The
existence, uniqueness and continuous dependence of the weak solution of the Black-Scholes model with
transaction cost are established.The continuity of weak solution of the parameters was discussed and
similar solution as in literature obtained.
Entity Linking via Graph-Distance MinimizationRoi Blanco
Entity-linking is a natural-language--processing task that consists in identifying strings of text that refer to a particular
item in some reference knowledge base.
One instance of entity-linking can be formalized as an optimization problem on the underlying concept graph, where the quantity to be optimized is the average distance between chosen items.
Inspired by this application, we define a new graph problem which is a natural variant of the Maximum Capacity Representative Set. We prove that our problem is NP-hard for general graphs; nonetheless, it turns out to be solvable in linear time under some more restrictive assumptions. For the general case, we propose several heuristics: one of these tries to enforce the above assumptions while the others try to optimize similar easier objective functions; we show experimentally how these approaches perform with respect to some baselines on a real-world dataset.
Integration by substitution allows difficult integrals to be evaluated by making a substitution of variables that simplifies the integrand. This technique involves changing both the variable of integration and the limits of integration. Key steps include identifying an appropriate substitution, determining the differential, and performing the integration with respect to the new variable before substituting back to the original. Extensive practice is important to master integration by substitution.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document discusses equivariant estimation, which involves finding estimators that are invariant or equivariant under transformations of the data. Specifically:
- A statistical model is invariant under a transformation if applying the transformation to the data results in a model with the same form. This induces a transformation on the parameter space.
- For a model to be invariant, the loss function should also be invariant under a corresponding transformation of the decisions. This induces a transformation on the decision space.
- A decision problem is invariant under a transformation if the model and loss are invariant under the induced transformations on the parameter and decision spaces.
- Often a problem will be invariant under not just one transformation but a group of related transformations
The document discusses linear partial differential equations (PDEs) with constant coefficients. It defines such PDEs and provides examples. It describes how to find the general solution of homogeneous linear PDEs with constant coefficients by finding the roots of the auxiliary equation. The general solution consists of the complementary function plus a particular integral. Methods for finding the particular integral when the right side consists of powers of x and y are also presented.
Let f(z) be a function continuous at a point z0.
To show that f(z) is also continuous at z0, we need to show:
limz→z0 f(z) = f(z0)
Since f(z) is given to be continuous at z0, by the definition of continuity:
limz→z0 f(z) = f(z0)
Therefore, if f(z) is continuous at a point z0, it automatically satisfies the condition for continuity at z0. Hence, f(z) is also continuous at z0.
So in summary, if a function f(z) is continuous at a
This document summarizes Yitang Zhang's proof that the difference between consecutive prime numbers is bounded above by a constant. Zhang improves on prior work by Goldston, Pintz, and Yildirim by showing this difference is less than 7×10^7. The key ideas are to consider a stronger version of the Bombieri-Vinogradov theorem and to refine the choice of a weighting function λ(n) in evaluating sums related to prime numbers. Zhang is able to efficiently bound error terms arising in this evaluation by exploiting the factorization of integers relatively free of large prime factors. This allows him to show the necessary inequalities hold and thus deduce his main result on bounded gaps between primes.
Entity Linking via Graph-Distance MinimizationRoi Blanco
Entity-linking is a natural-language--processing task that consists in identifying strings of text that refer to a particular
item in some reference knowledge base.
One instance of entity-linking can be formalized as an optimization problem on the underlying concept graph, where the quantity to be optimized is the average distance between chosen items.
Inspired by this application, we define a new graph problem which is a natural variant of the Maximum Capacity Representative Set. We prove that our problem is NP-hard for general graphs; nonetheless, it turns out to be solvable in linear time under some more restrictive assumptions. For the general case, we propose several heuristics: one of these tries to enforce the above assumptions while the others try to optimize similar easier objective functions; we show experimentally how these approaches perform with respect to some baselines on a real-world dataset.
Integration by substitution allows difficult integrals to be evaluated by making a substitution of variables that simplifies the integrand. This technique involves changing both the variable of integration and the limits of integration. Key steps include identifying an appropriate substitution, determining the differential, and performing the integration with respect to the new variable before substituting back to the original. Extensive practice is important to master integration by substitution.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document discusses equivariant estimation, which involves finding estimators that are invariant or equivariant under transformations of the data. Specifically:
- A statistical model is invariant under a transformation if applying the transformation to the data results in a model with the same form. This induces a transformation on the parameter space.
- For a model to be invariant, the loss function should also be invariant under a corresponding transformation of the decisions. This induces a transformation on the decision space.
- A decision problem is invariant under a transformation if the model and loss are invariant under the induced transformations on the parameter and decision spaces.
- Often a problem will be invariant under not just one transformation but a group of related transformations
The document discusses linear partial differential equations (PDEs) with constant coefficients. It defines such PDEs and provides examples. It describes how to find the general solution of homogeneous linear PDEs with constant coefficients by finding the roots of the auxiliary equation. The general solution consists of the complementary function plus a particular integral. Methods for finding the particular integral when the right side consists of powers of x and y are also presented.
Let f(z) be a function continuous at a point z0.
To show that f(z) is also continuous at z0, we need to show:
limz→z0 f(z) = f(z0)
Since f(z) is given to be continuous at z0, by the definition of continuity:
limz→z0 f(z) = f(z0)
Therefore, if f(z) is continuous at a point z0, it automatically satisfies the condition for continuity at z0. Hence, f(z) is also continuous at z0.
So in summary, if a function f(z) is continuous at a
This document summarizes Yitang Zhang's proof that the difference between consecutive prime numbers is bounded above by a constant. Zhang improves on prior work by Goldston, Pintz, and Yildirim by showing this difference is less than 7×10^7. The key ideas are to consider a stronger version of the Bombieri-Vinogradov theorem and to refine the choice of a weighting function λ(n) in evaluating sums related to prime numbers. Zhang is able to efficiently bound error terms arising in this evaluation by exploiting the factorization of integers relatively free of large prime factors. This allows him to show the necessary inequalities hold and thus deduce his main result on bounded gaps between primes.
The document discusses partial differential equations (PDEs) and numerical methods for solving them. It begins by defining PDEs as equations involving derivatives of an unknown function with respect to two or more independent variables. PDEs describe many physical phenomena involving variations across space and time, such as fluid flow, heat transfer, electromagnetism, and weather prediction. The document then focuses on solving elliptic, parabolic, and hyperbolic PDEs numerically using finite difference and finite element methods. It provides examples of discretizing and solving the Laplace, heat, and wave equations to estimate unknown functions.
