5. Stochastic Processes I 

The best teachers do a great job of introducing problems and then showing you the tools to solve them. With these teachers, you always know why you're doing something, you always have a sense of intuition for the problem, and you easily build a sense of experience having worked with these tools in future similar scenarios. There are so many instances wherein this professor does just that and it's a huge blessing to have access to this content for free

Recursive argument at 28:00: Call p the probability you hit -50 first. There’s a 50% chance you hit -50 before you hit 50, by symmetry. Once you hit 50, the game is reversed, by stationary property. Hence p = 0.5 + 0.5 * (1 - p), from which p is 1/3.

Some notable Timestamps 0:00:33 Stochastic Process 0:10:57 (Simple) Random Walk 0:32:43 Markov Chain 0:58:41 Martingale 1:06:47 Stopping time / Optional Stopping Theorem

when you don't wanna read or write anymore but still wanna do some math, well you've got to the right place.

This guy has the most elegant writing style and manner of presentation.

This guy is absolutely fantastic. Could not have been explained more clearly, with a sound logical structure. People complaining about him should probably try lecturing themselves before offering their criticism.

insane lecture, tried so many different online materials, this one is clear af!

I am currently working on understanding the stochastic processes, and I am very confused by the concept of “a collection of random variables”, but the trajectory thing given by the lecturer helped me understand the concept a lot easier. For a continuous random process, if I sample at very high frequency, I will get several curves in the “x(t)-t” plain (the curve depending on the setting of the random process).

0:00:33 Stochastic Process 0:10:57 (Simple) Random Walk 0:32:43 Markov Chain 0:58:41 Martingale 1:06:47 Stopping time / Optional Stopping Theorem For my reference

There is a reason he teaches at MIT this guy explains things so clearly and with ease! Im in H.S and can understand this! Absolutely amazing

49:03 ahh the "click" moment, seeing all the maths pieces coming together is really satisfying

Wow ! What a clear and concise lecturer. His ability in minimizing excess data to keep to the pure path of understanding is excellent. He is a star.

Thanks for ur efforts, I was just preparing for my first class about stochastic.

Bravo for the stopping time definition . Very helpful

you sir are a gift! Thankyou for your clear lecturing!

OMFG! This guy is genius in explaining and presenting concepts.

In the 1st and 2nd cases he's talking about delta hedge parity (in trading/market practice) as reflected by trend lines. In the 3rd case he's referring to the vol of vol, in this situation one must employee stochastic volitility models.

48:45 In order to predict the future using transition probability matrix A, we need to use transpose of A. (A^T^3650 * [1;0]) (this is fixed at 51:41). Since the eigenvalue of A^T is equal to A, the following theorems also hold. Hope this help. Thanks for the great lecture.

Absolutely fantastic video, presented with such clarity. Extremely helpful. Thank you.

This guy is amazing. His explanation is clear.

27:00 The argument to make it work the way the intuition of the student worked is via markov chains. Set up the states -50, 0, 50 and 100, write the transition probabilities, then calculate the absorption probabilities of the two recurrent states (-50 and 100) from 0 which give 1/4 and 1/2. The probability to end up with $100 is the probability of ending up becomes 1/4 / (1/4 + 1/2) (since the two other states will eventually bleed into either one of these states we know their steady state probability will be 0) which indeed gives 1/3.

To get variance, applied variance to both sides, var(sum(yi) over i). because yis are iid variance becomes sum(var(yi)). Var of each yi is one, and so variance is t. Var of each yi is one by computational formula of variance, E[yi^2]-E[yi]=1

Amazing lecture. Made it A LOT easier to understand the concepts and applications. Books on subject don't usually give examples, which makes it that much harder to understand.

I mean i dont get all these praises. The guy gives an overview of the topic but not rigorously at all. This is not the level of depth I would have expected but it serves me well in my preparations. It feels like I have to dive deeper on my own to get real understanding of the topic.

