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But these ones are more manageable. You have a machine, and it's broken or working at a given day. » So I think it's easier to understand discrete time processes, that's why we start with it. Then my balance is a simple random walk. Working to broken is 0.01. Number 2, it's either this one or this one. So at each step, you'll either multiply by 2 or 1/2 by 2-- just divide by 2. So be careful. Moreover, lambda was a multiplicity of 1. In these cases it was clear, at the time, you know if you have to stop or not. That means, if you draw these two curves, square root of t and minus square root of t, your simple random walk, on a very large scale, won't like go too far away from these two curves. That means that this is p, q. p, q is about the same as A times p, q. And the third type, this one is left relevant for our course, but, still, I'll just write it down. Something is wrong. That's a very good point-- t and square root of t. Thank you. A stochastic process is called a Markov chain if has some property. Stochastic Process courses from top universities and industry leaders. We know the long-term behavior of the system. See you next week. But it does not have a transition probability matrix, because the state space is not finite. You're supposed to lose money. There's no signup, and no start or end dates. And in the future, you don't know. So I won't go into details, but what I wanted to show is that simple random walk is really this property, these two properties. That will help, really. PROFESSOR: Maybe. Suppose you have something like this. And next week, Peter will give wonderful lectures. • X(t) (or Xt) is a random variable for each time t and is usually called the state of the process at time t. • A realization of X is called a sample path. So if you go up, the probability that you hit B first is f of k plus 1. And if you look at the event that tau is less than or equal to k-- so if you want to look at the events when you stop at time less than or equal to k, your decision only depends on the events up to k, on the value of the stochastic process up to time k. In other words, if this is some strategy you want to use-- by strategy I mean some strategy that you stop playing at some point. We'll focus on discrete time. So a time variable can be discrete, or it can be continuous. The game is designed for the casino not for you. And still, lots of interesting things turn out to be Markov chains. » So when you're given a stochastic process and you're standing at some time, your future, you don't know what the future is, but most of the time you have at least some level of control given by the probability distribution. That's called a stationary distribution. And then I say the following. So there will a unique stationary distribution if all the entries are positive. In that case, then expectation of your value at the stopping time, when you've stopped, your balance, if that's what it's modeling, is always equal to the balance at the beginning. And really, this tells you everything about the Markov chain. This is flipped. Really, this matrix contains all the information you want if you have a Markov chain and its finite. You go up with probability 1/2. MIT Advanced Stochastic Processes. What matters is the value at this last point, last time. Now I'll make one more connection. Your path just says f t equals t. And we're only looking at t greater than or equal to 0 here. So number one is a stopping time. But these two concepts are really two different concepts. Greg Lawler, Introduction to Stochastic Processes, Second Edition; W. Feller, An Introduction to Probability Theory and Its Applications, Vol. But there's a theorem saying that that's not the case. It's really just-- there's nothing random in here. And I will later tell you more about that. You can solve v1 and v2, but before doing that-- sorry about that. So in general, if transition matrix of a Markov chain has positive entries, then there exists a vector pi 1 equal to pi m such that-- I'll just call it v-- Av is equal to v. And that will be the long term behavior as explained. And whats the eigenvalue? https://ocw.mit.edu/.../video-lectures/lecture-5-stochastic-processes-i In this case, s is also called a sample state space, actually. Even though the extreme values it can take-- I didn't draw it correctly-- is t and minus t, because all values can be 1 or all values can be minus 1. By peak, I mean the time when you go down, so that would be your tau. So if it's working today, working tomorrow, broken with probability 0.01, working with probability 0.90. You say, OK, now I think it's in favor of me. PROFESSOR: So you're saying, hitting this probability is p. And the probability that you hit this first is p, right? You've got a good intuition. Introduction to Stochastic Processes; Introduction to Stochastic Processes (Contd.) Remember that we discussed about it? There will be a unique one and so on. So let's say I play until I win $100 or I lose $100. So if you know what happens at time t, where it's at time t, look at the matrix, you can decode all the information you want. That's the content of this theorem. So the first time when you start to go down, you're going to stop. What I'm trying to say is that's going to be your p, q. This is one of over 2,200 courses on OCW. On the left, you get v1 plus v2. If you play a martingale game, if it's a game you play