We extend a model of positive feedback and contagion in large mean-field systems by introducing a common source of noise driven by Brownian motion. Although the dynamics in the model are continuous, the feedback effect can lead to jump discontinuities in the solutions, which we refer to as `blow-ups'. We prove existence of solutions to the corresponding conditional McKean--Vlasov equation, by realising them as suitable limit points of the finite-dimensional particle system, and we show that the pathwise realisation of the common noise can both trigger and prevent blow-ups.

In this article three models of firms interaction on the market are described. One of these models is described by using a differential equation and by Lotka-Volterra model, where the equation has a different form. Also, there are models of non-competing and competing firms. The article presents an algorithm for solving the interaction of competing firms in taxation and the calculation of a compromise point. Besides, the article presents a compromise between the interests of a state and an enterprise.

This paper introduces a new correction scheme to a conventional regression-based event study method: a topological machine-learning approach with a self-organizing map (SOM).We use this new scheme to analyze a major market event in Japan and find that the factors of abnormal stock returns can be easily can be easily identified and the event-cluster can be depicted.We also find that a conventional event study method involves an empirical analysis mechanism that tends to derive bias due to its mechanism, typically in an event-clustered market situation. We explain our new correction scheme and apply it to an event in the Japanese market --- the holding disclosure of the Government Pension Investment Fund (GPIF) on July 31, 2015.

We apply the recently developed reduced Google matrix algorithm for the analysis of the OECD-WTO world network of economic activities. This approach allows to determine interdependences and interactions of economy sectors of several countries, including China, Russia and USA, properly taking into account the influence of all other world countries and their economic activities. Within this analysis we also obtain the sensitivity of economy sectors and EU countries to petroleum activity sector. We show that this approach takes into account multiplicity of network links with economy interactions between countries and activity sectors thus providing more rich information compared to the usual export-import analysis.

Employment gratuity is the money companies typically give to their employees at the end of their contracts as a legal requirement. Like pension, it is a form of retirement plan and is often given as an alternative to a pension plan. Nonetheless, there is now a new pattern whereby companies give their employees the option to receive their gratuity at various stages before the end of their contracts. In Botswana, for instance, some companies give their employees an option to receive their gratuity on a monthly basis rather than having them wait for a year or more. Many employees find this option attractive, but is it economically sound? This paper sheds light on this question, highlighting the need for analytically informed decisions. It quantifies the economic benefits of the tax relief provided by government relative to investing the monthly-received funds in a risk-free savings account or in helping repay a loan. The principles and methods used herein can be adapted and applied to different circumstances.

We study a class of optimal stopping games (Dynkin games) of preemption type, with uncertainty about the existence of competitors. The set-up is well-suited to model, for example, real options in the context of investors who do not want to publicly reveal their interest in a certain business opportunity. We show that there exists a Nash equilibrium in randomized stopping times which is described explicitly in terms of the corresponding one-player game.

We consider a fractional version of the Heston volatility model which is inspired by [16]. Within this model we treat portfolio optimization problems for power utility functions. Using a suitable representation of the fractional part, followed by a reasonable approximation we show that it is possible to cast the problem into the classical stochastic control framework. This approach is generic for fractional processes. We derive explicit solutions and obtain as a by-product the Laplace transform of the integrated volatility. In order to get rid of some undesirable features we introduce a new model for the rough path scenario which is based on the Marchaud fractional derivative. We provide a numerical study to underline our results.

In this article, I will present a paradox whose purpose is to draw your attention to an important topic in finance, concerning the non-independence of the financial returns (non-ergodic hypothesis). In this paradox, we have two people sitting at a table separated by a black sheet so that they cannot see each other and are playing the following game: the person we call A flip a coin and the person we'll call B tries to guess the outcome of the coin flip. At the end of the game, both people are asked to estimate the compound probability of the result obtained. The two people give two different answers, one estimates the events as independent and the other one considers the events as dependent therefore they calculate the conditional probability differently. This paradox show how the erroneous estimation of conditional probability implies a strong distortion of the forecasting skill that can lead us to bear excessive risks.

Hyperinflation and price volatility in virtual economies has the potential to reduce player satisfaction and decrease developer revenue. This paper describes intuitive analytical methods for monitoring volatility and inflation in virtual economies, with worked examples on the increasingly popular multiplayer game Old School Runescape. Analytical methods drawn from mainstream financial literature are outlined and applied in order to present a high level overview of virtual economic activity of 3467 price series over 180 trading days. Six-monthly volume data for the top 100 most traded items is also used both for monitoring and value estimation, giving a conservative estimate of exchange trading volume of over {\pounds}60m in real value. Our worked examples show results from a well functioning virtual economy to act as a benchmark for future work. This work contributes to the growing field of virtual economics and game development, describing how data transformations and statistical tests can be used to improve virtual economic design and analysis, with applications in real-time monitoring systems.