I have a following problem:
"We play a game with a symmetric coin. Before $n$-th round, possibly relying on the results of previous games, we set the stake in the $n$-th game: we choose $b_n$, $1 \le b_n \le a$, and if there is heads, get $b_n$, if tails - we pay $b_n$. Let $(S_n)$ be the amount of money we have after $n$ games. Prove that $(S_n)_{n\ge 1}$ is a martingale."
So let $S_n = W_1 + W_2 + \ldots + W_n$, where $W_i$ is the won (or lost) money in the $i$-th game. My goal is to show $\mathbb{E}(S_n | \ \sigma(S_1,S_2,\ldots ,S_{n-1})) = S_{n-1}$, which reduces to showing that $\mathbb{E}(W_n | \sigma(W_1,W_2,\ldots ,W_{n-1})) = 0$.
Also, let $V_i$ be the stake we have decided on just before $i$-th game. So I can write $\mathbb{P}(W_i = b_i | V_i = b_i) = \frac{1}{2}$ and $\mathbb{P}(W_i = -b_i | V_i = b_i) = \frac{1}{2}$, right?
So now, I have a problem to
a) write nicely this information that "relying on the results of previous games, we set the stake in the $n$-th game". Does that mean that for some $b_i$ we have $\mathbb{P}(V_i = b_i | W_1, W_2, \ldots W_{n-1}) = 1$?
b) if so, how to connect it with the desired expectation? I was thinking about conditioning it more:
$\mathbb{E}(W_n | W_1,W_2,\ldots ,W_{n-1}) = \mathbb{E}\left(\mathbb{E}\left[W_n | W_1,W_2,\ldots ,W_{n-1}, V_n\right] \ | \ W_1,W_2,\ldots ,W_{n-1}\right]$
But I'm not sure what to do next. Intuitively, the exercise is obvious, but I have a problem to write it down nicely and formally.
Let $\mathscr{F}_n=\sigma(Z_k,k\leq n)$ where $(Z_k)_{k \in \mathbb{N}}$ is a sequence of IID Rademacher random variables. These represent the coin flips. The betting stake strategy $(b_n)_{n \in \mathbb{N}}$ is $\mathscr{F}_n$-predictable. $b_n$ is what we put at stake at $n$, which is known at $n-1$. Our wealth process is given by $$S_n=S_0+\sum_{1\leq k\leq n}Z_kb_{k}$$ where $S_0$ is a known real number (our initial wealth). It is $\mathscr{F}_n$-adapted. So we have $$E[S_{n+1}|\mathscr{F}_n]=S_n+E[Z_{n+1}|\mathscr{F}_n]b_{n+1}=S_n+0\cdot b_{n+1}=S_n$$ This shows the martingale property of $S_n$.