived in the continuous case and existing in the literature.

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