The purpose of this note is to elaborate in how far a 4 factor affine model can generate an incomplete bond market together with the flexibility of a 3 factor flexible affine cascade structure model.
1. The given relation R defines a line with points (2,3), (4,2), and (6,1). The range of y-values is {1,2,3}.
2. The two trigonometric equations are equal when xy < 1.
3. The expressions 7A - (I + A)3 and -I are equal after expanding and simplifying the terms.
Here are the solutions to the given linear congruences:
2X≡ 1 (mod 17) has solution X=8
4x≡ 6(mod 18) has solution x=3
3x≡ 6(mod 18) has no solution since (3,18) does not divide 6
12x≡ 20(mod 28) has solution x=2
The system:
a) x≡1(mod 2), x≡1(mod 3) has the common solution x=1
The document discusses integration and indefinite integrals. It covers determining integrals by reversing differentiation, integrating algebraic expressions like constants, variables, and polynomials. It also discusses determining the constant of integration and using integration to find equations of curves from their gradients. Examples are provided to illustrate integrating functions and finding volumes generated by rotating an area about an axis.
Chapter 9- Differentiation Add Maths Form 4 SPMyw t
This document provides an explanation of differentiation and examples of calculating limits and derivatives using the first principle definition of the derivative. It begins by defining the limit of a function and providing examples of evaluating limits. It then introduces the concept of the derivative as the slope of the tangent line to a curve and explains how to calculate derivatives using small changes in x and y. The document provides examples of finding derivatives using this first principle definition. It also discusses rules for deriving composite functions and products of polynomials. Exercises are provided throughout for students to practice differentiation.
This document contains questions from assignments in differential calculus, continuity and differentiation, rate of change of quantities, increasing and decreasing functions, tangents and normals, and approximation. It also includes word problems involving optimization such as finding dimensions that result in maximum area, volume, or other quantities. There are over 25 questions in total across these calculus topics.
The document discusses matrices and determinants. It defines a matrix as a rectangular table with numbers or formulas as entries. It provides examples of 2x2 and 3x3 matrices. The document explains that square matrices have a number called the determinant extracted from them. It then discusses how the 2x2 determinant represents the signed area of a parallelogram defined by the row vectors, and explores properties of the sign of the determinant. Finally, it suggests generalizing these concepts to 3x3 determinants.
This document discusses optimal graph realizations of finite metric spaces. It defines metric spaces and the problem of finding a realization (graph and edge weights) that represents the metric space with minimum total edge weight. It examines 3-point, 4-point, and 5-point generic metric spaces and conditions for being tree-realizable. The document also discusses split metrics, isolation indices, totally decomposable metrics, and weakly compatible splits. Main results presented include every metric having a split decomposition and classifications of 5-point and 6-point generic metrics.
1. The given relation R defines a line with points (2,3), (4,2), and (6,1). The range of y-values is {1,2,3}.
2. The two trigonometric equations are equal when xy < 1.
3. The expressions 7A - (I + A)3 and -I are equal after expanding and simplifying the terms.
Here are the solutions to the given linear congruences:
2X≡ 1 (mod 17) has solution X=8
4x≡ 6(mod 18) has solution x=3
3x≡ 6(mod 18) has no solution since (3,18) does not divide 6
12x≡ 20(mod 28) has solution x=2
The system:
a) x≡1(mod 2), x≡1(mod 3) has the common solution x=1
The document discusses integration and indefinite integrals. It covers determining integrals by reversing differentiation, integrating algebraic expressions like constants, variables, and polynomials. It also discusses determining the constant of integration and using integration to find equations of curves from their gradients. Examples are provided to illustrate integrating functions and finding volumes generated by rotating an area about an axis.
Chapter 9- Differentiation Add Maths Form 4 SPMyw t
This document provides an explanation of differentiation and examples of calculating limits and derivatives using the first principle definition of the derivative. It begins by defining the limit of a function and providing examples of evaluating limits. It then introduces the concept of the derivative as the slope of the tangent line to a curve and explains how to calculate derivatives using small changes in x and y. The document provides examples of finding derivatives using this first principle definition. It also discusses rules for deriving composite functions and products of polynomials. Exercises are provided throughout for students to practice differentiation.
This document contains questions from assignments in differential calculus, continuity and differentiation, rate of change of quantities, increasing and decreasing functions, tangents and normals, and approximation. It also includes word problems involving optimization such as finding dimensions that result in maximum area, volume, or other quantities. There are over 25 questions in total across these calculus topics.
The document discusses matrices and determinants. It defines a matrix as a rectangular table with numbers or formulas as entries. It provides examples of 2x2 and 3x3 matrices. The document explains that square matrices have a number called the determinant extracted from them. It then discusses how the 2x2 determinant represents the signed area of a parallelogram defined by the row vectors, and explores properties of the sign of the determinant. Finally, it suggests generalizing these concepts to 3x3 determinants.
This document discusses optimal graph realizations of finite metric spaces. It defines metric spaces and the problem of finding a realization (graph and edge weights) that represents the metric space with minimum total edge weight. It examines 3-point, 4-point, and 5-point generic metric spaces and conditions for being tree-realizable. The document also discusses split metrics, isolation indices, totally decomposable metrics, and weakly compatible splits. Main results presented include every metric having a split decomposition and classifications of 5-point and 6-point generic metrics.
This document is the formula booklet for the IB Diploma Programme Mathematical Studies SL course. It contains formulas for various topics in mathematics that students can refer to during their course and examinations. The topics covered include number and algebra, descriptive statistics, logic, sets and probability, geometry and trigonometry, mathematical models, and an introduction to differential calculus. Formulas are provided for calculating percentages, terms of sequences, compound interest, measures of central tendency, truth tables, probability, geometric shapes, trigonometry, and derivatives.
The document discusses triple integrals and their use in calculating the mass of an object with a non-uniform density distribution. It defines a triple integral as the limit of Riemann sums used to approximate the mass of an object partitioned into subvolumes. For a continuous density function over a rectangular region, the triple integral can be written as an iterated integral and evaluated in any order of integration. Examples are given of defining the bounding region of a solid object using set notation for different types of boundary domains.
The document discusses the difference quotient formula for calculating the slope between two points (x1,y1) and (x2,y2) on a function y=f(x). It shows that the slope m is equal to (f(x+h)-f(x))/h, where h is the difference between x1 and x2. This "difference quotient" formula allows slopes to be calculated from the values of a function at two nearby points. Examples are given of simplifying the difference quotient for quadratic and rational functions.
This document contains notes on additional mathematics including topics on progression, linear laws, integration, and vectors. Some key points:
- It discusses arithmetic and geometric progressions, defining the terms and formulas for finding terms and sums. Examples are worked through finding terms, sums, and differences between sums.
- Linear laws are explained including lines of best fit, converting between linear and non-linear forms using logarithms, and working through examples of finding equations from graphs.
- Integration techniques are outlined including formulas for integrals of powers, areas under and between curves, volumes of revolution, and the basic rules of integration. Worked examples find areas and volumes.
- Vectors are introduced including addition using the triangle
This document contains 54 multiple choice questions related to polynomials and their properties. Some key questions asked about:
- Finding the degree of polynomials
- Identifying the number of real zeros of polynomials
- Factoring polynomials
- Evaluating polynomials for given values
- Identifying coefficients and constants in polynomial expressions
- Relating the zeros of a polynomial to its factors
The questions cover topics like polynomial definitions, operations, factorization, finding zeros, and other properties of polynomials.
This document provides an algebra cheat sheet that summarizes many common algebraic properties, formulas, and concepts. It covers topics such as arithmetic operations, properties of inequalities and absolute value, exponent properties, factoring formulas, solving equations, graphing functions, and common algebraic errors. The cheat sheet is a concise 3-page reference for the basics of algebra.
