From 64077323e885b0122c8bd570716cf40932cef150 Mon Sep 17 00:00:00 2001
From: GiuseppeMascolo <32720158+GiuseppeMascolo@users.noreply.github.com>
Date: Mon, 27 Nov 2017 16:48:50 +0000
Subject: [PATCH 1/3] Exercise 2.2.3

This exercise has been copied from Meucci code and commented in order to understand it.
---
 exercise223/LognormalCopulaPDF.m | 23 ++++++++++++++++++
 exercise223/exercise223_main.m   | 40 ++++++++++++++++++++++++++++++++
 2 files changed, 63 insertions(+)
 create mode 100644 exercise223/LognormalCopulaPDF.m
 create mode 100644 exercise223/exercise223_main.m

diff --git a/exercise223/LognormalCopulaPDF.m b/exercise223/LognormalCopulaPDF.m
new file mode 100644
index 0000000..f0b6d4e
--- /dev/null
+++ b/exercise223/LognormalCopulaPDF.m
@@ -0,0 +1,23 @@
+function F_U = LognormalCopulaPDF(u,Mu,Sigma)
+% this function computes the pdf of the copula of the lognormal distribution
+% at the generic point u in the unit hypercube
+% see formula 2.30
+% we need the joint distribution of X as numerator
+% and product of marginal distributions as denominator
+
+N=length(u);         % dimension of the hypercube
+s=sqrt(diag(Sigma)); % st. deviations 
+
+x=logninv(u,Mu,s);   %from uniform we generate a lognormal sample with the 
+                     % inverse of lognormal cdf
+
+%numerator is the joint distribution of a lognormal multivariate variable as defined in 2.213 and 2.156                  
+Numerator = (2*pi)^(-N/2) * (  (det(Sigma))^(-.5) ) /prod(x) * exp(-.5*(log(x)-Mu)'*inv(Sigma)*(log(x)-Mu));
+
+%denominator is the product of marginal distributions
+fs=lognpdf(x,Mu,s); %returns values of the lognormal pdf at x 
+Denominator = prod(fs);
+
+F_U = Numerator/Denominator; %copula pdf as in 2.30
+
+
diff --git a/exercise223/exercise223_main.m b/exercise223/exercise223_main.m
new file mode 100644
index 0000000..49065bd
--- /dev/null
+++ b/exercise223/exercise223_main.m
@@ -0,0 +1,40 @@
+% Exercise 2.2.3 - main
+% *Author: Giuseppe Mascolo*
+% Logonrmal copula pdf
+
+close all; 
+clc; clear;
+
+% input parameters
+Mu=[1  -1]';   % exp values   
+r=0.7;              % correlation
+sigmas=[1 1]';   % st. deviations 
+nu=1;               % degree of freedom
+
+%if we want a specific Sigma we just have to change sigmas and r
+Sigma=diag(sigmas)*[1 r;r 1]*diag(sigmas);
+
+%% compute and display results
+
+%creates grid
+GridSide1=[.05:.05:.95];
+GridSide2=GridSide1;
+NumGrid=length(GridSide1);
+
+f_U=zeros(NumGrid); %in order to plot
+
+%loop inside the grid
+for j=1:NumGrid
+    for k=1:NumGrid
+        u=[GridSide1(j)
+            GridSide2(k)];
+        f_U(j,k)=LognormalCopulaPDF(u,Mu,Sigma); %function returns copula pdf
+    end
+end
+    
+
+[G1,G2]=meshgrid(GridSide1,GridSide2); %creates 2D grid
+surf(G1,G2,f_U)  %3D surface plot
+xlabel('U_1')
+ylabel('U_2')
+zlabel('Copula pdf')

From 2e71aee62b709f568fa61a1b4217fafe58e6697a Mon Sep 17 00:00:00 2001
From: GiuseppeMascolo <32720158+GiuseppeMascolo@users.noreply.github.com>
Date: Mon, 27 Nov 2017 17:51:58 +0000
Subject: [PATCH 2/3] deleted useless nu

---
 exercise223/exercise223_main.m | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/exercise223/exercise223_main.m b/exercise223/exercise223_main.m
index 49065bd..fb117fc 100644
--- a/exercise223/exercise223_main.m
+++ b/exercise223/exercise223_main.m
@@ -8,8 +8,7 @@
 % input parameters
 Mu=[1  -1]';   % exp values   
 r=0.7;              % correlation
-sigmas=[1 1]';   % st. deviations 
-nu=1;               % degree of freedom
+sigmas=[1 1]';   % st. deviations
 
 %if we want a specific Sigma we just have to change sigmas and r
 Sigma=diag(sigmas)*[1 r;r 1]*diag(sigmas);

From cd603e50addcd611503ea23a7872d02076d86ec9 Mon Sep 17 00:00:00 2001
From: GiuseppeMascolo <32720158+GiuseppeMascolo@users.noreply.github.com>
Date: Mon, 27 Nov 2017 17:54:17 +0000
Subject: [PATCH 3/3] added a note

---
 exercise223/exercise223_main.m | 5 ++++-
 1 file changed, 4 insertions(+), 1 deletion(-)

diff --git a/exercise223/exercise223_main.m b/exercise223/exercise223_main.m
index fb117fc..bcb765e 100644
--- a/exercise223/exercise223_main.m
+++ b/exercise223/exercise223_main.m
@@ -7,7 +7,7 @@
 
 % input parameters
 Mu=[1  -1]';   % exp values   
-r=0.7;              % correlation
+r=-0.7;              % correlation
 sigmas=[1 1]';   % st. deviations
 
 %if we want a specific Sigma we just have to change sigmas and r
@@ -37,3 +37,6 @@
 xlabel('U_1')
 ylabel('U_2')
 zlabel('Copula pdf')
+
+%notice that when r is zero we have a flat copula pdf, which means
+%independence