Neveu is elegant beyond belief. I keep my copy close. I was aimed at Neveu by a star student of E. Cinlar, long at Princeton and before that at Northwestern -- long editor in chief of Mathematics of Operations Research. Neveu was a student of M. Loeve at Berkeley. So was the current darling of machine learning, L. Breiman, because of his Classification and Regression Trees (CART). Breiman's Probability as published by SIAM is generally easier reading than Neveu.
For stochastic processes, there are several relatively different directions to go.
Martingale theory is gorgeous, astounding, amazing, with one of the most powerful inequalities in math, the astounding, tough to believe, martingale convergence theorem, and likely the shortest proof of the strong law of large numbers.
Then can do Markov processes more generally. The discrete state space version is important and not too difficult -- Cinlar has a nice introductory text.
A high end direction for Markov processes is potential theory. There are claims that that is the math for exotic options on Wall Street, but I doubt that there have ever been any applications.
There is a big role for second order stationary stochastic processes in electronic engineering. I ran into that for processing ocean wave data for the US Navy. Here the fast Fourier transform added a lot of interest.
And there's more.
Generally long Russia, France, and Japan seemed to have emphasized stochastic processes more than the US. But by now I suspect that the US is well caught up.
I'd have a tough time believing that very many people with money to hire know enough about high end stochastic processes, or even just Neveu, to hire in those fields. US national security may be about the only hope, that is, outside of academics.
Yes it appears that some of the quantum field theorists in physics are interested in path integrals.
Uh, I'm disorganized here: There is the field of stochastic optimal control!
As usual for advanced applied math, my suggestion is, outside of academics or US national security, find a valuable application and start a business to make money. That is, don't expect to be hired.
Incidentally, I have a copy of Loeve's old probability textbook - I wonder if it is too out of date to be any use.
Basically we're talking graduate probability based heavily on measure theory and some on functional analysis.
Last I heard of Neveu he was back in France and at least in part working on probability for stock markets.
There are other more recent authors. It seems that a big fraction of the best researchers sometimes take out a year and write yet another book comparable with Neveu. Personally, I did enough with Loeve, Neveu, Breiman, and Chung and for more attention to such material want to use that background to move on. For now, I'm concentrating on my startup; it's based one some applied math I derived, but I've done that and programmed it. Tonight I got four hard disks installed in my first server. Three of them are partitioned and formatted, and as I type this the fourth is being partitioned and formatted.