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galvinator
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Feasible(?) summer holiday self-study project

March 25th, 2019, 9:02 am

Hi Wilmott forums,

I was thinking of self-studying Measure, Integral and Probability - Marek Capinski, Springer Undergraduate Mathematics Series, over the summer holidays (~13 weeks). However, I would like to know whether this proposal is feasible and would like to hear your opinions!

Some background info:
1) I finish my first year of undergraduate studies in summer
2) Formally, I have taken courses in linear algebra, single and multi-variable calculus, mathematical analysis (Introduction to Real Analysis - Bartle), calculus-based probability (A First Course in Probability - Sheldon Ross). I have previously done some self-study on Riemann integral (Elementary Analysis -  Kenneth Ross) and ordinary differential equations (Paul’s notes - Lamar University) though I will have to revisit the latter two formally.

The long-term goal in mind is to learn stochastic calculus in finance by the third/ fourth year of undergraduate studies. But I know there are many prerequisites to pick up: measure theory, measure-theoretic probability, discrete stochastic processes, continuous stochastic processes and then finally stochastic calculus. As such, I have decided to start by first learning measure theory and further probability.
 
However, I know that this plan will not be easy, as a level of mathematical maturity is required.
So if you feel that my project is not feasible (maybe because I will find myself struggling at every step), feel free to suggest any alternatives that are along the line of mathematical finance!  PS: I may be doing this project with a mathematics professor from my university over the summer, albeit unsure to what extent of supervision will I have (let's assume little).

I hope to hear from you soon! :) 

Galvinator
 
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bearish
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Joined: February 3rd, 2011, 2:19 pm

Re: Feasible(?) summer holiday self-study project

March 25th, 2019, 11:47 am

Different people have different learning styles. Personally, I found lectures and enforced homework to be particularly helpful in highly abstract topics like measure and integration. I would recommend instead trying to do an empirical research project that brings you in direct contact with financial data. Those prices that you would like to model prospectively with stochastic processes manifest themselves ex post as time series that can be analyzed with statistical tools. Is volatility stochastic? Is the correlation between stock prices stable? Does the shape of the yield curve predict the stock market? At this stage, the exact questions you ask aren’t that critical. Who knows, you may even discover that you don’t care that much for actual finance, which can be a bit messy.
 
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bearish
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Joined: February 3rd, 2011, 2:19 pm

Re: Feasible(?) summer holiday self-study project

March 25th, 2019, 12:26 pm

As for studying stochastic calculus for finance applications, I’m a fan of the approach taken by Steve Shreve in his two volume book. The first volume presents both the probability theory and the finance fairly rigorously in a discrete time/state setting, which greatly simplifies the mathematics. Volume two then addresses the continuous time/state version of the theory, with great benefit of analogies to the material in volume one.
 
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Cuchulainn
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Re: Feasible(?) summer holiday self-study project

March 25th, 2019, 1:46 pm

Since it is the summer (roll out lazy, hazy days..) it might be useful to scope on  Measure and Lebesgue theory separately from applications as it is quite difficult. Looking at finance books that pretend to do measure will confuse you more. The question is also whether you have the mathematical sophistication to learn it on your own (I did MT only in 2nd year). Maybe the MT book by Paul  Halmos and possibly Rudin Mathematical  Analysis.  Also Real Variables by Spiegel (Schaum) because it has lots of numeric examples.

Bartle's book looks very good; if you can learn everything in it in 13 weeks is a better investment than MT in the short term. And I think it is feasible.

An option would be Riemann and Riemann-Stieltjes integrals before Lebesgue.

Here was a discussion.
viewtopic.php?f=8&t=95199&p=681744&hili ... re#p681744

BTW I hope your are undergrad in a decent math program.
 
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Cuchulainn
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Re: Feasible(?) summer holiday self-study project

March 26th, 2019, 11:28 am

Sanity 101 check, test for Lebesgue understanding..

Integrate [$]f(x) = x(1-x)[$] on [$](0,1)[$] by hand from 1st principles (upper and lower sums based on disjoint(?) measurable sets) with say 4 divisions of [$]0 \leq f(x) \leq 1[$].
Using pen and paper.

It even can be automated but you need to find all roots [$]x[$] of [$]Y = f(x)[$] where [$]Y[$] is a given function value. In this case it is a quadratic equation.