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I've always enjoyed probability and stochastic processes (took two courses in stochastic models in undergrad, and a PhD level intro to stochastic processes course for my master's). Someday I'd like to go back to school, and most likely I'll study applied probability/stochastic processes. Now, the vast majority of work in this area these days seems to be motivated by quantitative finance; while I have nothing against this field, it is not really what I'm interested in. I took a course in accounting and one in engineering economics and was bored out of my mind.

Additionally, I still have no idea specifically what I want to research. This of course makes it hard to write a coherent personal statement for graduate school applications. I will be working for the next 3-4 years to pay off my loans, so I have some time to think about this. I work in revenue management, and the most interesting thing I've seen so far is a probabilistic overbooking model. That model isn't overly sophisticated but it's a cool example of using probability to maximize expected revenues.

Any ideas on areas to research (and books/references) would be greatly appreciated!

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Any field that involves some sort of processes with noise, uncertainty or probabilistic behavior will make use of concepts from stochastic processes.

Practically speaking, I've seen the associated theory applied to spacecraft dynamics (in the context of attitude estimation), celestial mechanic (in the context of tracking), meteorological phenomena (in the context of data assimilation), analytical mechanics (in the context of vibrational behavior), and in electrical engineering (in the context of computer vision and stochastic signals).

These methods can also be important in data science/machine learning, although I have no experience with either.

Often times, the ideas from stochastic processes are used in estimation schemes, such as filtering. These are method which are used to propagate the moments of a probabilistic dynamical system. Since many systems can be probabilistic (or have some associated uncertainty), these methods are applicable to a varied class of problems.

When framing some problems, quantities which in reality are deterministic, can be treated as probabilistic. This is sometimes done in parameter estimation (or system identification, which is similar), which seeks to estimate some deterministic parameter based on some input data.

You can look up more references based on the fields and contexts, or ask me, but here are two examples as a proof of concept:

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Quantitative finance is not accounting, nor is it engineering economics. How it possible to show a disinterest in something when your only experience with it is based on your experience with unrelated (or, not completely related) fields? That is the first question I would ask.

There are applications to so many fields - neuroscience and physics, for example. See the book Numerical Methods for Stochastic Differential Equations by P. Kloeden. They have a whole section on applications of stochastic processes to different fields.

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  • $\begingroup$ Well of course those are entirely different fields. I'm just saying that I've never particularly been interested in finance, and my only formal exposure to anything involving calculations with money has left me extremely bored. I like the theory of stochastic processes. I don't really have any interest in finance. $\endgroup$ – Math1000 Mar 5 '15 at 1:34
  • $\begingroup$ Do you know how probability theory was invented? $\endgroup$ – AXH Mar 5 '15 at 21:01
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It is probably marketed as an entry point into financial mathematics, since people in finance have realized that engineers and mathematicians can help them make computer programs to make money.

However, lots of other fields exist: Signal processing, Image processing, Machine Learning, "Big Data" are just the ones I can think of straight away.

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Mathematical biology (ranging from population dynamics to newer systems biology) makes wide use of both deterministic and stochastic models.

See, for instance, http://www.math.wisc.edu/~anderson/RecentTalks/2010/CIBM.pdf by David Anderson for some readable slides.

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