5. Stochastic Processes I

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visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Today
we're going to study stochastic processes and,
among them, one type of it, so discrete time. We'll focus on discrete time. And I'll talk about
what it is right now.

So a stochastic
process is a collection of random variables indexed by
time, a very simple definition. So we have either– let's
start from 0– random variables like this, or we have random
variables given like this. So a time variable
can be discrete, or it can be continuous. These ones, we'll call
discrete-time stochastic processes, and these
ones continuous-time. So for example, a
discrete-time random variable can be something
like– and so on. So these are the values, X_0,
X_1, X_2, X_3, and so on. And they are random variables. This is just one–
so one realization of the stochastic process. But all these variables
are supposed to be random.

And then a continuous-time
random variable– a continuous-time
stochastic process can be something like that. And it doesn't have to be
continuous, so it can jump and it can jump and so on. And all these values
are random values. So that's just a very
informal description. 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. So you have all a
bunch of possible paths that you can take.

And you're given some
probability distribution over it. And then that will
be one realization. Another realization will look
something different and so on. So this one– it's more
intuitive definition, the first one, that it's a
collection of random variables indexed by time. But that one, if you want
to do some math with it, from the formal point of view,
that will be more helpful. And you'll see why
that's the case later. So let me show you
some more examples. For example, to describe
one stochastic process, this is one way to describe
a stochastic process. t with– let me show you
three stochastic processes, so number one, f(t) equals t. And this was probability 1. Number 2, f(t) is
equal to t, for all t, with probability 1/2, or f(t)
is equal to minus t, for all t, with probability 1/2. And the third one
is, for each t, f(t) is equal to t or minus
t, with probability 1/2. The first one is
quite easy to picture. It's really just– there's
nothing random in here.

This happens with probability 1. Your path just
says f(t) equals t. And we're only looking at t
greater than or equal to 0 here. So that's number 1. Number 2, it's either
this one or this one. So it is a stochastic process. If you think about it this
way, it doesn't really look like a stochastic process. But under the
alternative definition, you have two possible
paths that you can take. You either take this path, with
1/2, or this path, with 1/2. Now, at each point,
t, your value X(t) is a random variable. It's either t or minus t. And it's the same for all t. But they are dependent
on each other. So if you know one
value, you automatically know all the other values. And the third one is
even more interesting. Now, for each t, we get
rid of this dependency.

So what you'll have is
these two lines going on. I mean at every
single point, you'll be either a top one
or a bottom one. 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. 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. Any questions? 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. And in the future,
you don't know.

But you want to know
something about it. You want to have some
intelligent conclusion, intelligent information about
the future, based on the past. For this stochastic
process, it's easy. No matter where you
stand at, you exactly know what's going to
happen in the future. For this one, it's
also the same. 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. But that one is
slightly different. 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.

Though it's not true if I
say any information at all. We know that each value
has to be t or minus t. You just don't know what it is. 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. Here, it was, you can
really determine the line. 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. 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. So there are three
types of questions that we mainly study here.

So (a), first type, is
what are the dependencies in the sequence of values. 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 (b) is what is the long
term behavior of the sequence? So think about the
law of large numbers that we talked about last
time or central limit theorem. And the third type, this one is
less relevant for our course, but, still, I'll
just write it down. What are the boundary events? 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.

How often will that happen? 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 that's was an introduction. Any questions? Then there are really lots
of stochastic processes. One of the most important ones
is the simple random walk. So today, I will focus on
discrete-time stochastic processes. Later in the course, we'll go
on to continuous-time stochastic processes. And then you'll see
like Brownian motions and– what else– Ito's
lemma and all those things will appear later. Right now, we'll
study discrete time.

And later, you'll see that
it's really just– what is it– they're really parallel. So this simple
random walk, you'll see the corresponding thing
in continuous-time stochastic processes later. So I think it's
easier to understand discrete-time processes,
that's why we start with it. But later, it will really help
if you understand it well. Because for continuous
time, it will just carry over all the knowledge. What is a simple random walk? Let Y_i be IID, independent
identically distributed, random variables, taking
values 1 or minus 1, each with probability 1/2.

