@xashans9754
On sets of measure zero, always bet on Lebesgue
@RajivSambasivan
This is the best summary of the types of integration. Awesome.
@jamiepianist
I'm so interested in Itô integrals now, Dr. Sean is the GOAT
@QuillPGall
another day, another Dr. Sean banger
@alexbstl
Re: peeking into the future, that’s not entirely true. The Stratonovich Integral (midpoint rule) is also an adapted process (meaning it can’t see the future) and even results in the standard chain rule when taken in differential form. The real reason people use an Ito integral is that it is a martingale. Admittedly, this is a bit technical.
@SeanRaleigh
I didn't know about the Itô integral, so I learned something new today. Nice!
@kindreddarkness
You are a fantastic teacher. I am new to calculus, and have really wondered why we would need more than a "general integral." This video not only answered the question, but also justified what the hell derivatives are actually measuring (and why we bother taking them). Thank you.
@castagnos509
you forgot the Kurzweil Henstock integral
@awesomesam101
Great video, but at 9:55 it's more correct to say that we get an irrational number almost every time (i.e. with probability one, but it's still technically possible to get a rational number).
@harriehausenman8623
Brilliant video!👍 Exactly the right amount of depth for me 🤗 Thanks for putting effort into the production and having great audio and video, and double thanks for not using negatively biased graphics. 🙏
@jesuseduardobanosgonzalez8116
Woah, I am astounded by how easy to understand you made the concepts of the more complicated integrals!
@fdileo
How to explain integral calculus in 12 minutes... You nailed it perfectly!
@jdp9994
Thank you. Very nice to see this info being simply explained. In the Ito integral, it's interesting how 2nd order terms are important (in a standard deviation sense) because of the nature of the random process, whereas in the other types of integration presented the 2nd order terms are considered insignificant (zero). Would have been nice (a luxury) to include the Generalized Riemann Integral (uses a different type of partitioning).
@agnelomascarenhas8990
Beautiful exposition with just the right dose/exposure of a new topic.
@Cpt.Zenobia
Love the visualizations always makes it easier.
@harjooni
If I'm not mistaken, by "random process" you mean a stochastic process, i.e. a random variable with an index number (often, though not always, interpreted as time) attached? In the case you showed, the index number has an interpretable direction so talking about it as time is meaningful and some authors would call the process causal. But what happens if we give up on the causality assumption (as some do in time series analysis, though that is in discrete time) and let the "future" (i.e. events with a high index number) affect "today"? Is the Ito integral still valid?
@ennyiszizlak7131
Hi Dr Sean! The video was very insighftul and easy to comprehend. Thank you very much for your work. I am looking forward to lean more maths in an MBA program than in my undergrad in finance. I was wondering about what measury theory is and how brownian is relevant to stohastic analysis. Thanks again, looking forward for more videos in the future :)
@taranmellacheruvu2504
I’d just like to say on thing regarding Itô integrals: one of the reasons it’s mathematically tricky to work with is that you’re integrating wrt a fractal function (Brownian motion is a fractal) meaning that you can’t really make your partitions infinitely thin in the same way as Riemann integrals. This results in some interesting rules regarding differentials
@fightocondria
Define a process for generating a random number that takes less than infinite time which can demonstrate only irrational numbers. Also, can you go into further depth on the higher level integrals?
@mtaur4113