I'm reading the paper by Zhao et al (2008) and have a problem with used definitions in the text on the page 1535.
First, we generate a sample, $R$, of a given size from the distribution (21). Let $\hat{\mu}$ and $\hat{\sigma}^2$ be the sample mean vector and the sample variance covariance matrix. Then, we modify the generated sample to $$\hat{R}=\mu + (R-\hat{\mu})\hat{\sigma}^{-1}\sigma.$$ Then, the modified sample $\hat{R}$ has the same first and second moments as the original distribution. To further test, whether the modified sample is arbitrage free, we use the Matlab backslash function to examine whether the solution to $(1 + R)^{\top} \backslash 1$ is componentwise positive.
Question. What means $1$ in the last line? Is it an identity matrix or a column vector of ones? And what is dimensions of this $1$?
I have tried to examine the solution from Table 1.
library(pracma)
n <- 5
R<-matrix(
c(
0.0025, 0.0377, 0.0110, 0.0769, 0.0047,
0.0025, 0.0431, 0.0001, 0.0045, 0.0562,
0.0025, 0.0469, 0.0643, 0.0400, 0.0370,
0.0025, 0.0504, 0.0422, 0.0169, 0.0333,
0.0025, 0.0596, 0.0038, 0.1896, 0.0663), ncol=n)
#I<-ones(n)
I <-diag(n)
mldivide(t(I+R),I)
Add after JejeBelfort's comment. If the $1$ is a column vector of ones then what is $+$? Union operation or Kronecker product operator?
I <- rep(1, n)
cbind(I, R)
Reference.
Yonggan Zhao, William T. Ziemba (2008) Calculating risk neutral probabilities and optimal portfolio policies in a dynamic investment model with downside risk control. European Journal of Operational Research 185 (2008) 1525–1540.