| 1. |
- Boichenko, Viktoria, et al.
(author)
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Interpolation problems for random fields on Sierpinski’s carpet
- 2023
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In: Mohyla Mathematical Journal. - : National University of Kyiv-Mohyla Academy. - 2617-7080 .- 2663-0648. ; 6, s. 28-34
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Journal article (other academic/artistic)abstract
- The prediction of stochastic processes and the estimation of random fields of different natures is becoming an increasingly common field of research among scientists of various specialties. However, an analysis of papers across different estimating problems shows that a dynamic approach over an iterative and recursive interpolation of random fields on fractal is still an open area of investigation. There are many papers related to the interpolation problems of stationary sequences, estimation of random fields, even on the perforated planes, but all of this still provides a place for an investigation of a more complicated structure like a fractal, which might be more beneficial in appliances of certain industry fields. For example, there has been a development of mobile phone and WiFi fractal antennas based on a first few iterations of the Sierpinski carpet. In this paper, we introduce an estimation for random fields on the Sierpinski carpet, based on the usage of the known spectral density, and calculation of the spectral characteristic that allows an estimation of the optimal linear functional of the omitted points in the field. We give coverage of an idea of stationary sequence estimating that is necessary to provide a basic understanding of the approach of the interpolation of one or a set of omitted values. After that, the expansion to random fields allows us to deduce a dynamic approach on the iteration steps of the Sierpinski carpet. We describe the numerical results of the initial iteration steps and demonstrate a recurring pattern in both the matrix of Fourier series coefficients of the spectral density and the result of the optimal linear functional estimation. So that it provides a dependency between formulas of the different initial sizes of the field as well as a possible generalizing of the solution for N-steps in the Sierpinski carpet. We expect that further evaluation of the mean squared error of this estimation can be used to identify the possible iteration step when further estimation becomes irrelevant, hence allowing us to reduce the cost of calculations and make the process viable.
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| 2. |
- Boluh, Kateryna, et al.
(author)
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Simulating Stochastic Diffusion Processes and Processes with “Market” Time
- 2020
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In: Mohyla Mathematical Journal. - : National University of Kyiv-Mohyla Academy. - 2617-7080 .- 2663-0648. ; 3, s. 25-30
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Journal article (peer-reviewed)abstract
- The paper focuses on modelling, simulation techniques and numerical methods concerned stochastic processes in subject such as financial mathematics and financial engineering. The main result of this work is simulation of a stochastic process with new market active time using Monte Carlo techniques.The processes with market time is a new vision of how stock price behavior can be modeled so that the nature of the process is more real. The iterative scheme for computer modelling of this process was proposed. It includes the modeling of diffusion processes with a given marginal inverse gamma distribution. Graphs of simulation of the Ornstein-Uhlenbeck random walk for different parameters, a simulation of the diffusion process with a gamma-inverse distribution and simulation of the process with market active time are presented.To simulate stochastic processes, an iterative scheme was used:xk+1 = xk + a(xk, tk) ∆t + b(xk, tk) √ (∆t) εk,,where εk each time a new generation with a normal random number distribution.Next, the tools of programming languages for generating random numbers (evenly distributed, normally distributed) are investigated. Simulation (simulation) of stochastic diffusion processes is carried out; calculation errors and acceleration of convergence are calculated, Euler and Milstein schemes. At the next stage, diffusion processes with a given distribution function, namely with an inverse gamma distribution, were modelled. The final stage was the modelling of stock prices with a new "market" time, the growth of which is a diffusion process with inverse gamma distribution. In the proposed iterative scheme of stock prices, we use the modelling of market time gains as diffusion processes with a given marginal gamma-inverse distribution.The errors of calculations are evaluated using the Milstein scheme. The programmed model can be used to predict future values of time series and for option pricing.
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| 3. |
- Drin, Svitlana, 1977-, et al.
(author)
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A Model of a System of Simultaneous Equations with a Lag Effect for Estimating the Quality of an Advertising Campaign
- 2022
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In: Mohyla Mathematical Journal. - МОДЕЛЬ : National University of Kyiv-Mohyla Academy. - 2617-7080 .- 2663-0648. ; 5, s. 33-37
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Journal article (peer-reviewed)abstract
- This article describes the creation of a more generalized system of simultaneous equations for forecasting the level of sales depending on advertising campaigns on different channels and other factors. RStudio (R programming language) and Google Colab (Python programming language) environments describe the creation of a model based on real data of a product. The Hausman specification test was applied to determine the model estimation method. As a conclusion, the indicators of advertising campaigns turned out to be endogenous variables, which indicates the importance of using the 2MNK method. It was found that the volume of advertising is the cause of the volume of sales according to Granger, which cannot be said about the reverse assumption - the causality of the volume of advertising from sales according to Granger. The ”depth” of lags is also determined, namely, one lag for both advertising channels. The dependence of sales volumes on various factors, including product distribution, the price index, the influence of advertising and its lags, and the influence of competitors’ advertising activities, was evaluated. The coefficients of the resulting more generalized system of simultaneous equations were estimated using the two-step least squares method. All statistical indicators testify to the adequacy of the model. Performance indicators (ROI - return on investment) of advertising campaigns showed that advertising both on television and on the Internet is profitable for the company’s product in question. The relevance of this article lies in the creation of a more general system of simultaneous equations with the inclusion of a product sales forecast model taking into account the influence of advertising.
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| 4. |
- Drin, Yaroslav, et al.
