I am a beginner studying Heston Model. I notice that people are almost all talking about the closed form solution for options of the Heston Model. However, I am only interested in the closed form solution (or the density) of the basic two stochastic equations as shown below.dS(t) = mu*S(t)*dt + sqrt(v(t))*S(t)*dZ(t) (1)dv(t) = k*(theta - v(t))*dt + sigma*sqrt(v(t))*dW(t) (2)Can someone tell me what is the closed form solution (or the density) of the above stochastic model?I know (2) is a CIR process, we know the closed form solution and it follows a non-central chi-square distribution. How about the equation (1)? Equation (1) is similar to the geometric process, so I guess the log of S(t) might follow normal distribution? Does that make sense? Then how about the whole model (1) and (2)? Can we say the log(S(t)) and v(t) follows some joint distribution, where v(t) is non-central chi-square and log(S(t)) is normal? Then given the observed data S(t) and v(t), how to estimate the model? Thanks so much.

Just because you're a beginner shouldn't exempt you from doing a quick Google searchYou should start by reading articles such as thishttp://arxiv.org/pdf/1502.02963.pdf

See my Oct 19, 2015 in this thread for the Heston joint transition density.