Fuzzy Logic And Application Jntu Model Paper{Www.Studentyogi.Com}guest3f9c6b
This document contains an exam for a course on Fuzzy Logic and Applications. It includes 8 questions covering topics such as operations on crisp and fuzzy sets using Venn diagrams, fuzzy relations, membership functions, fuzzy logic connectives, defuzzification methods, and decision making under fuzzy conditions. Students are instructed to answer any 5 of the 8 questions.
(DL hacks輪読) Variational Inference with Rényi DivergenceMasahiro Suzuki
This document discusses variational inference with Rényi divergence. It summarizes variational autoencoders (VAEs), which are deep generative models that parametrize a variational approximation with a recognition network. VAEs define a generative model as a hierarchical latent variable model and approximate the intractable true posterior using variational inference. The document explores using Rényi divergence as an alternative to the evidence lower bound objective of VAEs, as it may provide tighter variational bounds.
Murphy: Machine learning A probabilistic perspective: Ch.9Daisuke Yoneoka
This document summarizes key concepts about the exponential family and generalized linear models (GLMs). It defines the exponential family and provides examples like the Bernoulli, multinomial, and Gaussian distributions. The exponential family has important properties like finite sufficient statistics, existence of conjugate priors, and convexity. Maximum likelihood estimation for the exponential family involves matching sample moments to population moments. Conjugate priors allow tractable Bayesian inference for the exponential family. The document outlines maximum entropy derivation of the exponential family and how GLMs can generate classifiers.
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSTahia ZERIZER
In this article, we study boundary value problems of a large
class of non-linear discrete systems at two-time-scales. Algorithms are given to implement asymptotic solutions for any order of approximation.
18 Machine Learning Radial Basis Function Networks Forward HeuristicsAndres Mendez-Vazquez
This document discusses radial basis function networks and forward selection heuristics for neural networks. It begins by outlining topics to be covered, including predicting the variance of weights and outputs, selecting the regularization parameter, and forward selection algorithms. It then derives an expression for the variance of the weight vector w when noise is assumed to be normally distributed. Next, it discusses how to calculate the variance matrix and selects the regularization parameter λ. Finally, it introduces how to determine the number of dimensions and provides an overview of forward selection algorithms.
This document outlines the plan for a seminar on kernel methods. The seminar will cover four main topics: kernel methods, random projections, deep learning, and more about kernels. The overall goal is to provide enough theoretical background to understand several related papers on convolutional kernel networks, fastfood kernel approximations, and deep fried convolutional neural networks. The document includes an agenda, definitions, theorems, and references to support understanding kernel methods and their applications.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
In this paper we introduce the notions of Fuzzy Ideals in BH-algebras and the notion
of fuzzy dot Ideals of BH-algebras and investigate some of their results.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit. It then discusses estimating integrals using the midpoint rule and properties of integrals such as integrals of nonnegative functions being nonnegative and integrals being "increasing" if one function is greater than another. An example is worked out using the midpoint rule to estimate an integral. The document provides an outline of topics and notation for integrals.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
This document discusses fuzzy relations, reasoning, and linguistic variables. It defines fuzzy relations as membership functions between elements of Cartesian product spaces. It describes the extension principle for mapping fuzzy sets through functions. Max-min and max-product composition are defined for combining fuzzy relations. Linguistic variables allow information to be expressed using fuzzy linguistic terms rather than numerical values. Operations on linguistic variables like concentration and dilation are discussed. Fuzzy if-then rules are defined using implication functions to model "if A then B" statements where A and B are linguistic values. Fuzzy reasoning uses these rules and facts to derive conclusions.
Optimal Prediction of the Expected Value of Assets Under Fractal Scaling Expo...mathsjournal
In this paper, the optimal prediction of the expected value of assets under the fractal scaling exponent is considered. We first obtain a fractal exponent, then derive a seemingly Black-Scholes parabolic equation. We further obtain its solutions under given conditions for the prediction of expected value of assets given the fractal exponent.
This document presents a numerical solution and comparison of linear Black-Scholes models using finite difference and finite element methods. It begins with an introduction to the Black-Scholes partial differential equation and previous analytical and numerical solutions in the literature. The document then transforms the Black-Scholes equation into a heat equation and presents the finite element formulation and discretization. Numerical results are obtained for the European call and put options and compared between finite difference and finite element methods.
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
- The document presents a probabilistic algorithm for computing the polynomial greatest common divisor (PGCD) with smaller factors.
- It summarizes previous work on the subresultant algorithm for computing PGCD and discusses its limitations, such as not always correctly determining the variant τ.
- The new algorithm aims to determine τ correctly in most cases when given two polynomials f(x) and g(x). It does so by adding a few steps instead of directly computing the polynomial t(x) in the relation s(x)f(x) + t(x)g(x) = r(x).
This document summarizes the relationship between the Unique Games Conjecture (UGC) and the Small Set Expansion Hypothesis (SSEH). It discusses how unique games instances can be mapped to small set expansion instances, though this mapping is not a valid reduction. The SSEH implies the UGC, and both problems have similar known algorithmic results. Small set expansion can be viewed as finding vectors that are "analytically sparse" by optimizing quadratic forms subject to structural constraints. Norm ratios like the l1/l2 ratio are good proxies for sparsity, and the document proves theorems relating the lp/l2 ratio to graph expansion.
This document describes an uncertain volatility model for pricing equity option trading strategies when the volatilities are uncertain. It uses the Black-Scholes Barenblatt equation developed by Avellaneda et al. to derive price bounds. The model is implemented in C++ using recombining trinomial trees to discretize the asset prices over time and space. The code computes the upper and lower price bounds by solving the Black-Scholes Barenblatt PDE using numerical techniques, with the volatility set based on the sign of the option gamma.
sublabel accurate convex relaxation of vectorial multilabel energiesFujimoto Keisuke
This document summarizes a presentation on the paper "Sublabel-Accurate Convex Relaxation of Vectorial Multilabel Energies". It discusses how the paper proposes a method to efficiently solve high-dimensional, nonlinear vectorial labeling problems by approximating them as convex problems. Specifically, it divides the problem domain into subregions and approximates each subregion with a convex function, yielding an overall approximation that is still non-convex but with higher accuracy. This lifting technique transforms the variables into a higher-dimensional space to formulate the data and regularization terms in a way that allows solving the problem as a convex optimization.
This document introduces simulation using MATLAB. It discusses how to generate random numbers from both discrete and continuous probability distributions using MATLAB commands or by converting uniformly distributed random numbers. For discrete distributions, the document shows how to generate random variables from the Bernoulli, binomial, Poisson, and geometric distributions. For continuous distributions, it explains the inverse transform method to generate random variables from exponential, gamma, normal, and other distributions by transforming uniformly distributed random numbers. The document also provides examples of MATLAB code to simulate coin tosses, generate random numbers from various distributions, and apply the Box-Muller transform to generate normal random variables. It concludes by reviewing useful MATLAB commands for commonly used discrete and continuous probability distributions.
