Analysis and Linear Algebra for Finance: Part II Patrick Roger Download free books at Patrick Roger Analysis and Linear Algebra for Finance: Part II Patrick ROGER LaRGE Research Center EM Strasbourg Business School University of Strasbourg Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part II First edition © 2013 Patrick Roger & bookboon.com (Ventus Publishing ApS) ISBN 978-87-403-0429-9 Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part II Contents Contents Introduction Vector spaces and linear mappings 1.1 Vector spaces: definitions and general properties 1.2 Linear mappings 25 1.3 Finite-dimensional spaces and matrices 33 1.4 Norms and inner products 55 1.5 Hilbert spaces 61 1.6 Separation theorems and Farkas lemma 64 Functions of several variables 73 2.1 Metric spaces 74 2.2 Continuity and differentiability 84 2.3 Implicit and homogeneous functions 106 Fast-track your career Masters in Management Stand out from the crowd Designed for graduates with less than one year of full-time postgraduate work experience, London Business School’s Masters in Management will expand your thinking and provide you with the foundations for a successful career in business The programme is developed in consultation with recruiters to provide you with the key skills that top employers demand Through 11 months of full-time study, you will gain the business knowledge and capabilities to increase your career choices and stand out from the crowd London Business School Regent’s Park London NW1 4SA United Kingdom Tel +44 (0)20 7000 7573 Email mim@london.edu Applications are now open for entry in September 2011 For more information visit www.london.edu/mim/ email mim@london.edu or call +44 (0)20 7000 7573 www.london.edu/mim/ Download free eBooks at bookboon.com Click on the ad to read more Analysis and Linear Algebra for Finance: Part II Contents Optimization without constraints 113 3.1 Preliminaries 114 3.2 Optimizing a single-variable function 120 3.3 Optimizing a function of two variables 124 3.4 Functions of n variables 131 Constrained optimization 137 4.1 Functions of two variables and equality constraint 138 4.2 Functions of p variables with m equality constraints 145 4.3 Functions of p variables with mixed constraints 150 Index 154 Download free eBooks at bookboon.com Click on the ad to read more Analysis and Linear Algebra for Finance: Part II                                                                                                                                                                                                                                                                                                       Download free eBooks at bookboon.com Introduction Analysis and Linear Algebra for Finance: Part II Vector spaces and linear mappings                                                                                                                                                                                                               Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part II  Vector spaces and linear mappings                                                                                                                                                                                                                                              Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part II Vector spaces and linear mappings                                                                                                                                                     your chance to change the world Here at Ericsson we have a deep rooted belief that the innovations we make on a daily basis can have a profound effect on making the world a better place for people, business and society Join us In Germany we are especially looking for graduates as Integration Engineers for • Radio Access and IP Networks • IMS and IPTV We are looking forward to getting your application! To apply and for all current job openings please visit our web page: www.ericsson.com/careers Download free eBooks at bookboon.com Click on the ad to read more Analysis and Linear Algebra for Finance: Part II  Vector spaces and linear mappings                                                                                                                                                                                                                                                                                                          Download free eBooks at bookboon.com 10      Constrained optimization Analysis and Linear Algebra for Finance: Part II                                                                                                                                                                                                                                               Brain power By 2020, wind could provide one-tenth of our planet’s electricity needs Already today, SKF’s innovative knowhow is crucial to running a large proportion of the world’s wind turbines Up to 25 % of the generating costs relate to maintenance These can be reduced dramatically thanks to our systems for on-line condition monitoring and automatic lubrication We help make it more economical to create cleaner, cheaper energy out of thin air By sharing our experience, expertise, and creativity, industries can boost performance beyond expectations Therefore we need the best employees who can meet this challenge! The Power of Knowledge Engineering Plug into The Power of Knowledge Engineering Visit us at www.skf.com/knowledge Download free eBooks at bookboon.com 142 Click on the ad to read more Analysis and Linear Algebra for Finance: Part II Constrained optimization                                                                                                                                                                                      Download free eBooks at bookboon.com 143 Analysis and Linear Algebra for Finance: Part II  Constrained optimization                                                                                                                                                                                                                                                                                       