This document provides notes on various mathematics topics for the IGCSE including: decimals and standard form, accuracy and error, powers and roots, ratio and proportion, and trigonometry. It includes examples and practice problems for each topic. The notes are intended to help with revision for IGCSE mathematics question papers and assessments.
The document defines relative and absolute maxima and minima for functions z=f(x,y). A relative maximum occurs when f(a,b) is greater than f(x,y) within a neighborhood circle of (a,b). An absolute maximum occurs when f(a,b) is greater than f(x,y) over the entire domain. Similarly for minima with the inequalities reversed. Extrema refer to maxima and minima. For a continuous function over a closed and bounded domain, absolute extrema exist and occur either in the interior or on the boundary. Examples find and classify extrema of functions.
This document discusses numerical solutions of partial differential equations. It contains an introduction and four chapters:
1. Preliminaries - Defines basic concepts like differential equations, partial derivatives, order of a differential equation.
2. Partial Differential Equations of Second Order - Classifies second order PDEs and provides examples.
3. Parabolic Equations - Discusses explicit and implicit finite difference methods like Schmidt's method and Crank-Nicolson method to solve heat equation.
4. Hyperbolic Equations - Will discuss numerical methods to solve hyperbolic PDEs like the wave equation.
A study on number theory and its applicationsItishree Dash
A STUDY ON NUMBER THEORY AND ITS APPLICATIONS
Applications
Modular Arithmetic
Congruence and Pseudorandom Number
Congruence and CRT(Chinese Remainder Theorem)
Congruence and Cryptography
The chapter discusses numerical methods for solving the 1D and 2D heat equation. Four methods are described for the 1D equation: Schmidt, Crank-Nicolson, iterative (Jacobi and Gauss-Seidel), and Du Fort-Frankel. The Schmidt method is explicit but conditionally stable, while Crank-Nicolson is implicit and unconditionally stable. Examples are solved using each method and compared to analytical solutions. The alternating direction explicit (ADE) method is described for the 2D equation.
The document summarizes key aspects of quadratic graphs:
1) A quadratic function takes the form of ax2 + bx + c, with examples given.
2) When plotted, a quadratic function produces a smooth curve called a parabola.
3) There are two ways to solve quadratic graphs - using a table of values to find coordinates, or directly replacing x-values into the function.
Steps for each method are outlined along with an example.
This document discusses elliptic curves in Weierstrass normal form and finding torsion points on elliptic curves. It defines Weierstrass normal form, discusses uses of elliptic curves including Andrew Wiles' proof of Fermat's Last Theorem. It also defines the group structure of elliptic curves, discusses how points are added, and defines the torsion subgroup as points of finite order. Methods for finding the torsion subgroup include reduction modulo primes and applying theorems like Nagell-Lutz. Examples are worked through on specific elliptic curves.
This document contains notes and formulae on solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, algebraic formulae, linear inequalities, statistics, significant figures and standard form, quadratic expressions and equations, sets, mathematical reasoning, straight lines, and trigonometry. The key concepts covered include formulas for calculating the volume and surface area of various 3D shapes, properties of angles in circles and polygons, factorising and expanding algebraic expressions, solving linear and quadratic equations, set notation and Venn diagrams, types of logical arguments, equations of straight lines, and defining the basic trigonometric ratios.
The document provides the steps to solve a multi-part calculus problem. It involves finding the derivative of two functions, determining the equations of tangent lines to those functions at given points, finding the intersection points of those tangent lines, and using those intersection points to write the equations of three circles with a radius of 5.
This document provides a summary of core mathematics concepts including:
1) Linear graphs and equations such as y=mx+c and finding the equation of a line.
2) Quadratic equations and graphs including using the quadratic formula, completing the square, and finding the vertex and axis of symmetry.
3) Simultaneous equations and interpreting their solutions geometrically as the intersection of graphs.
4) Other topics covered include surds, polynomials, differentiation, integration, and areas under graphs.
The document provides the steps to solve a multi-part calculus problem involving derivatives, tangent lines, and circles. It determines the derivative of two functions f(x) and g(x), finds the equations of the tangent lines at specific x-values, identifies the intersection points of the tangent lines, and uses those intersection points as the centers of three circles with a radius of 5 to write the equations of the circles.
On Triplet of Positive Integers Such That the Sum of Any Two of Them is a Per...inventionjournals
In this article we discussed determination of distinct positive integers a, b, c such that a + b, a + c, b + c are perfect squares. We can determine infinitely many such triplets. There are such four tuples and from them eliminating any one number we obtain triplets with the specific property. We can also obtain infinitely many such triplets from a single triplet.
(1) The document discusses various integration techniques including: review of integral formulas, integration by parts, trigonometric integrals involving products of sines and cosines, trigonometric substitutions, and integration of rational functions using partial fractions.
(2) Examples are provided to demonstrate each technique, such as using integration by parts to evaluate integrals of the form ∫udv, using trigonometric identities to reduce powers of trigonometric functions, and using partial fractions to break down rational functions into simpler fractions.
(3) The key techniques discussed are integration by parts, trigonometric substitutions to transform integrals involving quadratic expressions into simpler forms, and partial fractions to decompose rational functions for integration. Various examples illustrate the
The document summarizes the brachistochrone problem from calculus of variations. It introduces the brachistochrone curve as the curve of fastest descent under gravity between two points. The problem is then solved using tools from calculus of variations, arriving at the Euler-Lagrange equation. This equation shows that the brachistochrone curve between two points is a cycloid. Additionally, the document discusses that the cycloid is a tautochronic curve, meaning an object will take the same amount of time to slide from any point on it to the lowest point.
IIT Jam math 2016 solutions BY TrajectoryeducationDev Singh
The document contains a mathematics exam question paper with 10 single mark questions (Q1-Q10) and 20 two mark questions (Q11-Q30). The questions cover topics like sequences, linear transformations, integrals, permutations, differential equations etc. Some key questions asked about the nature of a sequence involving sines, order of a permutation, evaluating a limit, checking if a differential equation is exact etc. and provided solutions to them.
This document is the formula booklet for the IB Diploma Programme Mathematical Studies SL course. It contains formulas for various topics in mathematics that students can refer to during their course and examinations. The topics covered include number and algebra, descriptive statistics, logic, sets and probability, geometry and trigonometry, mathematical models, and an introduction to differential calculus. Formulas are provided for calculating percentages, terms of sequences, compound interest, measures of central tendency, truth tables, probability, geometric shapes, trigonometry, and derivatives.
The document discusses triple integrals and their use in calculating the mass of an object with a non-uniform density distribution. It defines a triple integral as the limit of Riemann sums used to approximate the mass of an object partitioned into subvolumes. For a continuous density function over a rectangular region, the triple integral can be written as an iterated integral and evaluated in any order of integration. Examples are given of defining the bounding region of a solid object using set notation for different types of boundary domains.
The document discusses the difference quotient formula for calculating the slope between two points (x1,y1) and (x2,y2) on a function y=f(x). It shows that the slope m is equal to (f(x+h)-f(x))/h, where h is the difference between x1 and x2. This "difference quotient" formula allows slopes to be calculated from the values of a function at two nearby points. Examples are given of simplifying the difference quotient for quadratic and rational functions.
This document contains notes on additional mathematics including topics on progression, linear laws, integration, and vectors. Some key points:
- It discusses arithmetic and geometric progressions, defining the terms and formulas for finding terms and sums. Examples are worked through finding terms, sums, and differences between sums.
- Linear laws are explained including lines of best fit, converting between linear and non-linear forms using logarithms, and working through examples of finding equations from graphs.
- Integration techniques are outlined including formulas for integrals of powers, areas under and between curves, volumes of revolution, and the basic rules of integration. Worked examples find areas and volumes.
