Love such type of lecturers who develop everything from the intuitive idea, and derive everything on a blackboard. The only shortcoming of the course was the small size of the blackboard.
This guy is great! My lecturer in Utrecht is awful, she basically just writes a 2 line proof and somehow we're supposed to figure it all out. Really appreciate that he went through all the steps, especially when he slowed down a bit at 22:55
At 24:30 he had exp(mu t + d x) twice in the Taylor expression. But the d x wasn't the differential dx. Instead it should have been exp( mu t + sigma x), which is the same as f(t,x) . It didn't matter though because the next step brought the correct result.
A minor point. Dr. Lee mentioned a few times "sum of normal variables is normal." This is true for *independent* normal variables but not in general. In the context of the video, the variables were independent (changes of Brownian motions over distinct intervals), so the statements remain correct.
NO. Sum of dependent normal distribution is also normal. You need to add the co-variance part into the variance of the new distribution, but it is still normal.
@@abcde9421 NO. This is only true if they are multivariate Gaussian. (Cramer Wold device) The canonical counterexample is two univariate Gaussians which is not a bivariate Gaussian
23:00 it's exp{ µt + σx }, not exp{ µt + dx } 26:40 it's dSt / St = 1/2 • σ^2dt + σdBt, not dSt = 1/2 • σ^2dt + σdBt 1:06:10 P and P_tilda share the same rank in terms of matrix
I agree that he's a fine speaker -- and one of the things he has going for him is he, or perhaps his technician, has bothered to get the sound very close to right. This is unusual: a lot of educational tech folks still seem to think we're in their debt for their existing at all, and thinking they should actually do their jobs competently is far far more than we're entitled to. So well done, fella.
Rachel Watsky mit may not be poor, but that doesnt mean they give the ocw team an endless supply of money. More so since ocw doesn't actually benefit MIT
You love very strange things Rachel... People have dedicated a lot of time to what's in and behind all these videos. And you get it for free... Something tells me you think the same about your parents. Thank you MIT
@@fluxpistol3608 It really doesn't take much effort or money for MIT to make courses publicly available. You would think for an institution that's so "woke" and into social justice, they'd care quite a bit about making education available to everyone for free, as this is especially beneficial for people in developing countries
um.. my personal opinion and it ain't be correct but he simply assumed or that he let f(x)=x^2 and f(t,x) =x^2 the same. I took this time does not affect and is constant, so in the stochastic process dXt = mu*dt + sigma*dBt, time has no influence to the function f(t,x), so the drift term should be '0' and mu becomes zero. However, sigma term(brownian motion term) must be exist, but I am not sure why the sigma value should be '1'. or.... did he probably think the stochastic process follows standard normal distribution which has mu and variance (0, 1)?
Let's start at the basics: why do the 2nd order terms and above vanish so that f(x+dx) = f(x)+f'(x)dx (here I'm treating dx like an infintesimal, but if you want to make it standard just add the limits)? Answer: because for a smooth function, the linear approximation using the slope is good enough in the limit. It also has bounded variation as a function. So why isn't that good enough for Ito's lemma? Because the Brownian motion isn't smooth. In fact, as a stochastic process it has quadratic variation. This means that (dB)^2 and dt grow at roughly the same rate. So if we're going to keep the first order terms, then we also have to keep (dB)^2 as well. So it's not that we're not getting rid of the second order terms and above- we are- it's that we're forced to keep one of them as it contributes as much as a first-order term. But since it is only quadratic variation, it's safe to throw all the higher order terms away. If your question was "why do the higher order terms vanish in general", then that's just because x+(x^n) is basically equal to x for very small x and n>1.
The lecture is helpful in one sense. In another sense it is very bad maths. Teaching others how NOT TO think like mathematicians. Extremely non-rigorous and almost anti scientific. Comments are positive only because of the lack of serious maths background of reviewers. Of course there are advanced materials he can't cover here. But this should have been immediately clarified when appropriate instead of giving affirmations that are fundamentally wrong. It is better to tell "we admit this result" than to give an explanation that is not correct. In particular (dBt)^2=dt.....what , why ?! In spite of all that, I would like to stress how useful and profound is this lecture, there are incredible insights that I could not find anywhere else not even in Shreve books. MANY TANKS MIT .
because if you take the generale form of dX_t = udt + sigmadB_t and you are applying the function f directly only on Bt then considering the general form , u must be 0 and sigma 1 so that dX_t = dB_t or X_t = B_t
Sorry, we were not given the problem set solutions to publish. What materials we were given can be found on MIT OpenCourseWare at: ocw.mit.edu/18-S096F13.
