If a continuous-time process $x_t$ is [geometric brownian motion](http://en.wikipedia.org/wiki/Geometric_Brownian_motion) it would have this property, or the discrete-time equivalent (geometric random walk).

A difference in logs is is (for $u_t$ small at least) effectively a percentage change. 

See also the connection to the *[force of mortality](http://en.wikipedia.org/wiki/Force_of_mortality)* (what actuaries used to call the hazard function, they seem to be using it less these days) and the [force of interest](http://en.wikipedia.org/wiki/Compound_interest#Force_of_interest), which are 'instantaneous' equivalents of your annualized (or more generally, *periodized*) discrete measure.