- Vectors are introduced including addition using the triangle
This document contains 54 multiple choice questions related to polynomials and their properties. Some key questions asked about:
- Finding the degree of polynomials
- Identifying the number of real zeros of polynomials
- Factoring polynomials
- Evaluating polynomials for given values
- Identifying coefficients and constants in polynomial expressions
- Relating the zeros of a polynomial to its factors
The questions cover topics like polynomial definitions, operations, factorization, finding zeros, and other properties of polynomials.
This document provides an algebra cheat sheet that summarizes many common algebraic properties, formulas, and concepts. It covers topics such as arithmetic operations, properties of inequalities and absolute value, exponent properties, factoring formulas, solving equations, graphing functions, and common algebraic errors. The cheat sheet is a concise 3-page reference for the basics of algebra.
This document provides notes on various mathematics topics for the IGCSE including: decimals and standard form, accuracy and error, powers and roots, ratio and proportion, and trigonometry. It includes examples and practice problems for each topic. The notes are intended to help with revision for IGCSE mathematics question papers and assessments.
The document defines relative and absolute maxima and minima for functions z=f(x,y). A relative maximum occurs when f(a,b) is greater than f(x,y) within a neighborhood circle of (a,b). An absolute maximum occurs when f(a,b) is greater than f(x,y) over the entire domain. Similarly for minima with the inequalities reversed. Extrema refer to maxima and minima. For a continuous function over a closed and bounded domain, absolute extrema exist and occur either in the interior or on the boundary. Examples find and classify extrema of functions.
This document discusses numerical solutions of partial differential equations. It contains an introduction and four chapters:
1. Preliminaries - Defines basic concepts like differential equations, partial derivatives, order of a differential equation.
2. Partial Differential Equations of Second Order - Classifies second order PDEs and provides examples.
3. Parabolic Equations - Discusses explicit and implicit finite difference methods like Schmidt's method and Crank-Nicolson method to solve heat equation.
4. Hyperbolic Equations - Will discuss numerical methods to solve hyperbolic PDEs like the wave equation.
A study on number theory and its applicationsItishree Dash
A STUDY ON NUMBER THEORY AND ITS APPLICATIONS
Applications
Modular Arithmetic
Congruence and Pseudorandom Number
Congruence and CRT(Chinese Remainder Theorem)
Congruence and Cryptography
The chapter discusses numerical methods for solving the 1D and 2D heat equation. Four methods are described for the 1D equation: Schmidt, Crank-Nicolson, iterative (Jacobi and Gauss-Seidel), and Du Fort-Frankel. The Schmidt method is explicit but conditionally stable, while Crank-Nicolson is implicit and unconditionally stable. Examples are solved using each method and compared to analytical solutions. The alternating direction explicit (ADE) method is described for the 2D equation.
The document summarizes key aspects of quadratic graphs:
1) A quadratic function takes the form of ax2 + bx + c, with examples given.
2) When plotted, a quadratic function produces a smooth curve called a parabola.
3) There are two ways to solve quadratic graphs - using a table of values to find coordinates, or directly replacing x-values into the function.
Steps for each method are outlined along with an example.
This document discusses elliptic curves in Weierstrass normal form and finding torsion points on elliptic curves. It defines Weierstrass normal form, discusses uses of elliptic curves including Andrew Wiles' proof of Fermat's Last Theorem. It also defines the group structure of elliptic curves, discusses how points are added, and defines the torsion subgroup as points of finite order. Methods for finding the torsion subgroup include reduction modulo primes and applying theorems like Nagell-Lutz. Examples are worked through on specific elliptic curves.
This document contains notes and formulae on solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, algebraic formulae, linear inequalities, statistics, significant figures and standard form, quadratic expressions and equations, sets, mathematical reasoning, straight lines, and trigonometry. The key concepts covered include formulas for calculating the volume and surface area of various 3D shapes, properties of angles in circles and polygons, factorising and expanding algebraic expressions, solving linear and quadratic equations, set notation and Venn diagrams, types of logical arguments, equations of straight lines, and defining the basic trigonometric ratios.
The document provides the steps to solve a multi-part calculus problem. It involves finding the derivative of two functions, determining the equations of tangent lines to those functions at given points, finding the intersection points of those tangent lines, and using those intersection points to write the equations of three circles with a radius of 5.
This document provides a summary of core mathematics concepts including:
1) Linear graphs and equations such as y=mx+c and finding the equation of a line.
2) Quadratic equations and graphs including using the quadratic formula, completing the square, and finding the vertex and axis of symmetry.
3) Simultaneous equations and interpreting their solutions geometrically as the intersection of graphs.
4) Other topics covered include surds, polynomials, differentiation, integration, and areas under graphs.
The document provides the steps to solve a multi-part calculus problem involving derivatives, tangent lines, and circles. It determines the derivative of two functions f(x) and g(x), finds the equations of the tangent lines at specific x-values, identifies the intersection points of the tangent lines, and uses those intersection points as the centers of three circles with a radius of 5 to write the equations of the circles.
On Triplet of Positive Integers Such That the Sum of Any Two of Them is a Per...inventionjournals
In this article we discussed determination of distinct positive integers a, b, c such that a + b, a + c, b + c are perfect squares. We can determine infinitely many such triplets. There are such four tuples and from them eliminating any one number we obtain triplets with the specific property. We can also obtain infinitely many such triplets from a single triplet.
(1) The document discusses various integration techniques including: review of integral formulas, integration by parts, trigonometric integrals involving products of sines and cosines, trigonometric substitutions, and integration of rational functions using partial fractions.
(2) Examples are provided to demonstrate each technique, such as using integration by parts to evaluate integrals of the form ∫udv, using trigonometric identities to reduce powers of trigonometric functions, and using partial fractions to break down rational functions into simpler fractions.
(3) The key techniques discussed are integration by parts, trigonometric substitutions to transform integrals involving quadratic expressions into simpler forms, and partial fractions to decompose rational functions for integration. Various examples illustrate the
The document summarizes the brachistochrone problem from calculus of variations. It introduces the brachistochrone curve as the curve of fastest descent under gravity between two points. The problem is then solved using tools from calculus of variations, arriving at the Euler-Lagrange equation. This equation shows that the brachistochrone curve between two points is a cycloid. Additionally, the document discusses that the cycloid is a tautochronic curve, meaning an object will take the same amount of time to slide from any point on it to the lowest point.
IIT Jam math 2016 solutions BY TrajectoryeducationDev Singh
The document contains a mathematics exam question paper with 10 single mark questions (Q1-Q10) and 20 two mark questions (Q11-Q30). The questions cover topics like sequences, linear transformations, integrals, permutations, differential equations etc. Some key questions asked about the nature of a sequence involving sines, order of a permutation, evaluating a limit, checking if a differential equation is exact etc. and provided solutions to them.
This document provides examples of solving various population models using differential equations. It introduces the logistic population model P'(t) = kP(t)[M - P(t)] and shows how to derive the solution P(t) = M/(1 + Ce-kt) given initial conditions. Several examples are worked through to find the time needed for a population to reach a certain level using this model. Other models involving birth and death rates are also introduced and solved.
The Multivariate Gaussian Probability DistributionPedro222284
The document discusses the multivariate Gaussian probability distribution. It defines the distribution and provides its probability density function. It then discusses various properties including: functions of Gaussian variables such as linear transformations and addition; the characteristic function and how to calculate moments; marginalization and conditional distributions. It also provides some tips and tricks for working with Gaussian distributions including how to calculate products.
This document discusses Frullani integrals, which are integrals of the form ∫01 f(ax)−f(bx)x dx = [f(0)−f(∞)]ln(b/a). It provides 11 examples of integrals from Gradshteyn and Ryzhik that can be reduced to this Frullani form by appropriate choice of the function f(x). It also lists 9 examples found in Ramanujan's notebooks. One example, involving logarithms of trigonometric functions, requires a more complex approach. The document concludes by deriving the solution to this more delicate example.