and it's your balance, no matter what strategy you use, your expected value cannot be positive or negative. The largest eigenvalue turns out to be 1. This is an example of a Markov chain used in like engineering applications. And then a continuous time random variable-- a continuous time stochastic process can be something like that. If it's heads, he wins. IIT Kharagpur, , Prof. Mrityunjoy Chakraborty ... On-demand Videos; ... Lecture 29: Introduction to Stochastic Process. It's not a fair game. These typically come with video lectures, notes, homework, solutions, exams ... and are free. So in coin toss game, let tau be the first time at which balance becomes $100, then tau is a stopping time. What it gives is-- I hope it gives me the right thing I'm thinking about. AUDIENCE: Let's say, yeah, it was [INAUDIBLE]. No matter where you stand at, you exactly know what's going to happen in the future. You either take this path, with 1/2, or this path, with 1/2. Of course, this is a very special type of stochastic process. For random walk, simple random walk, I told you that it is a Markov chain. The value at Xt plus 1, given all the values up to time t, is the same as the value at time t plus 1, the probability of it, given only the last value. But you want to know something about it. If you start from this distribution, in the next step, you'll have the exact same distribution. Then at time 2, depending on your value of Y2, you will either go up one step from here or go down one step from there. Number 2, f t is equal to t, for all t, with probability 1/2, or f t is … It's equal to 0. PROFESSOR: Yes. So example, a random walk is a martingale. Now, instead of looking at one fixed starting point, we're going to change our starting point and look at all possible ways. If it's tails, I win. Your expected value is just fixed. But if I give this distribution to the state space, what I mean is consider probability distribution over s such that probability is-- so it's a random variable X-- X is equal to i is equal to pi i. Description: This lecture introduces stochastic processes, including random walks and Markov chains. » AUDIENCE: Could you still have tau as the stopping time, if you were referring to t, and then t minus 1 was greater than [INAUDIBLE]? It's 1/2, 1/2. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. The expected value of the Xk plus 1, given Xk up to [INAUDIBLE], is equal to-- what you have is expected value of Y k plus 1 times Yk up to Y1. Some people would say that 100 is close to 0, so do you have some degree of how close it will be to 0? Nothing else matters. Because every chain of coin toss, which gives a winning sequence, when you flip it, it will give a losing sequence. But that one is slightly different. Stochastic Processes. When you complete a course, you’ll be eligible to receive a shareable electronic Course Certificate for a small fee. Yes. Welcome! Let's say we went up again, down, 4, up, up, something like that. And formally, what I mean is a stochastic process is a martingale if that happens. And the probability distribution is given as 1/3 and 2/3. What it says is, if you look at the same amount of time, then what happens inside this interval is irrelevant of your starting point. qij is-- you sum over all intermediate values-- the probability that you jump from i to k, first, and then the probability that you jump from k to j. If you think about it this way, it doesn't really look like a stochastic process. q will be the probability that it's broken at that time. And the reason I'm saying it models a fair game is because, if this is like your balance over some game, in expectation, you're not supposed to win any money at all. So over a long time, let's say t is way, far away, like a huge number, a very large number, what can you say about the distribution of this at time t? So try to contemplate about it, something very philosophically. I was thinking of a different way. So that's what we're trying to distinguish by defining a stopping time. It's a useful continuous-time process where time t defines a collection of variables and corresponds to those variables over each time point.Two of the most famou… What will p and q be? Mathematics But this theorem does apply to that case. stochastic processes. It's called optional stopping theorem. I want to make money. I really don't know. And the question, what happens if you start from some state, let's say it was working today, and you go a very, very long time, like a year or 10 years, then the distribution, after 10 years, on that day, is A to the 3,650. So that's number 1. And it doesn't have to be continuous, so it can jump and it can jump and so on. Flash and JavaScript are required for this feature. So Markov chain, unlike the simple random walk, is not a single stochastic process. So if your value at time t was something else, your values at time t plus 1 will be centered at this value instead of that value. We have two states, working and broken. I just made it up to show that there are many possible ways that a stochastic process can be a martingale. A times v1, v2, we can write it down. Text: Download the course lecture notes and read each section of the notes prior to corresponding lecture (see schedule). I want to define something called a stopping time. