Tsvetelin Zaevski

Probability distributions defined on a bounded interval are essential tools in data science, engineering, and risk management. The most important practical reason for their existence is the need to model real-world phenomena that possess strict physical, logical, or economic boundaries. This paper introduces a new class of probability distributions based on the fractional linear transform applied to a distribution defined on the positive real half-line. Consequently, the distribution’s infinite support is mapped onto a bounded interval. We can assume without loss of generality that this interval is (0, 1). We first derive some general theoretical results for the probability density functions, quantile functions, and other probability properties. A duality between distributions is established. Consequently, we investigate several particular cases based on the gamma, inverse gamma, Rayleigh, and generalized inverse Gaussian distributions. The so-called saturation in the Hausdorff sense is discussed. It provides a first-order approximation of the cumulative distribution function to the Γ-shaped curve {(0, 0) → (0, 1) → (1, 1)}. Thus, it can be viewed as a break-even point and can be useful in economics, finance, risk management, and other natural sciences, such as chemistry, physics, meteorology, etc. We also investigate the maximum likelihood estimator and check its asymptotic properties, consistency, and stability. We apply this method, as well as the least-squares approach, to two empirical datasets: one for yield curves and another for unemployment insurance issues. The statistical results strongly confirm the usefulness of the proposed model.

The purpose of this paper is to obtain a duality between the game put and call options assuming three component penalties – proportion of the usual option payoff, shares of the underlying asset, and a fixed amount. We examine separately the cases of finite and infinite maturities. For the perpetual options, we need to derive a polynomial-style equations for the optimal boundaries. We prove the existence and uniqueness of their solutions as well as provide a method for their deriving. This result is important in itself since the current literature in the field is based on inverting of several functions or on solving of non-linear systems which may lead to some computational difficulties and significant errors for extreme parameter values. Furthermore, the duality is established under a finite time horizon too. It is important to note that this duality does not hold under the classical assumption of a fixed penalty.

Many authors analyze the prediction of chaos in a Josephson junction with quadratic damping by the Melnikov technique. Due to the lack of an explicit presentation of the Melnikov integral, the researchers apply numerical methods and illustrative examples to verify a good agreement between the numerical method and the analytical one. The reader has difficulty navigating and touching upon Melnikov’s elegant theory and, in particular, the formulation of the Melnikov criterion for the possible occurrence of chaos in a dynamical system, based solely on the provided illustrations of dependencies between the main parameters of the model under consideration. The statements in a number of publications devoted to this interesting topic, such as “It is easy to see that Melnikov’s integrals are finite and not zero. It is possible to see that the transverse zeros of the Melnikov function”, do not shed enough light on the origin of the “horseshoe”-type chaos. In this paper we will try to shed additional light on this important problem. A new planar system corresponding to the N-generalized Josephson junction with quadratic damping with many free parameters is considered, which may be of interest to specialists in the field of engineering sciences. Prediction of chaos in the proposed model by the Melnikov technique is closely related to the problem of approximately simultaneously finding all roots (simple or multiple) of generalized trigonometric polynomials. Several simulations are composed. We also demonstrate some specialized modules for investigating the dynamics of the model. One application about generating stochastic construction for possible control over oscillations is also discussed.

Many authors analyze the chaotic motion of the driven and damped double sine-Gordon equations and compute the Melnikov functions by numerical methods, taking an example to verify good agreement between numerical methods and analytical ones. Unfortunately, due to the lack of an explicit presentation of the Melnikov integral, the reader has difficulty navigating and touching upon Melnikov’s elegant theory and, in particular, the formulation of the Melnikov criterion for the occurrence of chaos in a dynamical system, based solely on the provided illustrations of dependencies between the main parameters of the model under consideration. In this paper we will try to shed additional light on this important problem. A new planar system corresponding to the generalized double sine-Gordon model with many free parameters is considered. We also look at the modeling of radiation diagrams and antenna factors as potential uses for the Melnikov functions. A number of simulations are created. We also show off a few specific modules for examining the model’s behavior. There is also discussion of one use for potential oscillation control.

The cancellable American options, also known as game options, are financial instruments that give a canceling right to the option’s writer in addition to the existing such holder’s right. The writer owes some penalty above the usual option payoff for using this right. We assume that this penalty consists of three parts – a proportion of the usual payoff, some number of shares of the underlying asset, and a fixed amount. It turns out that a cancellable option can be of one of the following three types – a regular American option, an American-style derivative that expires either at the maturity or when the underlying asset reaches the strike, or a real cancellable option. In this paper, the impact of the penalty on the option’s type is investigated. The perpetual case is only explored having in mind that it determines the kind of the finite maturity instruments in some sense.