Then define, for each time
t, X sub t as the sum of Y_i, from i equals 1 to t. Then the sequence of
random variables– and X_0 is equal to 0– X0,
X1, X2, and so on is called a one-dimensional
simple random walk. But I'll just refer to
it as simple random walk or random walk. And this is a definition. It's called simple random walk. Let's try to plot it. At time 0, we start at 0. And then, depending
on the value of Y1, you will either
go up or go down. Let's say we went up. So that's at time 1. 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. Let's say we went up
again, down, 4, up, up, something like that. And it continues. Another way to look at it– the
reason we call it a random walk is, if you just plot your values
of X_t, over time, on a line, then you start at 0, you go to
the right, right, left, right, right, left, left, left.

So the trajectory is like a
walk you take on this line, but it's random. And each time you
go to the right or left, right or
left, right or left. So that was two representations. This picture looks a
little bit more clear. Here, I just lost
everything I draw. Something like that
is the trajectory. So from what we
learned last time, we can already say
something intelligent about the simple random walk. For example, if you apply
central limit theorem to the sequence, what is
the information you get? 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? AUDIENCE: Is it close to 0? PROFESSOR: Close to 0. But by close to 0,
what do you mean? There should be a scale. I mean some would say
that 1 is close to 0. Some people would say
that 100 is close to 0, so do you have some degree
of how close it will be to 0? Anybody? AUDIENCE: So variance
will be small.

PROFESSOR: Sorry? AUDIENCE: The variance
will be small. PROFESSOR: Variance
will be small. About how much will
the variance be? AUDIENCE: 1 over n. PROFESSOR: 1 over n. 1 over n? AUDIENCE: Over t. PROFESSOR: 1 over t? Anybody else want
to have a different? AUDIENCE: [INAUDIBLE]. PROFESSOR: 1 over square
root t probably would. AUDIENCE: [INAUDIBLE]. AUDIENCE: The variance
you're right, sorry. 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. AUDIENCE: [INAUDIBLE]. Are you looking at the sums
or are you looking at the? PROFESSOR: I'm
looking at the X_t. Ah. That's a very good point. t and square root of t. Thank you. AUDIENCE: That's very different. PROFESSOR: Yeah,
very, very different. I was confused. Sorry about that. The reason is because X_t, 1
over the square root of t times X_t– we saw last
time that this, if t is really,
really large, this is close to the normal
distribution, 0,1.

So if you just look at it,
X_t over the square root of t will look like
normal distribution. 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. It's close to 0. And the scale you're looking at
is about the square root of t. So it won't go too
far away from 0. 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.

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. 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. You're going to play
within this area, mostly. AUDIENCE: I think
that [INAUDIBLE]. PROFESSOR: So, yeah, that
was a very vague statement. You won't deviate too much. So if you take 100
square root of t, you will be inside this
interval like 90% of the time. If you take this to be 10,000
times square root of t, almost 99.9% or
something like that.

And there's even
a theorem saying you will hit these two
lines infinitely often. So if you go over time, a
very long period, for a very, very long, you live long enough,
then, even if you go down here. Even, in this picture, you
might think, OK, in some cases, it might be the
case that you always play in the negative region. But there's a theorem saying
that that's not the case. With probability 1,
if you go to infinity, you will cross this
line infinitely often. And in fact, you will meet these
two lines infinitely often. So those are some
interesting things about simple random walk. Really, there are lot
more interesting things, but I'm just giving an
overview, in this course, now. Unfortunately, I can't talk
about all of these fun stuffs. But let me still try to show
you some properties and one nice computation on it. So some properties of a random
walk, first, expectation of X_k is equal to 0. That's really easy to prove.

Second important property is
called independent increment. So if look at these times,
t_0, t_1, up to t_k, then random variables X sub
t_i+1 minus X sub t_i are mutually independent. 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 that can easily be
shown by the definition. I won't do that, but we'll
try to do it as an exercise. Third one is called stationary,
so it has the property. 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 X_(t+h)
minus X_t is the same as the distribution of X sub h.

And again, this easily
follows from the definition. 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. The distribution is the same. And moreover, from
the first part, if these intervals do not
overlap, they're independent. So those are the two properties
that we're talking here. And you'll see these properties
appearing again and again. Because stochastic processes
having these properties are really good, in some sense. They are fundamental
stochastic processes. And simple random walk is like
the fundamental stochastic process. So let's try to see
one interesting problem about simple random walk. So example, you play a game. It's like a coin toss game. I play with, let's say, Peter. So I bet $1 at each turn. And then Peter tosses
a coin, a fair coin. It's either heads or tails.

