requirements for simulating a covariance matrix

I was wondering, is any positive semidefinite matrix a valid covariance matrix?

My problem is the following. I want to simulate a stochastic covariance matrix where the log-volatility (log of square root of variance) and the correlation are simulated separately according to some stochastic process. If I can ensure that the resulting covariance matrix is at all times positive semidefinite, is it a valid covariance matrix process?

To make things clearer, let’s assume I want to simulate a $2 times 2$ covariance matrix process. I would proceed by simulating two log-volatility processes and one correlation process:
$$logsigma^1_t = f(theta^1, t)$$
$$logsigma^2_t = f(theta^2, t)$$
$$rho_t = g(theta^3, t)$$
where the $theta$‘s are some parameters. Then, given $sigma^1_t = e^{f(theta^1, t)}$, $sigma^2_t = e^{f(theta^2, t)}$, $cv_t = rho_t sigma^1_t sigma^2_t$, I build the covariance matrix process
$$
X_t =
left[begin{array}{cccc}
(sigma^1_t)^2 & cv_t \
cv_t & (sigma^2_t)^2 \
end{array}right]$$
My question: if by choosing proper $theta$, I can ensure that $X_t$ is at all times positive semidefinite, is it a valid covariance matrix process?