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We work on a probability space $(\Omega,\mathcal{N},\mathfrak{F})$ with filtration $(\mathfrak{F}_t)_{0\leq t\leq T}$ and $\mathfrak{F}_T:=\mathfrak{F}$. Let $\xi$ be a $\mathfrak{F}_T$-mesurable contingent claim, and $N_t$ and $M_t$ two assets with positive prices. We assume the process $M_t/N_t$ is a martingale under the probability measure $\mathcal{N}$. ...
Let \begin{align*} \mathrm{d}S_t&=\mu S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}, \\ \mathrm{d}v_t&=\kappa(\bar{v}-v_t)\mathrm{d}t+\xi\sqrt{v_t}\mathrm{d}B_{v,t}, \end{align*} where $\mathrm{d}B_{S,t}\mathrm{d}B_{v,t}=\rho\mathrm{d}t$. The market price of risk (or Girsanov kernel or Sharpe ratio) is ${\varphi}_t=\left(\frac{\mu-r}{\sqrt{v_t}},\... 3 (I might not be answering your question, but I feel this clarification is needed.) A random variable$X$of$(\Omega, \mathcal{F})$is a$\mathcal{F}$-measurable function$X : \Omega → \mathbf{R}$. So,$X$depends on$\Omega$and$\mathcal{F}$, but does not depend on the probability measure put on$(\Omega, \mathcal{F})$. It is the distribution of$X\$ that ...