To Örebro University

The test for the location of the tangency portfolio on the set of feasible portfolios is proposed when both the population and the sample covariance matrices of asset returns are singular. The particular case of investigation is when the number of observations, n, is smaller than the number of assets, k, in the portfolio, and the asset returns are i.i.d. normally distributed with singular covariance matrix Σ such that rank(Σ) = r < n < k + 1. The exact distribution of the test statistic is derived under both the null and alternative hypotheses. Furthermore, the high-dimensional asymptotic distribution of that test statistic is established when both the rank of the population covariance matrix and the sample size increase to infinity so that r/n → c ∈ (0, 1). Theoretical findings are completed by comparing the high-dimensional asymptotic test with an exact finite sample test in the numerical study. A good performance of the obtained results is documented. To get a better understanding of the developed theory, an empirical study with data on the returns on the stocks included in the S&P 500 index is provided.

This article presents a comprehensive study on developing a predictive product pricing model using LightGBM, a machine learning method optimized for regression challenges in situations with limited historical data. It begins by detailing the core principles of LightGBM, including decision trees, boosting, and gradient descent, and then delves into the method’s unique features like Gradient-based One-Side Sampling (GOSS) and Exclusive Feature Bundling (EFB). The model’s efficacy is demonstrated through a comparative analysis with XGBoost, highlighting Light- GBM’s enhanced efficiency and slight improvement in prediction accuracy. This research offers valuable insights into the application of LightGBM in developing fast and accurate product pricing models, crucial for businesses in the rapidly evolving data landscape.

This article presents a comprehensive study on developing a predictive product pricing model using LightGBM, a machine learning method optimized for regression challenges in situations with limited historical data. It begins by detailing the core principles of LightGBM, including gradient descent, and then delves into the method's unique features like Gradient-based One-Side Sampling (GOSS) and Exclusive Feature Bundling (EFB). The model's efficacy is demonstrated through a comparative analysis with XGBoost, highlighting Light-GBM's enhanced efficiency and slight improvement in prediction accuracy. This research offers valuable insights into the application of LightGBM in developing fast and accurate product pricing models, crucial for businesses in the rapidly evolving data landscape.

The article focuses on Value-at-risk measuring for options in situations characterized by the lack of liquidity when the underlying stock price has motionless periods. A similar behavior can be observed in physical systems exhibiting sub-diffusion. In the considered sub-diffusive model, the bond movement and stock process are time-changed by the stochastic clock with gamma subordinator. In the model, the two techniques for option pricing were considered. The first very common approach for the time-changed model is to find option prices as the discounted expected payoff under the risk-neutral measure. The second technique for option pricing is based on a fractional version of what is called Dupire's equation. The Value-at-Risk evaluating procedure for the proposed model was discussed and we show that this procedure is based on the Fractional Fokker-Planck equation (FFPE).

Data analysis and the ability to preserve user privacy has become very important nowadays. So the data is coded without losing important information and made available for research. Gradient-boosted decision trees provide competitive, robust regression and classification procedures that can be interpreted and implemented to predict cold-start processes and are particularly suitable for working with less clean data. The LightGBM method is generally used for classification problems, both for open and encoded data. It is proposed to apply this method for forecasting the demand price of a product without history. We propose to use LightGBM as a safe and fast implementation of a loaded decision tree algorithm, which is widely used in data mining and machine learning tasks. For research, it is important to have a fairly wide base of goods with certain categories and goods with detailed characteristics.

In this paper, we propose the test for the location of the tangency portfolio on the set of feasible portfolios when both the population and the sample covariance matrices of asset returns are singular. We derive the exact distribution of the test statistic under both the null and alternative hypotheses. Furthermore, we establish the high-dimensional asymptotic distribution of that test statistic when both the portfolio dimension and the sample size increase to infinity. We complement our theoretical findings by comparing the high-dimensional asymptotic test with an exact finite sample test in the numerical study. A good performance of the obtained results is documented.

The article focuses on developing a predictive product pricing model using LightGBM. Also, the goal was to adapt the LightGBM method for regression problems and, especially, in the problems of forecasting the price of a product without history, that is, with a cold start.

The article contains the necessary concepts to understand the working principles of the light gradient boosting machine, such as decision trees, boosting, random forests, gradient descent, GBM (Gradient Boosting Machine), GBDT (Gradient Boosting Decision Trees). The article provides detailed insights into the algorithms used for identifying split points, with a focus on the histogram-based approach.

LightGBM enhances the gradient boosting algorithm by introducing an automated feature selection mechanism and giving special attention to boosting instances characterized by more substantial gradients. This can lead to significantly faster training and improved prediction performance. The Gradient-based One-Side Sampling (GOSS) and Exclusive Feature Bundling (EFB) techniques used as enhancements to LightGBM are vividly described. The article presents the algorithms for both techniques and the complete LightGBM algorithm.

This work contains an experimental result. To test the lightGBM, a real dataset of one Japanese C2C marketplace from the Kaggle site was taken. In the practical part, a performance comparison between LightGBM and XGBoost (Extreme Gradient Boosting Machine) was performed. As a result, only a slight increase in estimation performance (RMSE, MAE, R-squard) was found by applying LightGBM over XGBoost, however, there exists a notable contrast in the training procedure’s time efficiency. LightGBM exhibits an almost threefold increase in speed compared to XGBoost, making it a superior choice for handling extensive datasets.

This article is dedicated to the development and implementation of machine learning models for product pricing using LightGBM. The incorporation of automatic feature selection, a focus on highgradient examples, and techniques like GOSS and EFB demonstrate the model’s versatility and efficiency. Such predictive models will help companies improve their pricing models for a new product. The speed of obtaining a forecast for each element of the database is extremely relevant at a time of rapid data accumulation.

In this article, for the first time, the first boundary value problem for the equation of thermal conductivity with a variable diffusion coefficient and with a nonlinear term, which depends on the sought function with the deviation of the argument, is solved. For such equations, the initial condition is set on a certain interval. Physical and technical reasons for delays can be transport delays, delays in information transmission, delays in decision-making, etc. The most natural are delays when modeling objects in ecology, medicine, population dynamics, etc. Features of the dynamics of vehicles in different environments (water, land, air) can also be taken into account by introducing a delay. Other physical and technical interpretations are also possible, for example, the molecular distribution of thermal energy in various media (solid bodies, liquids, etc.) is modeled by heat conduction equations. The Green’s function of the first boundary value problem is constructed for the nonlinear equation of heat conduction with a deviation of the argument, its properties are investigated, and the formula for the solution is established.