Explicit Analytical Formulas Research Articles

DENSITY OF SKEW BROWNIAN MOTION AND ITS FUNCTIONALS WITH APPLICATION IN FINANCE

AbstractWe derive the joint density of a Skew Brownian motion, its last visit to the origin, its local and occupation times. The result enables us to obtain explicit analytical formulas for pricing European options under both a two‐valued local volatility model and a displaced diffusion model with constrained volatility.

Back in the saddle: large-deviation statistics of the cosmic log-density field

We present a first principle approach to obtain analytical predictions for spherically-averaged cosmic densities in the mildly non-linear regime that go well beyond what is usually achieved by standard perturbation theory. A large deviation principle allows us to compute the leading-order cumulants of average densities in concentric cells. In this symmetry, the spherical collapse model leads to cumulant generating functions that are robust for finite variances and free of critical points when logarithmic density transformations are implemented. They yield in turn accurate density probability distribution functions (PDFs) from a straightforward saddle-point approximation valid for all density values. Based on this easy-to-implement modification, explicit analytic formulas for the evaluation of the one- and two-cell PDF are provided. The theoretical predictions obtained for the PDFs are accurate to a few percent compared to the numerical integration, regardless of the density under consideration and in excellent agreement with N-body simulations for a wide range of densities. This formalism should prove valuable for accurately probing the quasi-linear scales of low redshift surveys for arbitrary primordial power spectra.

The diver with a rotor

We present and analyse a simple model for the twisting somersault. The model consists of a rigid body with a rotor attached that can be switched on and off. This makes it simple enough to devise explicit analytical formulas whilst still maintaining sufficient complexity to preserve the shape-changing dynamics essential for twisting somersaults performed in springboard and platform diving. With “rotor on” and with “rotor off” the corresponding Euler-type equations can be solved and the essential quantities characterising the dynamics, such as the periods and rotation numbers, can be computed in terms of complete elliptic integrals. We arrive at explicit formulas for how to achieve a dive with m somersaults and n twists in a given total time. This can be thought of as a special case of a geometric phase formula due to Cabrera (2007).

Effect of yerba mate (Ilex paraguariensis) on bone tissue

In the article, we propose an incremental algorithm for calculating the inversion of the confluent Vandermonde matrix and triangular factorization of this inversion. We implemented all the incremental operations, i.e. adding, deleting and changing the single matrix parameter to avoid repeating the same calculations again and again. Besides, contrary to other works in this field, the article derives an explicit analytic formula for the calculation of the triangular factorization of the confluent Vandermonde matrix inversion. Additionally, we propose a solution to these two problems with the use of a system of linear recursive equations. The results of this article do not require any symbolic calculations. Therefore they can be performed by a numerical algorithm implemented in any general-purpose programming language.

DESIGN OF MULTI-STAGE POWER DIVIDER BASED ON THE THEORY OF SMALL REFLECTIONS

This paper presents a novel multi-way multi-stage power divider design method based on the theory of small reflections. Firstly, the application of the theory of small reflections is extended from transmission line to microwave network. Secondly, an explicit closed-form analytical formula of the input reflection coefficient, which consists of the scattering parameters of power divider elements and the lengths of interconnection lines between each element, is derived. Thirdly, the proposed formula is applied to determine the lengths of interconnection lines. A prototype of a 16-way 4-stage power divider working at 4 GHz is designed and fabricated. Both the simulation and measurement results demonstrate the validity of the proposed method.

Distributed Detection over Adaptive Networks: Refined Asymptotics and the Role of Connectivity

We consider distributed detection problems over adaptive networks, where dispersed agents learn continually from streaming data by means of local interactions. The requirement of adaptation allows the network of detectors to track drifts in the underlying hypothesis. The requirement of cooperation allows each agent to deliver a performance superior to what would be obtained if it were acting individually. The simultaneous requirements of adaptation and cooperation are achieved by employing diffusion algorithms with constant step-size $\mu$ . By conducting a refined asymptotic analysis based on the mathematical framework of exact asymptotics, we arrive at a revealing understanding of the universal behavior of distributed detection over adaptive networks : as functions of $1/\mu$ , the error (log-)probability curves corresponding to different agents stay nearly-parallel to each other (as already discovered in [1] and [2] ), however, these curves are ordered following a criterion reflecting the degree of connectivity of each agent. Depending on the combination weights, the more connected an agent is, the lower its error probability curve will be. The analysis provides explicit analytical formulas for the detection error probabilities and these expressions are also verified by means of extensive simulations. We further enlarge the reference setting from the case of doubly-stochastic combination matrices considered in [1] and [2] to the more general and demanding setting of right-stochastic combination matrices; this extension poses new and interesting questions in terms of the interplay between the network topology, the combination weights, and the inference performance. The potential of the proposed methods is illustrated by application of the results to canonical detection problems, to typical network topologies, for both doubly-stochastic and right-stochastic combination matrices. Interesting and somehow unexpected behaviors emerge, and the lesson learned is that connectivity matters .

