Thank you. Do you know what does the $\bar{P}^{(i-1)}$ mean? Based on the author's definition of $\bar{P}$, can i say that the first column of $\bar{P}^{(i-1)}$ means the probability distribution of $X_{i}$ if $X_{i-1} = ln(u)$ and the second column means the probability distribution of $X_{i}$ if $X_{i-1} = ln(d)$