I am new in Martingale Theory and i am struggling to understand this stochastic process.
As far as i know (if i am right) that a martingale is process of two random variables $X$ and $Y$, such that $E[X|Y=y]$ remains the same in any time of the process.
My question is: How the formula for the discrete conditional expectation is : $E[X|Y=y]=\frac{E[X_{ \mathbf{I}}(Y)]}{P(Y)}$ is being used in order to calculate thing like :
Let $X_{n},n= 1,\dots $ be an i.i.d. sequence with mean $μ$ and variance $\sigma^{2} < \infty$. Let $\mathcal{F}_{n}$ be the Borel $\sigma$-algebra on $\mathbb{R}_{n}$. Then $$S_{n}−\mu n=\sum_{0 \leq k \leq n }X_{k}−\mu n$$ is a martingale.
And then he/she states that "Indeed $S_{n}$ is adapted to $\mathcal{F}_{n}$", and :
\begin{align*}
E[Sn+1−(n+1)μ|\mathcal{F}_{n}]&=E[Xn+1−μ+Sn−nμ|\mathcal{F}_{n}]\\
&=E[Xn+1−μ|\mathcal{F}_{n}] +E[Sn−nμ|\mathcal{F}_{n}]a\\
&=E[Xn+1−μ] +Sn−nμ\\
&=Sn−nμ.
\end{align*}
More specific:
Question 1 : How the conditional expectation has been used in line 1?
Question 2 : Again in line 1 : From where the $Sn+1−(n+1)μ$ came from ?
I am sorry for my poor understanding of martingale. Hope someone makes me understand.
Best Answer
The way you have defined a martingale in the first pargaraph doesn't make sense. You have to index the process using a 'time' variable before you can write down the definition.
Let $Y_n=S_n-n\mu$. By definition of a martingale $(Y_n)$ is a martingale if $E(Y_{n+1}|\mathcal F_n)=Y_n$ for each $n$.
$E(Y_{n+1}|\mathcal F_n)=E(S_{n+1}-\mu (n+1)|\mathcal F_n)$ by defintion of $Y_{n+1}$. Now $S_{n+1}-\mu (n+1)$ can be written as $S_n-n\mu +(X_{n+1}-\mu)$. So $E(Y_{n+1}|\mathcal F_n)=E(S_{n+1}-\mu (n+1)|\mathcal F_n)=E(S_n-n\mu|\mathcal F_n)+E(X_{n+1}-\mu|\mathcal F_n) $. In the first term use the fact that $S_n-n\mu$ is already measurable w.r.t $\mathcal F_n$ so it is its own conditional expectation. In the second term use the fact that $X_{n+1}-\mu$ is independent of $\mathcal F_n$ so the conditional expectation is $E(X_{n+1}-\mu)=\mu-\mu=0$.