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Revista Economica

THE USE OF NEGATIVE PROBABILITIES IN ECONOMICS

Sorin-Nicolae Curcă

Published Apr 1, 2022• 0 citations
Publisher

Abstract

The concept of „negative probability” has become one of interest in research. Initially developed in connection with physics (negative probabilities are important in the analysis of quantum phenomena), nowadays it has applications in many other scientific disciplines. Its main advantage is that it extends the classical probability theory, which involves values between 0 and 1, allowing negative or supraunitary values. From this point of view, negative probabilities are a useful tool that facilitates the calculation and makes analysis more flexible through models. This paper aims to present the theoretical framework, the principles in relation to which the use of negative probabilities in economics, in general, can be understood. The existing literature highlights that the negative probabilities are appropriate in the analysis of stochastic processes and, from this perspective, has developed applications especially in the field of finance. In this sense, negative probabilities are used in models such as the binomial CRR model (The Cox-Ross-Rubinstein Market Model), that evaluate the price of derivatives on the capital market, those that include hidden variable (non-observable) etc. In our analysis we describe these models and, based on them, we try to identify the features of the negative probabilities that can be used in the economic field. The research is a theoretical one, leading the way for new approaches in the direction of using negative probabilities in macroeconomic modelling.

DOI
10.56043/reveco-2022-0005

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The concept of „negative probability” has become one of interest in research. Initially developed in connection with physics (negative probabilities are important in the analysis of quantum phenomena), nowadays it has applications in many other scientific disciplines. Its main advantage is that it extends the classical probability theory, which involves values between 0 and 1, allowing negative or supraunitary values. From this point of view, negative probabilities are a useful tool that facilitates the calculation and makes analysis more flexible through models. This paper aims to present the theoretical framework, the principles in relation to which the use of negative probabilities in economics, in general, can be understood. The existing literature highlights that the negative probabilities are appropriate in the analysis of stochastic processes and, from this perspective, has developed applications especially in the field of finance. In this sense, negative probabilities are used in models such as the binomial CRR model (The Cox-Ross-Rubinstein Market Model), that evaluate the price of derivatives on the capital market, those that include hidden variable (non-observable) etc. In our analysis we describe these models and, based on them, we try to identify the features of the negative probabilities that can be used in the economic field. The research is a theoretical one, leading the way for new approaches in the direction of using negative probabilities in macroeconomic modelling.
There are important problems in physics related to the concept of probability. One of these problems is related to negative probabilities used in physics from 1930s. In spite of many demonstrations of usefulness of negative probabilities, physicists looked at them with suspicion trying to avoid this new concept in their theories. The main question that has bothered physicists is mathematical grounding and interpretation of negative probabilities. In this paper, we introduce extended probability as a probability function, which can take both positive and negative values. Defining extended probabilities in an axiomatic way, we show that classical probability is a positive section of extended probability.
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Quantum economics is an interdisciplinary field that combines the principles of quantum mechanics and economic theories. The research examines the main theoretical and practical aspects of quantum economics, starting with its theoretical foundations based on the principles of quantum mechanics, such as superposition, uncertainty and entanglement, and ending with practical applications and perspectives for further research. Special attention was paid to the potential impact of quantum technologies on finance, marketing, supply chain management, as well as on macroeconomic planning and policy. It discusses how quantum algorithms can transform traditional approaches in economic research and practice, providing new methods for analyzing complex data, improving forecasting of economic trends, and optimizing decisions under uncertainty. Quantum economics opens up new perspectives for understanding dynamic and interconnected processes in the global economy, which can fundamentally change our perception of economic laws and strategies. It is emphasized that the quantum economy is at the initial stage of its development, and scientists and practitioners face many challenges and questions that need to be solved. In the future, we can expect the development of new tools for economic analysis and decisionmaking based on quantum computing, which will allow us to cope with current and future economic challenges more effectively. The quantum economy can also offer new approaches to risk management and financial security, especially in light of growing cyber threats and the need to protect economic information. Quantum economics is an exciting and promising direction that can significantly enrich both theoretical and applied aspects of economic science.
This paper is devoted to foundations of probability theory. We discuss interpretations of probability, corresponding mathematical formalisms, and applications to quantum physics. One of the aims of this paper is to show that the probability model based on Kolmogorov's axiomatics cannot describe all stochastic phenomena, i.e., that quantum physics induces natural restrictions of the use of Kolmogorov's theory and that we need to develop non-Kolmogorov models for describing some quantum phenomena. The physical motivations are presented in a clear and brief manner. Thus the reader does not need to have preliminary knowledgeof quantum physics. Our main idea is that we cannot develop non-Kolmogorov models by the formal change of Kolmogorov's axiomatics. We begin with interpretations (classical, frequency, and proportional). Then we present a class of non-Kolmogorov models described by so-called p-adic numbers. Here, in particular, we obtain a quite natural realization of negative probabilities. These negative probability distributions might provide a solution of some quantum paradoxes.
Quantum economics postulates ideas about quantum money, quantum finance, and quantum social science and shows that the economy is an archetypal example of a quantum social system, complete with its versions of measurement, i.e., uncertainty, entanglement, superposition, and so on.A sizeable literature on quantum economics has grown in recent periods.This paper reviews some theoretical developments and empirical studies on Quantum economics to provide a comprehensive analysis of the potential implications of Quantum Physics and its philosophical ideas in the domains of economics, finance, and social science.Moreover, this studies also focused Why the field of Quantum Economics arises" how Quantum economics links physics, economics, and social science.All reviews and research papers examine the use of learning strategies in Quantum Economics.The overall results shows the following Possible applications of quantum Physics' in Economics: (i) enhanced computational capabilities for big data and complex problems, (ii) secure quantum-based currency, (iii) innovative tools for policymaking, and (iv) advanced behavioral and decisionmaking analysis under uncertainty.This paper provides an overview of the Quantum economics literature, background history, conceptual insights, economic policy, and conceptual and methodological critique.Finally, we present an agenda for future research in the field.
In this paper, we present a quantum version of some portions of Mathematical Finance, including theory of arbitrage, asset pricing, and optional decomposition in financial markets based on finite dimensional quantum probability spaces. As examples, the quantum model of binomial markets is studied. We show that this quantum model ceases to pose the paradox which appears in the classical model of the binomial market. Furthermore, we re-deduce the Cox-Ross-Rubinstein binomial option pricing formula by considering multi-period quantum binomial markets.