(author)
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About the Approximate Solutions to Linear and Non-Linear Pseudodifferential Reaction Diffusion Equations
- 2019
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In: Mohyla Mathematical Journal. - : National University of Kyiv-Mohyla Academy. - 2617-7080 .- 2663-0648. ; 2, s. 41-45
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Journal article (peer-reviewed)abstract
- Background: The concept of fractal is one of the main paradigms of modern theoretical and experimental physics, radiophysics and radar, and fractional calculus is the mathematical basis of fractal physics, geothermal energy and space electrodynamics. We investigate the solvability of the Cauchy problem for linear and nonlinear inhomogeneous pseudodifferential diffusion equations. The equation contains a fractional derivative of a Riemann–Liouville time variable defined by Caputo and a pseudodifferential operator that acts on spatial variables and is constructed in a homogeneous, non-negative homogeneous order, a non-smooth character at the origin, smooth enough outside. The heterogeneity of the equation depends on the temporal and spatial variables and permits the Laplace transform of the temporal variable. The initial condition contains a restricted function.Objective: To show that the homotopy perturbation transform method (HPTM) is easily applied tolinear and nonlinear inhomogeneous pseudodifferential diffusion equations. To prove the solvability and obtain the solution formula for the Cauchy problem series for the given linear and nonlinear diffusion equations.Methods: The problem is solved by the NPTM method, which combines a Laplace transform with a time variable and a homotopy perturbation method (HPM). After the Laplace transform, we obtain an integral equation which is solved as a series by degrees of the entered parameter with unknown coefficients. Substituting the input formula for the solution into the integral equation, we equate the expressions to equal parameter degrees and obtain formulas for unknown coefficients. When solving the nonlinear equation, we use a special polynomial which is included in the decomposition coefficients of the nonlinear function and allows the homotopy perturbation method to be applied as well for nonlinear non-uniform pseudodifferential diffusion equation.Results: The result is a solution of the Cauchy problem for the investigated diffusion equation, which is represented as a series of terms whose functions are found from the parametric series.Conclusions: In this paper we first prove the solvability and obtain the formula for solving the Cauchy problem as a series for linear and nonlinear inhomogeneous pseudodifferential equations.
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| 5. |
- Pauk, Viktoriia, et al.
(author)
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Two Approaches for Option Pricing under Illiquidity
- 2022
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In: Mohyla Mathematical Journal. - : National University of Kyiv-Mohyla Academy. - 2617-7080 .- 2663-0648. ; 5, s. 38-45
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Journal article (other academic/artistic)abstract
- The paper focuses on option pricing under unusual behaviour of the market, when the price may not be changed for some time what is quite a common situation on the modern financial markets. There are some patterns that can cause permanent price gaps to form and lead to illiquidity. For example, global changes that have a negative impact on financial activity, or a small number of market participants, or the market is quite young and is just in the process of developing, etc.In the paper discrete and continuous time approaches for modelling market with illiquidity and evaluation option pricing were considered.Trinomial discrete time model improves upon the binomial model by allowing a stock price not only to move up, down but stay the same with certain probabilities, what is a desirable feature for the illiquid modelling. In the paper parameters for real financial data were identified and the backward induction algorithm for building call option price trinomial tree was applied.Subdiffusive continuous time model allows successfully apply the physical models for describing the trapping events to model financial data stagnation's periods. In this paper the Inverse Gaussian process IG was proposed as a subordinator for the subdiffusive modelling of illiquidity and option pricing. The simulation of the trajectories for subordinator, inverse subordinator and subdiffusive GBM were performed. The Monte Carlo method for option evaluation was applied.Our aim was not only to compare these two models each with other, but also to show that both models adequately describe the illiquid market and can be used for option pricing on this market. For this purpose absolute relative percentage (ARPE) and root mean squared error (RMSE) for both models were computed and analysed.Thanks to the proposed approaches, the investor gets a tools, which allows him to take into account the illiquidity.
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| 6. |
- Solomanchuk, Georgiy, et al.
(author)
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Risk Modelling Approaches for Student-like Models with Fractal Activity Time
- 2021
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In: Mohyla Mathematical Journal. - : National University of Kyiv-Mohyla Academy. - 2617-7080 .- 2663-0648. ; 4, s. 28-33
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Journal article (other academic/artistic)abstract
- The paper focuses on value at risk (V@R) measuring for Student-like models of markets with fractal activity time (FAT). The fractal activity time models were introduced by Heyde to try to encompass the empirically found characteristics of real data and elaborated on for Variance Gamma, normal inverse Gaussian and skewed Student distributions. But problem of evaluating an value at risk for this model was not researched. It is worth to mention that if we use normal or symmetric Student`s models than V@R can be computed using standard statistical packages. For calculating V@R for Student-like models we need Monte Carlo method and the iterative scheme for simulating N scenarios of stock prices. We model stock prices as a diffusion processes with the fractal activity time and for modeling increments of fractal activity time we use another diffusion process, which has a given marginal inverse gamma distribution.The aim of the paper is to perform and compare V@R Monte Carlo approach and Markowitz approach for Student-like models in terms of portfolio risk. For this purpose we propose procedure of calculating V@R for two types of investor portfolios. The first one is uniform portfolio, where d assets are equally distributed. The second is optimal Markowitz portfolio, for which variance of return is the smallest out of all other portfolios with the same mean return.The programmed model which was built using R-statistics can be used as to the simulations for any asset and for construct optimal portfolios for any given amount of assets and then can be used for understanding how this optimal portfolio behaves compared to other portfolios for Student-like models of markets with fractal activity time.Also we present numerical results for evaluating V@R for both types of investor portfolio. We show that optimal Markovitz portfolio demonstrates in the most of cases the smallest possible Value at Risk comparing with other portfolios. Thus, for making investor decisions under uncertainty we recommend to apply portfolio optimization and value at risk approach jointly.
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