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815 KiB

Open AccessArticle

Analytical Solutions of the Black–Scholes Pricing Model for European Option Valuation via a Projected Differential Transformation Method

by Sunday O. Edeki, Olabisi O. Ugbebor and Enahoro A. Owoloko

Entropy 2015, 17(11), 7510-7521; https://doi.org/10.3390/e17117510 - 30 Oct 2015

Cited by 53 | Viewed by 7141

Abstract

In this paper, a proposed computational method referred to as Projected Differential Transformation Method (PDTM) resulting from the modification of the classical Differential Transformation Method (DTM) is applied, for the first time, to the Black–Scholes Equation for European Option Valuation. The results obtained [...] Read more.

In this paper, a proposed computational method referred to as Projected Differential Transformation Method (PDTM) resulting from the modification of the classical Differential Transformation Method (DTM) is applied, for the first time, to the Black–Scholes Equation for European Option Valuation. The results obtained converge faster to their associated exact solution form; these easily computed results represent the analytical values of the associated European call options, and the same algorithm can be followed for European put options. It is shown that PDTM is more efficient, reliable and better than the classical DTM and other semi-analytical methods since less computational work is involved. Hence, it is strongly recommended for both linear and nonlinear stochastic differential equations (SDEs) encountered in financial mathematics. Full article

(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

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633 KiB

Open AccessArticle

Local Fractional Homotopy Perturbation Method for Solving Non-Homogeneous Heat Conduction Equations in Fractal Domains

by Yu Zhang, Carlo Cattani and Xiao-Jun Yang

Entropy 2015, 17(10), 6753-6764; https://doi.org/10.3390/e17106753 - 5 Oct 2015

Cited by 108 | Viewed by 6327

Abstract

In this article, the local fractional Homotopy perturbation method is utilized to solve the non-homogeneous heat conduction equations. The operator is considered in the sense of the local fractional differential operator. Comparative results between non-homogeneous and homogeneous heat conduction equations are presented. The [...] Read more.

In this article, the local fractional Homotopy perturbation method is utilized to solve the non-homogeneous heat conduction equations. The operator is considered in the sense of the local fractional differential operator. Comparative results between non-homogeneous and homogeneous heat conduction equations are presented. The obtained result shows the non-differentiable behavior of heat conduction of the fractal temperature field in homogeneous media. Full article

(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

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17474 KiB

Open AccessArticle

Computing and Learning Year-Round Daily Patterns of Hourly Wind Speed and Direction and Their Global Associations with Meteorological Factors

by Hsing-Ti Wu, Hsieh Fushing and Laurence Z.H. Chuang

Entropy 2015, 17(8), 5784-5798; https://doi.org/10.3390/e17085784 - 11 Aug 2015

Cited by 1 | Viewed by 5003

Abstract

Daily wind patterns and their relational associations with other metocean (oceanographic and meteorological) variables were algorithmically computed and extracted from a year-long wind and weather dataset, which was collected hourly from an ocean buoy located in the Penghu archipelago of Taiwan. The computational [...] Read more.

Daily wind patterns and their relational associations with other metocean (oceanographic and meteorological) variables were algorithmically computed and extracted from a year-long wind and weather dataset, which was collected hourly from an ocean buoy located in the Penghu archipelago of Taiwan. The computational algorithm is called data cloud geometry (DCG). This DCG algorithm is a clustering-based nonparametric learning approach that was constructed and developed implicitly based on various entropy concepts. Regarding the bivariate aspect of wind speed and wind direction, the resulting multiscale clustering hierarchy revealed well-known wind characteristics of year-round pattern cycles pertaining to the particular geographic location of the buoy. A wind pattern due to a set of extreme weather days was also identified. Moreover, in terms of the relational aspect of wind and other weather variables, causal patterns were revealed through applying the DCG algorithm alternatively on the row and column axes of a data matrix by iteratively adapting distance measures to computed DCG tree structures. This adaptation technically constructed and integrated a multiscale, two-sample testing into the distance measure. These computed wind patterns and pattern-based causal relationships are useful for both general sailing and competition planning. Full article

(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

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347 KiB

Open AccessArticle

Consistency of Learning Bayesian Network Structures with Continuous Variables: An Information Theoretic Approach

by Joe Suzuki

Entropy 2015, 17(8), 5752-5770; https://doi.org/10.3390/e17085752 - 10 Aug 2015

Cited by 8 | Viewed by 5792

Abstract

We consider the problem of learning a Bayesian network structure given n examples and the prior probability based on maximizing the posterior probability. We propose an algorithm that runs in O(n log n) time and that addresses continuous variables and discrete variables without [...] Read more.

