Our First Ito Integral

In this video, we walk slowly through our first Ito Integral, as an introduction to stochastic calculus. Really, really slowly. I know how hard this stuff is to learn on your own, so I try to be as explicit as possible and skip NOTHING!
Ito Calculus isn't that hard once you get to know it ;)
Like, subscribe, have a great day!

[@junaidjawaid547]

The explanation is really good. Thank you for putting so much thought and effort.

[@andrewbenson8842]

Thank you so much! This makes so much more sense! The Ito Integral was such a mystery to me even after reading 3 separate textbooks on this topic all of which did a bunch of hand waving for even this exact same integral problem. Now I now how to actually to compute an Ito integral which is was I was was looking to learn.

[@johnthequant2571]

Me too, man. Textbook explanations of Ito Integrals are needlessly complicated. Glad I could help!

[@ghostwhowalks5623]

this is AWESOME!! Thanks for explaining this so clearly!

[@remarljosephdamalerio8921]

Thank you! You really helped me a lot!

[@pierrebesouhof9987]

beautiful explanation! thank you

[@stochasticxalid9853]

In history, there is John the Baptist and John the Quant. John the Baptist made the path easy to people to get to know the messiah. John the quant is showing the very shortcut to getting acquainted with quant finance. Thank you so much, John :)

[@johnthequant2571]

This is an incredible comment; the most flattering compliment I've ever received. Thank you!

[@tteej]

Brilliant, thank you!

[@atmonotes]

noice. this reminds me of number theory class, the professor would always use an ipad to write long page proofs exactly like that..

[@andrewbenson8842]

Why do you omit the 1/2 from the variance computation on the second summation?

[@xXDamolaXx]

Hello, the video helped a lot. Thank you
How did you use the property of Brownian Motion at 13:15? Thanks again

[@johnthequant2571]

Thanks for watching! Brownian motion is a Wiener process, and one of the defining characteristics of a Wiener process is that each time-step is an independent increment. In the case that increments are Gaussian, like in Brownian motion, each group of timesteps is a sum of independent Gaussian increments. The sum of independent Gaussian increments is also normally distributed; the mean and the variance of the sum is the sum of the means/variances of the increments.
So,
B(s)~N(0, s) and B(t)~N(0, t) => B(s) - B(t) ~ N(0, s-t),
and B(s-t)~N(0, s-t).
Since B(s) - B(t) and B(s-t) have exactly the same distribution, we can substitute one for the other inside the expected value.
I hope that helps!

[@xXDamolaXx]

@@johnthequant2571 That's amazing! Thank you very much, all the best.