This is neither how a Monte-Carlo process works (it's sampling the possible outcomes randomly, not simulating all outcomes), nor what is described here (building a physical system where an observable value is described by the differential equation one is trying to solve).

In practice the number of simulated scenarii is quite large which is what I meant by "every". Not rigorous though, I concur.

Monte Carlo methods are used to solve partial differential equations, I found both approaches similar in the sense that they look at how the system actually/could behave, instead of looking at the equations solely.

This is a pedantic response. There isn't an important difference between building a physical system and building a system in code whose generating process is that which you're trying to model. And everyone knew he didn't mean every outcome literally.

Worked in the securities division of a leading investment bank for 8 years including on the equities exotics and hybrids trading desk. And I can confirm that we used to use MC whenever we had something we couldn't price using a closed form method. That said, it's a lot slower and more compute intensive so when you could use a grid pricer or pde solver or whatever people would use that instead.

You're not really simulating every outcome, just a "large enough" representative sample.

As an aside, my dad was a chemical engineer and Newton-Raphson was his equivalent everything-is-a-nail numerical method. We used to use brent root and monte carlo simulation in the same way.

Numerical methods are awesome in general.

There is actually even a connection between PDEs and SDEs (stochastic differential equations), namely the Feynman-Kac formula which is used a lot in financial mathematics, but it generally allows you to switch between the expectation operator and solving a PDE.

You’re talking more about goal-seeking functions right? That’s the one I always see in a finance context.