However, finite Markov chains, really, there's one matrix that describes everything. PROFESSOR: Oh, you're right, sorry. Home Well, I know it's true, but that's what I'm telling you. Essentially, that kind of behavior is transitionary behavior that dissipates. So sum these two values, and you get lambda of this, v1, v2. Common usages include option pricing theory to modeling the growth of bacterial colonies. Let me conclude with one interesting theorem about martingales. Because stochastic processes having these properties are really good, in some sense. It's a good point. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. Something like that is the trajectory. So this is called the stationary distribution. Description: This lecture introduces stochastic processes, including random walks and Markov chains. That means that this is p, q. p, q is about the same as A times p, q. PROFESSOR: So that time after peak, the first time after peak? And your stopping time always ends before that time. So example, random walk probability that Xt plus 1 equal to s, given t is equal to 1/2, if s is equal Xt plus 1 or Xt minus 1, and 0 otherwise. Another realization will look something different and so on. Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. Moreover, lambda was a multiplicity of 1. And if that strategy only depends on the values of the stochastic process up to right now, then it's a stopping time. About how much will the variance be? 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]? 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's a very good point-- t and square root of t. Thank you. Topics in Mathematics with Applications in Finance. And it really enforces your intuition, at least intuition of the definition, that martingale is a fair game. It's either heads or tails. But yeah, there might be a way to make an argument out of it. A highlight will be the first functional limit theorem, Donsker's invariance principle, that establishes Brownian motion as a scaling limit of random walks. 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. Even, in this picture, you might think, OK, in some cases, it might be the case that you always play in the negative region. So this is a matrix. But you're saying from here, it's the same. The expectation is equal to that. The Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments. Probability Theory Refresher. They really are just two separate things. If it's some strategy that depends on future values, it's not a stopping time. Stochastic Processes • A stochastic process X = {X(t)} is a time series of random variables. All you have to know is this single value. All Courses. https://ocw.mit.edu/.../video-lectures/lecture-5-stochastic-processes-i What we know is f of B is equal to 1, f of minus A is equal to 0. PROFESSOR: Yes, that will be a stopping time. It's 1/2, 1/2. So if you just look at it, Xt over the square root of t will look like normal distribution. NPTEL provides E-learning through online Web and Video courses various streams. In other words, I look at the random walk, I look at the first time that it hits either this line or it hits this line, and then I stop. I play with, let's say, Peter. That should be the right one. That's not 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. Here, because of probability distribution, at each point, only gives t or minus t, you know that each of them will be at least one of the points, but you don't know more than that. • A sample path defines an ordinary function of t. Very good question. If it's heads, he wins. Really, this matrix contains all the information you want if you have a Markov chain and its finite. But it does not have a transition probability matrix, because the state space is not finite. Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. Courses > Stochastic Processes. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.. No enrollment or registration. No matter where you stand at, you exactly know what's going to happen in the future. Because of this-- which one is it-- stationary property. So you call it state set as well. It has these properties and even more powerful properties. Toggle navigation. udemy course statistics probability Delivered by . So we put Pij at [INAUDIBLE] and [INAUDIBLE]. Offered by National Research University Higher School of Economics. Stochastic Processes (Video) Syllabus; Co-ordinated by : IIT Delhi; Available from : 2013-06-20. And the reason simple random walk is a Markov chain is because both of them are just 1/2. Related Courses. Even though it's random, once you know what happened at some point, you know it has to be this distribution or this line, if it's here, and this line if it's there. We stop at either at the time when we win $100 or lose $50. PROFESSOR: I'm looking at the Xt. Learn Stochastic Process online with courses like Stochastic processes and Data Science Math Skills. There are some stuff which are not either of them. All right. IIT Kharagpur, , Prof. Mrityunjoy Chakraborty ... On-demand Videos; ... Lecture 29: Introduction to Stochastic Process. There should be a scale. I'm going to cheat a little bit and just say, you know what, I think, over a long period of time, the probability distribution on day 3,650 and that on day 3,651 shouldn't be that different. I want to make money. Second one, now let's say you're in a casino and you're playing roulette. One of the most important ones is the simple random walk. It's close to 0. So in the limit, they're 0, but until you get to the limit, you still have them. If you take this to be 10,000 times square root of t, almost 99.9% or something like that. So that's just a neat application. Then Xk is a martingale. And then actually, there's one recursive formula that matters to us. What is the probability that it will be at-- let's say, it's at 0 right now. But this theorem does apply to that case. Video Lectures Your path just says f t equals t. And we're only looking at t greater than or equal to 0 here. And now it starts again. Then, first of all, if the sum over all j and s, Pij, that is equal to 1. I win $1. » Actually, I made a mistake. And next week, Peter will give wonderful lectures. FREE. He wins the $1. What matters is the value at this last point, last time. 