Distribution of $(X(t_1),X(t_2))$ of a diffusion process $X(t)$

Question

I am very new to SDE's and diffusion processes, I came across this diffusion process given by $dX(t)=[\alpha-(\alpha + \beta)X(t)]dt + \sqrt{2X(t)(1-X(t))}dB(t)$ where $B(t)$ is a continuous Brownian Motion. Its stationary distribution is Beta$(\alpha,\beta)$ when $X(0)\sim$Beta$(\alpha,\beta)$. I suspect that the joint distribution of $(X(t_1),X(t_2))$ will be Dirichlet in this case. Is there a way to find such joint distributions?