The document discusses partial differential equations (PDEs) and numerical methods for solving them. It begins by defining PDEs as equations involving derivatives of an unknown function with respect to two or more independent variables. PDEs describe many physical phenomena involving variations across space and time, such as fluid flow, heat transfer, electromagnetism, and weather prediction. The document then focuses on solving elliptic, parabolic, and hyperbolic PDEs numerically using finite difference and finite element methods. It provides examples of discretizing and solving the Laplace, heat, and wave equations to estimate unknown functions.
Fuzzy Logic And Application Jntu Model Paper{Www.Studentyogi.Com}guest3f9c6b
This document contains an exam for a course on Fuzzy Logic and Applications. It includes 8 questions covering topics such as operations on crisp and fuzzy sets using Venn diagrams, fuzzy relations, membership functions, fuzzy logic connectives, defuzzification methods, and decision making under fuzzy conditions. Students are instructed to answer any 5 of the 8 questions.
(DL hacks輪読) Variational Inference with Rényi DivergenceMasahiro Suzuki
This document discusses variational inference with Rényi divergence. It summarizes variational autoencoders (VAEs), which are deep generative models that parametrize a variational approximation with a recognition network. VAEs define a generative model as a hierarchical latent variable model and approximate the intractable true posterior using variational inference. The document explores using Rényi divergence as an alternative to the evidence lower bound objective of VAEs, as it may provide tighter variational bounds.
Murphy: Machine learning A probabilistic perspective: Ch.9Daisuke Yoneoka
This document summarizes key concepts about the exponential family and generalized linear models (GLMs). It defines the exponential family and provides examples like the Bernoulli, multinomial, and Gaussian distributions. The exponential family has important properties like finite sufficient statistics, existence of conjugate priors, and convexity. Maximum likelihood estimation for the exponential family involves matching sample moments to population moments. Conjugate priors allow tractable Bayesian inference for the exponential family. The document outlines maximum entropy derivation of the exponential family and how GLMs can generate classifiers.
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSTahia ZERIZER
In this article, we study boundary value problems of a large
class of non-linear discrete systems at two-time-scales. Algorithms are given to implement asymptotic solutions for any order of approximation.
18 Machine Learning Radial Basis Function Networks Forward HeuristicsAndres Mendez-Vazquez
This document discusses radial basis function networks and forward selection heuristics for neural networks. It begins by outlining topics to be covered, including predicting the variance of weights and outputs, selecting the regularization parameter, and forward selection algorithms. It then derives an expression for the variance of the weight vector w when noise is assumed to be normally distributed. Next, it discusses how to calculate the variance matrix and selects the regularization parameter λ. Finally, it introduces how to determine the number of dimensions and provides an overview of forward selection algorithms.
This document outlines the plan for a seminar on kernel methods. The seminar will cover four main topics: kernel methods, random projections, deep learning, and more about kernels. The overall goal is to provide enough theoretical background to understand several related papers on convolutional kernel networks, fastfood kernel approximations, and deep fried convolutional neural networks. The document includes an agenda, definitions, theorems, and references to support understanding kernel methods and their applications.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
In this paper we introduce the notions of Fuzzy Ideals in BH-algebras and the notion
of fuzzy dot Ideals of BH-algebras and investigate some of their results.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit. It then discusses estimating integrals using the midpoint rule and properties of integrals such as integrals of nonnegative functions being nonnegative and integrals being "increasing" if one function is greater than another. An example is worked out using the midpoint rule to estimate an integral. The document provides an outline of topics and notation for integrals.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
This document discusses fuzzy relations, reasoning, and linguistic variables. It defines fuzzy relations as membership functions between elements of Cartesian product spaces. It describes the extension principle for mapping fuzzy sets through functions. Max-min and max-product composition are defined for combining fuzzy relations. Linguistic variables allow information to be expressed using fuzzy linguistic terms rather than numerical values. Operations on linguistic variables like concentration and dilation are discussed. Fuzzy if-then rules are defined using implication functions to model "if A then B" statements where A and B are linguistic values. Fuzzy reasoning uses these rules and facts to derive conclusions.
Optimal Prediction of the Expected Value of Assets Under Fractal Scaling Expo...mathsjournal
In this paper, the optimal prediction of the expected value of assets under the fractal scaling exponent is considered. We first obtain a fractal exponent, then derive a seemingly Black-Scholes parabolic equation. We further obtain its solutions under given conditions for the prediction of expected value of assets given the fractal exponent.
This document presents a numerical solution and comparison of linear Black-Scholes models using finite difference and finite element methods. It begins with an introduction to the Black-Scholes partial differential equation and previous analytical and numerical solutions in the literature. The document then transforms the Black-Scholes equation into a heat equation and presents the finite element formulation and discretization. Numerical results are obtained for the European call and put options and compared between finite difference and finite element methods.
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
- The document presents a probabilistic algorithm for computing the polynomial greatest common divisor (PGCD) with smaller factors.
- It summarizes previous work on the subresultant algorithm for computing PGCD and discusses its limitations, such as not always correctly determining the variant τ.
- The new algorithm aims to determine τ correctly in most cases when given two polynomials f(x) and g(x). It does so by adding a few steps instead of directly computing the polynomial t(x) in the relation s(x)f(x) + t(x)g(x) = r(x).
This document summarizes the relationship between the Unique Games Conjecture (UGC) and the Small Set Expansion Hypothesis (SSEH). It discusses how unique games instances can be mapped to small set expansion instances, though this mapping is not a valid reduction. The SSEH implies the UGC, and both problems have similar known algorithmic results. Small set expansion can be viewed as finding vectors that are "analytically sparse" by optimizing quadratic forms subject to structural constraints. Norm ratios like the l1/l2 ratio are good proxies for sparsity, and the document proves theorems relating the lp/l2 ratio to graph expansion.
This document describes an uncertain volatility model for pricing equity option trading strategies when the volatilities are uncertain. It uses the Black-Scholes Barenblatt equation developed by Avellaneda et al. to derive price bounds. The model is implemented in C++ using recombining trinomial trees to discretize the asset prices over time and space. The code computes the upper and lower price bounds by solving the Black-Scholes Barenblatt PDE using numerical techniques, with the volatility set based on the sign of the option gamma.
sublabel accurate convex relaxation of vectorial multilabel energiesFujimoto Keisuke
This document summarizes a presentation on the paper "Sublabel-Accurate Convex Relaxation of Vectorial Multilabel Energies". It discusses how the paper proposes a method to efficiently solve high-dimensional, nonlinear vectorial labeling problems by approximating them as convex problems. Specifically, it divides the problem domain into subregions and approximates each subregion with a convex function, yielding an overall approximation that is still non-convex but with higher accuracy. This lifting technique transforms the variables into a higher-dimensional space to formulate the data and regularization terms in a way that allows solving the problem as a convex optimization.
