Stochastic Calculus Self-Study Background required I am a CS graduate with working knowledge of calculus, probability and stats. I passed my CFA 2. I currently work 9-5 at a bulge bracket bank. 
I plan to attend a good MFE program in the fall of next year. As part of my pre-MFE prep, I am devoting my evenings to study the math pre-requisites with gusto. 
Currently, I am doing the following:


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*Solved and worked through Differential and integral calculus by N.Piskunov

*Completed chapters 1-3 & reading further, elements of real analysis, Bartle

*Reading and solving Elements of Integration and Lebesgue measure


Given that, I have time only until May '17, it would be extremely helpful, if someone could recommend just the precise set of topics to study from the below areas


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*Real Analysis

*Measure theory essentials

*measure-theoretic probability & renewals, queues and martingales

*ODEs and PDEs essentials


before my classes start. While I am not a math major, the goal is at the end of the course, I want to be really good at my stuff. 
I am enjoying the real analysis proofs and I think that's a good rigorous start.
Thanks,
Quasar.
 A: Advanced Probability Theory (Probability with Martingales by David Williams of course!)


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*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*)


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*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)


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*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