Excellent remark. and the chalk is huge !! I guess its more visible because the lecture amphitheater is big. white boards are only suitable for small class rooms not amphis
I don't understand something. How is it possible that in a top uni like MIT, the teacher says "I don't know why" or "I could not explain why..." about things that are very well known to anyone who seriously studied stochastic processes? I mean, you don't have strong teachers who did research? This teacher explains very well the basics but I am amazed that he says he doesn't know why driftless process is martingale or why the 2 BM are equivalent, and so on...
+Morpho32 I thought the same but then I looked it up. First of all, Mr. Lee is clearly from a different field of Mathematics. From what I saw on the internet, his research subjects are in the field of Discrete Mathematics. Moreover, this course was originially not planned to be a pure math lecture. It is also open to students from other majors than Mathematics and is supposed to give students a picture of what they might do after graduation. Mr. Lee does a great job here in explaning the basics, despite his background and I believe some of those technical details really are not that important for that purpose. I am pretty sure MIT got some good lectures on pure Stochastic Calculus as well for graduate students and if not, there are still ETH or UT Vienna. Him admitting he does not know some specific property or theorem that is beyond his field of research is just totally understandable and makes him human imho. I might add, that I only watched 2.5 episodes of this lecture tho.
+Morpho32 Please understand purposes of the course. It's more likely application of important stochastic calculus theories especially for undergraduate students. That is, it's not pure theory course for grad. students. That's why he became one of lecturers of the course though the topic is not his specialized field. I think the contents he dealt with in the course are pretty well organized in a concise manner.
There are many fields in math - NO ONE, including Fields medalists can possibly know much about other specialties. BTW, you would not understand pure "ITO calculus" if taught at the graduate level. MIT has plenty of experts in this field at their Math/FInance/Econ departments. Lee is one of the rising stars in the world, btw.
In 8:27 I didn't get why (dBt)(dt) and (dt)^2 are neglectable, square root of a tiny number gives a bigger number (sqrt(1/4)=1/2) so sqrt(dt) should be bigger than dt. I can't accept that explanation. The argument "they´re tending to zero faster than dt and dBt" isn't good enough. Can someone tell me whats going on.
I think the explanation goes beyond the level of this course. (I also study stochastic calculus this year and yep, this remained as a "just memorize this".
The explanation is basic I think , it is a shame he did not clarify it. When you write "dx" you exactly mean: the limit of (x+t )-(x) as t-->infinity. Similarly "at the limit" (dBt)(dt) is neglectable compared to just dt. Example: Let " ~" mean EQUIVALENT TO, then : (3x^2 + 5x) ---> zero as x-->0, yet (3x^2 + 5x)/5x --->one as x-->0 therefore (3x^2 + 5x) ~ 5x as x-->0 because x would be decimal at the neighbourhood of 0. dt and dB and any d"something" is close to zero by definition of a limit tending to zero ( think raising several limits that tends to zero to different powers and then summing them, only the that with the smallest power will remain after summation) .
I agree with both sides in this argument. The blackboard certainly seems like a ver-ree olde kludge, but the process of the guy walking through the development of any line of thought -- slowed to the speed at which he can write it -- is pretty good. Anything projected is going to be prepared beforehand, and some "slide" or screenshot that the guy spends hours on would or could flash by in an instant. All sense of development of the thought would be lost. Yes, there's got to be a better way. I have no idea what it might be.
Some notable Timestamps:
0:00:25 Itō Calculus
0:12:33 Ito’s lemma
0:40:57 Adapted processes
1:00:18 Change of Measure
1:05:32 Equivalence of probability distributions
1:13:18 Girsanov’s theorem
Thanks a lot👍
Love such type of lecturers who develop everything from the intuitive idea, and derive everything on a blackboard. The only shortcoming of the course was the small size of the blackboard.
8 years later,still saving lives!🔥🔥🔥
This guy is great! My lecturer in Utrecht is awful, she basically just writes a 2 line proof and somehow we're supposed to figure it all out. Really appreciate that he went through all the steps, especially when he slowed down a bit at 22:55
you are beautiful
ook in maastricht
are you single?
Haha! I had her last year. Awful indeed
He's Korean
this dude is a legend. Thank you so much Choongbum Lee and MIT.