This document introduces simulation using MATLAB. It discusses how to generate random numbers from both discrete and continuous probability distributions using MATLAB commands or by converting uniformly distributed random numbers. For discrete distributions, the document shows how to generate random variables from the Bernoulli, binomial, Poisson, and geometric distributions. For continuous distributions, it explains the inverse transform method to generate random variables from exponential, gamma, normal, and other distributions by transforming uniformly distributed random numbers. The document also provides examples of MATLAB code to simulate coin tosses, generate random numbers from various distributions, and apply the Box-Muller transform to generate normal random variables. It concludes by reviewing useful MATLAB commands for commonly used discrete and continuous probability distributions.
This document discusses partial ordering in the context of soft sets. It begins with basic definitions of soft sets and soft set operations like complement, Cartesian product, and composition of soft set relations. It then defines what a partial order is in terms of being reflexive, antisymmetric, and transitive. A partially ordered soft set is one where the soft set elements have a partial order defined on them. Linear (total) ordering is also discussed, where all elements in the soft set are comparable. Examples are provided to illustrate these concepts of ordering in soft sets.
This document discusses methods for solving algebraic and transcendental equations. It begins by defining key terms like roots, simple roots, and multiple roots. It then distinguishes between direct and iterative methods. Direct methods provide exact solutions, while iterative methods use successive approximations that converge to the exact root. The document focuses on iterative methods and describes how to obtain initial approximations, including using Descartes' rule of signs and the intermediate value theorem. It also discusses criteria for terminating iterations. One iterative method described in detail is the method of false position, which approximates the curve defined by the equation as a straight line between two points.
This document discusses methods for solving algebraic and transcendental equations. It begins by defining key terms like roots, simple roots, and multiple roots. It then distinguishes between direct and iterative methods. Direct methods provide exact solutions, while iterative methods use successive approximations that converge to the exact root. The document focuses on iterative methods and describes how to obtain initial approximations, including using Descartes' rule of signs and the intermediate value theorem. It also discusses criteria for terminating iterations. One iterative method described in detail is the method of false position, which approximates the curve defined by the equation as a straight line between two points.
Notes on intersection theory written for a seminar in Bonn in 2010.
Following Fulton's book the following topics are covered:
- Motivation of intersection theory
- Cones and Segre Classes
- Chern Classes
- Gauss-Bonet Formula
- Segre classes under birational morphisms
- Flat pull back
Fixed point result in probabilistic metric spaceAlexander Decker
This document presents a fixed point theorem for four self-mappings in a probabilistic metric space. It begins with definitions related to probabilistic metric spaces, including Menger spaces. It then proves a common fixed point theorem for four self-mappings defined on a complete Menger space, where the mappings satisfy certain conditions. Specifically, the mappings have nested range properties and some pairs are weakly or semi-compatible. Additionally, there is a contraction condition involving a rational function. The theorem guarantees the existence and uniqueness of a common fixed point for the four mappings. An example is also provided to support the theorem.
This document summarizes solutions to odd-numbered homework problems from Chapter 4 of a statistics textbook. It covers topics like discrete vs. continuous random variables, probability distributions, the normal and binomial distributions, and how to calculate probabilities using the z-table. Examples include determining the type of random variable, finding probabilities of intervals for different distributions, and approximating binomial probabilities with the normal distribution for large n.
X01 Supervised learning problem linear regression one feature theorieMarco Moldenhauer
1. The document describes supervised learning problems, specifically linear regression with one feature. It defines key concepts like the hypothesis function, cost function, and gradient descent algorithm.
2. A data set with one input feature and one output is defined. The goal is to learn a linear function that maps the input to the output to best fit the training data.
3. The hypothesis function is defined as h(x) = θ0 + θ1x, where θ0 and θ1 are parameters to be estimated. Gradient descent is used to minimize the cost function and find the optimal θ values.
The document discusses membrane harmonics and the Helmholtz equation. It begins by considering the one-dimensional Helmholtz equation on an interval, finding the eigenfunctions and eigenvalues. It then extends this to the two-dimensional case on a rectangle using separation of variables, obtaining eigenfunctions that are products of sine waves and eigenvalues that are sums of the one-dimensional eigenvalues.
1. The document discusses linearizing differential equation models to approximate their behavior near equilibrium points. Linearization involves ignoring higher-order terms to obtain a linear system that is solvable.
2. It provides two methods for linearizing a model - a non-calculus method involving rewriting the equations in terms of differences from the equilibrium, and a calculus-based Taylor expansion method. Both result in a system of linear differential equations.
3. Solutions to the linearized system can be found by computing eigenvalues and eigenvectors of the Jacobian matrix. The eigenvalues indicate stability - negative eigenvalues correspond to stable equilibria while positive eigenvalues correspond to unstable equilibria.
- The document outlines a BSc research project on pricing financial derivatives using the Black-Scholes model.
- The project aims to learn established financial models, compare pricing techniques, and see how newer models relate to existing ones.
- It provides background on the student's motivation and experience, and introduces key concepts like options, the Black-Scholes equation, and its derivation and solution.
- The student will present their work on applying and extending the Black-Scholes model to price derivatives.
Perspectives and application of fuzzy initial value problemsRAJKRISHNA MONDAL
This document discusses fuzzy differential equations and fuzzy initial value problems. It introduces fuzzy sets and fuzzy differential equations as a way to model dynamical systems involving uncertainties. It then examines three different fuzzy initial value problems and their solutions. The solutions exhibit very different behaviors despite being fuzzy representations of equivalent crisp differential equations. This shows that different fuzzy representations of the same crisp problem can lead to different outcomes.
Perspectives and application of fuzzy initial value problemsRAJKRISHNA MONDAL
This document discusses fuzzy differential equations and fuzzy initial value problems. It introduces fuzzy sets and fuzzy differential equations as extensions of classical set theory and differential equations to account for uncertainty. Several examples of fuzzy initial value problems are analyzed, comparing their behaviors under different types of differentiability. The solutions exhibit very different properties, even though the original crisp equations were equivalent, showing that different fuzzy representations can model the same real-world problem very differently.
Similar to THE WEAK SOLUTION OF BLACK-SCHOLE’S OPTION PRICING MODEL WITH TRANSACTION COST (20)
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively
A Positive Integer 𝑵 Such That 𝒑𝒏 + 𝒑𝒏+𝟑 ~ 𝒑𝒏+𝟏 + 𝒑𝒏+𝟐 For All 𝒏 ≥ 𝑵mathsjournal
According to Bertrand's postulate, we have 𝑝𝑛 + 𝑝𝑛 ≥ 𝑝𝑛+1. Is it true that for all 𝑛 > 1 then 𝑝𝑛−1 + 𝑝𝑛 ≥𝑝𝑛+1? Then 𝑝𝑛 + 𝑝𝑛+3 > 𝑝𝑛+1 + 𝑝𝑛+2where 𝑛 ≥ 𝑁, 𝑁 is a large enough value?
