Variance Accumulation in a Symmetric Random Walk

# Accumulation Rate of Variance in Random Walk


# Accumulation Rate of Variance in Random Walk












I am slightly confused with the terminology Shreve (2008), he states:

> "The variance of the symmetric random walk accumulates at rate one per unit time, so that the variance of the increment over any time interval $k$ to $l$ for nonnegative integerst $k<l$ is $l-k$."

I understand the latter half of the statement, but I don't understand the variance's accumulation rate. This is something not familar to me. The way I thought of variance's unit is that it is squared, so it is usually not easily interpreted.

The context in which this symmetric random walk is defined is think of a random variable that pays you \$1 if a coin comes up heads and -\$1 otherwise. Consider a cumulative gain or loss of this dollar bet at $k$-th bet, and we call this process a symmetric random walk.

So what does he mean here about the rate at which the variance accumulates? The rate at which the dispersion of the cumulative bet accumulates? Confused.

Reference: Shreve, Steven E. \textit{Stochastic Calculus for Finance II : Continuous-Time Models}. Springer, 2008.

## Answer by Cettt (score 1, accepted)

https://quant.stackexchange.com/a/49802

If we denote the random walk with $(X_k)_{k \in \mathbb{N}}$ than for all $k$ the random variable $\Delta X_k := X_{k} - X_{k-1}$ has mean zero and variance one: \begin{align} \mathbb{E}[\Delta X_k] = \frac 12\cdot 1 + \frac 12 \cdot (-1) = 0, \quad \text{Var}(\Delta X_k) = \mathbb{E}[(\Delta X_k)^2] - \bigl(\mathbb{E}[\Delta X_k]\bigr)^2 = 1 \end{align}

This is what is meant by the second part.

The first part deals with the variance of $X_k$ which can be computed like this: \begin{align} \text{Var}(X_k) &= \mathbb{E}\Bigl[\text{Var}(X_k \mid X_{k-1}) \Bigr] + \text{Var}\Bigl(\mathbb{E}[X_k \mid X_{k-1}] \Bigr) = 1 + \text{Var}(X_{k-1}), \\ \text{Var}(X_1) &= \mathbb{E}[(X_1)^2] - \bigl(\mathbb{E}[X_1]\bigr)^2 =1. \end{align}

Therefore $\text{Var}(X_k) = k$ for all $k \in \mathbb{N}$. In particular, the variance of the symmetric random walk accumulates at rate one per unit time.