Incremental numerical recipes for the high efficient inversion of the confluent Vandermonde matrices

In the article, we propose an incremental algorithm for calculating the inversion of the confluent Vandermonde matrix and triangular factorization of this inversion. We implemented all the incremental operations, i.e. adding, deleting and changing the single matrix parameter to avoid repeating the same calculations again and again. Besides, contrary to other works in this field, the article derives an explicit analytic formula for the calculation of the triangular factorization of the confluent Vandermonde matrix inversion. Additionally, we propose a solution to these two problems with the use of a system of linear recursive equations. The results of this article do not require any symbolic calculations. Therefore they can be performed by a numerical algorithm implemented in any general-purpose programming language.

Saddlepoint approximations to option price in a regime-switching model

In this paper we consider the saddlepoint approximation for the valuation of a European-style call option in a Markovian, regime-switching, Black–Scholes–Merton economy, where the price process of an underlying risky asset is assumed to follow a Markov-modulated geometric Brownian motion. The standard option pricing procedure under this model becomes problematic as the occupation time of chains for a given state cannot be evaluated easily. In the case of two-state Markov chains, we present an explicit analytic formula of the cumulant generating function (CGF). When the process has more than two states, an approximate formula of the CGF is provided. We adopt a splitting method to reduce the complexity of computing the matrix exponential function. Then we use these CGFs to develop the use of the saddlepoint approximations. The numerical results show that the saddlepoint approximation is an efficient and reliable approach for option pricing under a multi-state regime-switching model.

Binary Events Loading for Solvency II Technical Provisions: Practical Approximations

Under the Solvency II Directive insurers are required to allow for all possible events when setting their technical provisions, including those that may not have been historically realised before. Such events not presented in a set of observable historical loss data are often called Binary Events. Here, the term ‘Binary’ is traditionally used to define loss-generating events with low frequency and high severity impact, whereas an alternative name, Events Not In Data (ENID), may also be used to denote the much broader set of unobservable loss events.In its Technical Provisions Guidance issued in March 2011, whilst suggesting several approaches to allowing for Binary Events, Lloyd’s encourages actuarial functions to use one specific approach that offers uplifting reserve best estimates to allow for a limited range of historical (observable) data. This approach is called the Truncated Statistical Distribution approach. Although the Lloyd’s guidance does not provide any explicit analytical formulae for calculating the uplift factor, it seems the industry is in favour of two particular analytical approximations of the load of reserve best estimate, assuming log-normality of the true distribution of reserves. This paper examines the quality of the two Lloyd’s approximations of Binary Events loading for technical provisions, and also presents a distribution-free approach to estimating ENID load for reserve mean and variability using the following key determinants of reserve risk profile:• the level of variability measured by Coefficient of Variation (CoV); and• the degree of Skewness per unit of CoV, i.e. Skewness-to-CoV (SC) ratio.

Metric of two balancing Kerr particles in physical parametrization

The present paper aims at elaborating a completely physical representation for the general 4-parameter family of the extended double-Kerr spacetimes describing two spinning sources in gravitational equilibrium. This involved problem is solved in a concise analytical form by using the individual Komar masses and angular momenta as arbitrary parameters, and the simplest equatorially symmetric specialization of the general expressions obtained by us yields the physical representation for the well-known Dietz-Hoenselaers superextreme case of two balancing identical Kerr constituents. The existence of the physically meaningful "black hole-superextreme object" equilibrium configurations permitted by the general solution may be considered as a clear indication that the spin-spin repulsion force might actually be by far stronger than expected earlier, when only the balance between two superextreme Kerr sources was thought possible. We also present the explicit analytical formulas relating the equilibrium states in the double-Kerr and double-Reissner-Nordstr\"om configurations.