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Moving Target Detection Using CA, SO and GO-CFAR detectors in Nonhomogeneous ...mathsjournal
systems in complex situations. A fundamental problem in radar systems is to automatically detect targets while maintaining a
desired constant false alarm probability. This work studies two detection approaches, the first with a fixed threshold and the
other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
intelligent techniques. With these objectives in mind, implementing standard CFAR detectors is applied to nonhomogeneous
environment data. The primary focus is understanding the reason for the false detection when applying standard CFAR
detectors in a nonhomogeneous environment and how to avoid it using intelligent approaches.
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
The Impact of Allee Effect on a Predator-Prey Model with Holling Type II Func...mathsjournal
There is currently much interest in predator–prey models across a variety of bioscientific disciplines. The focus is on quantifying predator–prey interactions, and this quantification is being formulated especially as regards climate change. In this article, a stability analysis is used to analyse the behaviour of a general two-species model with respect to the Allee effect (on the growth rate and nutrient limitation level of the prey population). We present a description of the local and non-local interaction stability of the model and detail the types of bifurcation which arise, proving that there is a Hopf bifurcation in the Allee effect module. A stable periodic oscillation was encountered which was due to the Allee effect on the
prey species. As a result of this, the positive equilibrium of the model could change from stable to unstable and then back to stable, as the strength of the Allee effect (or the ‘handling’ time taken by predators when predating) increased continuously from zero. Hopf bifurcation has arose yield some complex patterns that have not been observed previously in predator-prey models, and these, at the same time, reflect long term behaviours. These findings have significant implications for ecological studies, not least with respect to examining the mobility of the two species involved in the non-local domain using Turing instability. A spiral generated by local interaction (reflecting the instability that forms even when an infinitely large
carrying capacity is assumed) is used in the model.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Moving Target Detection Using CA, SO and GO-CFAR detectors in Nonhomogeneous ...mathsjournal
Modernization of radar technology and improved signal processing techniques are necessary to improve detection systems in complex situations. A fundamental problem in radar systems is to automatically detect targets while maintaining a
desired constant false alarm probability. This work studies two detection approaches, the first with a fixed threshold and the
other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
intelligent techniques. With these objectives in mind, implementing standard CFAR detectors is applied to nonhomogeneous
environment data. The primary focus is understanding the reason for the false detection when applying standard CFAR
detectors in a nonhomogeneous environment and how to avoid it using intelligent approaches
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints
Modified Alpha-Rooting Color Image Enhancement Method on the Two Side 2-D Qua...mathsjournal
Color in an image is resolved to 3 or 4 color components and 2-Dimages of these components are stored in separate channels. Most of the color image enhancement algorithms are applied channel-by-channel on each image. But such a system of color image processing is not processing the original color. When a color image is represented as a quaternion image, processing is done in original colors. This paper proposes an implementation of the quaternion approach of enhancement algorithm for enhancing color images and is referred as the modified alpha-rooting by the two-dimensional quaternion discrete Fourier transform (2-D QDFT). Enhancement results of this proposed method are compared with the channel-by-channel image enhancement by the 2-D DFT. Enhancements in color images are quantitatively measured by the color enhancement measure estimation (CEME), which allows for selecting optimum parameters for processing by thegenetic algorithm. Enhancement of color images by the quaternion based method allows for obtaining images which are closer to the genuine representation of the real original color.
An Application of Assignment Problem in Laptop Selection Problem Using MATLABmathsjournal
The assignment – selection problem used to find one-to- one match of given “Users” to “Laptops”, the main objective is to minimize the cost as per user requirement. This paper presents satisfactory solution for real assignment – Laptop selection problem using MATLAB coding.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
Cubic Response Surface Designs Using Bibd in Four Dimensionsmathsjournal
Response Surface Methodology (RSM) has applications in Chemical, Physical, Meteorological, Industrial and Biological fields. The estimation of slope response surface occurs frequently in practical situations for the experimenter. The rates of change of the response surface, like rates of change in the yield of crop to various fertilizers, to estimate the rates of change in chemical experiments etc. are of
interest. If the fit of second order response is inadequate for the design points, we continue the
experiment so as to fit a third order response surface. Higher order response surface designs are sometimes needed in Industrial and Meteorological applications. Gardiner et al (1959) introduced third order rotatable designs for exploring response surface. Anjaneyulu et al (1994-1995) constructed third order slope rotatable designs using doubly balanced incomplete block designs. Anjaneyulu et al (2001)
introduced third order slope rotatable designs using central composite type design points. Seshu babu et al (2011) studied modified construction of third order slope rotatable designs using central composite
designs. Seshu babu et al (2014) constructed TOSRD using BIBD. In view of wide applicability of third
order models in RSM and importance of slope rotatability, we introduce A Cubic Slope Rotatable Designs Using BIBD in four dimensions.
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if quantum variations are allowed. An action is developed such that the variation yields the field equations
and the geodesic condition, and its quantization provides a method for determining the extent of the wave packet around the classical path.
Approximate Analytical Solution of Non-Linear Boussinesq Equation for the Uns...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.
Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalize...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings
under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy
metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
Table of Contents - September 2022, Volume 9, Number 2/3mathsjournal
Applied Mathematics and Sciences: An International Journal (MathSJ ) aims to publish original research papers and survey articles on all areas of pure mathematics, theoretical applied mathematics, mathematical physics, theoretical mechanics, probability and mathematical statistics, and theoretical biology. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics.
Code of the multidimensional fractional pseudo-Newton method using recursive ...mathsjournal
The following paper presents one way to define and classify the fractional pseudo-Newton method through a group of fractional matrix operators, as well as a code written in recursive programming to implement this
method, which through minor modifications, can be implemented in any fractional fixed-point method that allows
solving nonlinear algebraic equation systems.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
THE WEAK SOLUTION OF BLACK-SCHOLE’S OPTION PRICING MODEL WITH TRANSACTION COST
1. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
43
THE WEAK SOLUTION OF BLACK-SCHOLE’S OPTION
PRICING MODEL WITH TRANSACTION COST
Bright O. Osu and Chidinma Olunkwa
Department of Mathematics, Abia State University, Uturu, Nigeria
ABSTRACT
This paper considers the equation of the type
− + + = , ( , ) ∈ ℝ × (0, );
which is the Black-Scholes option pricing model that includes the presence of transaction cost. The
existence, uniqueness and continuous dependence of the weak solution of the Black-Scholes model with
transaction cost are established.The continuity of weak solution of the parameters was discussed and
similar solution as in literature obtained.
