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Consider the Heston (1993) model under the real world measure ($\mathbb{P}$) \begin{align*} \mathrm{d}S_t&=\mu^\mathbb{P} S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}^\mathbb{P}, \\ \mathrm{d}v_t&=\kappa^\mathbb{P} (\bar{v}^\mathbb{P}-v_t)\mathrm{d}t+\sigma^\mathbb{P}\sqrt{v_t}\mathrm{d}B_{v,t}^\mathbb{P}, \end{align*} where $\mathrm{d}B_{S,t}^\... 5 A very interesting topic ! Black-Scholes originally did not make use of the Girsanov theorem and arrived at the equation the way you described. Later theoretical work on arbitrage pricing uncovered the concepts of the risk-neutral measure and derivatives pricing as an expectation under that measure. That work relies on stochastic calculus far more and one ... 4 (Bloomberg and Reuters News are fond is reporting that some name is trading at some such CDS spread, "which implies N% probability of default". They neglect to mention what recovery assumption they used, and that this is risk-neutral probability, not physical.) For corporate names, Ed Altman published the well-known paper on Z-score. His basic idea ... 3 (I might not be answering your question, but I feel this clarification is needed.) A random variable$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 ... 3 Nothing new in this answer - I have just consolidated what others have said in the answer and the comments, and put the explanation next to each step. I have to move the original equations so as to have one equation on each line: 2 I don't think that you can apply Girsanov's theorem in this way, noting that I don't understand your (short) argument in the comments. This is how I would proceed, which makes sense mathematically, but the economic interpretation is then a bit strange. Let us write the SDE's a bit different, noting that it still preservers the same correlation structure \... 1 Intuition The stochastic discount factor (SDF) really has two jobs to do: it needs to incorporate the time value of money (discount) and take the riskiness of cash flows into account (stochastic). Thus, it can make sense to split the SDF into its two components, $$M_t=e^{-rt}A_t,$$ where$A_t$does the risk compensation. Girsanov's theorem is concerned with$...