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: $$\log\sigma^1_t = f(\theta^1, t)$$ $$\log\sigma^2_t = f(\theta^2, t)$$ $$\rho_t = g(\theta^3, t)$$ 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?