Continue the reasoning from 27:22: Assume the probability of the game ends at 100 is x. As probability of the game reaches 50 is 0.5; The probability from 50 to 100 is actually (1-x). So x=0.5*(1-x) --> x=1/3

very good presentation, enjoyed it!

the best intuition behind stochastic processes !, really good

28:00 Following the thought process of the student from the audience, after the balance reaches $50, there is a 1/2 chance for the balance to reach $100 (overall probability = 1/4) or fall back to $0 (overall probability = 1/4). If the balance falls back to zero, we can consider that as the start of the second cycle, where the distribution of the conditional probability is the same as the first cycle (1/2 chance to reach $-50, 1/4 chance to reach $100, and 1/4 to reach $50 first then return to $0). Same for the third cycle, forth cycle, etc. Therefore, we can express the overall probability for the balance to reach $100 as the infinite series of 1/4 + (1/4)^2 + (1/4)^3... which gives us 1/3.

They have the audacity to call Choongbum Lee an instructor, when he can give a presentation so complete, elegant, and accessible that he could (and maybe should) teach ALL of the other professors at MIT a thing or two about how to give a lecture.and communicate ideas throughout it. This guy is @#$%ing amazing! What a beast. God, I feel stupid in comparison.

Great lecture. Learned a lot.

This is great and simple stuff for students studying the particle theory and Brownian motion

Good presentation

All you people praising this lecturer, saying how easy and simple he makes everything, are not helping. He's making tons of mistakes, and I'm thinking I must be going crazy since everybody else seems to think this is the best lecture ever.

Thanks a lot. Very clear explanation.

Would be a honor to be part of your class, professor. Your content is just awesome and your care with the understanding of the students can be noticed by your looks. Thank you.

Very easy solution for 28:00. P(B), P(A) be probabilities that B,A occur first respectively. Probability that we hit 50$ before -50$ is 1/2 and also probability that we hit -50$ before 50$ is 1/2. If we reach 50$ first, we see problem is flipped now, we are 50$ closer to B and -100$ closer to A. So P(B/start at 50$) = P(A/start at 0$) So we can write P(B) = 1/2(P(A)) = 1/2(1-P(B)) Solving this simple equation we get P(B) = 1/3 In fact for any A,B there is a point where we can flip the problem, so try to generalize this and come up with a proof.

stopping time 개념이 헷갈렸었는데, 정말 직관적으로 이해가 가네요. 감사합니다!

a completely different level can not be compared with the first lectures

Top universities have the best lecturers, making it easier for the students. It’s like a “poverty trap” for higher education.

Awesome lecture. Just found out he went to the same college for his undergrad as me

thank so much for MIT...it very helpful for my short time study at University.

56:13 I think the confusion here comes from the fact that for the other eigenvalue, which actually is less than 1 and greater than 0, the corresponding eigenvector will converge to the 0 vector. The “sum trick” he did earlier wouldn’t work because v_1 + v_2 = \lambda (v_1 + v_2) doesn’t imply that \lambda = 1 when both v_1 and v_2 are 0. Hope I didn’t overlook anything!

Thanks a lot!!! Very good teacher :)

Very good explanation.

1:15:16 you might want to say that E(X_\tau) = E(X0). Remember that X0 is a random variable too.

Don't spend your time for another channels. It is the best one!

At 41:35, it should be P_m1 instead of P_2m.

OMG you are a genius stochastic process never looked this simple and intuitive

the last corollary is neat indeed, but the assumption of the theorem seems not be fulfilled. there does not exist T>tau, since it's possible for the random walker to bump between the lines -50 and 100 as long as it likes... can sb clarify?

I don't understand how 2 and 3 are different? They seem same to me. 6:00

Does anyone knows about more basic content so as to form a stonger intuition and be able fathom this deeper? Also recommendations for time series analysis will be appreciated as I'm basically working on that.

ank you for lecture

He is sooo good!

Very clear!

simply amazing

In 47:42 Multiplying a 2x2 matrix with a vector (1,0) will give back the p11 and p21 which stands for working today and working tomorrow(p11) and broken today but working tomorrow(p21) not the probability working and not working.

Can someone explain why tau would be bounded in the case (i) at 1:12:23 ?

@42:00 Isn't the transition prob matrix incorrect. Where the lower left corner should be P_{m,1} instead of P_{2,m}

@48:24 "probability distribution of day 3651 and day 3650 are the same." @54:04 if av=v, day 3651=day3650, then the machine of his example last forever?

great job

There is a mistake at 1;15:23 . An Expectation of a random variable is a number not a random variable. So E(Xtau)=E(X0).

