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# Notes04 - Lecture 4 Multivariate Distributions Scalar...

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1 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Lecture 4 – Multivariate Distributions Scalar functions of vectors and their derivatives Vector functions of vectors and their derivatives Multivariate probability theory Eigenvalues and Eigenvectors Quadratic forms 2 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions Let x be an n-dimensional vector such that >@ 12 n x xx c c ! " Let y = f(x 1 ,x 2 ,…,x n ) be a continuous, twice- differentiable function of the n real variables x 1 , x 2 , …, x n We will write this function as follows ± ² yf ! 3 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions Example: ±² , 11 1 , 22 2 nn ii j j ij x ax c ¦¦ !! " ! ! " ! Gradient Vector ± ² f y x x w w ww !

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4 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions The gradient is the n by 1 vector of first partial derivatives ±² 1 2 n y x y y x f y x w ªº «» w w w w ³ w w w ¬¼ ! ! " 5 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions Example: Consider the quadratic form defined before 1 2 f c ³ ´ !" " ! Homework assignment! Hessian Matrix This is the matrix of second partial derivatives 6 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions 22 11 ii i in ni yy x xx x y x x x x x ww §· ¨¸ ©¹ ! ""
7 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions 22 2 2 12 1 1 2 2 2 2 2 2 2 2 n n nn n yy y x xx y x x x y x ªº ww w «» w w w w §· w w ¨¸ w ©¹ w w ¬¼ !! ! " " "" " " " 8 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions Example: ±² 13 23 1 2 3 yf x x y x y x y x ³ ³ w ³ w w ³ w w ³ w ! 9 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions 222 2 1 3 1 22 2 2 2 32 2 3 011 101 110 fx x yyy x x x yyy y x x ³ ´ ³ ³ www w w w www w w ww ww w ! !

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10 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions Vector Valued Functions of a Vector Consider the following n functions each of which is a function of an n vector ± ² ±² 11 22 nn yg ! ! ! " 11 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions We can represent these functions as a vector ± ² 1 2 n g g g ªº «» ¬¼ ! ! #\$ ! ! " 12 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions Jacobian of g(x) 1 12 2 n n n n g gg x xx g x g x ww w w w w w w w w \$ %\$ ! " " """" "
13 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions Example ±² 11 2 3 21 2 3 31 2 1 3 2 3 23 13 12 231312 111 gx x x x x x x x x x x xx x J xxxxxx ³³ ªº «» ³³³ ¬¼ ! ! ! \$ 14 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Multivariate Functions The multivariate functions in portfolio applications arise in the following contexts 1. Multivariate Probability Theory 2. Portfolio Optimization 3. Risk Engineering We first develop the necessary tools for

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Notes04 - Lecture 4 Multivariate Distributions Scalar...

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