If it's heads, he wins. He wins the $1. If it's tails, I win. I win $1. So from my point of view,
in this coin toss game, at each turn my balance
goes up by $1 or down by $1. And now, let's say I
started from $0.00 balance, even though that's not possible. Then my balance will exactly
follow the simple random walk, assuming that the coin it's
a fair coin, 50-50 chance. Then my balance is a
simple random walk. And then I say the following. You know what? I'm going to play.

I want to make money. So I'm going to play until
I win $100 or I lose $100. So let's say I play until
I win $100 or I lose $100. What is the probability that I
will stop after winning $100? AUDIENCE: 1/2. PROFESSOR: 1/2 because? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yes. So happens with 1/2, 1/2. And this is by symmetry. Because every chain
of coin toss which gives a winning sequence,
when you flip it, it will give a losing sequence. We have one-to-one
correspondence between those two things. That was good. Now if I change it. What if I say I will
win $100 or I lose $50? What if I play until
win $100 or lose $50? 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. What is the probability that I
will stop after winning $100? AUDIENCE: [INAUDIBLE]. PROFESSOR: 1/3? Let me see. Why 1/3? AUDIENCE: [INAUDIBLE]. PROFESSOR: So you're saying,
hitting this probability is p.

And the probability that you
hit this first is p, right? It's 1/2, 1/2. But you're saying from
here, it's the same. So it should be 1/4
here, 1/2 times 1/2. You've got a good intuition. It is 1/3, actually. AUDIENCE: [INAUDIBLE]. PROFESSOR: And then
once you hit it, it's like the same afterwards? I'm not sure if there is a way
to make an argument out of it. I really don't know. There might be or
there might not be. I'm not sure. I was thinking of
a different way. But yeah, there might be a way
to make an argument out of it. I just don't see it right now. 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.

And the probability of
hitting this line– minus A– is B over A plus B. 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. This can be proved. It's actually not that
difficult to prove it. I mean it's hard to find
the right way to look at it. So fix your B and A. And
for each k between minus A and B define f of k as the
probability that you'll hit– what is it–
this line first, and the probability that
you hit the line B first when you start at k. So it kind of points
out what you're saying. 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. What we are interested
in is computing f(0). What we know is f of B is
equal to 1, f of minus A is equal to 0.

And then actually, there's
one recursive formula that matters to us. If you start at f(k), you
either go up or go down. You go up with probability 1/2. You go down with
probability 1/2. And now it starts again. Because of this– which one
is it– stationary property. So starting from
here, the probability that you hit B first is
exactly f of k plus 1. So if you go up, the
probability that you hit B first is f of k plus 1. If you go down,
it's f of k minus 1. And then that gives
you a recursive formula with two boundary values. If you look at it,
you can solve it. When you solve it,
you'll get that answer. 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. It has these properties and
even more powerful properties. So it's really easy to control. And at the same time
it's quite universal. It can model– like it's
not a very weak model.

It's rather restricted, but
it's a really good model for like a mathematician. From the practical
point of view, you'll have to twist some
things slightly and so on. But in many cases,
you can approximate it by simple random walk. And as you can see, you
can do computations, with simple random
walk, by hand. So that was it. I talked about the
most important example of stochastic process. Now, let's talk about
more stochastic processes. The second one is
called the Markov chain. Let me write that
part, actually. So Markov chain, unlike
the simple random walk, is not a single
stochastic process. A stochastic process is
called a Markov chain if has some property. And what we want to
capture in Markov chain is the following statement. 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.

That's quite a vague statement. 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. Then, if it's a Markov
chain, what it's saying is, you don't even have
know all about this. Like this part is
really irrelevant. What matters is the value at
this last point, last time. So if it's a Markov
chain, you don't have to know all this history. All you have to know
is this single value. And all of the effect of
the past on the future is contained in this value. Nothing else matters. Of course, this is
a very special type of stochastic process. Most other stochastic
processes, the future will depend on
the whole history. And in that case, it's
more difficult to analyze. But these ones are
more manageable. And still, lots of
interesting things turn out to be Markov chains. So if you look at
simple random walk, it is a Markov chain, right? So simple random walk, let's
say you went like that.