KEYWORDS
Black-Scholes Model, Option pricing, Transaction costs, Weak solution,Sobolev space
1. INTRODUCTION
Options are financial instrument that convey the right but not the obligation to engage in a
future transaction on the underlying assets.In a complete financial market without transaction
costs, the celebrated Black-Sholes no-arbitrage argument provide not only a rational option
pricing formula but also a hedging portfolio that replicate the contingent claim [8] .However the
Black-Scholes hedging portfoliorequires trading at all-time instants and the total turnover of
stock in the time interval [0, ] is infinite. Accordingly, when transaction cost directly
proportional to trading is incorporated in the Black –Scholes model the resulting hedging
portfolio is quite expensive. The condition under which hedging can take place has to be relaxed
such that the portfolio only dominate rather than replicate the value of the European call option
at maturity .The first model in that direction was presented in [4] . Here it was assumed that the
portfolio is rebalanced at a discrete time and that the transaction cost in buying and selling the
asset are proportional to the monetary value of the transaction. At a price S and a constant K
depending on an individual’s aversion to risk, the transaction costs are ∕ ∕ ∕, where N is
the number of shares bought ( > 0) or sold ( < 0). In [7], the existence, uniqueness and
continuity of the Black –Scholes model was discussed. Also in [6], option pricing with
transaction costs that leads to a nonlinear equation was investigated.In a related paper [1],the
discretetime, dominating policies was presented. In [3] further work on this in the presence of
transaction cost was presented...By applying the theorem of central limit, they show that as the
time step ∆ and transaction cost ∅ tend to zero. The price of discrete option converged to a
Black –Sholes price with adjusted volatility (. ). Here ∆ represent the mean time length for a
change in the value of the stock instead of transaction frequency. Here our adjusted volatility is
given by;
= (1 − ( ). (1.1)
2. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
44
If
= ( ), = −
∅
√
then we have the adjusted volatility as in[7], where is the time lag between consecutive
transaction.
Let ( , ) be the value of the option and be the value of the hedge portfolio. We
assumeinstead that the value of the underlying follows the random work
= + ∅
with∅ drowned from a normal distribution, is a measure of the average rate of growth of the
asset price also known as the drift where = − and is a measure of the diffusion
coefficient . Then the change in the value of the portfolio over the time step is given by
(using (1.1))
∆ = − ∆ ∅ + ∅ (1 − ( )) + ( − ) + − ∆ − / / .
Making a digression and investigating the nature of the number of assets bought or sold given
that we posit the number of asset (the delta of our option) as = .
Conventionally, thedelta of an option is represented by ∆ . Given that is evaluated at the asset
value s and time ,we have = ( , ).
Rehedging after finite time ∆ leads to a change in the value of assets as below
= ( + ∆ , + ∆ ).
This of course evaluated at the new asset price and time .therefore the number of assets to be
traded after ∆ is given by
= ( + ∆ , + ∆ ) − ( , ) ≈ ∅ .
Hence the expected transaction cost over a time step is
[ / / ] = [| |] = ∅
= [|∅|] ,
Where
|∅| = .
The expected change in the value of the portfolio is
3. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
45
(∆ ) = ( + ( , , ) − )
If the holder of the option expects to make as much from his portfolio as from a bank account at
a riskless interest rate (no arbitrage), then
(∆ ) = − .
Hence following[9] for option pricing with transaction costs is given by
+ ( , , ) + − − = 0, ( , ) ∈ (0, ∞) × (0, ), (1.2)
and the final condition
( , ) = max( − , 0) , ∈ (0, ∞),
for European call option with strike price E.
Note that equation (1.2) contents the usual Black-Scholes terms with additional nonlinear term
modeling the presence of transaction costs. Setting
= log , = −
1
2
, = ( , ),
equation (1.2) becomes
− + + ( − 1) − = , ( , ) ∈ (0, ∗), (1.3)
with initial condition
( , 0) = max( − 1,0) , ∈ ℝ,
Where
= ( 2⁄ )⁄ , = 8 ( )⁄ ∗
= 2⁄ .
Set
= ( , ) = ( , ).
Then (1.3)gives
− + + ( + 1) = + , ( , ) ∈ ℝ × (0, ), (1.4)
With the initial condition
( , 0) = (1 − , 0)
Let
= + 1
4. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
46
The previous discussion motivates us to consider the following problem that includes cost
structures that go beyond proportional transaction costs.
− + + = , ( , ) ∈ ℝ × (0, ), (1.5)
and
( , 0) = ( ), ∈ ℝ. (1.6)
In this paper we looked into parameters that are governing the Black-Scholes option pricing
model with the present of transaction costs such that equation (1.5) exhibits the desired
behaviour. More precisely, let
= = , ∈ [ , ] × , ,
where
> 0 > 0.
Defined a functional ( ) by
( ) = ‖ ( , ) − ‖ ( , ; ), (1.7)
where the dalta can be thought of as the desired value of ( ; ). The parameter identification
problem for (1.5) with the objective function (1.7) is to find
∗
= ∗
, ∗
∈
Satisfying
( ∗) = ∈ ( ). (1.8)
Let
→ ( )
from, in to ([0, ]; be the solution map . In what follows, the existence and uniqueness of
the weak solution of (1.5) is established in the next section. Continuity of the solution with
respect to data is established in section 3.
2. EXISTENCE AND UNIQUENESS OF WEAK SOLUTION
Since the type of equation in (1.5) do not belong to (ℛ) we introduce weighted lebessgue and
sobolev spaces
(ℛ)and (ℛ) for > 0
as follows.
(ℛ) = ∈ (ℛ): | |
∈ (ℛ) (2.1)
H (ℛ) = ∈ (ℛ): | |
∈ (ℛ), | |
∈ (ℛ) .(2.2)
5. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
47
The respective inner products and norms are defined by
( , ) (ℛ) = ∫ℛ
| |
(2.3)
( , ) (ℛ) = ∫ℛ
| |
+ ∫ℛ
| |
(2.4)
‖ ‖ (ℛ) = ∫ℛ
| | | |
(2.5)
‖ ‖ (ℛ) = ∫ℛ
| | | |
+ ∫ℛ
| | | |
(2.6)
We define the dual space of H (ℛ) as
(ℛ)
∗
= : (ℛ) → ℛ (2.7)
The duality pairing between (ℛ) and (ℛ)
∗
is given by
〈 , 〉 = ∫ℛ
| | | |
(2.8)
In what follows, we state,
LEMMA1: Let = (ℛ).For ∅ ∈ , ∅ = (−1,1), ∫ ∅ ( ) = 1, ∅ =ℛ
∅ ,
then
∅ ∗ → (ℛ). (2.9)
PROOF: Suppose = | |
,
then we have
(∅ ∗ ). = ∅ ∗ ( . ) + (∅ ∗ ). − ∅ ∗ ( . ) . (2.10)
since
. ∈ ∅ ∗ ( . )
it suffices to show that
‖ ‖ = (‖(∅ ∗ ). − ∅ ∗ ( . )‖) → 0 → 0 (2.11)
The fundamental theory of calculus for give
( ) = ∫ℛ∅ ( − ) ( )( ( ) − ( )) (2.12)
using
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48
∅ = (− , )
We get
| ( )| ≤ ∫ℛ
|∅ ( − )|| ( )|(2 | ( )|) = ∫ℛ
|∅ ( − )|| ( )|(2 | ( + )|) = ( )(2.13)
Since
( ) =
uniformly and
| ( )| ≤ 2 | ( )|,
thus
‖ ( )‖ → 0 as → 0
LEMMA 2: (ℛ)the space of test function in ℛ,is densein H (ℛ).