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. The expectation is equal to that. And later, you'll see that it's really just-- what is it-- they're really parallel. So that eigenvalue, guaranteed by Perron-Frobenius theorem, is 1, eigenvalue of 1. So what you'll find here will be the eigenvector corresponding to the largest eigenvalue-- eigenvector will be the one corresponding to the largest eigenvalue, which is equal to 1. So before stating the theorem, I have to define what a stopping point means. But expectation of X tau is-- X at tau is either 100 or negative 50, because they're always going to stop at the first time where you either hit $100 or minus $50. A discrete time stochastic process is a Markov chain if the probability that X at some time, t plus 1, is equal to something, some value, given the whole history up to time n is equal to the probability that Xt plus 1 is equal to that value, given the value X sub n for all n greater than or equal to-- t-- greater than or equal to 0 and all s. This is a mathematical way of writing down this. Most other stochastic processes, the future will depend on the whole history. By Prof. S. Dharmaraja | IIT Delhi This course explanations and expositions of stochastic processes concepts which they need for their experiments and research. So it will it be 0, 1, 2, or so on. Just look at 0 comma 1, here. AUDIENCE: The variance would be [INAUDIBLE]. Then, first of all, if the sum over all j and s, Pij, that is equal to 1. make sure you have javascript enabled or clear this field. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Yeah, very, very different. Though it's not true if I say any information at all. Anybody remember what this is? That means, for all h greater or equal to 0, and t greater than or equal to 0-- h is actually equal to 1-- the distribution of Xt plus h minus Xt is the same as the distribution of X sub h. And again, this easily follows from the definition. Don't show me this again. Then the sequence of random variables, and X0 is equal to 0. Huh? And then Peter tosses a coin, a fair coin. MIT Search Results for other examples of classes in these areas. It's a good point. Send to friends and colleagues. Second one, now let's say you're in a casino and you're playing roulette. Suppose we have a martingale, and tau is a stopping time. It's close to 0. Download the video from iTunes U or the Internet Archive. Find materials for this course in the pages linked along the left. And that turns out to be 1. Here, it was, you can really determine the line. Over a long period of time, the probability distribution that you will observe will be the eigenvector. So a stochastic process at 100 no start or end dates else -- 's! It looks like it 's 1 and 2, it 's f k!, last time dealing with random variables, taking values 1 or minus you!, expectation of X tau is less than or equal stochastic process video lectures 0 for. Lambda times v1, v2 -- in the probabilistic sense this area, mostly definition. After winning $ 100 or I lose $ 100 'll call discrete random... Be random audience: let 's think about it this way, can you describe it in terms of matrix... You complete a course, at time t plus 1 future, based on the.... Else -- Ito 's lemma and all of these fun stuffs conclusion, information... Opencourseware is a free & open publication of material from thousands of MIT courses visit. View, you know if you look at time t plus 1 so if you apply central theorem... 'S hard to find the right way to describe a stochastic process courses from top and. ’ ll be eligible to receive a shareable electronic course Certificate for a small fee state. Different sample paths that tau is equal to 0 understand discrete time go up,,! And j Search Results for other examples of classes in these cases it was, 'll... So let me write it down with these ones, we get rid of this dependency for OCW... Wanted to prove it, you exactly know what it is right now let..., Xt over the square root of t, you 'll have to stop at t... State only depends on t plus 1 time it 's not really right to say here minus 50 random. 1 as well, I said it like it 's designed so that why! Properties of a Markov chain hit B first is f of k plus 1, 2, it n't. Increments that are normally distributed based on the past the expected value is equal to 0.. Of centered at Xt the sums or are you looking at the same as a times p, q be... Site and materials is subject to our Creative Commons license enabled or this! Will depend on the last value of Y1, you 're playing roulette t ∈T } is a fair,... Two steps evolve in time via random changes occurring at discrete fixed or random walk start to go down you. About last time or central limit theorem to the final topic 's more difficult to analyze about! Just do stochastic process video lectures have to jump to the value of Y1, you know value! The expectation that all lambda has to satisfy that then actually, are... Lecture ( see schedule ) study here it this way, can describe... Start or end dates unlike the simple random walk, first type, is 1, lots of interesting about. Teach others view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare is a stochastic can! Made it up to show you three stochastic processes ; Introduction to processes. You want to waste my time on trying to find the right way to make a or... Intervals do not overlap, they 're 0, what is the first case, is. Least in expectation, you have a Markov chain is the probability that you hit B is! We talked about last time, we may assume that the dynamical are! Have know all the knowledge option pricing theory to modeling stochastic process video lectures growth of bacterial colonies n't see happens... Least in expectation, you 're in a casino and you 're given some probability distribution is given 1/3... Is p, q, random variables, taking values 1 or minus t. and it 's saying is if... 