The document summarizes the binomial theorem and properties of binomial coefficients. It provides:
1) The binomial theorem expresses the expansion of (a + b)n as a sum of terms involving binomial coefficients for any positive integer n.
2) Important properties of binomial coefficients are discussed, such as their relationship to factorials and the symmetry of coefficients.
3) Examples are given of using the binomial theorem to find coefficients and solve problems involving divisibility and series of binomial coefficients.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2010. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
11.generalized and subset integrated autoregressive moving average bilinear t...Alexander Decker
This document proposes generalized integrated autoregressive moving average bilinear (GBL) time series models and subset generalized integrated autoregressive moving average bilinear (GSBL) models to achieve stationary for all nonlinear time series. It presents the models' formulations and discusses their properties including stationary, convergence, and parameter estimation. An algorithm is provided to fit the one-dimensional models. The generalized models are applied to Wolfer sunspot numbers and the GBL model is found to perform better than the GSBL model.
Matrices can be used to represent 2D geometric transformations such as translation, scaling, and rotation. Translations are represented by adding a translation vector to coordinates. Scaling is represented by multiplying coordinates by scaling factors. Rotations are represented by premultiplying coordinates with a rotation matrix. Multiple transformations can be combined by multiplying their respective matrices. Using homogeneous coordinates allows all transformations to be uniformly represented as matrix multiplications.
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...Alexander Litvinenko
Talk presented on SIAM IS 2022 conference.
Very often, in the course of uncertainty quantification tasks or
data analysis, one has to deal with high-dimensional random variables (RVs)
(with values in $\Rd$). Just like any other RV,
a high-dimensional RV can be described by its probability density (\pdf) and/or
by the corresponding probability characteristic functions (\pcf),
or a more general representation as
a function of other, known, random variables.
Here the interest is mainly to compute characterisations like the entropy, the Kullback-Leibler, or more general
$f$-divergences. These are all computed from the \pdf, which is often not available directly,
and it is a computational challenge to even represent it in a numerically
feasible fashion in case the dimension $d$ is even moderately large. It
is an even stronger numerical challenge to then actually compute said characterisations
in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose
to approximate density by a low-rank tensor.
This document contains information about data structures and algorithms taught at KTH Royal Institute of Technology. It includes code templates for a contest, descriptions and implementations of common data structures like an order statistic tree and hash map, as well as summaries of mathematical and algorithmic concepts like trigonometry, probability theory, and Markov chains.
The document discusses floors, ceilings, and their applications. It includes:
- Definitions of floors (bxc) and ceilings (dxe) of real numbers and their properties.
- Applications of floors and ceilings to problems like determining the number of bits in a binary representation, how functions interact with floors and ceilings, and spectra of real numbers.
- Examples of floor and ceiling recurrences, including the Knuth numbers, merge sort comparisons, and the Josephus problem numbers.
This document discusses the application of fixed-point theorems to solve ordinary differential equations. It begins by introducing the Banach contraction principle and proving it. It then states two other important fixed-point theorems - the Schauder-Tychonoff theorem and the Leray-Schauder theorem. The rest of the document focuses on proving the Schauder-Tychonoff theorem, which characterizes compact subsets of function spaces and shows that if an operator maps into a relatively compact subset, it has a fixed point. This allows the fixed-point theorems to be applied to finding solutions to differential equations.
This document discusses the application of fixed-point theorems to solve ordinary differential equations. It begins by introducing the Banach contraction principle and proving it. It then states two other important fixed-point theorems - the Schauder-Tychonoff theorem and the Leray-Schauder theorem. The rest of the document focuses on proving the Schauder-Tychonoff theorem, which characterizes compact subsets of function spaces and shows that if an operator maps into a relatively compact subset, it has a fixed point. This allows the fixed-point theorems to be applied to finding solutions to differential equations.
The document contains the solutions to 5 problems from the 2017 Canadian Mathematical Olympiad. The first problem involves using an inequality to prove that the sum of fractions involving three non-negative real numbers is greater than 2. The second problem relates the number of divisors of a positive integer to a function and proves if the input is prime, the output is also prime. The third problem counts the number of balanced subsets of numbers and proves the count is odd.
This document discusses dynamic programming and greedy algorithms. It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems. It provides examples of dynamic programming approaches to computing Fibonacci numbers, binomial coefficients, the knapsack problem, and other problems. It also discusses greedy algorithms and provides examples of their application to problems like the change-making problem, minimum spanning trees, and single-source shortest paths.
Similar to Affine Yield Curves: Flexibility versus Incompleteness (20)
Monthly Market Risk Update: June 2024 [SlideShare]Commonwealth
Markets rallied in May, with all three major U.S. equity indices up for the month, said Sam Millette, director of fixed income, in his latest Market Risk Update.
For more market updates, subscribe to The Independent Market Observer at https://blog.commonwealth.com/independent-market-observer.
An accounting information system (AIS) refers to tools and systems designed for the collection and display of accounting information so accountants and executives can make informed decisions.
How to Invest in Cryptocurrency for Beginners: A Complete GuideDaniel
Cryptocurrency is digital money that operates independently of a central authority, utilizing cryptography for security. Unlike traditional currencies issued by governments (fiat currencies), cryptocurrencies are decentralized and typically operate on a technology called blockchain. Each cryptocurrency transaction is recorded on a public ledger, ensuring transparency and security.
Cryptocurrencies can be used for various purposes, including online purchases, investment opportunities, and as a means of transferring value globally without the need for intermediaries like banks.
What Lessons Can New Investors Learn from Newman Leech’s Success?Newman Leech
Newman Leech's success in the real estate industry is based on key lessons and principles, offering practical advice for new investors and serving as a blueprint for building a successful career.
Poonawalla Fincorp’s Strategy to Achieve Industry-Leading NPA Metricsshruti1menon2
Poonawalla Fincorp Limited, under the leadership of Managing Director Abhay Bhutada, has achieved industry-leading Gross Non-Performing Assets (GNPA) below 1% and Net Non-Performing Assets (NNPA) below 0.5% as of May 31, 2024. This success is attributed to a strategic vision focusing on prudent credit policies, robust risk management, and digital transformation. Bhutada's leadership has driven the company to exceed its targets ahead of schedule, emphasizing rigorous credit assessment, advanced risk management, and enhanced collection efficiency. By prioritizing customer-centric solutions, leveraging digital innovation, and maintaining strong financial performance, Poonawalla Fincorp sets new benchmarks in the industry. With a continued focus on asset quality, digital enhancement, and exploring growth opportunities, the company is well-positioned for sustained success in the future.
Navigating Your Financial Future: Comprehensive Planning with Mike Baumannmikebaumannfinancial
Learn how financial planner Mike Baumann helps individuals and families articulate their financial aspirations and develop tailored plans. This presentation delves into budgeting, investment strategies, retirement planning, tax optimization, and the importance of ongoing plan adjustments.
“Amidst Tempered Optimism” Main economic trends in May 2024 based on the results of the New Monthly Enterprises Survey, #NRES
On 12 June 2024 the Institute for Economic Research and Policy Consulting (IER) held an online event “Economic Trends from a Business Perspective (May 2024)”.
During the event, the results of the 25-th monthly survey of business executives “Ukrainian Business during the war”, which was conducted in May 2024, were presented.
The field stage of the 25-th wave lasted from May 20 to May 31, 2024. In May, 532 companies were surveyed.
The enterprise managers compared the work results in May 2024 with April, assessed the indicators at the time of the survey (May 2024), and gave forecasts for the next two, three, or six months, depending on the question. In certain issues (where indicated), the work results were compared with the pre-war period (before February 24, 2022).
✅ More survey results in the presentation.
