The story so far...
Despite the nature of my blog so far, I am in Boston this summer to take part in a UROP where I try my hand at research in Mathematics. A research project in summer can be difficult to organise. Summer is a popular time of year for academic conferences, with (almost) all of those pesky students away. Due to these conferences and the fact that I have two supervisors for my project, as well as a well deserved holiday to Italy, I had my first chat with one of my supervisors two weeks into my time here. In preparation for this meeting, I explored all the things I have learned over the last two weeks which I decided to share with you here!
As I mentioned in an earlier post my research journey started out with a whole lot of reading. On my very first day, I was handed a small pile of books, luckily only one that I needed a complete understanding of. This was 'An Introduction to Stochastic Differential Equations' by L.Evans which gives an undergraduate friendly welcome into the world of SDE's. Stochastic Differential Equations are related to Ordinary Differential Equations, in fact, the two are almost the same but with an added 'noise' term that makes it a lot more realistic when it comes to modelling finance, chemical interactions and the like. This noise term is often defined as Brownian motion, this could be likened to a path in which the distance (and direction, when taking signs into consideration) of every step you take is modelled by normal distribution independent to all other steps. Despite being smooth this motion is non-differentiable meaning we don't use the traditional 'dx by dt' notation, it also means we need to consider what really happens when integrating over 'dW'. Mathematicians defined two different types of integration to solve how we integrate with respect to something stochastic, which converge to two different results (helpful!) These are called Ito's Integral and the Stratonovich integral. I should also mention that because of the random noise term which is defined rigorously in terms of statistics, the solution to an SDE is a random variable. This meant that my project has to consider a lot of probability theory too.
Having learnt about Brownian motion, Stochastic integrals as well as the appropriate probability theory I was ready to solve some SDE's. One of my supervisors gave me a problem to look at that models drug delivery. So far I have attempted to solve the equation and subsequently went on a journey in order to check if this solution was correct. There are a few methods that can be used to estimate the solution of an SDE on a computer, a few of these stemming from a Taylor series approximation of the problem. After spending a few days on MATLAB I managed to render a computational result for the SDE which I put up against my calculated solution, my results were ... inconclusive. Apparently, MATLAB isn't the best language for SDE's, especially as such problems have multiple solutions that somehow I need to represent. This brings me up to now, where I have started to play around with the programming language, Julia.
My time so far has been very good for me to learn about which parts of the research process resonate most with me. The biggest challenge for me so far has been the reading, especially as my approach may not be representative of how reading for research is meant to be done. I was completely in the dark about my chosen subject so had no choice but to read up, meaning I spent long hours learning through books, videos and online forums in order to even touch an SDE. This part had its rewards such as being able to learn about some fascinating new material, but it also had its downsides such as trying to be switched on mathematically for very long periods of time. A difficult start then leads on to a more enjoyable experience later on where I was able to implement what I learnt in problem-solving and programming. Of course, I still read up but breaking it up with more engaging type of work makes it effective.
As I mentioned in an earlier post my research journey started out with a whole lot of reading. On my very first day, I was handed a small pile of books, luckily only one that I needed a complete understanding of. This was 'An Introduction to Stochastic Differential Equations' by L.Evans which gives an undergraduate friendly welcome into the world of SDE's. Stochastic Differential Equations are related to Ordinary Differential Equations, in fact, the two are almost the same but with an added 'noise' term that makes it a lot more realistic when it comes to modelling finance, chemical interactions and the like. This noise term is often defined as Brownian motion, this could be likened to a path in which the distance (and direction, when taking signs into consideration) of every step you take is modelled by normal distribution independent to all other steps. Despite being smooth this motion is non-differentiable meaning we don't use the traditional 'dx by dt' notation, it also means we need to consider what really happens when integrating over 'dW'. Mathematicians defined two different types of integration to solve how we integrate with respect to something stochastic, which converge to two different results (helpful!) These are called Ito's Integral and the Stratonovich integral. I should also mention that because of the random noise term which is defined rigorously in terms of statistics, the solution to an SDE is a random variable. This meant that my project has to consider a lot of probability theory too.
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| Inconclusive MATLAB results. Computer's solution is in Red, Mine is in Pink. |
My time so far has been very good for me to learn about which parts of the research process resonate most with me. The biggest challenge for me so far has been the reading, especially as my approach may not be representative of how reading for research is meant to be done. I was completely in the dark about my chosen subject so had no choice but to read up, meaning I spent long hours learning through books, videos and online forums in order to even touch an SDE. This part had its rewards such as being able to learn about some fascinating new material, but it also had its downsides such as trying to be switched on mathematically for very long periods of time. A difficult start then leads on to a more enjoyable experience later on where I was able to implement what I learnt in problem-solving and programming. Of course, I still read up but breaking it up with more engaging type of work makes it effective.
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| A Stochastic Differential Equation |