Download free eBooks at bookboon.com 144 Analysis and Linear Algebra for Finance: Part II Constrained optimization                                                                                                                                                                                                                                                                  Download free eBooks at bookboon.com 145 Analysis and Linear Algebra for Finance: Part II  Constrained optimization                                                                                        The financial industry needs a strong software platform That’s why we need you SimCorp is a leading provider of software solutions for the financial industry We work together to reach a common goal: to help our clients succeed by providing a strong, scalable IT platform that enables growth, while mitigating risk and reducing cost At SimCorp, we value commitment and enable you to make the most of your ambitions and potential Are you among the best qualified in finance, economics, IT or mathematics? Find your next challenge at www.simcorp.com/careers www.simcorp.com MITIGATE RISK REDUCE COST ENABLE GROWTH Download free eBooks at bookboon.com 146 Click on the ad to read more Analysis and Linear Algebra for Finance: Part II Constrained optimization                                                                                                                                                                                                                                                                  Download free eBooks at bookboon.com 147 Analysis and Linear Algebra for Finance: Part II  Constrained optimization                                                                                                                                                                                                                                                                                                                                                                                  Download free eBooks at bookboon.com 148 Analysis and Linear Algebra for Finance: Part II Constrained optimization                                                                                                                                                                                                                                                                                       Download free eBooks at bookboon.com 149 Analysis and Linear Algebra for Finance: Part II  Constrained optimization                                                                                                                                                                                                                                                Download free eBooks at bookboon.com 150 Analysis and Linear Algebra for Finance: Part II Constrained optimization                                                                                                                                                                                                                                                                                                                                        Download free eBooks at bookboon.com 151 Analysis and Linear Algebra for Finance: Part II  Constrained optimization                                                                                                                                                                                                                                                                                 Download free eBooks at bookboon.com 152 Analysis and Linear Algebra for Finance: Part II Constrained optimization                                                                                                                 Download free eBooks at bookboon.com 153 Click on the ad to read more Analysis and Linear Algebra for Finance: Part II Index Index convex, 104 homogeneous, 109 Arrow-Debreu security, 22 gradient, 88 ball Hessian, 97 closed, 79 open, 78 identity basis, 20 implicit function change, 45 element, theorem, 109 closed, 79 Implicit function theorem, 106 combination indifference curve, 93 convex, 103 inner product linear, 15 norm, 56 curve interior, 79 isomorphism, 28 indifference, 93 derivative Lagrangian, 140 partial, 86 multiplier, 140 determinant, 41, 42 lemma differential, 91 alternative, 66 distance Farkas, 66 limit, 84 metric, 74 linear eigenspace, 50 combination, 15 eigenvalue, 49 linear functional, 31 eigenvector, 49 endomorphism, 40 mapping linear, 25 Farkas lemma, 66 compound, 35 form functional, 32 isomorphism, 28 quadratic, 60 formula kernel, 26 matrix, 33 Taylor, 99 function C1, 86 marginal rate of substitution, 93 concave, 104 matrix continuous, 85 Download free eBooks at bookboon.com 154 quadratic form, 60 change-of-basis, 45 Analysis and Linear Algebra for Finance: Part II Index cofactor, 41 sequence definite, 57 Cauchy, 84 diagonalizable, 54 convergent, 84 diagonalization, 49 set eigenvalue, 49 bounded, 83 eigenvector, 49 closed, 79 Hessian, 97 closure, 81 identity, 40 compact, 83 orthogonal, 55 convex, 62, 103 principal minor, 41 dense, 82 semidefinite, 57 exterior, 81 maximum frontier, 81 global, 121, 150 interior, 81 local, 121, 122, 129, 145, 149 open, 79 metric solution distance, 74 Euclidian, 75 space space, 75 Hilbert, 61 minimum isomorphic, 29 global, 150 metric, 75 local, 129, 145, 149 separable, 82 corner, 116 multiplier sum direct, 13 vectors, 12 Kuhn-Tucker, 152 net present value, 108 norm, 55 theorem notation Euler, 111 mean value, 97 Landau, 100 numéraire, 71 vector optimum Arrow-Debreu, 22 global, 150 direct sum, 13 local, 126, 132, 147 family option rank, 24 spanning, 19 call, 15 linearly polynomial dependent, 16 independent, 17 characteristic, 50 Download free eBooks at bookboon.com 155 orthogonal, 59 Analysis and Linear Algebra for Finance: Part II rank, 24 space, Index basis, 20 finite-dimensional, 22 identity element, subspace, 11 supplementary, 13 sum, 12 vector space Hilbert, 61 inner product, 56 norm, 55 156