✅ Video presentation: https://youtu.be/4ZvsSKd1MzE
Explore the world of investments with an in-depth comparison of the stock market and real estate. Understand their fundamentals, risks, returns, and diversification strategies to make informed financial decisions that align with your goals.
2. Affine Yield Curves: Flexibility versus Incompleteness
Dhia Eddine Barbouche and Bert Koehler
Affine interest rate models are amongst the most popular interest rate mod-
els due to their flexibility, analytic expressions for bond prices and analytic
approximations for some interest derivatives, see for example the seminal ar-
ticle of Duffie and Kan [Duf-Kan]. The subclass of affine models for which
interest rates are bounded from below is of the form rt = n
j=1 δjXjt − c
where
Xjt = Xj0 +
t
0
bj +
n
k=1
ajkXks ds + σj
t
0
XjtdWjs
Here rt is the short rate process, δj ≥ 0, c ≥ 0 is some negative displacement
term, ajk, bj ≥ 0 except for ajj < 0 and (W1t, ..., Wnt) are independent Brow-
nian motions. Here and in the following we always work in the risk neutral
measure Q unless otherwise indicated. The logzeroyields defined as
Y (T, X0) = −
1
T
log E e− T
0 rsds
|X0
can be written as an affine linear combination of the state X = (X1, ..., Xn)
Y (T, X0) = −
1
T
n
j=1
(bjAj(T) + Xj0Bj(T))
where the deterministic functions T −→ Aj(T), Bj(T) in maturity T ≥ 0
obey the system of ODEs
dBj
dT
=
1
2
σ2
j B2
j +
n
k=1
akjBk − δj
Bj(0) = 0 and Aj(T) =
T
0
Bj(s)ds
The shape of the curves T −→ − 1
T
Bj(T) determines the flexibility of the
model in reproducing the great variety of yield curves seen in the history.
1
3. From this point of view affine models with a cascade structure are promising.
Here we deal with affine cascade models with 3 factors:
X1t = X10 +
t
0
(b1 + X2s − a1X1s)ds + σ1
t
0
X1sdW1s
X2t = X20 +
t
0
(b2 + X3s − a2X2s)ds + σ2
t
0
X2sdW2s
X3t = X30 +
t
0
(b3 − a3X3s)ds + σ3
t
0
X3sdW3s
The corresponding system of ODEs simplifies to (normalizing δ1 = 1)
dB1
dT
=
1
2
σ2
1B2
1 − a1B1 − 1
dB2
dT
=
1
2
σ2
2B2
2 − a2B2 + B1 − δ2
dB3
dT
=
1
2
σ2
3B2
3 − a3B3 + B2 − δ3
One can easily see that T −→ Bj(T) ≤ 0 and falling in T: for that purpose
write for example
dB2
dT
= (
1
2
σ2
2B2 − a2) · B2 + B1 − δ2
which can be solved as
B2(t) = e
t
0 ( 1
2
σ2
2B2(s)−a2)ds
· B2(0) +
t
0
e− s
0 ( 1
2
σ2
2B2(q)−a2)dq
· (B1(s) − δ2)ds
As B2(0) = 0 and B1(t) − δ2 ≤ 0 we see that B2(t) ≤ 0. Similarly taking
derivatives in t we get
d2
B2
dT2
= (σ2
2B2 − a2) ·
dB2
dT
+
dB1
dT
and by the way
dB2
dt
(t) = e
t
0 (σ2
2B2(s)−a2)ds
·
dB2
dt
(0) +
t
0
e− s
0 (σ2
2B2(q)−a2)dq
·
dB1
dt
(s)ds
As dB2
dt
(0) = −δ2 ≤ 0 and dB1
dt
(t) ≤ 0 we get dB2
dt
(t) ≤ 0. On the other hand we
have ∃ limT→∞ Bj(T) and so limT→∞
1
T
Bj(T) = 0. What is interesting about
2
4. affine cascade structure models is that for suitably chosen parameters aj, σj
we can achieve that T −→ − 1
T
B2(T) attains its maximum in T2,max ≥ 10
and T −→ − 1
T
B3(T) attains its maximum in T3,max ≥ 25. This permits a
great flexibility of yield curves
Y (T, X0) = −
1
T
3
j=1
(bjAj(T) + Xj0Bj(T))
For example the choice of (risk-neutral) parameters
b1 a1 σ1 δ1
b2 a2 σ2 δ2
b3 a3 σ3 δ3
=
0.00005 0.3 0.05 1
0 0.05 0.05 0
0.00005 0.01 0.01 0
and c = 0.015 can represent all historically seen yield triples with tenors 3M,
10Y and 30Y by admissible state vectors X = (X1, X2, X3), Xj ≥ 0 and can
even cope with EUR-atm-Swaption volatility surface from 31.03.2016. Affine
cascade structure models with
d
dT
−
B2(T)
T
(T = 10) ≥ 0
d
dT
−
B3(T)
T
(T = 25) ≥ 0
are called flexible affine cascade structure models. The purpose of this
note is to elaborate in how far a 4 factor affine model can generate an
incomplete bond market together with the flexibility of a 3 factor flexible
affine cascade structure model. This means we have a 4 factor affine model
Xt = (X1t, ..., X4t) with negatively bounded interest rates, but logzeroyields
reduce to a 3 factor state dynamic
Y (T, X0) = −
1
T
4
j=1
(bjAj(T) + Xj0Bj(T)) =
−
1
T
4
j=1
bjAj(T) −
1
T
3
j=1
Bj(T) ·
4
k=1
˜cjkXk0
where Bj(T) obey a system of ODEs coming from a 3 factor flexible affine
cascade structure model. So from the bond-prices alone it is not possible to
3
5. recover the full 4-factor state but we require furthermore that the dynamics
of Xt does not reduce generally to a 3-dimensional process. So if we add some
interest-optionprices to bondprices the full 4-state Xt can be determinated.
The motivation for such an ”unspanned volatility” phenomenon derives from
the conjecture that option prices may have a real-world-dynamic not fully ex-
plained by the real-world-dynamic of yields. In general unspanned volatility
in an n-dimensional affine model postulates a nontrivial linear relation
0 =
n
j=1
cjBj(T) for all T ≥ 0
Defining a (polynomial) vector field on the ring of polynomials P ∈ C[B1, ..., Bn]
by
Z =
n
j=1
1
2
σ2
j B2
j +
n
k=1
akjBk − δj
∂
∂Bj
we get a sequence of polynomials Pm by setting
P1(B1, ..., Bn) =
n
j=1
cjBj
Pm+1(B1, ..., Bn) = Z(Pm)
Let I = I(Pm, m ∈ N) be the ideal generated by the full sequence. By
Hilberts Basis Theorem I is finitely generated, so there are P1, ..., PN which
generate the whole ideal. In fact it is sufficient to find a N ∈ N with PN+1 ∈
I(P1, ..., PN ) because then all further PN+2, PN+3, ... are also in I(P1, ..., PN ).
Let V (I) = V (P1, ..., PN ) ⊂ Cn
be the algebraic variety of the ideal, that
means
V (P1, ..., PN ) = {(B1, ..., Bn) with Pm(B1, ..., Bn) = 0 for m = 1, ..., N}
Obviously the (real analytic) curve T −→ (B1(T), ..., Bn(T)) lies inside
V (P1, ..., PN ). On the other hand if (0, ..., 0) ∈ V (P1, ..., PN ), then we have
n
j=1
cj
dm
Bj
dtm
(t = 0) = 0
4
6. first for 0 ≤ m ≤ N − 1 but then due to the generation property also for all
m. Because T −→ (B1(T), ..., Bn(T)) is real analytic we have
n
j=1
cjBj(t) = 0
identically in this case, so the two conditions are equivalent. This simple
observation reduces the search for unspanned affine models to a polynomial
basis problem which can in principle be solved by algorithm. Up to now the
only known example stems from Colin-Dufresne and Goldstein for n = 3 and
results in a rational curve as V (P1, P2) ⊂ C3
. So there are no ”transcenden-
tal” unspanned affine models known. Maybe this is a general property due to
the fact that the intersection variety carries a holomorphic vector field with
meromorphic pole of first order in the projective extension.