Then what happens after
time t really just depends on how high this point is at. What happened before
doesn't matter at all. Because we're just having
new coin tosses every time. But this value can
affect the future, because that's
where you're going to start your process from. Like that's where you're
starting your process. So that is a Markov chain. This part is irrelevant. Only the value matters. So let me define it a
little bit more formally. 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
X_(t+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. The value at X_(t+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. And the reason simple random
walk is a Markov chain is because both of
them are just 1/2. I mean, if it's for–
let me write it down. So example: random walk. Probability that X_(t+1)
equal to s, given– t is equal to 1/2, if s is equal
X_t plus 1, or X_t minus 1, and 0 otherwise.

So it really depends only
on the last value of X_t. Any questions? All right. If there is case
when you're looking at a stochastic
process, a Markov chain, and all X_i have values
in some set S, which is finite, a finite
set, in that case, it's really easy to
describe Markov chains. So now denote the
probability i, j 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. Then, first of all, if you
sum over all j in S, P_(i,j), that is equal to 1.

Because if you
start at i, you'll have to jump to somewhere
in your next step. So if you sum over all
possible states you can have, you have to sum up to 1. And really, a very
interesting thing is this matrix, called
the transition probability matrix, defined as. So we put P_(i,j) at
i-th row and j-th column. And really, this
tells you everything about the Markov chain. Everything about the
stochastic process is contained in this matrix. That's because a
future state only depends on the current state. So if you know what happens at
time t, where it's at time t, you look at the
matrix, you can decode all the information you want. What is the probability that
it will be at– let's say, it's at 0 right now. What's the probability
that it will jump to 1 at the next time? Just look at 0 comma 1, here. There is no 0, 1,
here, so it's 1 and 2. Just look at 1 and
2, 1 and 2, i and j.

Actually, I made a mistake. That should be the right one. Not only that,
that's a one-step. So what happened is
it describes what happens in a single
step, the probability that you jump from i to j. But using that,
you can also model what's the probability that you
jump from i to j in two steps. 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. Then the matrix,
defined this way, can you describe it in
terms of the matrix A? Anybody? Multiplication? Very good. So it's A square. Why is it? So let me write this
down in a different way. q_(i,j) 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. 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.

And that's exactly what occurs. 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. Really, this matrix
contains all the information you want if you have a
Markov chain and it's finite. That's very important. For random walk,
simple random walk, I told you that it
is a Markov chain. But it does not have a
transition probability matrix, because the state
space is not finite. So be careful. However, finite Markov
chains, really, there's one matrix that
describes everything. I mean, I said it like it's
something very interesting. But if you think
about it, you just wrote down all
the probabilities. So it should
describe everything. So an example. You have a machine,
and it's broken or working at a given day. That's a silly example. So if it's working
today, working tomorrow, broken with probability 0.01,
working with probability 0.99. If it's broken, the
probability that it's repaired on the next day is 0.8.

And it's broken at 0.2. Suppose you have
something like this. This is an example of a Markov
chain used in like engineering applications. In this case, S is also called
the state space, actually. 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. And all these things
that you model represent states a lot of time. So you call it
state set as well. So that's an example.

And let's see what
happens for this matrix. We have two states,
working and broken. Working to working is 0.99. Working to broken is 0.01. Broken to working is 0.8. Broken to broken is 0.2. So that's what we've
learned so far. 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 will be–
that times [1, 0] will be the probability [p, q]. p will be the probability that
it's working at that time. q will be the probability
that it's broken at that time. What will p and q be? What will p and q be? That's the question that
we're trying to ask. We didn't learn, so
far, how to do this, but let's think about it. 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. They should be about the same. Let's make that assumption.

I don't know if
it's true or not. Well, I know it's true, but
that's what I'm telling you. Under that assumption, now you
can solve what p and q are. So approximately, I hope,
p, q– so A^3650 * [1, 0] is approximately the same
as A to the 3651, [1, 0]. That means that this is [p, q]. [p, q] is about the
same as A times [p, q].

Anybody remember what this is? Yes. So [p, q] will be the
eigenvector of this matrix. Over a long period of time,
the probability distribution that you will observe
will be the eigenvector. And whats the eigenvalue? 1, at least in this case,
it looks like it's 1. Now I'll make one
more connection. Do you remember
Perron-Frobenius theorem? So this is a matrix.

All entries are positive. So there is a
largest eigenvalue, which is positive and real. And there is an all-positive
eigenvector corresponding to it. What I'm trying to say is
that's going to be your [p, q]. But let me not jump
to the conclusion yet. And one more thing we know
is, by Perron-Frobenius, there exists an eigenvalue,
the largest one, lambda greater than 0, and eigenvector
[v 1, v 2], where [v 1, v 2] are positive.