PROOF .Let ∈ (ℛ) Φ ∈
such that
( ) =
1, | | ≤ 1
0, | | ≥ 2
Now we show that
= . (. ) ∗ ∈
where
= , →
in (ℛ).ie
→ ∇ → (ℛ) (2.14)
= . ( (. )) ∗ + . (. ) ∗ (2.15)
It suffices to show
. ( (. )) ∗ → (ℛ) (2.16)
By the lebesgue Dominated convergence Theorem ,we get
. ( (. )) → (ℛ)(2.17)
Hence Lemma 1 concludes the proof.
Since (ℛ) is dense in (ℛ) (ℛ) , the following lemma follows immediately.
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49
LEMMA 3:
(ℛ) ⊂ (ℛ) ⊂ (ℛ)
∗
,
from Gelfand triple.
Note. Since (ℛ) is dense in (ℛ),the definition of 〈. , . 〉 allows us to interprete the operator
as a mapping from →
∗
.
for our simplicity,we use
= (ℛ), ∗
= (ℛ)
∗
and = (ℛ)
To use the variational formulation let us defined the following bilinear form on ×
( , )( , ) = ∫ℛ
| |
+ ∫ℛ
| |
− − ∫ℛ
| |
(2.18)
for
> 0 > 0 ,
One can show ( , )( , ) is bounded and coercive in .Define linear operator
( ≡, ): ( , ) = : ∈ , ( , ) ∈ ∗
into ∗
by
( , )( , ) = ( , ), ,
for all ∈ ( , ) for all ∈ .
DEFINATION 4.Let X be a Banach space and , ∈ with < , 1 ≤ < ∞.Then
(0, ; ) and (0, ; )denote the space of measurable functions defined on ( , ) with
values in such that the function → ‖ (. , )‖ is square integrable and essentially bounded.
The respective norms are defined by
‖ ‖ ( , ; ) = ∫ ‖ (. , )‖ (2.19)
‖ ‖ ( , ; ) = . ‖ (. , )‖ . (2.20)
For details on these function space ,see [10]
Definition 5.A function : [0, ] → is a weak solution of (1.5) if
(i) ∈ (0, ; )and ∈ (0, ; ∗);
(ii) For every ∈ , 〈 ( ), 〉 + ( , )( ( ), ) = 0 ,for t pointwise a.e.in [0, ];
(0) = .
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50
Note .The time derivative understood in the distribution sense.The following two lemmas are
of critical importance for the existence and uniqueness of the weak solutions.
LEMMA 6.Let ↪ ↪ ∗
∈ (0, ; ) , ∈ (0, ; ∗) ,then ∈ ([0, ]; ).
Moreover, for any ∈ ,the real –valued function → ‖ ( )‖ is weakly differentiable in
(0, ) and satisfies
{‖ ‖ } = 〈 , 〉 (2.21)
LEMMA 7. (Gronwall’s Lemma) Let ( ) be a nonnegative,summable function on [0, ]
which satisfies the integral inequality
( ) ≤ ∫ ( ) + (2.22)
for constant ≥ 0
almost everywhere ∈ [0. ].Then
( ) ≤ (1 + )a.e on 0 ≤ ≤ ( 2.23).
in particular, if
( ) ≤ ∫ ( ) a.e on 0 ≤ ≤ ,then ( ) = 0 . on [0, ] (2.24)
FOR PROOF SEE [6].
LEMMA 8.The weak solution of (1.5) is unique if it exists.
Proof. Let and be two weak solution of (1.5). Let = − .To prove Lemma 8
.suffices to show that = 0 pointwise a.e.on [0, ].since 〈 ( ), 〉 + ( , )( ( ), ) = 0 for
any ∈ ,we take = ∈ to get
〈 ( ), 〉 + ( , )( ( ), ) = 0 (2.25)
(2.25) is true point wisea.s .on [0, ].Using (2.1) and the coercivity estimate,we have
1
2
‖ ‖ ≤ ‖ ‖ , (0) = 0
For some > 0.By Lemma 7, ‖ ‖ = 0 for all ∈ [0, ].Thus = 0 pointwise a.e in [0, ].
To show existence of the weak solution of (1.5) .we first show existence and uniqueness of
approximation solution. Now we define the approximate solution of (1.5)
DEFINITION9.A function : [0, ] → is an approximate solutions of (1.5) if
(i) ∈ (0, , )and ) ∈ (0, , );
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51
(ii) for every ∈ and 〈 ( ), 〉 + ( , )( ( ), ) = 0 pointwise a.e in [ , ]
(iii) (0) =
To prove the existence of approximate solution ,we take = in
〈 ( ), 〉 + ( , )( ( ), ) = 0
to get following system of ODEs
+ ∑ = 0 , (0) = (2.26)
Where
∈ , ∈ ,for0 ≤ ≤ , ( ) = , ,and = , for : [0, ] → ℛ ,
equation 2.24 can be written as
⃗ + ( ) ⃗ = 0 , ⃗ (0) = ⃗ (2.27)
Since
∈ (0, ; ℛ ×
,for ⃗ = ⃗ .
2.25can be written as
⃗ ( ) = ⃗ − ∫ ( ) ⃗ ( ) (2.28)
The following lemma is immediate from contraction mapping theorem and (2.28)
LEMMA 10: For any ∈ ,there a unique approximate solution : [0, ] → of (2.28).
The following theorem provide the energy estimate for approximate solutions.
