I have some doubts about the derivation of the pricing PDE in stochastic volatility models like the Heston model. I read some books like The Volatility Surface by Jim Gatheral and there's a bit I'm confused about. Before I ask my question, I'll lay out the basic concepts as I understand them. I have a system of two SDE \begin{equation*} \begin{cases} dS_t = \mu S_t dt + \sqrt{V_t}S_t dW^1_t \ dV_t = p_t dt + q_t \left( \varrho dW^1_t + \sqrt{1-\varrho^2} dW^2_t \right) \end{cases} \end{equation*} where (Wt1)t(W^1_t)_t and (Wt2)t(W^2_t)_t are independent Brownian motions. For example, in Heston's model we have \begin{equation*} p_t = \kappa (m-V_t), \qquad q_t = \eta\sqrt{V_t}. \end{equation*} The volatility matrix is given by $$ \sigma_t = \left[ \sqrt{V_t} \quad 0 \right],

is a solution $(\theta_t)_t$ of the equation $$ \sigma_t \theta_t = \mu -r $$ where $r$ is the risk-free rate. For every process $(\lambda_t)_t$ , $$ \begin{bmatrix} \frac{\mu - r}{\sqrt{V_t}}\\ \lambda_t \end{bmatrix} $$ is a market price of risk. There's no uniqueness of the market price of risk, hence the risk-neutral measure is not unique and the market model is incomplete. The Girsanov theorem states that the processes $(W^{\lambda,1}_t)_t$ and $(W^{\lambda,2}_t)_t$ given by \begin{equation*} \begin{cases} dW^{\lambda,1}_t = dW^1_t + \frac{\mu - r}{\sqrt{V_t}} dt \\ dW^{\lambda,2}_t = dW^2_t + \lambda_t dt \end{cases}. \end{equation*} are independent Brownian motions (*). In Heston model's literaure I have read the correlated version: if we define the correlated Brownian motions \begin{equation*} \begin{cases} Z^{\lambda,1}_t = W^{\lambda,1}_t \\ Z^{\lambda,2}_t = \varrho W^{\lambda,1}_t + \sqrt{1 - \varrho^2} W^{\lambda,2}_t \end{cases}, \end{equation*} then after some calculations we can prove that \begin{equation*} \begin{cases} dZ^{\lambda,1}_t = dZ^1_t + \frac{\mu - r}{\sqrt{V_t}} dt \\ dZ^{\lambda,2}_t = dZ^2_t + \Lambda_t dt \end{cases}, \end{equation*} where \begin{equation*} \Lambda_t := \varrho \frac{\mu - r}{\sqrt{V_t}}+ \sqrt{1 - \varrho^2} \lambda_t \end{equation*} is the market price of volatility risk. We can easily prove that the risk-neutral dynamics become \begin{equation*} \begin{cases} dS_t=r S_t dt + \sqrt{V_t} S_t dZ^{\lambda,1}_t\\ dV_t = (p_t-\Lambda_t q_t) dt + q_t dZ^{\lambda,2}_t \end{cases}. \end{equation*} For what concerns the derivation of the PDE, I know that the market is incomplete and hence the arbitrage pricing is not unique. The idea is to complete the market by adding an option $U_t=(t,S_t,V_t)$ . Let $F(t,S_t,V_t)$ denote the arbitrage price in the market completion, with some calculation we obtain the equation \begin{align*} &\frac{\left(\partial_t F + \frac{1}{2}S^2 V \partial_{ss}F + \frac{1}{2} q^2 \partial_{vv}F + \varrho q S \sqrt{V}\partial_{sv}F \right) + rS \partial_s F - rF}{\partial_vF}=\\ &=\frac{\left(\partial_t U + \frac{1}{2} S^2 V \partial_{ss}U + \frac{1}{2} q^2 \partial_{vv}U + \varrho q S \sqrt{V} \partial_{sv}U \right) + rS \partial_s U - rU}{\partial_vU} . \end{align*} The left-hand side is a function of $F$ only and the right-hand side is a function of $U$ only. The only way that this can be is for both sides to be equal to some function $\varphi(t,S_t,V_t)$ . Here comes the tricky part: in the books I've read, the function $\varphi$ is assumed to be $$\varphi = p \Lambda -q$$ without loss of generality. What does this actually mean? The function $\varphi$ is fixed, so I guess we are fixing a process $\Lambda$ (and hence a price market of risk). If so, why are we choosing $\Lambda$ such that $\varphi = p \Lambda -q$ rather than $\varphi = \Lambda$ or $\varphi = g(\Lambda)$ for some other function $g$ ? I know that $p-\Lambda q$ is the drift of the process $(V_t)_t$ under the risk neutral measure, but I don't understand this choice. Is it maybe related to the (unique) market price of risk of the market completion? I guess that this choice of $\Lambda$ makes the arbitrage price in the (not completed) model equal to the (unique) arbitrage price in the market completion. Could someone clarify this for me, from a mathematical point of view? I also have 2 other questions: In order to apply the Girsanov theorem in (*), we must verify that the price market of risk $$ \begin{bmatrix} \frac{\mu - r}{\sqrt{V_t}}\\ \lambda_t \end{bmatrix} $$ is such that its exponential martingale is a true martingale (for instance, we can verify the Novikov condition). Is this proved somewhere? In the Heston model, the process $(\Lambda_t)_t$ is assumed to be $\Lambda_t = \xi \sqrt{V_t}$ for some constant $\xi$ . Why is it a good assumption?