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? Download files for later. And there's even a theorem saying you will hit these two lines infinitely often. What will p and q be? You have a machine, and it's broken or working at a given day. So it's not a martingale. And that turns out to be 1. You go down with probability 1/2. Does it make sense? That part is Xk. 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. So at each step, you'll either multiply by 2 or 1/2 by 2-- just divide by 2. Let me iterate it. PROFESSOR: Variance will be small. We don't offer credit or certification for using OCW. But I think it's better to tell you what is not a stopping time, an example. You go up with probability 1/2. » And each time you go to the right or left, right or left, right or left. Freely browse and use OCW materials at your own pace. We look at our balance. I mean it's hard to find the right way to look at it. So you have some strategy which is a finite strategy. Now let me make a note here. Sorry about that. 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. And one more thing we know is, by Perron-Frobenius, there exists an eigenvalue, the largest one, lambda greater than 0, and eigenvector v1, v2, where v1, v2 are positive. If you think about it this way, it doesn't really look like a stochastic process. By peak, I mean the time when you go down, so that would be your tau. So that's just a very informal description. With probability 1, if you go to infinity, you will cross this line infinitely often. So corollary, it applies not immediately, but it does apply to the first case, case 1 given above. So I bet $1 at each turn. That means it will be some time index. I don't know if it's true or not. That this value can affect the future, because that's where you're going to start your process from. Another way to look at it-- the reason we call it a random walk is, if you just plot your values of Xt, over time, on a line, then you start at 0, you go to the right, right, left, right, right, left, left, left. Let's say we went up again, down, 4, up, up, something like that. But let me not jump to the conclusion yet. The course will conclude with a first look at a stochastic process in continuous time, the celebrated Browning motion. So there will a unique stationary distribution if all the entries are positive. Try not to be confused between the two. Unfortunately, I can't talk about all of these fun stuffs. t with--let me show you three stochastic processes, so number one, f t equals t.And this was probability 1. So that's what we're trying to distinguish by defining a stopping time. It's 0.99 v1 plus 0.01 v2. In these cases it was clear, at the time, you know if you have to stop or not. But let me still try to show you some properties and one nice computation on it. Now, instead of looking at one fixed starting point, we're going to change our starting point and look at all possible ways. But let me show you one, very interesting corollary of this applied to that number one. But that one, if you want to do some math with it, from the formal point of view, that will be more helpful. So I'll just forget about that technical issue. I was confused. So that's was an introduction. There will be a unique one and so on. So you have all a bunch of possible paths that you can take. But you want to know something about it. Working to working is 0.99. 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. So let's say I play until I win $100 or I lose $100. Really, there are lot more interesting things, but I'm just giving an overview, in this course, now. Ah. Most other stochastic processes, the future will depend on the whole history. There are martingales which are not Markov chains. Now, for each t, we get rid of this dependency. And the third one is, for each t, f t is equal to t or minus t, with probability 1/2. It's not a fair game. There's some probability that you stop at 100. So from what we learned last time, we can already say something intelligent about the simple random walk. Like this part is really irrelevant. But there's a theorem saying that that's not the case. And you'll see these properties appearing again and again. It's not clear that there is a bounded time where you always stop before that time. So a stochastic process is a collection of random variables indexed by time, a very simple definition. A stochastic process is called a Markov chain if has some property. And then, depending on the value of Y1, you will either go up or go down. Then there are really lots of stochastic processes. Then my balance will exactly follow the simple random walk, assuming that the coin it's a fair coin, 50-50 chance. 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. These typically come with video lectures, notes, homework, solutions, exams ... and are free. So if you go up, the probability that you hit B first is f of k plus 1. In general, if you have a transition matrix, if you're given a Markov chain and given a transition matrix, Perron-Frobenius theorem guarantees that there exists a vector as long as all the entries are positive. So in coin toss game, let tau be the first time at which balance becomes $100, then tau is a stopping time. But still, in expectation, you will always maintain. It's really just-- there's nothing random in here. Are you looking at the sums or are you looking at the? It's a stopping time. So if you look at simple random walk, it is a Markov chain, right? Courses That is a stopping time. Of course, this is a very special type of stochastic process. ... video lectures, and community discussion forums. Of course, at one instance, you might win money. Such vector v is called. So let me define it a little bit more formally. The game is designed for the casino not for you. You won't deviate too much. And then Peter tosses a coin, a fair coin. And there is on all positive eigenvector corresponding to it. Then, if it's a Markov chain, what it's saying is, you don't even have know all about this. Only the value matters. That means your lambda is equal to 1. Made for sharing. … So try to contemplate about it, something very philosophically. And the third type, this one is left relevant for our course, but, still, I'll just write it down. And this is a definition. 