At 24:30 he had exp(mu t + d x) twice in the Taylor expression. But the d x wasn't the differential dx. Instead it should have been exp( mu t + sigma x), which is the same as f(t,x) . It didn't matter though because the next step brought the correct result.
Thank you for pointing that out...I was confused about it :)
same here
he is so great, when explaining these concepts so well,,i cry dont know what i could do without this !
A minor point. Dr. Lee mentioned a few times "sum of normal variables is normal." This is true for *independent* normal variables but not in general. In the context of the video, the variables were independent (changes of Brownian motions over distinct intervals), so the statements remain correct.
thanks man
NO. Sum of dependent normal distribution is also normal. You need to add the co-variance part into the variance of the new distribution, but it is still normal.
@@abcde9421 NO. This is only true if they are multivariate Gaussian. (Cramer Wold device) The canonical counterexample is two univariate Gaussians which is not a bivariate Gaussian
Sum of two normals is not normal?
Why?
@@andrewzhang5345 can you please explain more?
If we add a Covariance matrix, this apparently should work, shouldn't it?
Typo: Around 30:00, the line below (Q), dS_t should be dS_t/S_t.
드디어 이토 적분을 이해했다. 한국인의 자랑 이중범 교수님, 지금은 퀀트 쪽으로 가신 것 같던데.. 나중에 꼭 뵐 수 있는 날이 오기를. 좋은 강의 감사합니다.
Thank you MIT!
this guy is amazing, first time I learned ito formula so clear ,thank you bro! Hope I can go study in mit one day!
Best discourse on Ito's Lemma, amazing work Prof. Lee.
I love this lecture. Really helped grasp the practical use of Ito. Kuddos to prof. Lee and MIT !
That chalk is amazingly soothing and satisfying
U r a backbencher😂
Peace be upon you, thank you for this valuable lecture
23:00 it's exp{ µt + σx }, not exp{ µt + dx }
26:40 it's dSt / St = 1/2 • σ^2dt + σdBt, not dSt = 1/2 • σ^2dt + σdBt
1:06:10 P and P_tilda share the same rank in terms of matrix
exactly. My expectations were very high for an MIT professor.
I love this material, thanks MIT. I stared to study Fokker Plank Equation and this is so helpfull.
Omg, you spoke so clearly. Thanks a lot. I will try to send my son to MIT in future.
great speaker, explains very good, complex concepts become clear
I agree that he's a fine speaker -- and one of the things he has going for him is he, or perhaps his technician, has bothered to get the sound very close to right. This is unusual: a lot of educational tech folks still seem to think we're in their debt for their existing at all, and thinking they should actually do their jobs competently is far far more than we're entitled to.
So well done, fella.
was trying to find a lognormal variance’s dependence on time and a rigorous and long venture was answered in the first minute
pretty good explanation to the complicated topic
49:57 for Ito's isometry
Clearly I understood everything...
note1: 10:06
mark: 31:43
I love Choongbum
amazingly clear, a rarity
Many thanks for sharing. Really good lecture.
You save my exam, thank you ;)
Thank you so much to post this video! He explained everything very clear. Better than my teacher. It really helped me a lot.
That blackboard is clean as
Wish i could be in this class.
The stochastic integral should be defined first. The "differential" is just a notation.
51:00 LOL Hearing _il_ , which means one in Korean surprised me, professor!!! Yeah, I'm Korean!
Thank You Dr. Lee
I love how the opening includes "if you'd like to make a donation," something tells me MIT isn't hurting for money
MIT is very poor Rachel
Rachel Watsky mit may not be poor, but that doesnt mean they give the ocw team an endless supply of money. More so since ocw doesn't actually benefit MIT
You love very strange things Rachel... People have dedicated a lot of time to what's in and behind all these videos. And you get it for free... Something tells me you think the same about your parents. Thank you MIT
@@fluxpistol3608 It really doesn't take much effort or money for MIT to make courses publicly available. You would think for an institution that's so "woke" and into social justice, they'd care quite a bit about making education available to everyone for free, as this is especially beneficial for people in developing countries
@@williambrown8249 what’s your point? That’s what they do, do...
very very good lecture, thank you for providing such videos.
He is legend to me.
Change of measure: 1:00:20
and does anybody know why in @24:27 it's e^(mu*t + d*x) not e^(mu*t + sigma*x)?
at 21:05 why that one is 0 and the one in the middle equals 2 and ...