Time Evolving Fission Chain Theory and Fast Neutron and Gamma-Ray Counting Distributions

We solve a simple theoretical model of time evolving fission chains due to Feynman that generalizes and asymptotically approaches the point model theory. The point model theory has been used to analyze thermal neutron counting data. This extension of the theory underlies fast counting data for both neutrons and gamma rays from metal systems. Fast neutron and gamma-ray counting is now possible using liquid scintillator arrays with nanosecond time resolution. For individual fission chains, the differential equations describing three correlated probability distributions are solved: the time-dependent internal neutron population, accumulation of fissions in time, and accumulation of leaked neutrons in time. Explicit analytic formulas are given for correlated moments of the time evolving chain populations. The equations for random time gate fast neutron and gamma-ray counting distributions, due to randomly initiated chains, are presented. Correlated moment equations are given for both random time gate and triggered time gate counting. Explicit formulas for all correlated moments are given up to triple order, for all combinations of correlated fast neutrons and gamma rays. The nonlinear differential equations for probabilities for time dependent fission chain populations have a remarkably simple Monte Carlo realization. A Monte Carlo code was developed for this theory and is shown to statistically realize the solutions to the fission chain theory probability distributions. Combined with random initiation of chains and detection of external quanta, the Monte Carlo code generates time tagged data for neutron and gamma-ray counting and from these data the counting distributions.

Extension of bivariate means to weighted means of several arguments by using binary trees

Averaging aggregation functions are valuable in building decision making and fuzzy logic systems and in handling uncertainty. Some interesting classes of averages are bivariate and not easily extended to the multivariate case. We propose a generic method for extending bivariate symmetric means to n-variate weighted means by recursively applying the specified bivariate mean in a binary tree construction. We prove that the resulting extension inherits many desirable properties of the base mean and design an efficient numerical algorithm by pruning the binary tree. We show that the proposed method is numerically competitive to the explicit analytical formulas and hence can be used in various computational intelligence systems which rely on aggregation functions.

Scattering of electromagnetic waves by many small perfectly conducting or impedance bodies

A theory of electromagnetic (EM) wave scattering by many small particles of an arbitrary shape is developed. The particles are perfectly conducting or impedance. For a small impedance particle of an arbitrary shape, an explicit analytical formula is derived for the scattering amplitude. The formula holds as a → 0, where a is a characteristic size of the small particle and the wavelength is arbitrary but fixed. The scattering amplitude for a small impedance particle is shown to be proportional to a2−κ, where κ ∈ [0, 1) is a parameter which can be chosen by an experimenter as he/she wants. The boundary impedance of a small particle is assumed to be of the form ζ = ha−κ, where h = const, Reh ≥ 0. The scattering amplitude for a small perfectly conducting particle is proportional to a3, and it is much smaller than that for the small impedance particle. The many-body scattering problem is solved under the physical assumptions a ≪ d ≪ λ, where d is the minimal distance between neighboring particles and λ is the wavelength. The distribution law for the small impedance particles is N(Δ)∼1/a2−κ∫ΔN(x)dx as a → 0. Here, N(x) ≥ 0 is an arbitrary continuous function that can be chosen by the experimenter and N(Δ) is the number of particles in an arbitrary sub-domain Δ. It is proved that the EM field in the medium where many small particles, impedance or perfectly conducting, are distributed, has a limit, as a → 0 and a differential equation is derived for the limiting field. On this basis, a recipe is given for creating materials with a desired refraction coefficient by embedding many small impedance particles into a given material.

Transversal instability for the thermodiffusive reaction-diffusion system

The propagation of unstable interfaces is at the origin of remarkable patterns that are observed in various areas of science as chemical reactions, phase transitions, and growth of bacterial colonies. Since a scalar equation generates usually stable waves, the simplest mathematical description relies on two-by-two reaction-diffusion systems. The authors’ interest is the extension of the Fisher/KPP equation to a two-species reaction which represents reactant concentration and temperature when used for flame propagation, and bacterial population and nutrient concentration when used in biology. The authors study circumstances in which instabilities can occur and in particular the effect of dimension. It is observed numerically that spherical waves can be unstable depending on the coefficients. A simpler mathematical framework is to study transversal instability, which means a one-dimensional wave propagating in two space dimensions. Then, explicit analytical formulas give explicitely the range of paramaters for instability.

