Check whether this stochastic process is a martingale

Question

I'm writing a probability theory exam in two days but still have trouble with the martingale section. I have no idea how to solve the following problem:

Assume that $(F_n)_{n\in\mathbb{N}}$ is a filtration and $(A_n)_{n\in\mathbb{N}}\subset F$ a sequence of events s.t. $A_n \in F_n$. Show that $(M_n)_{n\in\mathbb{N}}$ (where $M_0=0$) $$M_n = \sum^n_{k=1}(1_{A_k}-P(A_k|F_{k-1})), n\geq 1$$ is a martingale for the filtration $(F_n)_{n\in\mathbb{N}}$.

I seek for a solution to this, since time is running out. Thx in advance!

Answer 1

\begin{align*} E[M_{n+1} | F_n] &= E\left[ \sum_{k=1}^{n+1} (1_{A_k} - P(A_k | F_{k-1})) | F_n\right] \\ &= E[1_{A_{n+1}} - P(A_{n+1} | F_n)|F_n] +\sum_{k=1}^{n} E\left[ (1_{A_k} - P(A_k | F_{k-1})) | F_n\right] \tag{1}\\ &= P(A_{n+1} | F_n) - P(A_{n+1} | F_n) + \sum_{k=1}^n (1_{A_k}-P(A_k | F_{k-1})) \tag{2}\\ &= 0 + M_n. \end{align*} The other property you have to check is integrability, i.e. $E[|M_n|] < \infty$ for all $n \geq 1$.

Two key properties were used that you should commit to memory for your exam:

• If $Y$ is $G$-measurable, then $E[Y|G] = Y$. This was used in the first term of the sum over $k=1,...,n$ when going from (1) to (2), with $Y = 1_{A_k}$ and $G=F_n$. Indeed, $1_{A_k}$ is $F_n$ measurable for all $k \leq n$, since $A_k \in F_k \subset F_n$.
• If $G_1 \subset G_2$, then for any random variable $Y$, $$E[E[Y|G_1] | G_2] = E[E[Y|G_2]|G_1] = E[Y|G_1].$$ This was used on $E[P(A_k|F_{k-1})|F_n]$ when going from (1) to (2). In words, when conditioning twice, you end up conditioning on the coarser $\sigma$-algebra. This is obvious once you take a simple example (such as two coinflips) and work it out, which you may not have time for.

Last, just in case, note that $P(A | F) = E[1_{A} | F]$, so you only need to remember the "rules" for one of them; choose $E$.