Though you haven’t shown me the sde function itself, if it’s similar to the one I wrote, error_s should be a normal distribution, not a random walk. That is, within the context of the SDE, you draw one innovation per time step, but it doesn’t drift through time: only Vt does.
In the equations you described:
Y_t=\sqrt{V_{t-1}}\epsilon_t^s
V_t=\alpha_v+\beta_vV_{t-1}+\sigma_v\sqrt{V_{t-1}}[\rho\epsilon_t^s+\sqrt{1-\rho^2}\epsilon_t^v]
\epsilon_t^s ~ Normal(0, \sigma_s), no?
or you would say that \epsilon_t^s = \epsilon_{t-1}^s + \epsilon_t^* ? (in which case, yes it would be a random walk)