Difference between random walk and martingale

Question

I am trying to understand the diffrence between random walk and martingale. According to my understanding, a random walk without drift is

$$ y_{t} = y_{t-1} + u_{t} $$

where $u_{t}$ is $i.i.d.(0, \sigma_{t}^2)$ where $\text{Cov}(y_{t},y_{t-s})=0$ for $t \neq s$.

However, a martingale has just one restiction:

$$ \mathbb{E}[y_{t}|y_{t-1},y_{t-2},\dotsc] = y_{t-1} $$

and there are no restrictions on higher order moments. The distribution of $u_{t}$ is just required to satisfy $\mathbb{E}(u_{t})=0$. Heterogeneity of $u_{i}$ and/or correlation between sequences of $u_{i}$ are allowed.

Is my understanding right?

Answer 1

For a random walk, I think $u_t\sim i.i.d.(0,\sigma^2$) (no time subscript for $\sigma^2_t$).

For a martingale,

$$ \mathbb{E}(y_{t}|y_{t-1},y_{t-2},\dotsc) = y_{t-1} $$

implies $u_t$ are non-autocorrelated, because if they were, the conditional expectation would no longer be zero:

• $\mathbb{E}(u_t|y_{t-1},y_{t-2},\dotsc)=0$ implies $\mathbb{E}(u_t)=0$;
• given $y_{t-1}$ and $y_{t-2}$ we have $u_{t-1}$; if it is nonzero and $\{u_t\}$ as a sequence are autocorrelated, then $\mathbb{E}(u_t|u_{t-1}) \neq 0$ and $\mathbb{E}(y_{t}|y_{t-1},y_{t-2},\dotsc) \neq y_{t-1}$, which is a contradiction.