Paragraph ended before \align was complete

Question

I have a problem that has already been reported here, but I checked and I don't see any \begin{align} not closed through \end{align}. Another thing that's bothering me is the E for the expected value, it looks different than the ones I saw on article or the one on Wikipedia.

\documentclass[a4paper,11pt,twoside] {book}
\usepackage{hyperref}
\usepackage[latin1]{inputenc}
\usepackage[french]{babel}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{array}
\usepackage{geometry}
\usepackage{xspace}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{bbold}

\begin{document}

\chapter[Nom court chapitre]{My chapter}

$P=\sum_{i=1}^{r} \sum_{j=1}^{N_{i}} \mathbb 1_{T_{j}^{i}\geqslant 1}X_{i}$
$F_{1}(1) =\sum_{i=1}^{r} M_{i}(S_{i}(1)-K_{i})$
\begin{align}
P-F_{1}(1) &=\sum_{i=1}^{r} \sum_{j=1}^{N_{i}} \mathbb 1_{T_{j}^{i}\geqslant 1}X_{i} -\sum_{i=1}^{r} M_{i}(S_{i}(1)-K_{i}) \\
        &= \sum_{i=1}^{r} (\sum_{j=1}^{N_{i}} \mathbb 1_{T_{j}^{i}\geqslant 1}X_{i}-M_{i}(S_{i}(1)-K_{i}))
\end{align}
donc
\begin{align}
Var(P-F_{1}(1)) &=Var(P) + Var(\sum_{i=1}^{r} M_{i}(S_{i}(1)-K_{i})) -2Cov(P,F_{1}(1)) \\
            &=Var(P) + \sum_{i=1}^{r} M_{i}^2Var(S_{i}(1)) -2Cov(\sum_{i=1}^{r} \sum_{j=1}^{N_{i}} \mathbb 1_{T_{j}^{i} \geqslant 1}X_{i},\sum_{i=1}^{r} M_{i}(S_{i}(1))) \\
            &=Var(P) + \sum_{i=1}^{r} M_{i}^2Var(S_{i}(1)) - 2(\mathbb{E}[\sum_{i=1}^{r} \sum_{j=1}^{N_{i}} \mathbb 1_{T_{j}^{i} \geqslant 1}X_{i}\sum_{l=1}^{r} M_{l}(S_{l}(1))]-\mathbb{E}[P]\mathbb{E}[\sum_{i=1}^{r} M_{i}(S_{i}(1))]) \\
            &=Var(P) + \sum_{i=1}^{r} M_{i}^2Var(S_{i}(1)) - 2(\sum_{i=1}^{r}\sum_{j=1}^{N_{i}}\sum_{l=1}^{r} X_{i}M_{l} \mathbb{E}[\mathbb 1_{T_{j}^{i} \geqslant 1}S_{l}(1)]-\mathbb{E}[P]\mathbb{E}[\sum_{i=1}^{r} M_{i}(S_{i}(1))]
\end{align}

Comme la fonction de survie est stochastique, les termes d'espérance et variance de P doivent être détaillés:

\begin{align}
\mathbb{E}[\mathbb 1_{T_{j}^{i}\geqslant 1}S_{l}(1)]&=\mathbb{E}[\mathbb{E}[\mathbb 1_{T_{j}^{i}\geqslant 1}S_{l}(1)/]\kappa_{1}^((1)),\kappa_{1}^((2))] \\
&=\mathbb{E}[S_{l}(l)\mathbb{E}[\mathbb 1_{T_{j}^{i}/\kappa_{1}^((1)),\kappa_{1}^((2))]
\end{align}
Nous avons également que

