probability - $mathbb{P}$ vs $mathbb{Q}$ Probabilities - Transitioning Between Measures

I'd like this question to definitively guide a practitioner to using both $\mathbb{P}$ vs $\mathbb{Q}$ probabilities in trading and research.

Let's take only one fact as given: if I have a risk-neutral probability distribution I can price and hedge any option.

1. Is the distinction more philosophical or practical? Does it have real impact on trading desks P/L? For example, is it a construct to remind us we're not in the "real world" when modeling?


2. This question says it is the difference in using $\mu$ vs $r$ when solving the S.D.E... which seems to say if I definitively knew $\mu$ and $r$ I would be able to transition with absolutely no loss of information. What edge would this provide me in the market?


3. This good paper and this good answer seems to divide them on the approach of their research. $\mathbb{P}$-quants vs $\mathbb{Q}$-quants... in this sense it seems to be that $\mathbb{P}$-Quants are concerned with modeling the future using historical data sets. Projection. $\mathbb{Q}$-quants are concered with relative valuation and making sure that their pricing shemes are consistent with exchange traded products that are observed in the market. Extrapolation. I see that these job functions are different, but I do not see why one could not apply $\mathbb{P}$ methods to the $\mathbb{Q}$ world (their effectiveness seems less important to me - it doesn't seem like a scientific violation).


4. Girsanov's Theorem shows its possible to switch between the two. Now I know I CAN draw conclusions from each other, but the method is not clear.

Is there a way on paper to move from $\mathbb{P}$ to $\mathbb{Q}$ and vice-versa if I have a closed form-solution or a parameterized model of either $\mathbb{P}$ or $\mathbb{Q}$? If my returns under $\mathbb{Q}$ is $X \sim \mathcal{N}(r,\,\sigma^{2})\,$. From here, how can I get to $\mathbb{P}$.

I'd prefer to stay out of a model-framework completely and let all results be in general. From what I've found I believe the connection is in putting a price on the market risk premium, but I have not found empirical estimations of this or attempts to use its estimation for moving between $\mathbb{P}$ and $\mathbb{Q}$. Any papers on $\lambda$ estimation or extraction would be appreciated.

I wanted to add this quote from Gary Hatfield:

Recall that the whole point of risk neutral pricing is to recover the price of traded options in a way that avoids arbitrage. As such, the probabilities of various paths are implied from the prices of various traded securities whose payoffs depend on those paths. Since investors are in aggregate risk averse, these prices imply higher probabilities to bad scenarios than they do to good scenarios. Hence, while everyone (almost!) agrees that stocks have a higher expected return than risk free bonds, the prices of stock and stock options imply the only difference between stocks and risk free bonds is that stocks are more volatile. Put another way, a risk neutral scenario set has many more really bad scenarios than a real world scenario set precisely because investors fear these scenarios. They therefore overweigh their probability when deciding how much a security is worth.

This provides intuitive context to the difference, but makes it seem impossible to every replicate the $\mathbb{P}$ world.