With that in mind - I will start with the very basics; introducing my first blog in over two years, I give a quick overview of Random Walks in $\mathbb{Z}$

Let $X_0 := 0$ be our starting position and for each $i \in \mathbb{N}$ let $$X_i:=X_{i-1}+dX_i$$ where $dX_i = \pm 1$ with equal probability - i.e is a Bernoulli distributed random variable with $p=q=\frac{1}{2}$. Therefore our position at step $N$ is just the sum of the

*independent*Bernoulli variables;

$$X_N= \sum_{i=0}^{N} dX_i$$.

We can calculated the expectation: $$\mathbb{E}[X_N]= \mathbb{E}\left[ \sum_{i=0}^{N} dX_i\right]= \sum_{i=0}^{N} \mathbb{E} [ dX_i]=0$$ by using the definition of the expectation of a discrete random variable and the fact that $dX_i$ and $dX_j$ are independent for $i \neq j$. Now

$$Var(X)=\mathbb{E}[X^2]- \mathbb{E}[X]^2$$ we can calculate $$\mathbb{E}[X_N^2] = \sum_{i=0}^N 1 = N$$That is, the variance of $X_N$ is the jump size times the total steps $N$.

Below I have simulated the aforementioned random walk for $N=100$ and $N=100000$ respectively.

Next we'll look at a specific scaling of these random walks to obtain a

*Wiener Process*and eventually talk about Stochastic Integrals and Ito's lemma.
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