Quadratic and Total Variation of Brownian Motions Paths, inc mathematical and visual illustrations

[@francoisallouin1865]

Forget about KhanAcademy. This IS the thing ! I love the robotic voice, it suits Maths sooo much and makes things very clear. the text underlying the voice is rigorously and smartly written. the mere idea of starting with a general definition of quadratic variation of a differentiable function is genius. This is a revolution in Maths Education.

[@quantpie]

Thank you for the kind words, Francois!

[@kimchi_taco]

as non native english speaker, robotic voice is more clear for me. XD

[@JaGWiREE]

Struggled with getting this concept 100% for a few weeks, thank you!

[@95MarioKaci95]

Thank you. A am basically studing your channel now :)

[@alexbougias2948]

Nice and complete video. Good job!

[@quantpie]

Thanks Alex!

[@salomking5796]

in 9:28 why did you put out b/2^n outside of the limit though it depends on n so not a constant ????and thanks a lot

[@quantpie]

True. Should have left the limit in front of b/2^n, as that is what makes the whole expression go to zero in the last step. So used implicitly, but you are right, should make it explicit.

[@quantpie]

Thanks for highlighting this. Much appreciated!

[@tramquangpho]

At 8:06 why after using the Mean Value theorem the total term from 2**n to 2**n *n , why is that ?

[@tramquangpho]

And why the total term is 2 **n when it should be just n

[@quantpie]

thanks that n after 2^n is a typo! well spotted!! many thanks!

[@quantpie]

Re-2^n, this is because we use refining partitions: at step 1, we have two partitions; at step 2, we have 2^2=4, and so forth. Reason this approach is used is because it helps us continuously refine the partition, keeping the points from the previous step. This helps eliminate the impact of changes in the set of evaluation points.

[@malyathashridharan8389]

Thank you for your videos. They are amazing!

[@quantpie]

thank you! glad you liked it!

[@vidarlarsen4090]

Excellent video. Thank you!

[@quantpie]

Thanks Vidar!

[@wowZhenek]

Very nice visual, extremely extremely helpful. The proof of the QV of BM could have been made easier without any extra theorems though. The Shreeve book has a very basic proof, but anyway, magnificent, truly.

[@quantpie]

Thanks!! Re-Shreve, please see the discussion in the comments section of this question: math.stackexchange.com/questions/80001/quadratic-variation-of-a-brownian-motion-up-to-time-t-converges-to-t-in-l2?rq=1 L^2 is indeed easier.

[@tim2138]

Hi, thanks for the video and it's really insightful! I am wondering for the last step using the Borel Cantelli lemma, how to get to lim(S_nk) = 0 from P(limsup(S_nk >= epsilon)) = 0?

[@NathanCrock]

I notice in later videos you mention that (dB_t)^2=t and explain that the square of the differential is 'basically' quadratic variation. So I came back to this video for clarity. However, I still have a point of confusion. The quadratic variation takes the square of the absolute deviation over small intervals and adds them all up. I was under the impression that dB_t was any one such interval. Therefore how is dB_t, one such interval, 'basically' the same as the sum of all such intervals? Perhaps I am thinking too much like ordinary calculus. What is the relationship between the stochastic differentials and variation?

[@quantpie]

Thanks @Nathan Crock! Great curiosity and great attention to detail! The few concepts are very interlinked. Let’s start in the Ito’s sense. The stochastic differential equation do not have meaning in the differential sense (e.g., Brownian is non-differentiable), it is just a fudge so that we can use the same/similar techniques when dealing with deterministic and stochastic differential equations. The SDE differential get their meaning only through the integral, and hence any interpretation of the SDE depends on the interpretation of the equation in the integral form. And that’s why Ito’s lemma (and Ito’s definition of stochastic integral) is so important. The extra term you get there is just linked to the quadratic variation. When you work with Stratonovich, then you don’t get this quadratic term, so normal rules of calculus work, but then you pay the price in terms of the properties of the increments. Hope this helps! And we are planning to do a short video on this topic once we have finished the in-progress work! many thanks!

[@NathanCrock]

Looking forward to the supplemental video! A few examples of how differentials and their powers are calculated would be very helpful (such as those you use in subsequent videos). You're so knowledgable and helpful I almost feel guilty abusing your generosity. Thanks for the quick and detailed reply.

[@quantpie]

@@NathanCrock thanks! no worries at all, we only learn through questions, keep it up!

[@Sobriquet80]

Just one clarification on your question (and thanks to the excellent quantpie videos. I am a fan!!), if my understanding is right, it doesn't make sense to say dB_t^2=t. But is more like summation over (say) from 0 to t for dB_s^2 = t. (so it depends on what is the integration interval). Its only in the short form of the SDE where you lose the integral signs you make a quick approximation of dB_t^2=t. But it really means that the integral signs sum from 0 to t.

[@arianj2863]

the robot voice gives me a headache, unfortunately, since I really like the explanations :-/

[@quantpie]

thanks! We don't use this voice anymore, so hopefully you will find the new videos more pleasant!

[@steventroup9061]

Why is the speaking like an automated scam phone call?

[@quantpie]

sorry about that, we have changed the voice to reflect feedback, but we have not redone the historical uploads. Many thanks for the feedback!