Working to broken is 0.01. All right. Broken to working is 0.8. So you have some strategy which is a finite strategy. So that was two representations. Under that assumption, now you can solve what p and q are.

We'll focus on discrete time. At time 0, we start at 0. 1, at least in this case, it looks like it's 1. Yeah, so everybody, it should have been flipped in the beginning. But in many cases, you can approximate it by simple random walk. The game is designed for the casino not for you. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. That's the content of this theorem.

You go up with probability 1/2. But all these variables are supposed to be random. 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. 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. What is a simple random walk? Because it's designed so that the expected value is less than 0. 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. There's some probability that you stop at 100. I'm not sure.

I mean at every single point, you'll be either a top one or a bottom one.

Just look at 0 comma 1, here. But that one is slightly different.

Even though, theoretically, you can be that far away from your x-axis, in reality, what's going to happen is you're going to be really close to this curve. A stochastic process is by nature continuous; by contrast a time series is a set of observations indexed by integersl.

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.

He wins the \$1. So what happened is it describes what happens in a single step, the probability that you jump from i to j.

They should be about the same.

To mention some applications: - hedging and pricing of options, - portfolio selection, - risk management, - real options and investment on energy markets, -  high frequency trading.

You have a pre-defined set of strategies. Now, let's talk about more stochastic processes. And then I say the following.

So it's really easy to control. 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.

And there is such a state.

And I will write it down more formally later, but the message is this. There's no signup, and no start or end dates. So before stating the theorem, I have to define what a stopping point means. And the probability distribution is given as 1/3 and 2/3. Hours - Lab: 0. Do you remember Perron-Frobenius theorem?

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. I was confused. In this case, s is also called a sample state space, actually.

And all these values are random values. PROFESSOR: Variance will be small. Download the video from iTunes U or the Internet Archive. ), Learn more at Get Started with MIT OpenCourseWare, MIT OpenCourseWare makes the materials used in the teaching of almost all of MIT's subjects available on the Web, free of charge.

Anybody remember what this is? That's not a stopping time. Téléchargez plus de 1 700 eBooks sur les soft skills et l'efficacité professionnelle. I mean, if it's for-- let me write it down. So it really depends only on the last value of Xt. Stochastic Optimal Control in Finance H. Mete Soner Ko¸c University Istanbul, Turkey msoner@ku.edu.tr. But under the alternative definition, you have two possible paths that you can take. And that turns out to be 1. You have a machine, and it's broken or working at a given day.

Essentially, that kind of behavior is transitionary behavior that dissipates.

And really, a very interesting thing is this matrix called the transition probability matrix, defined as. So I bet \$1 at each turn. I talked about the most important example of stochastic process. Then that is a martingale.

Thus, they could be modelized by stochastic processes, assuming theses prices are known in continuous time. So p, q will be the eigenvector of this matrix. Mathematics But what we're trying to capture here is-- now, look at some generic stochastic process at time t. You know all the history up to time t. You want to say something about the future. No matter where you stand at, you exactly know what's going to happen in the future.

It's really just-- there's nothing random in here.

Ceci est un eBook gratuit pour les étudiants, Téléchargez des manuels d’apprentissage au format PDF ou lisez-les en ligne. 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. 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.

Then my balance will exactly follow the simple random walk, assuming that the coin it's a fair coin, 50-50 chance. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. You have a strategy that is defined as you play some k rounds, and then you look at the outcome.

And the probability of hitting this line, minus A, is B over A plus B.

And still, lots of interesting things turn out to be Markov chains. So in coin toss game, let tau be the first time at which balance becomes \$100, then tau is a stopping time.

Third one is some funny example. X0, X1, X2, and so on is called a one-dimensional, simple random walk. But let me not jump to the conclusion yet.

PROFESSOR: Today we're going to study stochastic processes and, among them, one type of it, so discrete time. If you go down, it's f of k minus 1. It's equal to 0. Topics in Mathematics with Applications in Finance

Multiplication?

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.

Now, for each t, we get rid of this dependency. The reason is-- so you can compute the expected value. But let me still try to show you some properties and one nice computation on it.

That's what I'm trying to say here. 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. It's actually not that difficult to prove it. So if you just look at it, Xt over the square root of t will look like normal distribution. 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]?

Thématiques: statistique et finance, gestion des risques, méthodes numériques.

It's called optional stopping theorem. »

So that is a Markov chain. Even, in this picture, you might think, OK, in some cases, it might be the case that you always play in the negative region. Here, I just lost everything I draw. It describes the most important stochastic processes used in finance in a pedagogical way, especially Markov chains, Brownian motion and martingales.

That was good.

Some people would say that 100 is close to 0, so do you have some degree of how close it will be to 0? 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. Very good.

So you have all a bunch of possible paths that you can take. On the left, what you get is v1 plus v2, so sum two coordinates. Let me show you by example. Remember that coin toss game which had random walk value, so either win \$1 or lose \$1. It has known important developments over the last years inspired especially by problems in mathematical finance.

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