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. That's the question that we're trying to ask. The reason is because Xt, 1over the square root of t times Xt-- we saw last time that this, if t is really, really large, this is close to the normal distribution, 0,1. The second one is called the Markov chain. So I wanted to prove it, but I'll not, because I think I'm running out of time. Stochastic Processes. But in many cases, you can approximate it by simple random walk. That's called a stationary distribution. So in general, if you put a line B and a line A, then probability of hitting B first is A over A plus B. You're supposed to lose money. And what we're trying to model here is a fair game, stochastic processes which are a fair game. You want to have some intelligent conclusion, intelligent information about the future, based on the past. What is a simple random walk? And so, in this case, if it's 100 and 50, it's 100 over 150, that's 2/3 and that's 1/3. I don't want to waste my time on trying to find what's wrong. And this is by symmetry. Remember that coin toss game which had random walk value, so either win $1 or lose $1. Mathematics And I will write it down more formally later, but the message is this. Everything about the stochastic process is contained in this matrix. 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. If you look at it, you can solve it. And it doesn't have to be continuous, so it can jump and it can jump and so on. If something can be modeled using martingales, perfectly, if it really fits into the mathematical formulation of a martingale, then you're not supposed to win. And now, let's say I started from $0.00 balance, even though that's not possible. What we are interested in is computing f 0. They should be about the same. The lecture notes for this course can be found here. • 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. If you have watched this lecture and know what it is about, particularly what Mathematics topics are discussed, please help us by commenting on this video with your suggested description and title.Many thanks from, Third one is some funny example. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. This is an example of a Markov chain used in like engineering applications. And in fact, you will meet these two lines infinitely often. On the left, you get v1 plus v2. And it's broken at 0.2. AUDIENCE: Just kind of [INAUDIBLE] question, is that topic covered in portions of [INAUDIBLE]? I just made it up to show that there are many possible ways that a stochastic process can be a martingale. Nothing else matters. And that's exactly what occurs. Suppose we have a martingale, and tau is a stopping time. And the probability of hitting this line, minus A, is B over A plus B. Any questions? » But in expected value, you're designed to go down. So the random walk is an example which is both Markov chain and martingale. So the first time when you start to go down, you're going to stop. So simple random walk, let's say you went like that. But I'll just refer to it as simple random walk or random walk. Yeah, but Perron-Frobenius theorem say there is exactly one eigenvector corresponding to the largest eigenvalue. So that's number 1. And that's something very general. Stochastic processes are a standard tool for mathematicians, physicists, and others in the field. Because if you start at i, you'll have to jump to somewhere in your next step. On the right, you get lambda times v1 plus v2. I'm going to stop. What if I play until win $100 or lose $50? Working to broken is 0.01. All entries are positive. That's the concept of the theorem. So I hope this gives you some feeling about stochastic processes, I mean, why we want to describe it in terms of this language, just a tiny bit. Find materials for this course in the pages linked along the left. And then a continuous time random variable-- a continuous time stochastic process can be something like that. This is one of over 2,200 courses on OCW. So p, q will be the eigenvector of this matrix. So starting from here, the probability that you hit B first it exactly f of k plus 1. PROFESSOR: Yes. This part looks questionable. Over a long period of time, the probability distribution that you will observe will be the eigenvector. Required Text: None. On the left, what you get is v1 plus v2, so sum two coordinates. So if it converges, it will converge to that. » q will be the probability that it's broken at that time. So fix your B and A. What else? Stochastic Process courses from top universities and industry leaders. And by Perron-Frobenius theorem, we know that there is a vector satisfying it. So let me write this down in a different way. From the practical point of view, you'll have to twist some things slightly and so on. 15 . We have two states, working and broken. 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. So let's write this down. Not only that, that's a one-step. So this one-- it's more a intuitive definition, the first one, that it's a collection of random variables indexed by time. At time 0, we start at 0. I'm going to play. For example, if you apply central limit theorem to the sequence, what is the information you get? 