Explicit approximate analytic formulas for timer option pricing with stochastic interest rates

The interest rate risk is an important factor in the valuation of timer options. Since the valuation of timer options with interest rate risk is a four-dimensional problem, the dimensionality curse causes tremendous difficulty in finding analytic solutions to the pricing of timer options. In this paper, a fast approximate analytic method is developed to price power style timer options with Vasicek interest rate model. The valuation of timer options with interest rate risk is formulated as a four-dimensional partial differential equation (PDE) using Δ-hedging approach. A dimension-reduction technique is then proposed to reduce the four-dimensional PDE into a two-dimensional nonlinear PDE. A perturbation approach is developed to solve the reduced two-dimensional nonlinear PDEs and then an explicit approximate analytic formula for the timer option is obtained. In particular, explicit approximate analytic formulas for timer options under both Heston and Hull–White models are further derived. Numerical examples of pricing timer options under the above two models are provided. Both the approximate analytic method and the crude Monte Carlo simulation method are used for the examples. The numerical results show that prices of timer options by both methods are close and the approximate analytic method is much faster than the crude Monte Carlo method.

Catastrophe options with double compound Poisson processes

We study the following catastrophe option pricing model with double jump processes: (i) Stock process of an insurance company which sells catastrophe option are described through an exponential jump-diffusion process. (ii) All jump terms are modeled by two compound Poisson processes. One is correlated to the catastrophe loss process, and models the jumps of a stock due to catastrophe events. Another one models the jumps of the stock process caused by other financial market risks. For the model, we obtain explicit analytical formulas for the price of the put option, and then use several numerical examples based on Monte Carlo simulation to show its reasonability.

A $p$-adic construction of ATR points on $\mathbb{Q}$-curves

In this note we consider certain elliptic curves defined over real quadratic fields isogenous to their Galois conjugate. We give a construction of algebraic points on these curves defined over almost totally real number fields. The main ingredient is the system of Heegner points arising from Shimura curve uniformizations. In addition, we provide an explicit p-adic analytic formula which allows for the effective, algorithmic calculation of such points.

General distortional buckling formulae for both fixed-ended and pinned-ended C-section columns

This paper presents general explicit analytical formulae to provide distortional critical stress estimates for cold-formed steel C-section columns subjected to uniform compression. These formulae are derived by employing the Lau and Hancock model and by introducing a new factor for considering the web rotational restraint reduced by web bending. The results obtained from the general explicit analytical formulae are compared with the numerical results obtained from the computer programs CUFSM or/and GBTUL. For pinned-ended columns, these values are also compared with those yielded by the formulae developed by Lau and Hancock, Schafer, and Silvestre and Camotim. In particular, the distortional critical stress of fixed-ended C-section columns is validated by comparing the corresponding ultimate load based on Schafer’s DSM expressions with numerical results from the research literature. The formulae derived here are validated and their application, accuracy and capabilities are illustrated by these comparisons and validations.

Leaf-to-leaf distances and their moments in finite and infinite ordered m-ary tree graphs.

We study the leaf-to-leaf distances on one-dimensionally ordered, full and complete m-ary tree graphs using a recursive approach. In our formulation, unlike in traditional graph theory approaches, leaves are ordered along a line emulating a one-dimensional lattice. We find explicit analytical formulas for the sum of all paths for arbitrary leaf separation r as well as the average distances and the moments thereof. We show that the resulting explicit expressions can be recast in terms of Hurwitz-Lerch transcendants. Results for periodic trees are also given. For incomplete random binary trees, we provide first results by numerical techniques; we find a rapid drop of leaf-to-leaf distances for large r.

Conditions for formation and trapping of the two-ion Coulomb cluster in the dissipative optical superlattice

Conditions have been studied under which a polychromatic optical superlattice can form and trap the Coulomb cluster of two strongly interacting ions. In our previous work (Krasnov I V and Kamenshchikov L P 2014 Opt. Comm. 312 192) this new all-optical method of obtaining and confining the Coulomb clusters was demonstrated by numerical simulations for special values of the optical superlattice parameters and in the case of Yb ions. In the present paper the conditions are explicitly formulated, under which the long-lived two-ion cluster in the superlattice cell is formed. The peculiarity of these conditions is the renormalization of the ion–ion Coulomb interaction. Notably, the renormalized Coulomb force is determined by the effective charge which depends on the light field parameters and can strongly differ from the “bare” ion charge. This result can be accounted for by the combined manifestation of the quantum fluctuations of optical forces, nonlinear dependence of these forces on the velocity, and non-Maxwellian (Tsallis type) velocity distribution of the ions in the optical superlattice. Explicit analytical formulas are also obtained for the parameters of the optical two-ion cluster.