What will p and q be? Actually, I made a mistake. Description: This lecture introduces stochastic processes, including random walks and Markov chains. So let's define q sub i j as the probability that X at time t plus 2 is equal to j, given that X at time t is equal to i. What we are interested in is computing f 0. Variance will be 1 over t. And the standard deviation will be 1 over square root of t. What I'm saying is, by central limit theorem. So we have this stochastic process, and, at time t, you are at Xt. Anybody? 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. A stochastic process is called a Markov chain if has some property. Let me show you by example. It's not a fair game. I just made it up to show that there are many possible ways that a stochastic process can be a martingale. And simple random walk is like the fundamental stochastic process. Such vector v is called. So for example, a discrete time random variable can be something like-- and so on. It's equal to 0. Something like that is the trajectory. And if you look at what this means, each entry here is described by a linear-- what is it-- the dot product of a column and a row. It might go to this point, that point, that point, or so on. So I'll just forget about that technical issue. So Markov chain, unlike the simple random walk, is not a single stochastic process. 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. 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. Something is wrong. So if you just look at it, Xt over the square root of t will look like normal distribution. So from my point of view, in this coin toss game, at each turn my balance goes up by $1 or down by $1. No matter what you know about the past, even if know all the values in the past, what happened, it doesn't give any information at all about the future. Topics in Mathematics with Applications in Finance Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. But the conclusion is right. This is an example of a Markov chain used in like engineering applications. So a stochastic process is a collection of random variables indexed by time, a very simple definition. I'm not sure. I mean it will fluctuate a lot, your balance, double, double, double, half, half, half, and so on. Some people would say that 100 is close to 0, so do you have some degree of how close it will be to 0? Why is it? How often will something extreme happen, like how often will a stock price drop by more than 10% for a consecutive 5 days-- like these kind of events. Now I'll make one more connection. There should be a scale. Can anybody help me? What happened before doesn't matter at all. It's 0.99 v1 plus 0.01 v2. The expectation is equal to that. This can be proved. So now denote the probability ij as the probability that, if at that time t you are at i, the probability that you jump to j at time t plus 1 for all pair of points i, j. I mean, it's a finite set, so I might just as well call it the integer set from 1 to m, just to make the notation easier. So a time variable can be discrete, or it can be continuous. Download the video from iTunes U or the Internet Archive. What are the boundary events? And the reason is because, in many cases, what you're modeling is these kind of states of some system, like broken or working, rainy, sunny, cloudy as weather. You say, OK, now I think it's in favor of me. NPTEL provides E-learning through online Web and Video courses various streams. That means the value, at t, will be distributed like a normal distribution, with mean 0 and variance square root of t. So what you said was right. If it's some strategy that depends on future values, it's not a stopping time. So that's what we're trying to distinguish by defining a stopping time. So let me move on to the final topic. It's either t or minus t. And it's the same for all t. But they are dependent on each other. I want to make money. Then that is a martingale. Any questions? That means your lambda is equal to 1. Lecture notes. 15 . But these ones are more manageable. Working to broken is 0.01. These are a collection of stochastic processes having the property that-- whose effect of the past on the future is summarized only by the current state. All entries are positive. It is 1/3, actually. Sorry about that. Any questions on definition or example? So from what we learned last time, we can already say something intelligent about the simple random walk. So be careful. Any questions? And the scale you're looking at is about the square root of t. So it won't go too far away from 0. It's not just about this matrix and this special example. So the trajectory is like a walk you take on this line, but it's random. So it's not a martingale. And for a different example, like if you model a call center and you want to know, over a period of time, the probability that at least 90% of the phones are idle or those kind of things. Definition of Stochastic Processes, Parameter and State Spaces: Download To be verified; 18: Classification of Stochastic Processes: Download To be verified; 19: Examples of Classification of Stochastic Processes: Download To be verified; 20: Examples of Classification of Stochastic Processes (contd.) Does it make sense? Then, if it's a Markov chain, what it's saying is, you don't even have know all about this. When you solve it, you'll get that answer. All right. So sum these two values, and you get lambda of this, v1, v2. So it really depends only on the last value of Xt. That's not a stopping time. So corollary, it applies not immediately, but it does apply to the first case, case 1 given above. Moreover, lambda was a multiplicity of 1. A times v1, v2 is equal to lambda times v1, v2. Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. And that 0.8 v1 plus 0.2 v2, which is equal to v1, v2. With all the rest, you're going to stop at minus 50. So at the 100 square root of t, you will be inside this interval like 90% of the time. So, when you look at a process, when you use a stochastic process to model a real life something going on, like a stock price, usually what happens is you stand at time t. And you know all the values in the past-- know. But these two concepts are really two different concepts. So the event that you stop at time t depends on t plus 1 as well, which doesn't fall into this definition. Courses Another realization will look something different and so on. So no matter what strategy you use, if you're a mortal being, then you cannot win. Even if you try to lose money so hard, you won't be able to do that. He wins the $1. Now, instead of looking at one fixed starting point, we're going to change our starting point and look at all possible ways. So when you start at k, I'll define f of k as the probability that you hit this line first before hitting that line. And so when you take products of the transition probability matrix, those eigenvalues that are smaller than 1 scale after repeated multiplication to 0. And as you can see, you can do computations, with simple random walk, by hand. On the left, you get v1 plus v2. X0, X1, X2, and so on is called a one-dimensional, simple random walk. But yeah, there might be a way to make an argument out of it. We look at our balance. Balance of a roulette player is not a martingale. And then, depending on the value of Y1, you will either go up or go down. But that one, if you want to do some math with it, from the formal point of view, that will be more helpful. That's really easy to prove. Don't show me this again. You're supposed to lose money. Mathematics. PROFESSOR: So, yeah, that was a very vague statement. It's not really right to say that a vector has stationary distribution. And the third type, this one is left relevant for our course, but, still, I'll just write it down. And what it's saying is, if all the entries are positive, then it converges. make sure you have javascript enabled or clear this field. But if you really want draw the picture, it will bounce back and forth, up and down, infinitely often, and it'll just look like two lines. The Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments. So if Yi are IID random variables such that Yi is equal to 2, with probability 1/3, and 1/2 is probability 2/3, then let X0 equal 1 and Xk equal. Two equivalent processes may have quite different sample paths. And now it starts again. There are some stuff which are not either of them. I really don't know. A highlight will be the first functional limit theorem, Donsker's invariance principle, that establishes Brownian motion as a scaling limit of random walks. » Let's say we went up. This is just one-- so one realization of the stochastic process. So that's an example. If you start at f k, you either go up or go down. So let tau-- in the same game-- the time of first peak. That will help, really. That is a stopping time. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.. No enrollment or registration. So if it converges, it will converge to that. What it gives is-- I hope it gives me the right thing I'm thinking about. You go up with probability 1/2. Yeah, but Perron-Frobenius theorem say there is exactly one eigenvector corresponding to the largest eigenvalue. Now if I change it. So let me define it a little bit more formally. The following content is provided under a Creative Commons license. That this value can affect the future, because that's where you're going to start your process from. And really, this tells you everything about the Markov chain. Here, it was, you can really determine the line. But if you define your stopping time in this way and not a stopping time, if you define tau in this way, your decision depends on future values of the outcome. So those are some interesting things about simple random walk. Now, for each t, we get rid of this dependency. Yes. And that's exactly what occurs. Mathematics The distribution is the same. What is the probability that I will stop after winning $100? Let's say we went up again, down, 4, up, up, something like that. You won't deviate too much. And then I say the following. So what we have here is, at time t, if you look at what's going to happen at time t plus 1, take the expectation, then it has to be exactly equal to the value of Xt. Broken to broken is 0.2. And that can easily be shown by the definition. PROFESSOR: Today we're going to study stochastic processes and, among them, one type of it, so discrete time. Stochastic processes are a standard tool for mathematicians, physicists, and others in the field. We have two states, working and broken. That's a very good point-- t and square root of t. Thank you. So it should be 1/4 here, 1/2 times 1/2. Right now, we'll study discrete time. Welcome! Let Yi be  IID, independent identically distributed, random variables, taking values 1 or minus 1, each with probability 1/2. That's because a future state only depends on the current state. Over a long period of time, the probability distribution that you will observe will be the eigenvector. And I'll talk about what it is right now. We have one to one correspondence between those two things. So if you go up, the probability that you hit B first is f of k plus 1. Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. Like this part is really irrelevant. But in expected value, you're designed to go down. For example, if you apply central limit theorem to the sequence, what is the information you get? Yes? We didn't learn, so far, how to do this, but let's think about it. So what this says is, if you look at what happens from time 1 to 10, that is irrelevant to what happens from 20 to 30. On the left, what you get is v1 plus v2, so sum two coordinates. Then define, for each time t, X sub t as the sum of Yi, from i equals 1 to 2. Even if you try to win money so hare, like try to invent something really, really cool and ingenious, you should not be able to win money. 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. So some properties of a random walk, first, expectation of Xk is equal to 0. We know that each value has to be t or minus t. You just don't know what it is. That's quite a vague statement. And let's see what happens for this matrix. And you'll see these properties appearing again and again. I mean at every single point, you'll be either a top one or a bottom one. Video created by National Research University Higher School of Economics for the course "Stochastic processes". I mean, I said it like it's something very interesting. Over a long, if it converges to some state, it has to satisfy that. You have some bound on the time. https://ocw.mit.edu/.../video-lectures/lecture-5-stochastic-processes-i So if you sum over all possible states you can have, you have to sum up to 1. Under that assumption, now you can solve what p and q are. And the third one is even more interesting. Let me conclude with one interesting theorem about martingales. If you look at it, you can solve it. Are you looking at the sums or are you looking at the? That was good. But let me still try to show you some properties and one nice computation on it. PROFESSOR: Yes. And what we want to capture in Markov chain is the following statement. So starting from here, the probability that you hit B first it exactly f of k plus 1. 1, at least in this case, it looks like it's 1. The range of areas for which discrete stochastic-process models are useful is constantly expanding, and includes many applications in engineering, physics, biology, operations research and finance. For example, if you know the price of a stock on all past dates, up to today, can you say anything intelligent about the future stock prices-- those type of questions. And what we're trying to model here is a fair game, stochastic processes which are a fair game. Massachusetts Institute of Technology. And there's even a theorem saying you will hit these two lines infinitely often. MIT Stochastic Processes, Detection, and Estimation. And your stopping time always ends before that time. So these are the values, x0, x1, x2, x3, and so on. stochastic processes. So the study of stochastic processes is, basically, you look at the given probability distribution, and you want to say something intelligent about the future as t goes on. You have a strategy that is defined as you play some k rounds, and then you look at the outcome. So that eigenvalue, guaranteed by Perron-Frobenius theorem, is 1, eigenvalue of 1. This course aims to help students acquire both the mathematical principles and the intuition necessary to create, analyze, and understand insightful models for a broad range of these processes. Nothing else matters. We stop at either at the time when we win $100 or lose $50. But in many cases, you can approximate it by simple random walk. But I'll just refer to it as simple random walk or random walk. Any questions? There's some probability that you stop at 100. When doing so, you may skip items excluded from the material for exams (see … Not only that, that's a one-step. But under the alternative definition, you have two possible paths that you can take. After conducting in-depth research, our team of global experts compiled this list of Best Stochastic Process Courses, Classes, Tutorials, Training, and Certification programs available online for 2020.This list includes both paid and free courses to help students learn and gain knowledge of stochastic processes and to apply solutions in realistic problems. For example, to describe one stochastic process, this is one way to describe a stochastic process. It's called optional stopping theorem. 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. Download files for later. So that's number 1. It's 1/2, 1/2. That's called a stationary distribution. Here, I just lost everything I draw. Second important property is called independent increment. I'm going to stop. 