This guy just saved my life omygad
Thanks for good lecture
What is a drift term? Why is it necessary?
at first example min 20 why does mu = 0 and sigma = 1 ?
um.. my personal opinion and it ain't be correct but he simply assumed or that he let f(x)=x^2 and f(t,x) =x^2 the same. I took this time does not affect and is constant, so in the stochastic process dXt = mu*dt + sigma*dBt, time has no influence to the function f(t,x), so the drift term should be '0' and mu becomes zero. However, sigma term(brownian motion term) must be exist, but I am not sure why the sigma value should be '1'.
or.... did he probably think the stochastic process follows standard normal distribution which has mu and variance (0, 1)?
Perfect explanation!
Thanks Mr Lee!
great class. Thanks
Can someone just tell me the vibe of this? Thanks.
40:00 Damn, nobody ever told me this!
6:19 , 7:30 , 10:05
This guy has zero OCD
WHY do all the 3rd order and higher terms in the Taylor expansion for Ito's lemma all disappear??
Let's start at the basics: why do the 2nd order terms and above vanish so that
f(x+dx) = f(x)+f'(x)dx
(here I'm treating dx like an infintesimal, but if you want to make it standard just add the limits)?
Answer: because for a smooth function, the linear approximation using the slope is good enough in the limit. It also has bounded variation as a function.
So why isn't that good enough for Ito's lemma?
Because the Brownian motion isn't smooth. In fact, as a stochastic process it has quadratic variation. This means that (dB)^2 and dt grow at roughly the same rate. So if we're going to keep the first order terms, then we also have to keep (dB)^2 as well.
So it's not that we're not getting rid of the second order terms and above- we are- it's that we're forced to keep one of them as it contributes as much as a first-order term.
But since it is only quadratic variation, it's safe to throw all the higher order terms away. If your question was "why do the higher order terms vanish in general", then that's just because x+(x^n) is basically equal to x for very small x and n>1.
The lecture is helpful in one sense. In another sense it is very bad maths. Teaching others how NOT TO think like mathematicians. Extremely non-rigorous and almost anti scientific. Comments are positive only because of the lack of serious maths background of reviewers. Of course there are advanced materials he can't cover here. But this should have been immediately clarified when appropriate instead of giving affirmations that are fundamentally wrong. It is better to tell "we admit this result" than to give an explanation that is not correct. In particular (dBt)^2=dt.....what , why ?!
In spite of all that, I would like to stress how useful and profound is this lecture, there are incredible insights that I could not find anywhere else not even in Shreve books. MANY TANKS MIT .
the (dBt)^2=dt part is covered in the previous video of the course
Really cool! Thanks!
Very clear thank you
At 20:05, how did he obtain the values of mu and sigma?
because if you take the generale form of dX_t = udt + sigmadB_t and you are applying the function f directly only on Bt then considering the general form , u must be 0 and sigma 1 so that dX_t = dB_t or X_t = B_t
THM isn't equal lemma
where to find the solution of the problem set?
Sorry, we were not given the problem set solutions to publish. What materials we were given can be found on MIT OpenCourseWare at: ocw.mit.edu/18-S096F13.
even the definition of the riemann integral isn't correctly
can anyone tell me how to apply Fokker plank equation as filter? please, help me guys.
He is great
where is brownian bridge part?
38:30 i hope , someone need it
Kant was right.
lot of identity little thought
Tap tap tap goes the chalk after every bit
우왕 학교 선배다ㅋㅋㅋ 저도 카이스트 수학과에요.
Does someone know about the previous video where he proves (d(Bt)^2)=dt? - because of quadratic variance?
He explains it in lecture 17. Stochastic Processes II. ruclips.net/video/PPl-7_RL0Ko/видео.html
why was mu=0 and sigma =1 in the example at 21:10
Arka Bose initial setting is like that,indeed.
Fantastic
y use chalk instead of white board and pen?
Excellent remark. and the chalk is huge !! I guess its more visible because the lecture amphitheater is big. white boards are only suitable for small class rooms not amphis
thanks
I don't understand something. How is it possible that in a top uni like MIT, the teacher says "I don't know why" or "I could not explain why..." about things that are very well known to anyone who seriously studied stochastic processes? I mean, you don't have strong teachers who did research? This teacher explains very well the basics but I am amazed that he says he doesn't know why driftless process is martingale or why the 2 BM are equivalent, and so on...