We consider the problem of learning a Bayesian network structure given n examples and the prior probability based on maximizing the posterior probability. We propose an algorithm that runs in O(n log n) time and that addresses continuous variables and discrete variables without assuming any class of distribution. We prove that the decision is strongly consistent, i.e., correct with probability one as n ! 1. To date, consistency has only been obtained for discrete variables for this class of problem, and many authors have attempted to prove consistency when continuous variables are present. Furthermore, we prove that the “log n” term that appears in the penalty term of the description length can be replaced by 2(1+ε) log log n to obtain strong consistency, where ε > 0 is arbitrary, which implies that the Hannan–Quinn proposition holds. Full article

(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

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3901 KiB

Open AccessArticle

Gaussian Network’s Dynamics Reflected into Geometric Entropy

by Domenico Felice and Stefano Mancini

Entropy 2015, 17(8), 5660-5672; https://doi.org/10.3390/e17085660 - 6 Aug 2015

Cited by 3 | Viewed by 4659

Abstract

We consider a geometric entropy as a measure of complexity for Gaussian networks, namely networks having Gaussian random variables sitting on vertices and their correlations as weighted links. We then show how the network dynamics described by the well-known Ornstein–Uhlenbeck process reflects into [...] Read more.

We consider a geometric entropy as a measure of complexity for Gaussian networks, namely networks having Gaussian random variables sitting on vertices and their correlations as weighted links. We then show how the network dynamics described by the well-known Ornstein–Uhlenbeck process reflects into such a measure. We unveil a crossing of the entropy time behaviors between switching on and off links. Moreover, depending on the number of links switched on or off, the entropy time behavior can be non-monotonic. Full article

(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

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522 KiB

Open AccessArticle

Applications of the Fuzzy Sumudu Transform for the Solution of First Order Fuzzy Differential Equations

by Norazrizal Aswad Abdul Rahman and Muhammad Zaini Ahmad

Entropy 2015, 17(7), 4582-4601; https://doi.org/10.3390/e17074582 - 1 Jul 2015

Cited by 19 | Viewed by 5783

Abstract

In this paper, we study the classical Sumudu transform in fuzzy environment, referred to as the fuzzy Sumudu transform (FST). We also propose some results on the properties of the FST, such as linearity, preserving, fuzzy derivative, shifting and convolution theorem. In order [...] Read more.

In this paper, we study the classical Sumudu transform in fuzzy environment, referred to as the fuzzy Sumudu transform (FST). We also propose some results on the properties of the FST, such as linearity, preserving, fuzzy derivative, shifting and convolution theorem. In order to show the capability of the FST, we provide a detailed procedure to solve fuzzy differential equations (FDEs). A numerical example is provided to illustrate the usage of the FST. Full article

(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

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164 KiB

Open AccessArticle

Reliability Analysis Based on a Jump Diffusion Model with Two Wiener Processes for Cloud Computing with Big Data

by Yoshinobu Tamura and Shigeru Yamada

Entropy 2015, 17(7), 4533-4546; https://doi.org/10.3390/e17074533 - 26 Jun 2015

Cited by 26 | Viewed by 5711

Abstract

At present, many cloud services are managed by using open source software, such as OpenStack and Eucalyptus, because of the unification management of data, cost reduction, quick delivery and work savings. The operation phase of cloud computing has a unique feature, such as [...] Read more.

At present, many cloud services are managed by using open source software, such as OpenStack and Eucalyptus, because of the unification management of data, cost reduction, quick delivery and work savings. The operation phase of cloud computing has a unique feature, such as the provisioning processes, the network-based operation and the diversity of data, because the operation phase of cloud computing changes depending on many external factors. We propose a jump diffusion model with two-dimensional Wiener processes in order to consider the interesting aspects of the network traffic and big data on cloud computing. In particular, we assess the stability of cloud software by using the sample paths obtained from the jump diffusion model with two-dimensional Wiener processes. Moreover, we discuss the optimal maintenance problem based on the proposed jump diffusion model. Furthermore, we analyze actual data to show numerical examples of dependability optimization based on the software maintenance cost considering big data on cloud computing. Full article

(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

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2141 KiB

Open AccessArticle

Detecting Chronotaxic Systems from Single-Variable Time Series with Separable Amplitude and Phase

by Gemma Lancaster, Philip T. Clemson, Yevhen F. Suprunenko, Tomislav Stankovski and Aneta Stefanovska

Entropy 2015, 17(6), 4413-4438; https://doi.org/10.3390/e17064413 - 23 Jun 2015

Cited by 5 | Viewed by 5458

Abstract

The recent introduction of chronotaxic systems provides the means to describe nonautonomous systems with stable yet time-varying frequencies which are resistant to continuous external perturbations. This approach facilitates realistic characterization of the oscillations observed in living systems, including the observation of transitions in [...] Read more.