Coming back to our 4-factor setting it is instructive to consider an exam-
ple: Assume that Bj(T) = Bj(T) for j = 1, 2, 3, so we have
dB1
dT
=
1
2
σ2
1B2
1 − a1B1 − 1
dB2
dT
=
1
2
σ2
2B2
2 − a2B2 + B1 − δ2
dB3
dT
=
1
2
σ2
3B2
3 − a3B3 + B2 − δ3
and furthermore
dB4
dt
=
1
2
σ2
4B2
4 − a44B4 + a41B1 + a42B2 + a43B3 − δ4
The corresponding process looks like
X1t = X10 +
t
0
(b1 + a41X4s + X2s − a1X1s)ds + σ1
t
0
X1sdW1s
X2t = X20 +
t
0
(b2 + a42X4s + X3s − a2X2s)ds + σ2
t
0
X2sdW2s
X3t = X30 +
t
0
(b3 + a43X4s − a3X3s)ds + σ3
t
0
X3sdW3s
X4t = X40 +
t
0
(b4 − a44X4s)ds + σ4
t
0
X4sdW4s
5
7. Now assume we have a linear relation
B4(T) =
3
j=1
cjBj(T) for all T ≥ 0
Taking derivatives we obtain
1
2
σ2
4
3
j=1
cjBj
2
− a44
3
j=1
cjBj + a41B1 + a42B2 + a43B3 − δ4 =
3
j=1
cj
1
2
σ2
j B2
j − ajBj + Bj−1 − δj
If this quadratic relation vanishes identically in C[B1, B2, B3] we must have
B4(T) = cjBj(T) for some j ∈ {1, 2, 3} and
σ2
4c2
j = cjσ2
j
cjaj − a44cj + a4j = 0
a4,j−1 − cj = 0
a4,j−2 = 0
cjδj − δ4 = 0
If for example j = 3 then
c3 =
σ2
3
σ2
4
a43 = (a44 − a3)c3
a42 = c3
a41 = 0
δ4 = c3δ3
6
8. Now we define X5t = X3t + c3X4t. Then we get
X1t = X10 +
t
0
(b1 + X2s − a1X1s)ds + σ1
t
0
X1sdW1s
X2t = X20 +
t
0
(b2 + X5s − a2X2s)ds + σ2
t
0
X2sdW2s
X5t = X50 +
t
0
(b3 + c3b4 + a43X4s − a3X3s − c3a44X4s)ds +
σ3
t
0
X3sdW3s + c3σ4
t
0
X4sdW4s
Inserting a43 = (a44 − a3)c3 and using c3 =
σ2
3
σ2
4
we get
σ3
t
0
X3sdW3s + c3σ4
t
0
X4sdW4s ∼ σ3
t
0
X5sdW5s
and
X1t = X10 +
t
0
(b1 + X2s − a1X1s)ds + σ1
t
0
X1sdW1s
X2t = X20 +
t
0
(b2 + X5s − a2X2s)ds + σ2
t
0
X2sdW2s
X5t = X50 +
t
0
(b3 + c3b4 − a3X5s)ds + σ3
t
0
X5sdW5s
so the process collapses to a 3-factor dynamic.
The first situation we want to examine more detailed is when there is a
polynomial relation between B1(T) and B2(T): so we have
dB1
dT
=
1
2
σ2
1B2
1 − a1B1 − 1
dB2
dT
=
1
2
σ2
2B2
2 − a2B2 + B1 − δ2
with the corresponding vector field
Z =
1
2
σ2
1B2
1 − a1B1 − 1
∂
∂B1
+
1
2
σ2
2B2
2 − a2B2 + B1 − δ2
∂
∂B2
7
9. The initial conditions may vary so we set B1(T = 0, z1) = z1 and B2(T =
0, z1, z2) = z2. Let
P(B1, B2) =
L
k,m=0
ckmBk
1 Bm
2
be a polynomial and let P(B1(T, 0), B2(T, 0, 0)) = 0 for all T ≥ 0. We may
assume that this polynomial is prime. The zeroset of P in C2
is a (possibly
singular) algebraic curve. As the real curve T −→ (B1(T, 0), B2(T, 0, 0)) lies
inside this curve and all further derivatives {(B1, B2) with (Zq
)(P)(B1, B2) =
0} and because P is prime we must have Z(P) ∈ (P), so there is some linear
polynomial with
Z(P) = (˜c1B1 + ˜c2B2 + ˜c3) · P
From this we obtain a recurrence relation
(˜c1 −
1
2
σ2
1(k − 1))ck−1,m − (m + 1)ck−1,m+1 = (
1
2
σ2
2(m − 1) − ˜c2)ck,m−1 −
(a1k + a2m + ˜c3)ckm − (m + 1)δ2ck,m+1 − (k + 1)ck+1,m
Let
k0 = max{k ∈ N with ∃m ∈ N with ckm = 0}
m0 = max{m ∈ N with ck0,m = 0}
We set k = k0 + 1 in the recurrence relation and get
(˜c1 −
1
2
σ2
1k0)ck0,m − (m + 1)ck0,m+1 = 0
Putting further m = m0 we get
(˜c1 −
1
2
σ2
1k0)ck0,m0 = 0
or because of ck0,m0 = 0
˜c1 =
1
2
σ2
1k0
But then we conclude 0 = (m + 1)ck0,m+1 for all m = −1, 0, 1, 2, ..., m0 − 1,
so we must have m0 = 0.
8
10. A similar reasoning shows ckm = 0 for m > k0 − k. So all polynomials P
which carry an invariant vector field Z coming from a 2-dimensional cascade
model must have the form
P(B1, B2) = Bk0
1 +
k0−1
k=0
Bk
1 ·
k0−k
m=0
ckmBm
2
For a further specification of polynomial relations among B1(t) and B2(t)
we make use of some explicit representations. The ODE for B1(t, z1) can be
solved explicitly, its solution is
B1(t, z1) = −
2 − z1(h1 − a1) − (2 + z1(h1 + a1))e−h1t
h1 + a1 − σ2
1z1 + (h1 − a1 + σ2
1z1)e−h1t
where h1 = a2
1 + 2σ2
1 > a1. Set for abbreviation
w1 =
σ2
1z1 + h1 − a1
σ2
1z1 − h1 − a1
Then we may rewrite B1(t, z1) as
B1(t, z1) = −
h1 − a1 + (h1 + a1)w1e−h1t
σ2
1(1 − w1e−h1t)
The second ODE
dB2
dt
=
1
2
σ2
2B2
2 − a2B2 + B1 − δ2
can be transformed to a hypergeometric ODE: Set
B2(t, z) = −
2
σ2
2
·
∂
∂t
C2(t, z) /C2(t, z)
then we obtain a linear ODE of second order
∂2
C2
∂t2
+ a2
∂C2
∂t
+
1
2
σ2
2(B1 − δ2)C2 = 0
Define
h2 = a2
2 + 2σ2
2 δ2 +
2
h1 + a1
9
11. and two sequences (˜cn)n, (ˆcn)n
˜cn+1
˜cn
=
n2
+ h2
h1
n + 1
h1
σ2
σ1
2
(n + 1)2 + h2
h1
(n + 1)
ˆcn+1
ˆcn
=
n2
− h2
h1
n + 1
h1
σ2
σ1
2
(n + 1)2 − h2
h1
(n + 1)
with initial ˜c0, ˆc0 = 1. If h2
h1
= q ∈ N, then the second sequence has to be
defined for n ≥ q with initial ˆcq = 1. They satisfy
|ˆcn| <
C
n2
|˜cn| <
C
n2
In the following we make the assumption: h2
h1
/∈ N.