Theorem11.There exist a constant depending only on Ω such that the approximate
solution satisfies
‖ ‖ ( , ; ) + ‖ ‖ ( , ; ) + ( , ; )
≤ ‖ ‖ (2.29)
Proof: For every ∈ we have
〈 ( ), 〉 + ( , )( ( ), ) = 0.
take ∈ ( ),then we have
〈 ( ), 〉 + ( , )( ( ), ) = 0.,pointwisea.e in (0, ) (2.30)
using 2.30 and the coercivity estimate.We find that there exists constants
> 0, > 0
such that
( ‖ ‖ ) + ‖ ‖
≤ 0 (2.31)
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52
Integrating 2.31 with respect to t,using the initial condition (0) = ( ), ( 2.39).
and
‖ ( )‖ ≤ ‖ ‖ , we get
( ‖ ‖ ) + ‖ ‖
(2.32)
taking the supriemum over [0, ],we get
‖ ‖ ( , ; ) + ‖ ‖ ( , ; ) ≤ ‖ ‖ (2.33)
Since
( ) ∈ ∗
,
we have
( ) ∗ = sup ∈ ∗
( ),
‖ ‖
, ≠ 0 (2.34)
Using the notion of approximate solution and boundedness of A we have
‖ ‖ ( , ; ) + ‖ ‖ ( , ; ) + ( , ; )
≤ ‖ ‖ (2.35)
To complete the proof of weak solution, we now show the convergence of the approximate
solutions by using weak compactness argument.
DEFINITION 12: Let (0, ; ∗) be the dual space of (0, ; ).Let ∈ (0, ; ∗) and
∈ (0, ; ),then we say → (0, ; ) weakly if
∫ 〈 ( ), ( )〉 → ∫ 〈 ( ), ( )〉 ∀ ∈ (0, ; ∗) (2.36)
Lemma 13 .A subsequence { } of approximate solutions converge weakly in (0, ; ∗)
to a weak solution ∈ ([0, ]; ) ∩ (0, ; )of (1.5) with ∈ (0, ; ∗).Moreover,it
satisfies
‖ ‖ ( , ; ) + ‖ ‖ ( , ; ) + ‖ ‖ ( , ; ) ≤ ‖ ‖ (2.37)
PROOF.Theorem 11 implies that the approximate solutions { } are bounded in (0, ; )
and their derivatives are bounded in (0, ; ∗). By the Banach-Alaoglu theorem, we
can extract a subsequence { } such that
→ (0, ; ), → (0, ; ∗)weakly(2.38)
Let ∅ ∈ (0,T) be a real-valued test function and ∈ for some = .Replacing by
( ) 〈 ( ), 〉 + ( , )( ( ), ) = 0
and integrating from 0 toT, we get.
∫ 〈 ( ), ( ) 〉 + ∫ ( , )( ( ), ( ) ) = 0 ≥ 2.38
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53
taking the limit as → ∞,we get
∫ ( ), = ∫ 〈 , ∅ 〉 (2.39)
by using boundedness of ( , ),we get
∫ ( , )( ( ), ( ) ) = ∫ ( , )( ( ), ∅( ) ) (2.40)
using boundedness of ( , ),we get
〈 ( ), 〉 + ( , )( , ) = 0 (2.41)
pointwise a.e in (0, )
since 2.41 is true for all ∈
∈
and( 2.42)
is dense in V,so (2.42) holds for all ∈ . now it remains to show that (0) = .using
(2.42),integrating by parts and Galerkin approximation we have
〈 (0), 〉 = 〈 , 〉 → ∞
for every ∈ .Thus (0) =
3 EXISTENCEOF OPTIMALPARAMETER
Lemma 14 .Let ∈ .Then the mapping , → , from
= = , ∈ [ , ] × ,
into is continuous.
Proof .Suppose that → in ℛ as → ∞.We denote = , and = , .We claim
that
‖( − ) ‖ → 0
as → ∞. Let ∈ with‖ ‖ ≤ 1.Then
|〈( − ) , 〉| ≤ (| − || || | )
ℛ
+ − | || |
ℛ
+ − | || |
ℛ
+ (| − || || | )
ℛ
≤ 2| − | | |( )
ℛ
+ − | |( )
ℛ
+ − | |( )
ℛ
→ 0
→ ∞
Lemma 15.Suppose that , → , in ℛ , and → weakly in V as → ∞.Then →
weakly in .
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54
Proof.Let ∈ ,then.
〈 , , 〉 − 〈 , 〉 = 〈 , ,〉 − 〈 〉 ≤ |〈 − 〉 , | + |〈 , − 〉| (2.43)
Since a weakly convergent sequence is bounded, we have
|〈 − 〉 , | ≤ ‖ − ‖ ‖ ‖ ≤ ‖ − ‖ → 0
as → ∞ Lemma 14.The second term
〈 , , − 〉 → 0
since → weakly.
Lemma16.Let ∈ . Then the solution map → ( ) from into ([0, ]; ) is
continuous.
Proof.Let → in as → ∞.Since ( ; ) is the weak solution of (1.5) for any ∈
we have the following estimate.
‖ ( ; )‖ ( , ; ) + ‖ ( ; ), ‖ ( , ; ) + ( ; ) ( , ; )
≤ ‖ ‖ (2.44)
Where C is constant independent of ∈ . Estimate (2.44) shows that ( ; ) is bounded in
(0, ).Since (0, ) is reflexive.we can choose a sub-sequence ( ; ) weakly convergent
to a function in (0, ).The fact that ( ; ) is bounded in (0, ) implies that ( , )
is bounded in (0, ; ),so ( ; )weakly convergent to a function in (0, ; ).Since
is compactly imbedded in ,then by the classical compactness theorem[4] ( ; ) → in
(0, ; ),.By (2,44) the derivative ( ; ) and are uniformly bounded in
(0, ; ).Therefore functions ; , are equicontinuous in
([0, ]; )..Thus ( ; ) → in ([0, ]; ) …In particular ( ; ) → ( ) in H and
( ; ) → weakly in V for any ∈ [0, ].By lemma 15, ] ( ; ) → ( ) weakly in
.Now we see that z satisfies the equation given in definition 5,ie it is the weak solution ( )
The uniqueness of the weak solution implies that ( ) → ( ) → ∞ in
([0, ]; ) for the entire sequence ( ) and not for its subsequence. Thus that ( ; ) →
( ) in ([0, ]; ) as that → in as claimed.
3. CONCLUSIONS
The Black-Scholes option pricing model with transaction cost was discussed, where we use an
adjusted volatility given as = (1 − ( ) and a continuous random work which
generalizes the works in the literature.The parameters associated with the Black –Scholes option
pricing model with transaction cost was considered. Also the existence and uniqueness of weak
solution of Black-Scholes option price with transaction cost was studied. The continuity of
weak solution of the parameters was discussed and similar solution as in literature obtained. The
extra terms introduced in this paper is to directly model asset pricedynamics in the case when
the large trader chooses a givenstock-trading strategy.If transaction costs are taken into account
perfect replicationof the contingent claim is no longer possible. Hence, one can re-adjust the
volatility (when the investor’s preferences are characterized by anexponential utility function)in
the form;
13. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
55
= 1 + − ( − ) ’
and if a the weak solution can be obtained.
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