Moreover, lambda was
at multiplicity 1. I'll get back to it later. So let's write this down. A times [v 1, v 2] is equal
to lambda times [v 1, v2]. A times [v 1, v 2],
we can write it down. It's 0.99 v_1 plus 0.01 v_2. And that 0.8 v_1 plus 0.2 v_2,
which is equal to [v1, v2]. You can solve v_1 and
v_2, but before doing that– sorry about that.

This is flipped. Yeah, so everybody,
it should have been flipped in the beginning. So that's 8. So sum these two values, and
you get lambda times [v 1, v 2]. On the left, what you
get is v_1 plus v_2, you sum two coordinates. On the left, you
get v_1 plus v_2. On the right, you get
lambda times v_1 plus v_2. That means your
lambda is equal to 1. 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. And that's something
very general. It's not just about this matrix
and this special example. 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 general, if transition
matrix of a Markov chain has positive entries, then
there exists a vector pi_1 up 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. Over a long term, if it
converges to some state, it has to satisfy that. And by Perron-Frobenius
theorem, we know that there is a
vector satisfying it. So if it converges, it
will converge to that. And what it's saying is, if
all the entries are positive, then it converges. And there is such a state. We know the long-term
behavior of the system. So this is called the
stationary distribution. Such vector v is called. It's not really right
to say that a vector is stationary distribution. 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.

If you start from this
distribution, in the next step, you'll have the exact
same distribution. That's what I'm
trying to say here. That's called a
stationary distribution. Any questions? AUDIENCE: So [INAUDIBLE]? PROFESSOR: Yes. Very good question. Yeah, but Perron-Frobenius
theorem says there is exactly one
eigenvector corresponding to the largest eigenvalue. And that turns out to be 1. The largest eigenvalue
turns out to be 1. So there will a unique
stationary distribution if all the entries are positive. AUDIENCE: [INAUDIBLE]? PROFESSOR: This one? AUDIENCE: [INAUDIBLE]? PROFESSOR: Maybe. It's a good point. Huh? Something is wrong. Can anybody help me? This part looks questionable. AUDIENCE: Just kind of
[INAUDIBLE] question, is that topic covered in
portions of [INAUDIBLE]? The other eigenvalues in the
matrix are smaller than 1.

And so when you take products
of the transition probability matrix, those eigenvalues
that are smaller than 1 scale after repeated
multiplication to 0. So in the limit, they're 0,
but until you get to the limit, you still have them. Essentially, that
kind of behavior is transitionary
behavior that dissipates. But the behavior corresponding
to the stationary distribution persists. PROFESSOR: But,
as you mentioned, this argument seems to be
giving that all lambda has to be 1, right? Is that your point? You're right.

I don't see what the
problem is right now. I'll think about it later. I don't want to waste my time
on trying to find what's wrong. But the conclusion is right. There will be a
unique one and so on. Now let me make a note here. So let me move on
to the final topic. It's called martingale. And this is, there
is another collection of stochastic processes. And what we're trying to
model here is a fair game. Stochastic processes
which are a fair game. And formally, what I mean
is a stochastic process is a martingale if that happens. Let me iterate it. 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 X_t. So we have this stochastic
process, and, at time t, you are at X_t. At time t plus 1, lots
of things can happen. It might go to this point, that
point, that point, or so on.

But the probability
distribution is designed so that the
expected value over all these are exactly equal
to the value at X_t. So it's kind of centered
at X_t, centered meaning in the probabilistic sense. The expectation
is equal to that. 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. 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 And I will later tell
you more about that. So example, a random
walk is a martingale. What else? Second one, now let's
say you're in a casino and you're playing roulette.

Balance of a roulette
player is not a martingale. Because it's designed so
that the expected value is less than 0. You're supposed to lose money. Of course, at one instance,
you might win money. But in expected value,
you're designed to go down. So it's not a martingale. It's not a fair game. The game is designed for
the casino not for you. Third one is some funny example. I just made it up to show that
there are many possible ways that a stochastic process
can be a martingale. So if Y_i are IID
random variables such that Y_i is equal to 2, with
probability 1/3, and 1/2 is probability 2/3, then let
X_0 equal 1 and X_k equal. Then that is a martingale. So at each step, you'll
either multiply by 2 or 1/2 by 2– just divide by 2.