Who is this guy? His explanation on the subject is awesome

This is next level.

WOW... THANKS FOR THIS....

Can someone explains me whats the difference of the stochastic processes number 2 and 3 defined at minute 4:30 ? Thank you so much

58:10 Someone asked whether the algebraic manipulation led to the (seeming incorrect) conclusion that all eigenvalues lambda are 1. That was not true, since the assumption for that equation is that we are dealing with a stationary state, and therefore, the conclusion is for a stationary state, its eigenvalue must be 1, as stated by Lee.

Show me the lectures of the Poisson process

I wish my lecturers could lecture in such a well structured way :(

Interesting to see a proof that the simple random walk is expected to take t steps in order to move sqrt(t), which is relevant in Markov chain Monte Carlo theory.

what is the prerequisite for this course, does anywhere i can find a detail simplified version of all the explanation relating to this topic.

This is a good video. just that there is a little mistake under the transition matrix. With the matrix provided, the last entry under the first column should have been P subscript 3m and not 2m.

Amazing Lecture, I think at 57:56 , the equation v1 + v2 = lambda(v1+v2) only holds for lambda = 1(the only case where both v1 and v2 can be positive) , for the other eigenvalue v1+ v2 =0. This Should extend to any dimension.

I'm confused about the machine working/broken example. At 0:49:09 I believe it should be [1 0]*A^3650 = [p q]. Then for eigenvector at 1:17:40 it should be A(transpose)*[v1,v2] = [v1,v2], as you can see he modified the matrix from A to A transpose. With the way it is shown here p, q should have different meaning.

wonderful teacher, but I couldn't understand the last example. Why the probability is=0?

partial differential equations pde

47:52 Shouldn't we premultiply here,. i.e [1 0](A^3650) = [p q] pre-multiply (with [1,0] as 1x2 vector) instead of post-multiply.

I think that I don't understand the independence property of random walks, given around 21:00. His verbal explanation sounds a lot like the Markov property, but I doubt that he would define the same thing two different ways without saying that they're equivalent. Are there any systems with the independence property, but not the Markov property, or vice-versa?

Can’t the derivative be taken until 0 or check the order and negativity/positivity of the function be assessed to shortcut how many options to search for in each axis?

Anyone know some way to get the solutions for the assignments?

does stochastic process varies linearly with time? because in your first example function f(t) varies linearly with the time. in some books it is referred as random process. quite confusing ,guide me

Can technically everything be a markov chain if the history is included in the current state?

I think it should be N(0, 1/4) at 17:13

@23:00 how can simple random walk be stationary when variance grows with time ? Did he mean increments are stationary ?

1:02:00 Randomwalk is a martingale

Very interesting

this guy is a genius

確率方程式＝Stochastic Processes I

Plz... suggest the book for stochastic process

@17:13 Can anyone elaborate on 1/sqrt(t) X_t ~N(0,1)? I understood high level conecpt of C.L.T. however cannot really understand what X_t is referring to. is it mean of set of observations? or one random variable.

Around 32 mins time mark, why f(B) = 1 and f(-A) = 0? Thanks for the help.

Thanks man

i dont understand the difference between 2 and 3 at 4:39

In the definition of p_ij, was homogeneity assumed anywhere? Maybe I missed it, but it definitely needs to be a homogeneous process! That means, p_ij shouldn't depend on t.

At 19:12 , the probability of a N(0,1) to be between -1 and 1 is ~68%, not close to 90% or more as said. Otherwise, great lecture.

at 16:52, shouldn't mean 0 and standard deviation equals to 1. Am not sure about this, can anyone please explain??

the matrix at 0:49:09 was wrong. Also, the transition matrix is (p_{1j},p_{2j}....), not (p_{k1},p_{k2},....).

where can i find articles talk about "estimating Heston's model using MCMC"

17:15 if the variance is t, how's std equal to square root of t. Isn't supposed to be just 1 since you'd divide variance with t first?

At time before 44:40 he said random walk does not have finite set.. But he earlier said that the values are limited under a curve with standard deviation of root t.? Anyone please help

Thanks