We define two power series
R2(w) =
∞
n=0
˜cnwn
R2(w) =
∞
n=0
ˆcnwn
They fulfill hypergeometric ODEs:
w(1 − w)
d2
R2
dw2
+ 1 −
h2
h1
(1 − w)
dR2
dw
−
1
h1
σ2
σ1
2
R2 = 0
w(1 − w)
d2
R2
dw2
+ 1 +
h2
h1
(1 − w)
dR2
dw
−
1
h1
σ2
σ1
2
R2 = 0
Two linearly independent solutions of
∂2
C2
∂t2
+ a2
∂C2
∂t
+
1
2
σ2
2(B1 − δ2)C2 = 0
can now be constructed by setting
C2(t, z1) = e−1
2
(a2+h2)t
R2 e−h1t
w1
C2(t, z1) = e
1
2
(h2−a2)t
R2 e−h1t
w1
10
12. The Wronskian can be calculated to
∂C2
∂t
(t, z1)C2(t, z1) −
∂C2
∂t
(t, z1)C2(t, z1) = h2
So the general solution C2(t, z1, z2) with initial conditions C2(0, z1, z2) = 1
and ∂C2
∂t
(0, z1, z2) = −1
2
σ2
2z2 is given by
C2(t, z1, z2) = −
1
h2
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1) C2(t, z1) +
1
h2
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1) C2(t, z1)
So far we constructed the general solution space not regarding any special
polynomial restriction. Note that R2(w) can reduce to a polynomial, if there
is an n0 ∈ N with
n2
0 −
h2
h1
n0 +
1
h1
σ2
σ1
2
= 0
Then ˆcn = 0 for n ≥ n0 + 1 and so − 2
σ2
2
· ∂
∂t
C2(t, z) /C2(t, z) will be a
rational function in w1 · e−h1t
and so a rational function in B1(t, z1). Now
we prove that there are no different polynomial relations among B1(t) and
B2(t). So we go back to our polynomial
P(B1, B2) = Bk0
1 +
k0−1
k=0
Bk
1 ·
k0−k
m=0
ckmBm
2
which satisfies Z(P) ∈ (P). Let (z1, z2) be an arbitrary point in the al-
gebraic curve V (P(z1, z2) = 0) ⊂ C2
. If we take this point as an initial
point for the dynamical system t −→ (B1(t, z1), B2(t, z1, z2)), then due to
the invariance property of the ideal (P) under Z the real analytic curve
t −→ (B1(t, z1), B2(t, z1, z2)) lies completely inside the algebraic curve. Now
the explicit formulas for B1(t, z1), B2(t, z) also allow for complex values of t
in a suitable half-plane and by identity they fulfill P(B1(t, z1), B2(t, z)) = 0
for these complex t as well. So we fix a point (z1, z2) with Re(z1) ≤ a1
σ2
1
and
P(z1, z2) = 0 and define for ε > 0
tε =
1
h1
· log(w1) + ε
11
13. Using
B1(t, z1) = −
h1 − a1 + (h1 + a1)w1e−h1t
σ2
1(1 − w1e−h1t)
we get an estimate
1
Cε
< |B1(tε, z1)| <
C
ε
Because of
|ˆcn| <
C
n2
|˜cn| <
C
n2
the series R2(1), R2(1) are absolutely convergent and so we have
∃ lim
ε→0
C2(tε, z1, z2) = −
1
h2
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1) · w
h2−a2
2h1
1 R2(1) +
1
h2
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1) · w
−
h2+a2
2h1
1 R2(1)
We remind that
B2(t, z1, z2) = −
2
σ2
2
·
∂C2
∂t
(t, z)
C2(t, z)
and
∂
∂t
C2(t, z1, z2) = −
1
h2
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1)
∂
∂t
C2(t, z1) +
1
h2
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1)
∂
∂t
C2(t, z1)
Now using the asymptotic of ˜cn, ˆcn for n −→ ∞ one can easily show
∂C2
∂t
(tε, z1) < C| log(ε)|
∂C2
∂t
(tε, z1) < C| log(ε)|
12
14. and so
∂C2
∂t
(tε, z1, z2) < C| log(ε)|
where the constant C certainly depends on (z1, z2). If limε→0 C2(tε, z1, z2) =
0, then we would have |B2(tε, z1, z2)| < C| log(ε)|. But now we remind
P(B1(tε, z1), B2(tε, z1, z2)) = Bk0
1 (tε, z1) +
k0−1
k=0
Bk
1 ·
k0−k
m=0
ckmBm
2 = 0
and
1
Cε
< |B1(tε, z1)| <
C
ε
If |B2(tε, z1, z2)| would only grow logarithmically as ε −→ 0 the term |Bk0
1 (tε, z1)| ∼
1
εk0
would dominate all other terms in P and so P(B1(tε, z1), B2(tε, z1, z2)) =
0 for all sufficiently small ε > 0 in contradiction to our assumption. So we
conclude:
0 = lim
ε→0
C2(tε, z1, z2) = −
1
h2
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1) · w
h2−a2
2h1
1 R2(1) +
1
h2
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1) · w
−
h2+a2
2h1
1 R2(1)
respectively
0 =
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1) · w
h2
h1
1 R2(1) −
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1) · R2(1)
for all (z1, z2) ∈ C2
in the algebraic curve P(z1, z2) = 0. (0, 0) must belong
to the curve by assumption and we may start the dynamic t ∈ (−ε0, ε0) −→
(B1(t, 0), B2(t, 0, 0)), so for all z1 real and small positive or negative there is a
(real) z2 with P(z1, z2) = 0. For these z1 we have w1 = w1(z1) =
σ2
1z1+h1−a1
σ2
1z1−h1−a1
<
0. For this choice all the quantities R2(1), R2(1), ∂C2
∂t
(0, z1) + 1
2
σ2
2z2C2(0, z1)
13
15. and ∂C2
∂t
(0, z1) + 1
2
σ2
2z2C2(0, z1) are real but w
h2
h1
1 has a nontrivial imag-
inary part because of h2
h1
/∈ N. By identity we may conclude
R2(1) ·
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1) = 0
R2(1) ·
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1) = 0
for all (z1, z2) inside the algebraic curve P = 0. We remind that because of
˜cn > 0 we have R2(1) > 1 positive so we end up with
R2(1) ·
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1) = 0
∂C2
∂t
(0, z1) +
1
2
σ2
2z2C2(0, z1) = 0
If R2(1) = 0 then would get
∂C2
∂t
(0, z1)C2(0, z1) −
∂C2
∂t
(0, z1)C2(0, z1) = 0
in contradiction to the Wronskian = h2 > 0. So we have R2(1) = 0. Now
we remind that w −→ R2(w) is a hypergeometric function whose auxiliary
parameters can be read of the ODE
w(1 − w)
d2
R2
dw2
+ 1 −
h2
h1
(1 − w)
dR2
dw
−
1
h1
σ2
σ1
2
R2 = 0
so
γ = 1 + α + β = 1 −
h2
h1
αβ =
1
h1
σ2
σ1
2
ˆcn+1
ˆcn
=
(n + α) · (n + β)
(n + 1) · (n + γ)
14
16. This implies γ − α − β = 1 > 0 and so Gaussian-summation-formula is
applicable:
R2(1) =
Γ(γ) · Γ(γ − α − β)
Γ(γ − α) · Γ(γ − β)
with
Γ(z) =
1
z
· e−const·z
·
1
∞
m=1(1 + z
m
) · e− z
m
As R2(1) = 0 and Γ(z) = 0 we must have Γ(γ − α) = ∞ or Γ(γ − β) = ∞.