The recent introduction of chronotaxic systems provides the means to describe nonautonomous systems with stable yet time-varying frequencies which are resistant to continuous external perturbations. This approach facilitates realistic characterization of the oscillations observed in living systems, including the observation of transitions in dynamics which were not considered previously. The novelty of this approach necessitated the development of a new set of methods for the inference of the dynamics and interactions present in chronotaxic systems. These methods, based on Bayesian inference and detrended fluctuation analysis, can identify chronotaxicity in phase dynamics extracted from a single time series. Here, they are applied to numerical examples and real experimental electroencephalogram (EEG) data. We also review the current methods, including their assumptions and limitations, elaborate on their implementation, and discuss future perspectives. Full article

(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

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229 KiB

Open AccessArticle

Brownian Motion in Minkowski Space

by Paul O'Hara and Lamberto Rondoni

Entropy 2015, 17(6), 3581-3594; https://doi.org/10.3390/e17063581 - 1 Jun 2015

Cited by 3 | Viewed by 4832

Abstract

We construct a model of Brownian motion in Minkowski space. There are two aspects of the problem. The first is to define a sequence of stopping times associated with the Brownian “kicks” or impulses. The second is to define the dynamics of the [...] Read more.

We construct a model of Brownian motion in Minkowski space. There are two aspects of the problem. The first is to define a sequence of stopping times associated with the Brownian “kicks” or impulses. The second is to define the dynamics of the particle along geodesics in between the Brownian kicks. When these two aspects are taken together, the Central Limit Theorem (CLT) leads to temperature dependent four dimensional distributions defined on Minkowski space, for distances and 4-velocities. In particular, our processes are characterized by two independent time variables defined with respect to the laboratory frame: a discrete one corresponding to the stopping times when the impulses take place and a continuous one corresponding to the geodesic motion in-between impulses. The subsequent distributions are solutions of a (covariant) pseudo-diffusion equation which involves derivatives with respect to both time variables, rather than solutions of the telegraph equation which has a single time variable. This approach simplifies some of the known problems in this context. Full article

(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

251 KiB

Open AccessArticle

On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem

by Soheil Salahshour, Ali Ahmadian, Norazak Senu, Dumitru Baleanu and Praveen Agarwal

Entropy 2015, 17(2), 885-902; https://doi.org/10.3390/e17020885 - 16 Feb 2015

Cited by 146 | Viewed by 7550

Abstract

In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE) with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of [...] Read more.

In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE) with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville) or a solution with increasing length of their support (Hukuhara difference). Then, in order to solve the FFDE analytically, we introduce the fuzzy Laplace transform of the Caputo H-derivative. To the best of our knowledge, there is limited research devoted to the analytical methods to solve the FFDE under the fuzzy Caputo fractional differentiability. An analytical solution is presented to confirm the capability of the proposed method. Full article

(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

Review

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225 KiB

Open AccessReview

Entropic Dynamics

by Ariel Caticha

Entropy 2015, 17(9), 6110-6128; https://doi.org/10.3390/e17096110 - 1 Sep 2015

Cited by 36 | Viewed by 5812

Abstract

Entropic Dynamics is a framework in which dynamical laws are derived as an application of entropic methods of inference. No underlying action principle is postulated. Instead, the dynamics is driven by entropy subject to the constraints appropriate to the problem at hand. In [...] Read more.

Entropic Dynamics is a framework in which dynamical laws are derived as an application of entropic methods of inference. No underlying action principle is postulated. Instead, the dynamics is driven by entropy subject to the constraints appropriate to the problem at hand. In this paper we review three examples of entropic dynamics. First we tackle the simpler case of a standard diffusion process which allows us to address the central issue of the nature of time. Then we show that imposing the additional constraint that the dynamics be non-dissipative leads to Hamiltonian dynamics. Finally, considerations from information geometry naturally lead to the type of Hamiltonian that describes quantum theory. Full article

(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

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