In the paper "The Impact of Jumps in Volatility and Returns" by Nicholas Polson, Bjorn Eraker, and Michael Johannes (2003), the authors state in footnote 6 on page 1273 that, given an Ornstein-Uhlenbeck process of the form: $$dv_t = k(\theta - v_t) dt + \sigma \sqrt{v_t} dW_t$$ where $v_t$ is the process, k is the rate of mean reversion, $\theta$ is the long-run mean, and $\sigma$ is the volatility, we have that:
$$E[v_t] = E[v_0] + E \left[ \int_0^t k(\theta - v_u) du \right] + E \left[ \int_0^t \sigma \sqrt{v_u} dW_u \right]$$
from which they conclude that $$\mu_v = \mu_v + k(\theta - \mu_v)t$$ where $μ_v$ is the expected value of $v$.
I am confused by this conclusion. I remember that the expected value of an Ornstein-Uhlenbeck process is of the form
$$E[v_t] = v_0 e^{-kt} + \theta (1 - e^{-kt})$$ Even if we consider $v_0$ unknown, I do not see how the authors can conclude that $E[v_t]=\mu_v=E[v_0]$.
One possibility is that this is only true for $t$ close to $0$. In this case, the term $\mathbb{E} \left[ \int_0^t k(\theta - v_u) du \right]$ will be small, and the expected value of $v_t$ will be approximately equal to the initial value $v_0$. However, this does not seem to be what the authors are claiming.
Another possibility is that the authors have made a mistake. This is certainly possible, as the footnote is only a few lines long and is not very well-explained.
I would appreciate it if someone could clarify this for me.
The access to this paper is restricted so I cannot check if you quote these authors correctly. There is however no reason to assume that the Journal of Finance has printed that nonsensical equation $\mu_v = \mu_v + k(\theta - \mu_v)t\,.$ It should be $$ d\mu_v = k(\theta - \mu_v)\,dt $$ which has the solution that you remember: $$ \mu_v(t)=\mu_v(0)e^{-kt}+\theta(1-e^{-kt})\,. $$