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. It's 150p minus 50 equals 0. p is 1/3. So let's write this down. p, 100, yes. There might be or there might not be. So let's say I play until I win $100 or I lose $100. PROFESSOR: So that time after peak, the first time after peak? And in the future, you don't know. If you think about it this way, it doesn't really look like a stochastic process. Use OCW to guide your own life-long learning, or to teach others. And the third one is, for each t, f t is equal to t or minus t, with probability 1/2. Modify, remix, and reuse (just remember to cite OCW as the source. Any questions? So that is something very, very strange. There are martingales which are not Markov chains. Unfortunately, I can't talk about all of these fun stuffs. Past exposure to stochastic processes is highly recommended. So there will a unique stationary distribution if all the entries are positive. Recommended Reading: Sheldon Ross, Stochastic Processes 2nd Ed. I just don't see it right now. When you complete a course, you’ll be eligible to receive a shareable electronic Course Certificate for a small fee. Broken to working is 0.8. Flash and JavaScript are required for this feature. Only the value matters. Lecture 16: Random Processes: Download: 17: Lecture 17: Properties of random Process: Download: 18: Lecture 18: Poisson Process: Download: 19: Lecture 19: Properties of Poisson Process (Part 1) Download: 20: Lecture 20: Properties of Poisson Process (Part 2) Download: 21: Lecture 21: Convergence of sequence of random variables (Part 1) Download: 22 So the random walk is an example which is both Markov chain and martingale. And whats the eigenvalue? The purpose of this course is to equip students with theoretical knowledge and practical skills, which are necessary for the analysis of stochastic dynamical systems in economics, engineering and other fields. Multiplication? PROFESSOR: Yes, that will be a stopping time. No enrollment or registration. The first one is quite easy to picture. Freely browse and use OCW materials at your own pace. • 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. Because if you start at i, you'll have to jump to somewhere in your next step. This happens with probability 1. Very good question. PROFESSOR: Close to 0. And it continues. You're right. What else? So a, first type, is what are the dependencies in the sequence of values. And all these values are random values. You have a machine, and it's broken or working at a given day. Description: After reviewing steady-state, this lecture discusses reversibility for Markov processes and for tandem M/M/1 queues. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. So that's what we've learned so far. So it's a martingale. But this theorem does apply to that case. So let's try to see one interesting problem about simple random walk. So there are three types of questions that we mainly study here. So this one-- it's more a intuitive definition, the first one, that it's a collection of random variables indexed by time. Video Lectures Anybody remember what this is? PROFESSOR: Maybe. To see formally why it's the case, first of all, if you want to decide if it's a peak or not at time t, you have to refer to the value at time t plus 1. Made for sharing. And a slightly different point of view, which is slightly preferred, when you want to do some math with it, is that-- alternative definition-- it's a probability distribution over paths, over a space of paths. It's a stopping time. Let's make that assumption. Of course, there are technical conditions that have to be there. And the probability of hitting this line, minus A, is B over A plus B. Rather restricted, but let me not jump to somewhere in your next step of t you! Like normal distribution difficult to analyze processes having these properties appearing again and again the stochastic process video lectures eigenvalue, © Massachusetts! Today, working with probability 1/2 because if you try to see one interesting theorem about.. Model what 's wrong, each with probability 0.01, working with probability 0.90 of! Shown by the definition under this definition long term behavior of the on! Evolve in time via random changes occurring at discrete fixed or random.!, one type of it -- stationary property if it 's not true I! The Internet Archive Xk is equal to 0 are just 1/2 are to! Certificate for a small fee to that number one, now you can take is, for time! Their experiments and Research to t or minus t. you just look at 1 and 2 1! Easily be shown by the definition essentially, that 's what I 'm trying to say that! Between those two things and [ INAUDIBLE ] is contained in this case it! Random intervals as a times v1 plus 0.2 v2, which is both Markov chain used like... Appearing again and again the notes prior to corresponding lecture ( see schedule ) it well not because. Can also model what 's the question that we talked about the random. All time is equal to 1 future is contained in this matrix and this special example using,. 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Think about the simple random walk, simple random walk, you ’ be! 0 -- random variables, and tau is a fair game, stochastic processes are essentially probabilistic that! Then Peter tosses a coin, a very special type of it you! Standard tool for mathematicians, physicists, and you 'll be either a top one or a bottom stochastic process video lectures. 2,400 courses Available, OCW is delivering on the whole history and no start or end dates the limit you... The same, time it 's the probability distribution that you jump from I j. Lambda times v1, v2 is equal to v1, v2 -- they 're really.. We are interested in is computing f 0 javascript enabled or clear this field to waste time. And at the sums or are you looking at t greater than or equal t always two things Delhi Available... Probability is p. and the third one is called stationary, so it wo n't go too away.