I consider the following system of SDE's (Heston model): $$ dX_t = -\frac12V_t dt + \sqrt{V_t}dW_t\\ dV_t = \kappa (\theta - V_t)dt + \xi \sqrt{V_t}dZ_t \\ d\langle W,Z\rangle_t = \rho dt, $$ where Z, W are Brownian motions and $\kappa, \theta, \xi > 0$. If $(W^1,W^2)$ is a 2-dimensional standard Brownian motion I can define $W = \sqrt{1-\rho^2}W^1 + \rho W^2$ and $Z = W^2$ to get the desired covariation $\rho$.
Problem: Assume that for $p \in \mathbb{R}$ we have $\mathbb{E}[e^{pX_t}]<\infty$. Apparently the MGF of of $X_t$ has the following representation $$ \mathbb{E}[e^{pX_t}]=\mathbb{E}\left[\exp\left( p\rho\int_0^t\sqrt{V_s}dW^2_s - \frac{p^2\rho^2}2\int_0^tV_sds + \frac{p(p-1)}2\int_0^tV_s ds \right)\right]. $$ However, I am not able to see this. When I plug in $X_t$ and $W_t$ the MGF turns into $$ \mathbb{E}\left[\exp\left( p\rho\int_0^t\sqrt{V_s}dW^2_s + p \sqrt{1-\rho^2}\int_0^t\sqrt{V_s}dW^1_s- \frac p2\int_0^tV_s ds \right)\right]. $$ Looking at the SDE of $V$, I can see that it only depends on the randomness of $Z = W^2$, therefore, when I condition on $W^2$, I get $$ \mathbb{E}\left[\mathbb{E}\left[\exp\left( p\rho\int_0^t\sqrt{V_s}dW^2_s + p \sqrt{1-\rho^2}\int_0^t\sqrt{V_s}dW^1_s- \frac p2\int_0^tV_s ds \right)\Big|(W^2_s)_{s \in [0,t]}\right]\right] \\ = \mathbb{E}\left[\exp\left( p\rho\int_0^t\sqrt{V_s}dW^2_s - \frac p2\int_0^tV_s ds \right)\mathbb{E}\left[\exp\left( p \sqrt{1-\rho^2}\int_0^t\sqrt{V_s}dW^1_s \right)\Big|(W^2_s)_{s \in [0,t]}\right]\right] $$ Now, I am stuck. If I could conclude that the inner term is independent of $W^2$ (which I do not think is right, since the integrand depends on $V$ and therefore on $W^2$) I could evaluate the inner expected value.
Question: Can someone give me a hint on how to proceed, or if this is the right direction? Any references to the derivation of the MGF in the Heston model is greatly appreciated! (Since this is more about stochastic analysis, I did not post this on quant stack exchange.)
Thanks in advance!
Edit: I was able to boil down the problem to the following: When we consider $$ \mathbb{E}\left[\exp\left( p \sqrt{1-\rho^2}\int_0^t\sqrt{V_s}dW^1_s \right)\Big|(W^2_s)_{s \in [0,t]}\right], $$ it is apparently true that the integrand (conditional) on $W^2$ is measurable, and the integrator is independent. Hence, I can treat the integrand as a deterministic function, take the unconditional expectation and use the MGF of a normal distribution, yielding the desired result.
Although this sounds reasonable, is there a way to make this argument rigorous (maybe something like this)? Any references would be appreciated!