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:
Advanced Probability Theory (Probability with Martingales by David Williams of course!)
- Measure Spaces
- Events
- Random Variables
- Independence
- Integration
- Expectation
- WLLN, SLLN, CLT
- Conditional Expectation
- Martingales
- Convergence of Random Variables
- Uniform Integrability
- Characteristic Functions
Basic Real Analysis (Lay Analysis with an Introduction to Proof or Ross Elementary Analysis*)
- Real Numbers (inf, sup, Heine-Borel, Bolzano-Weierstrass)
- Functions, Limits, Continuity
- Definitions, Existence, Properties of Integrals
- Sequences of Real Numbers
- Sequences of Functions
Advanced Real Analysis (Royden Fitzpatrick - Real Analysis)
- Lebesgue Measure
- Lebesgue Measurable Functions
- Lebesgue Integral
ODE and PDE
These were barely touched in my stochastic calculus classes. I think the only thing relevant here is solving second order linear ODEs.
I guess there are/can be links between DE and stochastic calculus/analysis as you go deeper into certain areas, but I don't think these are required for basics of stochastic calculus/analysis.
Measure Theory
My advanced probability and stochastic calculus classes needed only real analysis classes as prerequisites. It seems the basics of measure theory are already covered in Lebesgue Measure and Lebesgue Measurable Functions in Real Analysis and Measure Spaces, Events, Random Variables and Integration in Advanced Probability.
Basic Probability Theory
Don't forget Basic Probability Theory. Be sure to understand basic set theory, independence of events, independence of random variables, moment generating functions, conditional probability and conditional expectation on events before going into independence of sigma-algebras, characteristic functions and conditional expectation on random variables or sigma-algebras.
*Trench Real Analysis was kinda hard for me to digest. idk. We used trench and lay in undergrad. I used Ross and Lay last year
Best Answer
My two cents...
Put aside all the measure/analysis textbooks. When you learnt calculus in college or high-school you did not have to learn analysis right then, isn't it? So focus on getting a good understanding of Ito calculus first. Focus on being able to compute stuff. Pay attention on the "how to" and not on the "why". Go to a library and check out these two books: