Why is the Martingale problem interesting, or useful to areas outside of math like economics, game theory, physics, etc.?
As a reminder, the Martingale problem is about finding a process so that when a given linear operator is applied to it, you get a martingale.
http://www.ma.utexas.edu/mediawiki/index.php/Martingale_Problem
Impreciseness is fine, I'm only looking for intuition and scope of knowledge here. (For instance, that website is too advanced for me to understand technically.)
I know Gaussian processes and Markov Chains are useful to physics. Are Martingales also useful to physics?
First of all, martingale problems often arise in the theory of Markov processes as they are related to the characterization. For example, let $(X_n)_{n\in \Bbb N}$ be a sequence of Markov processes with generators $\mathscr L_n$ and we would like to check whether they converge to a Markov process $X$ with a generator $\mathscr L$. In such a case, one can just check a certain convergence $\mathscr L_n\to\mathscr L$. However, to extend this result from $\mathscr L$ to $X$ one needs to show that a martingale problem for $\mathscr L$ has a unique solution. In general, one can define Markov processes as solutions of martingale problems.
Martingales themselves are not only (or almost not at all) about martingale problems (which themselves are more about operators). They often appear in optimal control problems thanks to their almost-constant structure, and are inevitable when dealing with SDEs or SPDEs. The latter often arise in evolutional dynamics and can have examples in all systems that evolve in time.
P.S. your reminder of what is a martingale problem, especially a linear operator applied to a process is misleading.