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. For just looking at values up to time t, you don't know if it's going to be a peak or if it's going to continue. When doing so, you may skip items excluded from the material for exams (see … I won't do that, but we'll try to do it as an exercise. stochastic processes online lecture notes and books This site lists free online lecture notes and books on stochastic processes and applied probability, stochastic calculus, measure theoretic probability, probability distributions, Brownian motion, financial mathematics, Markov … Introduction to Stochastic Processes - Lecture Notes (with 33 illustrations) Gordan Žitković Department of Mathematics The University of Texas at Austin Use OCW to guide your own life-long learning, or to teach others. If you go down, it's f of k minus 1. Your expected value is just fixed. I talked about the most important example of stochastic process. AUDIENCE: Let's say, yeah, it was [INAUDIBLE]. It's called martingale. 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. Learn more », © 2001–2018 What if I say I will win $100 or I lose $50? Any questions? So we have this stochastic process, and, at time t, you are at Xt. And formally, what I mean is a stochastic process is a martingale if that happens. Some people would say that 100 is close to 0, so do you have some degree of how close it will be to 0? Lec : 1; Modules / Lectures. I want to define something called a stopping time. Lecture 5: Stochastic Processes I. So some properties of a random walk, first, expectation of Xk is equal to 0. It's 150p minus 50 equals 0. p is 1/3. And simple random walk is like the fundamental stochastic process. Over a long, if it converges to some state, it has to satisfy that. There is no 0, 1, here, so it's 1 and 2. Text: Download the course lecture notes and read each section of the notes prior to corresponding lecture (see schedule). This happens with probability 1. We'll focus on discrete time. And if you want to look at the three-step, four-step, all you have to do is just multiply it again and again and again. And so when you take products of the transition probability matrix, those eigenvalues that are smaller than 1 scale after repeated multiplication to 0. 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. Two equivalent processes may have quite different sample paths. And I will later tell you more about that. Recommended Reading: Sheldon Ross, Stochastic Processes 2nd Ed. There are Markov chains which are not martingales. So that was it. So before stating the theorem, I have to define what a stopping point means. 1.3 Equivalence of Stochastic Processes Definition 1.3.1 A stochastic process {X t,t ∈T}is equivalent to another stochastic process {Y t,t ∈T}if for each t ∈T P {X t = Y t}= 1. Then my balance is a simple random walk. Find materials for this course in the pages linked along the left. 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. 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. But later, it will really help if you understand it well. But that one is slightly different. Don't show me this again. Because every chain of coin toss, which gives a winning sequence, when you flip it, it will give a losing sequence. So example, a random walk is a martingale. So be careful. But this is designed so that the expected value is equal to 1. I'm not sure if there is a way to make an argument out of it. It's not really right to say that a vector has stationary distribution. Because we're just having new coin tosses every time. Anybody remember what this is? Though it's not true if I say any information at all. Then what happens after time t really just depends on how high this point is at. But the probability distribution is designed so that the expected value over all these are exactly equal to the value at Xt. And what we want to capture in Markov chain is the following statement. AUDIENCE: [INAUDIBLE]. Even if you try to lose money so hard, you won't be able to do that. We have one to one correspondence between those two things. I mean some would say that 1 is close to 0. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. In this case, s is also called a sample state space, actually. It is 1/3, actually. So number one is a stopping time. And that 0.8 v1 plus 0.2 v2, which is equal to v1, v2. So it should describe everything. So if you're playing a martingale game, then you're not supposed to win or lose, at least in expectation. This is one of over 2,200 courses on OCW. Lecture notes. The following content is provided under a Creative Commons license. And then that will be one realization. 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. Lecture Notes and Homework Assignments will be posted here. I don't see what the problem is right now. So if you sum over all possible states you can have, you have to sum up to 1. A times v1, v2, we can write it down. It might go to this point, that point, that point, or so on. Because for continuous time, it will just carry over all the knowledge. Thanks for your information Have a nice dayStochastic Process online videos, Lecture 1: Introduction to Discrete Stochastic Processes and Probability Review, Lecture 2: More Review; The Bernoulli Process, Lecture 3: Law of Large Numbers, Convergence, Lecture 4: Poisson (The Perfect Arrival Process), Lecture 5: Poisson Combining and Splitting, Lecture 7: Finite-state Markov Chains; The Matrix Approach, Lecture 8: Markov Eigenvalues and Eigenvectors, Lecture 9: Markov Rewards and Dynamic Programming, Lecture 10: Renewals and the Strong Law of Large Numbers, Lecture 11: Renewals: Strong Law and Rewards, Lecture 12: Renewal Rewards, Stopping Trials, and Wald's Inequality, Lecture 18: Countable-state Markov Chains and Processes, Lecture 13: Little, M/G/1, Ensemble Averages, Lecture 16: Renewals and Countable-state Markov, Lecture 17: Countable-state Markov Chains, Lecture 19: Countable-state Markov Processes, Lecture 20: Markov Processes and Random Walks, Lecture 21: Hypothesis Testing and Random Walks, Lecture 23: Martingales (Plain, Sub, and Super), Lecture 24: Martingales: Stopping and Converging, Massachusetts Institute of Technology, MIT, Creative Commons Attribution Non-Commercial Share Alike (CC-BY-NC-SA). 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