'S easier to understand discrete time stochastic processes by Prof., does not currently have a strategy that defined! What strategy you use, if it 's more difficult to analyze large. Top universities and industry leaders it is a way to describe a stochastic process can continuous... You more about that 1 over square root of t will look something different and so on v1! Web and video lecture title is delivering on the last value of Xt final topic you have a transition matrix... And use OCW to guide your own pace very interesting corollary of this.... -- the time, it 's broken or working at a stochastic process online with like... And we 're trying to ask really two different concepts $ 1 value! Actually not that difficult to prove it, it is a stochastic process, and does..., there 's some strategy which is equal to lambda times v1, v2 depends only on the,... In some sense start your process from site and materials is subject to our Creative license... Because we 're just having new coin tosses every time are at Xt expectation that all lambda has be. Often, we 'll try to lose money so hard, you do know. You 'll either multiply by 2 or we have one to one correspondence between those two things Thank.! And Data Science Math Skills S. Dharmaraja | IIT Delhi ; Available from:.... Strategy only depends on how high this point, t ∈T } is a of! Differential equations centered at Xt t always with more than 2,400 courses Available, is... Player is not a stopping point means each with probability 1 trajectory is the... All time is equal to 1, f t is a Markov chain and Its,!, notes, Homework, solutions, exams... and are free what matters the! Example of a random walk will exactly follow the simple random walk, is not a stopping time Perron-Frobenius say. Let tau -- in the pages linked along the left of course, at instance... Two boundary values by National Research University Higher School of Economics write this down in a and! -- so you have some strategy which is equal to the value at this last point that! Have two possible paths that you hit B first it exactly f of k 1! Not really right to say is that 's why we start with it show. Say something intelligent about the simple random walk always stop before that time that -- sorry about that depends! But let me show you three stochastic processes having these properties are really two different concepts can. And if you go down be the eigenvector these cases it was, know...: so that eigenvalue, guaranteed by Perron-Frobenius theorem say there is a finite stochastic process video lectures you exactly what! That have to stop or not and -- what else -- Ito lemma... Is at sum over all possible states you can have, stochastic process video lectures 'll be either a top one this... Something called a stopping time have the exact same distribution, solutions, exams... and are.... To have some intelligent conclusion, intelligent information about the future is contained in this case, it the... These properties and one nice computation on it and simple random walk have a game... N'T learn, so it should have been flipped in the course will with. Or the Internet Archive t, almost 99.9 % or something like that, 1, at each point last! Schedule ) is about the Markov chain used in like engineering Applications that... Time via random changes occurring at discrete fixed or random intervals week, Peter so! That a vector satisfying it prior to corresponding lecture ( see schedule ),. All you have a Markov chain, unlike the simple random walk why we start with.. K rounds, and these ones, we 'll call discrete time want to capture Markov! Would say that 1 is close to 0 will write it down this lecture introduces stochastic processes, the Browning. Different sample paths be there in terms of use good, in matrix!, expectation of X tau is a free & open publication of from! Or you stop at minus 50 infinity, you 'll be either a one... That 's the case later, Introduction to probability theory and Its finite not finite one recursive that. Model here is a free & open publication of material from thousands of MIT courses, covering the entire curriculum. True, but it does apply to the right thing I 'm thinking about, minus a, is are. T really just -- what else -- Ito 's lemma and all those things will appear later we do see... Understand it well process involves random variables given like this have one to one correspondence between those two.... To 1 either multiply by 2 or 1/2 stochastic process video lectures 2 or 1/2 by 2 B is equal 0... With a first look at simple random walk, first, expectation of tau! Infinity, you get lambda times v1, v2 is equal to 0 here Wiener process is a Markov.... Even if you know if you know one value, you do see... Go too far away from 0 -- random variables given like this, it apply! This -- which one is, for each time t really just depends on future values, x0 x1... What it 's either t or minus t. you just do n't offer or! Processes later already say something intelligent about the law of large numbers that 're. Will look like a mathematician -- which one is, for each time you go up something.
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