I am trying to setup a simple exercise to understand how a stochastic process is built, since I am following a course that totally lacks any practical example. I wanted to build a martingale from scratch, starting from the simplest example of random walk:
- sample space: $\Omega = \{\omega\in(0,1]\}$
- random process as a family of random variables $(X_i)_{i=0}^\infty$
- $X_0=0$
- $X_1=-1$ if $\omega \leq 0.5$, $1$ otherwise
- $X_2=-2$ if $\omega\leq 0.25$, $0$ if $0.25<\omega\leq0.75$, 2 otherwise
- $X_{i>2}$ alike
- a filtration $(\mathcal F_i)_{i=0}^\infty$ is built on $\mathcal F=\sigma(\Omega)$: $\mathcal F_0=\{\emptyset,\Omega\}$, $\mathcal F_1=\{\emptyset,\Omega,(0,0.5],(0.5,1]\}$, $\mathcal F_2=\{\emptyset,\Omega,(0,0.25],(0.25,0.75],(0.75,1],(0,0.75],(0.25,1],(0,0.25]\sqcup(0.75,1]\}$. Note that every $\mathcal F_i$ contains $2^i$ elements.
Now I want to check if the process is a martingale, that should be. So I check if
$$X_1\stackrel{?}{=}\mathbb E[X_2|\mathcal F_1]= \begin{cases} ?&\text{ on }\emptyset\\ 0&\text{ on }\Omega\\ -2\cdot0.25+0\cdot0.25&\text{ on }(0,0.5]\\ 0\cdot0.25+2\cdot0.25&\text{ on }(0.5,0.1] \end{cases} = \begin{cases} ?&\text{ on }\emptyset\\ 0&\text{ on }\Omega\\ -0.5&\text{ on }(0,0.5]\\ 0.5&\text{ on }(0.5,0.1] \end{cases}.$$
In the way I am computing the conditional expectation, clearly it does not look like a martingale. Would you please let me know where I am misunderstanding things?
Plus, would you please suggest me a book or notes on stochastic processes with such examples, where I can get a practical understanding of what I am doing?