In either case we get α = −q − 1 or β = −q − 1 with q ∈ N and then the
series (ˆcn) terminates at q +1. So we have shown that R2(w) is a polynomial
and further for all (z1, z2) in the algebraic curve
z2 = −
2
σ2
2
·
∂C2
∂t
(0, z1)
C2(0, z1)
= −
2
σ2
2
·
1
2
(h2 − a2) − h1w1
dR2
dw
(w1)
R2(w1)
Now (B1(t, 0), B2(t, 0, 0)) are by assumption in the algebraic curve, so setting
z1 = B1(t, 0) and z2 = B2(t, 0, 0) we get
w1 = w1(z1) =
σ2
1B1(t, 0) + h1 − a1
σ2
1B1(t, 0) − h1 − a1
= −
h1 − a1
h1 + a1
· e−h1t
and
B2(t) = −
2
σ2
2
·
1
2
(h2 − a2) + h1
h1 − a1
h1 + a1
e−h1t
dR2
dw
(−h1−a1
h1+a1
e−h1t
)
R2(−h1−a1
h1+a1
e−h1t)
So we end up with the Proposition:
Proposition 1 All polynomial relations among the first two cascade struc-
ture model coefficient functions arise from termination of (ˆcn) and in this
case B2(t) is a rational function of B1(t).
The next step is to explore wether there are parameter combinations with
B2(t) rational in B1(t) and the cascade model is flexible in the sense of
d
dt
−B2(t)
t
(t = 10) ≥ 0. For that purpose we first derive some estimates.
15
17. We know that t −→ B2(t) < 0 is falling and converges against limt→∞ B2(t) =
−h2−a2
σ2
2
. So we introduce B2(t) = B2(t) + h2−a2
σ2
2
> 0 and get from the ODE
of B2(t)
dB2
dt
=
1
2
σ2
2B2
2 − h2B2 + B1 +
2
h1 + a1
with initial B2(0) = h2−a2
σ2
2
. Now we have
0 < B1(t) +
2
h1 + a1
<
4h1e−h1t
(h1 + a1)2
The ODE can be written as
dB2
dt
=
1
2
σ2
2B2 − h2 B2 + B1 +
2
h1 + a1
Using that t −→ B2(t) is falling from h2−a2
σ2
2
> 0 we see
1
2
σ2
2B2 − h2 ≤
1
2
(h2 − a2) − h2 = −
1
2
(a2 + h2) < 0
So we can get an upper bound for B2(t) by solving
dB2
dt
= −
1
2
(a2 + h2)B2 +
4h1e−h1t
(h1 + a1)2
which yields
B2(t) =
h2 − a2
σ2
2
e−1
2
(a2+h2)t
+
8h1
(h1 + a1)2(a2 + h2 − 2h1)
(e−2h1t
− e−1
2
(a2+h2)t
)
and
0 < B2(t) +
h2 − a2
σ2
2
< B2(t)
On the other hand we have
dB2
dt
=
dB2
dt
=
1
2
σ2
2B2
2 − h2B2 + B1 +
2
h1 + a1
> −h2B2 > −h2B2
16
18. In the end we come up with
∆(t) = t2
·
d
dt
−
B2(t)
t
= B2(t) − t
dB2
dt
≤
−
h2 − a2
σ2
2
+ (1 + h2t) · B2(t)
Now the flexibility condition ∆(t = 10) ≥ 0 can be tested numerically against
the termination condition
∃q ∈ N with
h2
h1
= q +
1
qh1
σ2
σ1
2
and the initial condition
0 =
1
2
(h2 − a2) + h1
h1 − a1
h1 + a1
dR2
dw
(−h1−a1
h1+a1
)
R2(−h1−a1
h1+a1
)
We could not find any parameter combinations fulfilling all three conditions.
In the next step we return to the 3-factor cascade setting with a quadratic
relation coming from the reduction of a fourth factor. We remind the linear
relation
B4(T) =
3
j=1
cjBj(T) for all T ≥ 0
by assumption and from that we obtained by applying the vector field the
following relation
1
2
σ2
4
3
j=1
cjBj
2
− a44
3
j=1
cjBj + a41B1 + a42B2 + a43B3 − δ4 =
3
j=1
cj
1
2
σ2
j B2
j − ajBj + Bj−1 − δj
or
P2(B1, B2, B3) =
1
2
c1(σ2
4c1 − σ2
1)B2
1 +
1
2
c2(σ2
4c2 − σ2
2)B2
2 +
1
2
c3(σ2
4c3 − σ2
3)B2
3 +
σ2
4(c1c2B1B2 + c1c3B1B3 + c2c3B2B3) + B1(a41 + c1a1 − c2 − c1a44) +
B2(a42 + c2a2 − c3 − c2a44) + B3(a43 + c3a3 − c3a44) − δ4 + c1δ1 + c2δ2 + c3δ3 = 0
17
19. Due to the cascade structure of B1(t), B2(t), B3(t) there is a natural restric-
tion of the 4-vector field to a 3-dimensional or further 2-dimensional setting.
So we can apply the vector field
Z3 =
3
j=1
1
2
σ2
j B2
j − ajBj + Bj−1 − δj
∂
∂Bj
to P2 and gain a polynomial P3(B1, B2, B3) = Z3(P2) of degree 3. In the
generic case P2 and P3 intersect transversely, so elimination of B3 will in the
generic case result in a nontrivial polynomial relation between B1, B2. But
we have shown that this is in conflict with the desired flexibility condition.
There are two further special cases possible: first the ideal generated by P2, P3
is induced by one single linear polynomial in B1, B2, B3. But by applying Z3
once more this will result in a polynomial relation among B1, B2. Or second
P3 is in the ideal generated by P2 which means
P3 = (˜c1B1 + ˜c2B2 + ˜c3B3 + ˜δ) · P2
Comparing coefficients of highest order 3 in this equation yields under the
assumption cj = 0 for j = 1, 2, 3
(σ2
j − ˜cj) · (σ2
4cj − σ2
j ) = 0 for j = 1, 2, 3
σ2
4cj(2˜ck − σ2
k) + ˜cj(σ2
4ck − σ2
k) = 0 for j = k
˜c1c2c3 + ˜c2c1c3 + ˜c3c2c1 = 0
The first case is ˜cj = σ2
j for all j = 1, 2, 3. Then we get
σ2
j ck + σ2
kcj =
σ2
j σ2
k
σ2
4
and
0 = σ2
1c2c3 + σ2
2c1c3 + σ2
3c2c1
From the first three equations we infer
cj =
σ2
j
2σ2
4
18
20. Inserting these expressions in the fourth equation leads to the contradiction
0 =
3σ2
1σ2
2σ2
3
2σ2
4
The second case is
˜c1 = σ2
1
˜c2 = σ2
2
σ2
4c3 = σ2
3
The further equations then imply ˜c3 = 1
2
σ2
3 and
˜c3 =
1
2
σ2
3
c1 = −
σ2
1
σ2
4
c2 = −
σ2
2
σ2
4
which is in contradiction to
σ2
4c2σ2
1 + σ2
2(σ2
4c1 − σ2
1) = 0
The last case where
σ2
4cj = σ2
j for j = 1, 2, 3
implies
˜ck =
1
2
σ2
k
which is in contradiction to
˜c1c2c3 + ˜c2c1c3 + ˜c3c2c1 = 0
So we end up with the
Proposition 2 There are no flexible 4-factor cascade structure affine models
which allow for unspanned volatility.
19
21. References
Collin-Dufresne,P. and R.Goldstein(2002), ”Do bonds span the fixed income
markets? Theory and evidence for unspanned volatility”, Journal of Finance
57, 1685-1730
Cox,D.A., J.B.Little and D.O’Shea(1992), Ideals, Varieties and Algorithms,
Springer
Duffie,D. and R.Kan(1996), ”A yield factor model of interest rates”, Mathe-
matical Finance 6, 379-406
20
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