An asset price $p_t$ at any time $t = 1,2, \ldots, n$ is given by $x_t = x_{t-1}e^{\mu + \epsilon_t}$, where $\epsilon \stackrel{iid}{\sim} N(0,1)$.
I was supposed to test $H_0: \mu = 0$ against $H_1: \mu \ne 0$, determining which is the appropriated test as well as the corresponding statistic.
What I tried so far:
- From $x_t = x_0e^{n\mu + \sum \epsilon_t}$, and consequently $\ln x_t = \ln x_0 + n\mu + \sum \epsilon_t$, I thought the z-test would fit, since $\ln x_t$ follows normal distribution. Haven't made any progress from this point.
- From $\ln x_t = \ln x_{t-1} + \mu + \epsilon_i$, I thought I'd have to test for the drift, since this is the form of a geometric random walk with drift $\mu$. Also haven't made any further progress.
I have never seen this kind of test.