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.
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 :)
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!
The explanation is really good. Thank you for putting so much thought and effort.
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.
Me too, man. Textbook explanations of Ito Integrals are needlessly complicated. Glad I could help!
Great video, thank you so much!
this is AWESOME!! Thanks for explaining this so clearly!
The explanation is really good
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 :)
This is an incredible comment; the most flattering compliment I've ever received. Thank you!
Thank you! You really helped me a lot!
Hello, the video helped a lot. Thank you
How did you use the property of Brownian Motion at 13:15? Thanks again
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!
@@johnthequant That's amazing! Thank you very much, all the best.
beautiful explanation! thank you
noice. this reminds me of number theory class, the professor would always use an ipad to write long page proofs exactly like that..
Brilliant, thank you!
Why do you omit the 1/2 from the variance computation on the second summation?