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. 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