+Morpho32 I thought the same but then I looked it up. First of all, Mr. Lee is clearly from a different field of Mathematics. From what I saw on the internet, his research subjects are in the field of Discrete Mathematics. Moreover, this course was originially not planned to be a pure math lecture. It is also open to students from other majors than Mathematics and is supposed to give students a picture of what they might do after graduation. Mr. Lee does a great job here in explaning the basics, despite his background and I believe some of those technical details really are not that important for that purpose. I am pretty sure MIT got some good lectures on pure Stochastic Calculus as well for graduate students and if not, there are still ETH or UT Vienna. Him admitting he does not know some specific property or theorem that is beyond his field of research is just totally understandable and makes him human imho. I might add, that I only watched 2.5 episodes of this lecture tho.
+Morpho32 Please understand purposes of the course. It's more likely application of important stochastic calculus theories especially for undergraduate students. That is, it's not pure theory course for grad. students. That's why he became one of lecturers of the course though the topic is not his specialized field. I think the contents he dealt with in the course are pretty well organized in a concise manner.
There are many fields in math - NO ONE, including Fields medalists can possibly know much about other specialties. BTW, you would not understand pure "ITO calculus" if taught at the graduate level. MIT has plenty of experts in this field at their Math/FInance/Econ departments. Lee is one of the rising stars in the world, btw.
In 8:27 I didn't get why (dBt)(dt) and (dt)^2 are neglectable, square root of a tiny number gives a bigger number (sqrt(1/4)=1/2) so sqrt(dt) should be bigger than dt. I can't accept that explanation.
The argument "they´re tending to zero faster than dt and dBt" isn't good enough. Can someone tell me whats going on.
I think the explanation goes beyond the level of this course. (I also study stochastic calculus this year and yep, this remained as a "just memorize this".
The explanation is basic I think , it is a shame he did not clarify it. When you write "dx" you exactly mean:
the limit of (x+t )-(x) as t-->infinity. Similarly "at the limit" (dBt)(dt) is neglectable compared to just dt.
Example: Let " ~" mean EQUIVALENT TO, then :
(3x^2 + 5x) ---> zero as x-->0, yet (3x^2 + 5x)/5x --->one as x-->0 therefore (3x^2 + 5x) ~ 5x as x-->0 because x would be decimal at the neighbourhood of 0. dt and dB and any d"something" is close to zero by definition of a limit tending to zero ( think raising several limits that tends to zero to different powers and then summing them, only the that with the smallest power will remain after summation) .
(dt)^2 is the square, not the square root. so, as per your example, if dt=1/4, then (dt)^2=1/16, and gets smaller and smaller as you decrease dt.
Why is (dt)^2=0 and (dt)(dB_t)=0?
check previous lecture stochastic proc II
Dt is small dt^2 is so minus that we can ignore it.
Disappears in the limit as dt approaches zero
Dudes and doddettes you forgot vector calculus.
figma times stigma ?
practice listening
I am a high school student I understand nothing about this.
I'm too dumb but still make 240k a year, and for that I'm out. I didn't even finish college.
❤❤❤❤❤❤❤❤ phong cách dạy hay
Perfect !
m*df/dx should be replaced by m*df/dt he bad knows math
loot at that chalk
MIT OpenCourseWare
18. Itõ Calculus
Instructor: Choongbum Lee
0:50 min ... Brownian ...
class!
Umbrella sampling
Good❤❤
51:01 / 한국인 웃고갑니다
🚀 thx!!
do they not have projectors at MIT?
for what?
nutteLau Hurnsohn so they dont have to write on a stupid blackboard.
10babiscar
most people prefer blackboard
I agree with both sides in this argument. The blackboard certainly seems like a ver-ree olde kludge, but the process of the guy walking through the development of any line of thought -- slowed to the speed at which he can write it -- is pretty good.
Anything projected is going to be prepared beforehand, and some "slide" or screenshot that the guy spends hours on would or could flash by in an instant. All sense of development of the thought would be lost.
Yes, there's got to be a better way. I have no idea what it might be.
Maybe a white board with dry erase pen would be better, but the sliding board structure is a very good idea.
always has big mistakes
24:18 eat box .... hey ladies ;)
Wow
Beside his good teaching, don't you think this guy is quite handsome?
SHA-256 Secure 👻🌖🐺🦞🌬🌪🌪🌪😇🗳💞✌️