Stochastic Calculus Self-Study Background required

Question

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:

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

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.

any particular reason the course begins in June? Most courses begin in September right? — BCLC, Oct 24 '16 at 22:23
You are right. They all begin in Sep. I am also applying to just one at Singapore, which begins in June. :) — Quasar, Oct 25 '16 at 01:09
I am doing fine. How are things at your end? Great to hear after long. — Quasar, Apr 08 '18 at 08:27
@BCLC Hope you are well. I would like to connect to you offline. I enrolled for and undergraduate mathematics course, currently in the first year — Quasar, Apr 08 '18 at 11:59
Does this answer your question? Will I have learned the prerequisites for self learning stochastic calculus and monte carlo method? — user95921, Dec 28 '23 at 01:35
@user95921, imho the answer below includes all topics on the critical path. The books I used were (i) Stephen Abbott's UA, (ii) selected topics from Capinski - measure/expectation, (iii) convergence theorems - Grimmett & Stirzaker and (iv) A first course in stochastic calculus written by quant & trader, L.P. Arguin — Quasar, Dec 28 '23 at 20:34

BCLC · Accepted Answer · 2017-07-21T09:08:43.787

2

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

edited Jul 21 '17 at 09:08

answered Sep 24 '16 at 01:31

BCLC

13,459

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The connection between multidimensional SDE and PDE involving the Laplacian is quite significant but otherwise it's not a big deal. – Ian Sep 24 '16 at 01:37
@Ian Feynman-Kac? – BCLC Oct 22 '16 at 10:26
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Feynman-Kac is one of those connections, yes. – Ian Oct 22 '16 at 11:39
@Ian what are some of the others please? – BCLC Oct 22 '16 at 12:22
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There are some connections where the probability side involves stopping times which arise through boundary conditions on the PDE side. For example, http://math.stackexchange.com/questions/1946397/probability-that-a-brownian-motion-with-drift-hits-1-before-hitting-1-before-t/1959667#1959667 – Ian Oct 22 '16 at 12:56
@Ian oh thanks. That looks more significant than FK because it involves series solutions – BCLC Oct 22 '16 at 13:12
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Nah, that's just me explicitly solving the PDE. The connection is still just "the solution to this probability problem is the solution to this PDE with these auxiliary conditions." And the PDE itself is the same as Feynman-Kac. – Ian Oct 22 '16 at 13:18
@Ian reread question and answer + your comments. The solution to the PDE gives the required probability right? If so it seems to suggest that probability theorists would be interested in solving PDEs for other/future probability problems? My context: In stochastic calculus class we used probability to solve a PDE. What you've shown me is that we've used PDE to compute a probability. So PDEs seem pretty significant to probability. Am I mistaking something here? – BCLC Oct 24 '16 at 22:22