The Paradigm of Complex Probability and Thomas Bayes’ Theorem

The mathematical probability concept was set forth by Andrey Nikolaevich Kolmogorov in 1933 by laying down a five-axioms system. This scheme can be improved to embody the set of imaginary numbers after adding three new axioms. Accordingly, any stochastic phenomenon can be performed in the set C of complex probabilities which is the summation of the set R of real probabilities and the set M of imaginary probabilities. Our objective now is to encompass complementary imaginary dimensions to the stochastic phenomenon taking place in the “real” laboratory in R and as a consequence to gauge in the sets R, M, and C all the corresponding probabilities. Hence, the probability in the entire set C = R + M is incessantly equal to one independently of all the probabilities of the input stochastic variable distribution in R, and subsequently the output of the random phenomenon in R can be evaluated totally in C. This is due to the fact that the probability in C is calculated after the elimination and subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic phenomenon. We will apply this novel paradigm to the classical Bayes’ theorem in probability theory.

Stochastic demand as an addition to the Solow Growth Model

The subject of this research is the Solow Growth Model. The relevance is substantiated by the fact that the Solow Growth Model is conceptually simple, and simultaneously it can be complicated with clarifications and additions. The authors believe that one of such clarification is consideration of the demand as a stochastic variable. The goal of this research is to propose a model that takes into account the random nature of consumer demand based on the Solow Growth Model. The article aims to examine the Solow Growth model; conduct a literature overview of the most common modifications of the model; analyze the well-known modifications and complications of the model; outline the methods of such modifications and complications; offer Solow Growth Model supplemented with microeconomic substantiation with consideration of the stochastic demand. The article employs the methods of analysis, synthesis, comparison, and differential calculus. The novelty lies in the statement  that consumption depends on demand; it is intuitively obvious that demand can be considered as stochastic variable. This is explained by the individual traits of the consumers. Therefore, the demand can be described via stochastic differential equation based on the standard Wiener process (analogy with Brownian motion). The article offers a stochastic differential equation of demand. The Solow Growth Model is supplemented with the stochastic differential equation of demand. In conclusion, the authors determine the key modification and complication trends of the Solow Growth Model; developed the model based on the Solow Growth Model with the stochastic differential equation of demand as its addition. Further research should be aimed at solution of the obtained mathematical model supplemented with the stochastic differential equation of demand.

Cooperative Stochastic Games with Mean-Variance Preferences

In stochastic games, the player’s payoff is a stochastic variable. In most papers, expected payoff is considered as a payoff, which means the risk neutrality of the players. However, there may exist risk-sensitive players who would take into account “risk” measuring their stochastic payoffs. In the paper, we propose a model of stochastic games with mean-variance payoff functions, which is the sum of expectation and standard deviation multiplied by a coefficient characterizing a player’s attention to risk. We construct a cooperative version of a stochastic game with mean-variance preferences by defining characteristic function using a maxmin approach. The imputation in a cooperative stochastic game with mean-variance preferences is supposed to be a random vector. We construct the core of a cooperative stochastic game with mean-variance preferences. The paper extends existing models of discrete-time stochastic games and approaches to find cooperative solutions in these games.

A Novel Integrated Profit Maximization Model for Retailers under Varied Penetration Levels of Photovoltaic Systems

In contemporary energy markets, the Retailer acts as the intermediate between the generation and demand sectors. The scope of the Retailer is to maximize its profits by selecting the appropriate procurement mechanism and selling price to the consumers. The wholesale market operation influences the profits since the mix of generation plants determines the system marginal price (SMP). In the related literature, the SMP is treated as a stochastic variable, and the wholesale market conditions are not taken into account. The present paper presents a novel methodology that aims at connecting the wholesale and retail market operations from a Retailer’s perspective. A wholesale market clearing problem is formulated and solved. The scope is to examine how different photovoltaics (PV) penetration levels in the generation side influences the profits of the Retailer and the selling prices to the consumers. The resulting SMPs are used as inputs in a retailer profit maximization problem. This approach allows the Retailer to minimize economic risks and maximize profits. The results indicate that different PV implementation levels on the generation side highly influences the profits and the selling prices.

Enhancing Stock Movement Prediction with Adversarial Training

This paper contributes a new machine learning solution for stock movement prediction, which aims to predict whether the price of a stock will be up or down in the near future. The key novelty is that we propose to employ adversarial training to improve the generalization of a neural network prediction model. The rationality of adversarial training here is that the input features to stock prediction are typically based on stock price, which is essentially a stochastic variable and continuously changed with time by nature. As such, normal training with static price-based features (e.g. the close price) can easily overfit the data, being insufficient to obtain reliable models. To address this problem, we propose to add perturbations to simulate the stochasticity of price variable, and train the model to work well under small yet intentional perturbations. Extensive experiments on two real-world stock data show that our method outperforms the state-of-the-art solution [Xu and Cohen, 2018] with 3.11% relative improvements on average w.r.t. accuracy, validating the usefulness of adversarial training for stock prediction task.