Analysis and Linear Algebra for Finance Part i

Patrick Roger

Analysis and Linear Algebra forFinance: Part I

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Patrick Roger

Analysis and Linear Algebra for Finance: Part I

Patrick ROGERLaRGE Research Center

EM Strasbourg Business SchoolUniversity of Strasbourg

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Analysis and Linear Algebra for Finance: Part IFirst edition© 2013 Patrick Roger & bookboon.com (Ventus Publishing ApS)ISBN 978-87-403-0412-1

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Contents

Contents

Introduction 6

1 Preliminaries 71.1 Sets and subsets 81.2 Binary relations 171.3 Mappings 201.4 Topology of

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2 Functions of one variable 432.1 Definitions and notations 442.2 Limits and continuity 502.3 Differentiation 612.4 Logarithms and exponential functions 702.5 Polynomial approximations and Taylor formula 75

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Contents

3 Integrals 813.1 Integral of a step function 813.2 General case 863.3 Computations 943.4 Improper integrals 98

4 Matrices 1014.1 Definitions 1014.2 Elementary algebra on matrices 1044.3 Linear equations 115

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Introduction




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Preliminaries




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Integrals




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Index

Indexbijection, 22bound infimum, 28 lower, 27, 49 supremum, 28 upper, 27, 49boundary, 30

cardinality, 9closure, 30compact, 31complement, 32Consol bond, 36

derivative, 62 higher order, 68 left, 63

right, 63 second-order, 68

exterior, 30

frontier, 30function, 44 affine, 44 bounded, 49 compound, 67 concave, 61 continuous, 55 convex, 61 decreasing, 46 derivative, 62 differentiable, 62 domain, 44 even, 45 increasing, 46 left-continuous, 56


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Index

limit, 50 linear, 44 monotonic, 57 odd, 45 range, 44 Riemann-integrable, 87 right-continuous, 55

image, 20inclusion, 11injection, 22integral improper, 98 of a step function, 83 Riemann, 87interior, 30intersection, 12interval, 28

limit, 50 infinite, 53 left, 52 right, 51

Mac Laurin formula, 78mapping, 20 bijective, 22 identity, 24 image, 20 injective, 22 inverse, 23 one-to-one, 22 range, 20 surjective, 22matrix, 101 diagonal, 102 identity, 110 inverse, 110 invertible, 110

null, 102 product, 107 product by a real number, 106 square, 102 substraction, 106 transpose, 104 triangular, 102maximum global, 50 local, 49method Gauss, 120 substitution, 118minimum global, 49 local, 49

ordering, 17 complete, 18 partial, 18

partition, 15pre-ordering, 17

range, 20 reciprocal, 22relation asymmetric, 17 binary, 17 equivalence, 19 graph, 20 ordering, 17 refl?exive, 17 symmetric, 17 transitive, 17

sequence, 32 adjacent, 41, 87 arithmetic, 36





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Index

Cauchy, 34 convergence, 33 decreasing, 37 geometric, 36 increasing, 37set, 8 boundary, 30 bounded, 28, 31 cardinality, 9 closed, 30 closure, 30 compact, 31, 35 complement, 12 element, 8 empty, 10 exterior, 30 frontier, 30 inclusion, 11 interior, 29, 30 intersection, 12

lower bound, 27 neighborhood, 29 null, 10 open, 30 partition, 14 subset, 11 union, 12 upper bound, 27step function, 82subdivision, 82subset, 11surjection, 22system Cramer, 117 homogeneous, 117 impossible, 117 indeterminate, 117 square, 117

Taylor

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127

Index

expansion, 79 formula, 78 Mac-Laurin, 78

union, 12

variable dependent, 44 independent, 44

Documents

Analysis and Linear Algebra for Finance Part i