\begin{align}
\mathbb{E}[P] &=\sum_{i=1}^{r} X_{i} \sum_{j=1}^{N_{i}} \mathbb{E}[\mathbb 1_{T_{j}^{i}\geqslant 1}] \\
&=\sum_{i=1}^{r} X_{i} \sum_{j=1}^{N_{i}} \mathbb{E}[\mathbb{E}[\mathbb 1_{T_{j}^{i}\geqslant 1}/S_{i}(1)]] \\
&=\sum_{i=1}^{r} X_{i} \sum_{j=1}^{N_{i}} \mathbb{E}[\mathbb{P}[T_{j}^{i}\geqslant 1/S_{i}(1)]] \\
&=\sum_{i=1}^{r} X_{i} \sum_{j=1}^{N_{i}} \mathbb{E}[S_{i}(1)]\\
&=\sum_{i=1}^{r} X_{i}N_{i}\mathbb{E}[S_{i}(1)]
\end{align}
Or nous savons que \\
\begin{center}
$Var(P)= \mathbb{E}[Var(P/S(1))] + Var(\mathbb{E}[P/S(1)])$
\end{center}
\end{document}

Answer 1

\mathbb 1_{T_{j}^{i}/\kappa_{1}^((1))

You are missing a brace } but also ^( is not a tex error but only raises the ( so you probably want

\mathbb 1_{T_{j}^{i}}/\kappa_{1}^{(1)}

In addition, never end a paragraph with \\ so

Or nous savons que \\

should be

 Or nous savons que

and never leave a blank line in the source before any display math environment such as align

\begin{center}
$Var(P)= \mathbb{E}[Var(P/S(1))] + Var(\mathbb{E}[P/S(1)])$
\end{center}

isn't exactly wrong but why centred inline math not a math display:

Or nous savons que 
\[
\operatorname{Var}(P)= \mathbb{E}[\operatornameVar(P/S(1))] + \operatornameVar(\mathbb{E}[P/S(1)])
\]

(never use math italic for multi-letter identifiers such as Var) ?

Answer 2

As others have said, you have an unbalanced brace in the second line of the third display. Also ^((1)) should be ^{(1)}.

I spent some time to fix also several other mistakes and reordering the input.

The hyperref package should be last. For variance and covariance, use operator names; for repetitive tasks, define macros. For instance, I defined \One to stand for \mathbb{1}.

I also tried and fix the size of parentheses and the overfull lines (using split).

A display should never be preceded by a blank line; one can follow provided a new paragraph starts.

\documentclass[a4paper,11pt,twoside] {book}
\usepackage[utf8]{inputenc}
\usepackage[french]{babel}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{array}
\usepackage{geometry}
\usepackage{xspace}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{bbold}
\usepackage{hyperref} % should be last

\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Cov}{Cov}
\DeclareMathOperator{\Ex}{\mathbb{E}}
\let\Pr\relax
\DeclareMathOperator{\Pr}{\mathbb{P}}
\newcommand{\One}{\mathbb{1}}

