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285 is a special case of the fractional differential Riccati equation [1]. We trans- formed it into an equivalent Volterra integral equation:
The reason behind this transformation being due to the fact that the compu- tation of the fractional integral is by far more accessible numerically than the fractional derivative (the fractional integral consists only of an integration, while the fractional derivative is a composition of an integration and a differentiation).
We look for an approximate solution of the form
, which is suggested by the analytical solution as fractional power series (this seems to speed up the training of the neural network).
The loss function is defined as follows:
∑(
)
The fractional integral in the loss function is approximated through the
Gauss-Legendre quadrature method [4], since it gave us by far the best results.
Similar method was applied in [5].
For we use a neural network with one hidden layer containing 10 nodes. The training interval [0,2] consisted of 65 equidistant points. Repeating the algorithm several times we noted that with at most 20 iterations with the
BFGS method we already got a loss below
. The average computation time was around 0.4 seconds. In Fig.
2 the result is compared with the analytical solution [1], which is expressed as fractional power series. The computation of the analytical solution maintains a very high accuracy [1], even though we trun- cate the series due to obvious machine limitations.
In conclusion, this approach seems to be working well, with advantages of low-code, and universality. The accuracy can be increased by increasing of the number of nodes, number of grid points, number of iterations, or by using more sophisticated approximation schemes. Certainly, the computation time will also increase.
The code is freely available at [7].
Research was supported by the Regional Mathematical Center of the
Southern Federal University with the Agreement 075-02-2022-893 of the Minis- try of Science and Higher Education of Russia.

286
References
1. Callegaro G., Grasselli M., Pagès G. Fast Hybrid Schemes for Fractional Ric- cati Equations (Rough is not so Tough), Mathematics of Operations Research,
Vol. 46, 221-254, 2021.
2. Lagaris I. E., Likas A., Fotiadis D. I., Artificial Neural Networks for Solving
Ordinary and Partial Differential Equations IEEE Transactions On Neural
Networks, vol. 9, nr. 5, p. 989-1000, September 1998.
3. Hassoun M., Fundamentals of Artificial Neural Networks, MIT Press 1995.
4. Kovvali N., Theory and Applications of Gaussian Quadrature Methods, Syn- thesis Lectures on Algorithms and Software in Engineering, September 2011,
Vol. 3, No. 2, Pages 1–65 5. Pashaie M., Sadeghi M., A. Jafarian; Artificial Neural. Networks with Nelder-
Mead Optimization Method for Solving Nonlinear Integral Equations, Journal of Computer Science and Applications. Volume 8, Number 1 (2016), pp. 1–20.
6. Nocedal J., Wright S. J., Numerical Optimization, Springer, New York, 2006.
7. https://github.com/nicolahcm/neural_net_sito_conference.

287

TWO ROBUST MODELS FOR OPTIMAL PORTFOLIO PROBLEM
Yao Keyu
Southern Federal University,
Rostov-on-Don
E-mail: iao@sfedu.ru
The problem of optimal portfolio finding has been relevant since the publi- cation of the work of Markowitz in 1952 to the present day [1]. In the interpre- tation of Markowitz the quality of the portfolio is determined by two parame- ters ‒ the average profitability and the risk. The portfolio should be chosen in such way that the average profitability is as high as possible and at the same time the risk is as low as possible. According to the structure, the optimal portfo- lio problem belongs to the optimization problems with vector criteria. The solu- tion of the problem with the vector criteria is usually understood as the Pareto set. One of the ways for calculating Pareto optimal portfolios is to scalarize the vector criteria. To calculate these vectors of sample means and sample covari- ance matrices we use the unsupervising learning. Therefore, to obtain the set of values of sample means and the set of values of sample covariance matrices it is proposed to divide the sample into clusters. As the algorithm of splitting the sample into two clusters, the maximum likelihood algorithm is proposed, as the algorithm of splitting the sample into several clusters the dichotomous algorithm is proposed [2].
However, there is the problem of calculating mean vector and covariance matrix. Let us consider two methods of finding them. In the first method, we will apply the MCD algorithm [3]. In the second method we will use the Was- serstein metric [4].
References
1. Markowitz H. Portfolio selection. J. Financ. vol. 7. n. 1. p. 77–91. 1952.
2. Beliavsky G. I., Danilova N. V., Logunov А. D. Unsupervised learning and robust optimization in the portfolio problem. Bulletin of higher education in- stitutes. North Caucasus region. Natural sciences. № 4. p.4-9. 2020. (in
Russian)
3. Khachiyan L. Rounding of polytopes in the real number model of computa- tion // Mathematics of Operations Research. vol. 21. n. 2. p. 307-320. 1996.
4. Blanchety G., Chenz L., Zhoux X. Distributionally robust mean-variance portfolio selection with Wasserstein distances // Management Science. 2021.

288
A NEURAL NETWORK SOLUTION FOR THE BANKRUPTCY
MODELING
Yazici M.
Southern Federal University,
Rostov-on-Don
E-mail: yazichi@sfedu.ru
An Artificial Neural Networks (ANN) is based on a collection of connected units or nodes called artificial neurons, which loosely model the neurons in a biological brain. Each connection, like the synapses in a biological brain, can transmit a signal to other neurons. An artificial neuron receives a signal then processes it and can signal neurons connected to it. The "signal" at a connection is a real number, and the output of each neuron is computed by some non-linear function of the sum of its inputs.
ANNs are becoming more and more popular in financial applications. They allow one to fully utilize the data and let the data determine the structure and parameters of a model without any restrictive parametric modeling assumptions. Thus, they are appealing in financial area.
The objective of this study is to represent an applications of ANNs in finance including a Neural Network (NN) study. In the Neural Network, a NN has been built that predicts the bankruptcy of companies in the UK. The results are compared with Altman's z-score, which is the main bankruptcy model. In the direction of the success of the prediction, NN model obtained more success than the Altman‘s Z-score model. The logistic sigmoid function is used on the output layer of the NN. It is also a cumulative distributed function because it give us probabilities. As a result, it is determined that the NN model has a nearly success rate of 80 percent, while the Altman‘s model has a predictive success rate of 23.45 percent.
References
1. Altman E. I., Financial ratios, discriminant analysis and the prediction of corporate bankruptcy, Journal of Finance, Vol. XXIII, No. 4, Sep- tember 1968.
2. AL Hodgkin., Evidence for electrical transmission in nerve: Part i. The Jour- nal of Physiology, 90(2):183, – 1937.
3. Boer K., Kaymak U., Spiering J., From discrete-time models to continuoustime, asynchronous modeling of financial markets, Computational
Intelligence, vol. 23(2), 2007, pp. 142–161.
4. Edelman GM (1987), Neural Darwinism: Theory if neuronal group selection,
Basic Books, New York.

289 5. Eden UT, Loren F., Brown E., Dynamic analysis of neural encoding by point process adaptive filtering, Neural Computation, vol. 16, pp. 971–998, 2004.
6. Janusz K. and Witold P., Springer Handbook of Computational Intelligence.
Springer, 2015.
7. Marr B., Key performance indicators, Financial Times Present, 2012.
8. Rist M. and Pizzica A. J., Financial ratios for executives, Apress 1st edition,
2014.
9. Wulfram G., Werner M. K., Richard N., and Liam P., Neuronal Dynamics: From single neurons to networks and models of cognition. Cambridge University Press,
2014.

5
Научное издание
XXIX научная конференция
«Современные информационные технологии:
тенденции и перспективы развития»
Компьютерная верстка: Багдасарян А. Л.
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