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12At the same time, the return on the use of the existing apparatus in applications turns out to be small.
There are many reasons. Many promising methods and criteria are not being applied due to their lack of pub-
licity. Due to the complexity, the use of many methods is impossible without the corresponding software,
which may not be available.
But more often the problems turn out to be more serious. The correct application of any criterion is condi-
tional on the fulfillment of certain assumptions that may not be fulfilled in a particular application. In this case,
the statistical conclusion formed on the basis of this criterion may turn out to be incorrect.
Under non-standard conditions of applications, the properties of the methods for estimating parameters
change, the distributions of statistics of the applied criteria change, corresponding to the validity of the hy-
pothesis being tested.
The properties of the estimates and the distributions of the statistics of the tests are reflected in the pres-
ence of round-off errors, as a result of which, in this case, the possibility of using the existing asymptotic re-
sults is also excluded.
Solving the problems of applying statistical methods and criteria in non-standard conditions of applica-
tions by analytical methods requires large intellectual costs and, in general, turns out to be unproductive.
At the same time, good results are demonstrated by the use of computer technologies based on the com-
puting apparatus of applied mathematical statistics and the Monte Carlo method. This approach makes it pos-
sible to successfully investigate probabilistic and statistical patterns in conditions of various violations of
standard assumptions. Moreover, the study of the required patterns can be carried out interactively (in the
course of the statistical analysis). And further, the found regularity or distribution of the criterion statistics can
be used to form the correct statistical inference.
Using this approach, the problems of applying chi-square goodness-of-fit tests in non-standard conditions
were investigated. When testing complex hypotheses, the problems of applying a set of nonparametric good-
ness-of-fit tests were investigated, and models of the statistical distributions of these tests were constructed.
Models of distributions of statistics for multi-sample criteria of homogeneity of laws were used. The properties
of a set of parametric and nonparametric criteria used to test the homogeneity of variances were investigated.
The properties of sets of criteria for normality, uniformity, exponentiality, criteria for testing hypotheses about
the absence of a trend, and others were investigated.
The software implements the possibility of correct application of the considered criteria, including in con-
ditions of influence on the distribution of rounding statistics
Estimation of the probability of two consecutive passages through the resonant regions in the descent
of an asymmetric rigid body in a rarefied atmosphere
V. V. Lyubimov
Samara National Research University
Email: vlubimov@mail.ru
DOI 10.24412/cl-35065-2021-1-00-83
A two-frequency nonlinear system of ordinary differential equations is considered, which describes the
perturbed motion of a rigid body with considerable asymmetry in the rarefied atmosphere. If the frequencies
of the system coincide, the occurrences of capture or passage through the principal resonance are random [1].
In addition, the application of well-known expression [1] made it possible to estimate the probability of a sin-
gle capture into the principal resonance during the descent of a spacecraft with small asymmetry in dense at-
mosphere [2]. The aim of this work is to obtain an estimate of the probability of two consecutive passages
through the resonance regions during the descent in the rarefied atmosphere of Mars of a rigid body with con-
siderable geometric and aerodynamic asymmetries. The veracity of the obtained estimate is confirmed by the
Monte-Carlo method.
References
1. Neishtadt A. Averaging, capture into resonances, and chaos in nonlinear system, in Chaos. New York: Amer. Inst.
Phys., 1990. P. 261�273.
2. Kurkina E.V., Lyubimov V.V. Estimation of the probability of capture into resonance and parametric analysis in the
descent of an asymmetric spacecraft in an atmosphere // J. of Applied and Industrial Mathematics. 2018. V. 12, N. 3.
P. 492-500.
Stochastic modelling of financial securities with a systemic risk component
R. N. Makarov
Wilfrid Laurier University, Waterloo, ON, Canada
Email: rmakarov@wlu.ca
DOI 10.24412/cl-35065-2021-1-00-84
We propose and study a new jump-diffusion model for pricing multiple assets, where systemic-risk securi-
ty is combined with several conditionally independent base assets. This approach allows for analyzing and
modelling a portfolio that integrates high-activity security, such as an Exchange Trading Fund (ETF) tracking a
major market index (e. g., S&P500 or TSX) with several low-activity assets. The latter may have missing and
asynchronous pricing data when the assets are not traded frequently on financial markets. The proposed
framework allows for constructing several models, including the following: (a) a diffusion-type model without
jumps where all asset price processes are Geometric Brownian Motions; (b) a jump-diffusion model with only
common jumps [1]; (c) a jump-diffusion model with both common and asset-specific jumps [2]. We discuss the
properties of the proposed model, the estimation of its parameters using the Maximum Likelihood Estimation
method, and the pricing of European-style basket options. The NSERC Discovery Grant program supported this
work.
References
1. Chen Y., Makarov R. N. Modelling asynchronous assets with jump-diffusion processes. In International Conference
on Applied Mathematics, Modeling and Computational Science, pp. 477�487. Springer, 2017.
2. Xu R., Makarov R.N. High-Frequency Statistical Modelling for Jump-Diffusion Multi-Asset Price Processes with a
Systemic Component, Springer, 2019 (accepted).
Monte Carlo simulation of halos in crystal clouds
Q. Mu1, E. G. Kablukova2, B. A. Kargin1,2, S. M. Prigarin1,2
1Novosibirsk State University
2Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Email: mutsyuev@gmail.com
DOI 10.24412/cl-35065-2021-1-00-86
In this paper, we try to answer the question: how multiple scattering, sun elevation, shape and orientation
of ice crystals in the cirrus clouds affect a halo pattern. To study the radiation transfer in optically anisotropic
clouds we have developed the software based on Monte Carlo method [1] and ray tracing. In addition to halos,
this software enables one to simulate �anti-halos�, which above the cloud layer can be seen by observers [2].