\begin{document}

\chapter[Nom court chapitre]{My chapter}

Quelques mots ici
\begin{align}
P-F_{1}(1)
  &=\sum_{i=1}^{r} \sum_{j=1}^{N_{i}} \One_{T_{j}^{i}\geqslant 1}X_{i} 
   -\sum_{i=1}^{r} M_{i}(S_{i}(1)-K_{i}) \\
  &= \sum_{i=1}^{r} \Bigl(\,\sum_{j=1}^{N_{i}} \One_{T_{j}^{i}\geqslant 1}X_{i}
   -M_{i}(S_{i}(1)-K_{i})\Bigr)
\end{align}
donc
\begin{align}
\Var(P-F_{1}(1))
  &=\Var(P) + \Var\Bigl(\,\sum_{i=1}^{r} M_{i}(S_{i}(1)-K_{i})\Bigr) -2\Cov(P,F_{1}(1)) \\
\begin{split}
  &=\Var(P) + \sum_{i=1}^{r} M_{i}^2\Var(S_{i}(1)) \\
  &\qquad -2\Cov\Bigl(\,\sum_{i=1}^{r} \sum_{j=1}^{N_{i}} \One_{T_{j}^{i} 
                 \geqslant 1}X_{i},\sum_{i=1}^{r} M_{i}(S_{i}(1))\Bigr)
\end{split}
\\
\begin{split}
  &=\Var(P) + \sum_{i=1}^{r} M_{i}^2\Var(S_{i}(1)) \\
  &\qquad - 2\Bigl(\Ex\Bigl[\,\sum_{i=1}^{r} \sum_{j=1}^{N_{i}} \One_{T_{j}^{i} \geqslant 1}X_{i}
                  \sum_{l=1}^{r} M_{l}(S_{l}(1))\Bigr]-
                  \Ex[P]\Ex\Bigl[\,\sum_{i=1}^{r} M_{i}(S_{i}(1))\Bigr]\Bigr)
\end{split}
\\
\begin{split}
  &=\Var(P) + \sum_{i=1}^{r} M_{i}^2\Var(S_{i}(1)) \\
  &\qquad - 2\Bigl(\,\sum_{i=1}^{r}\sum_{j=1}^{N_{i}}\sum_{l=1}^{r} X_{i}M_{l}
              \Ex[\One_{T_{j}^{i} \geqslant 1}S_{l}(1)]-
              \Ex[P]\Ex\Bigl[\,\sum_{i=1}^{r} M_{i}(S_{i}(1))\Bigr]\Bigr)
\end{split}
\end{align}
Comme la fonction de survie est stochastique, les termes d'espérance et variance de $P$ doivent être détaillés:
\begin{align}
\Ex[\One_{T_{j}^{i}\geqslant 1}S_{l}(1)]
&=\Ex[\Ex[\One_{T_{j}^{i}\geqslant 1}S_{l}(1)/]\kappa_{1}^{(1)},\kappa_{1}^{(2)}] \\
&=\Ex[S_{l}(l)\Ex[\One_{T_{j}^{i}}/\kappa_{1}^{(1)},\kappa_{1}^{(2)}]
\end{align}
Nous avons également que
\begin{align}
\Ex[P] &=\sum_{i=1}^{r} X_{i} \sum_{j=1}^{N_{i}} \Ex[\One_{T_{j}^{i}\geqslant 1}] \\
&=\sum_{i=1}^{r} X_{i} \sum_{j=1}^{N_{i}} \Ex[\Ex[\One_{T_{j}^{i}\geqslant 1}/S_{i}(1)]] \\
&=\sum_{i=1}^{r} X_{i} \sum_{j=1}^{N_{i}} \Ex[\Pr[T_{j}^{i}\geqslant 1/S_{i}(1)]] \\
&=\sum_{i=1}^{r} X_{i} \sum_{j=1}^{N_{i}} \Ex[S_{i}(1)]\\
&=\sum_{i=1}^{r} X_{i}N_{i}\Ex[S_{i}(1)]
\end{align}
Or nous savons que
\begin{equation*}
\Var(P)= \Ex[\Var(P/S(1))] + \Var(\Ex[P/S(1)])
\end{equation*}

\end{document}

Answer 3

You forgot to close an opening [ as well as an opening { I'm just quoting the relevant part of your code here:

Comme la fonction de survie est stochastique, les termes d'espérance et variance de P doivent être détaillés:

\begin{align}
   \mathbb{E}[\mathbb 1_{T_{j}^{i}\geqslant 1}S_{l}(1)]&=\mathbb{E}. [\mathbb{E}[\mathbb 1_{T_{j}^{i}\geqslant 1}S_{l}(1)/]\kappa_{1}^((1)),\kappa_{1}^((2))] \\
   &=\mathbb{E}[S_{l}(l)\mathbb{E}[\mathbb 1_{T_{j}^{i}}\kappa_{1}^((1)),\kappa_{1}^((2))]]
\end{align}
Nous avons également que

Notice the added closing ] at the end and the closing } in 1_{T_{j}^{i}}. I did not really bother to understand your calculations, so these may not actually be in the right places for the math to work out, but LaTeX compiles just fine for me after these changes.