Given the random walk : $P_t=P_{t-1} + \varepsilon_t$ where $\varepsilon_t$ is i.i.d normal with a mean $\mu$ and variance $\sigma^2$
- How do I calculate the conditional mean and variance of $P_t$, given that $P_{t-1}=x$
Edit : My working is below
Conditional mean:
$E[P_t|P_{t-1}=x]=E[P_{t-1}+\varepsilon_t|P_{t-1}=x]$
$=E[x+\varepsilon_t|P_{t-1}=x]$
$=x+\mu$
Conditional Variance: (most concerned about this result)
$Var[P_t|P_{t-1}=x]=E[P_t-E[(P_t|P_{t-1}=x])^2|P_{t-1}=x]$
$=E[(P_t -(x+\mu))^2|P_{t-1}=x]$
$=E[(x+\varepsilon_t -x-\mu)^2|P_{t-1}]$
$=E[(\varepsilon_t-\mu)^2|P_{t-1}]$
$=E[\varepsilon_t^2-2\varepsilon_t\mu+\mu^2|P_{t-1}]$
$=2\mu^2-2\mu^2=0$
- I am confused with the result of the conditional variance because I feel like it should depend on time 't', instead of being zero.
$$ \mathbf{E}\left[P_t|P_{t-1} = x\right] = \mathbf{E}\left[P_{t-1}+\varepsilon_t|P_{t-1} = x\right] = \mathbf{E}\left[x+\varepsilon_t\right] = \mathbf{E}\left[x\right] + \mathbf{E}\left[\varepsilon_t\right] = x + \mu. $$
$$ \mathbf{Var}\left[P_t|P_{t-1} = x\right] = \mathbf{Var}\left[P_{t-1}+\varepsilon_t|P_{t-1} = x\right] = \mathbf{Var}\left[x+\varepsilon_t\right] = \mathbf{Var}\left[x\right] + \mathbf{Var}\left[\varepsilon_t\right] = 0 + \sigma^2 = \sigma^2. $$