And the probability distribution
is given as 1/3 and 2/3. Then X_k is a martingale. The reason is– so you can
compute the expected value. The expected value of the
X_(k+1), given X_k up to X_0, is equal to– what you have is
expected value of Y_(k+1) times Y_k up to Y_1. That part is X_k. But this is designed so that the
expected value is equal to 1. So it's a martingale. I mean it will fluctuate
a lot, your balance, double, double, double,
half, half, half, and so on.

But still, in expectation,
you will always maintain. I mean the expectation at
all time is equal to 1, if you look at it
from the beginning. You look at time 1, then
the expected value of X_1 and so on. Any questions on
definition or example? So the random walk is an
example which is both Markov chain and martingale. But these two concepts are
really two different concepts. Try not to be confused
between the two. They're just two
different things. There are Markov chains
which are not martingales. There are martingales which
are not Markov chains. And there are somethings
which are both, like a simple random walk. There are some stuff which
are not either of them. They really are just
two separate things. Let me conclude with
one interesting theorem about martingales. And it really enforces
your intuition, at least intuition of the definition,
that martingale is a fair game.

It's called optional
stopping theorem. And I will write it down
more formally later, but the message is this. 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. Even if you try to
lose money so hard, you won't be able to do that. Even if you try to win
money so hard, like try to invent something really,
really cool and ingenious, you should not be
able to win money. Your expected value
is just fixed.

That's the content
of the theorem. Of course, there are
technical conditions that have to be there. So if you're playing
a martingale game, then you're not
supposed to win or lose, at least in expectation. So before stating
the theorem, I have to define what a
stopping point means. So given a stochastic process,
a non-negative integer valued random variable tau
is called a stopping time, if, for all integer k greater
than or equal to 0, tau, lesser or equal to k,
depends only on X_1 to X_k. So that is something
very, very strange. I want to define something
called a stopping time. It will be a non-negative
integer valued random variable. So it will it be
0, 1, 2, or so on. That means it will
be some time index. 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. You have a strategy
that is defined as you play some k rounds, and
then you look at the outcome. You say, OK, now I think
it's in favor of me. I'm going to stop. You have a pre-defined
set of strategies. And if that strategy
only depends on the values of the stochastic
process up to right now, then it's a stopping time. If it's some strategy that
depends on future values, it's not a stopping time. Let me show you by example. Remember that coin toss game
which had random walk value, so either win $1 or lose $1. So in coin toss game,
let tau be the first time at which balance becomes $100,
then tau is a stopping time.

Or you stop at either
$100 or negative $50, that's still
a stopping time. Remember that we
discussed about it? We look at our balance. We stop at either at the time
when we win $100 or lose $50. That is a stopping time. But I think it's better to
tell you what is not a stopping time, an example. That will help, really. So let tau– in the same
game– the time of first peak. By peak, I mean the
time when you go down, so that would be your tau. So the first time when
you start to go down, you're going to stop. That's not a stopping time. Not a stopping time. 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.

If you're 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. So the event that
you stop at time t depends on t plus 1
as well, which doesn't fall into this definition. So that's what we're
trying to distinguish by defining a stopping time. In these cases it was
clear, at the time, you know if you
have to stop or not. 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 it's not a stopping
time under this definition. Any questions? Does it make sense? Yes? 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]? PROFESSOR: So. AUDIENCE: Let's say,
yeah, it was [INAUDIBLE]. PROFESSOR: So that
time after peak, the first time after peak? AUDIENCE: Yes. PROFESSOR: Yes, that
will be a stopping time. So three, tau is tau_0 plus 1,
where tau 0 is the first peak, then it is a stopping time.

It's a stopping time. So the optional stopping
theorem that I promised says the following. Suppose we have a martingale,
and tau is a stopping time. And further suppose
that there exists a constant T such that tau is
less than or equal to T always. So you have some strategy
which is a finite strategy. You can't go on forever. You have some bound on the time. And your stopping time
always ends before that time. In that case, the 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 no matter what strategy you
use, if you're a mortal being, then you cannot win. That's the content
of this theorem. So I wanted to prove
it, but I'll not, because I think I'm
running out of time.

But let me show you one, very
interesting corollary of this applied to that number one. So number one is
a stopping time. It's not clear that there is a
bounded time where you always stop before that time. But this theorem does
apply to that case. So I'll just forget about
that technical issue. So corollary, it
applies not immediately, but it does apply to the first
case, case 1 given above. And then what it says
is expectation of X_tau is equal to 0. 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. So this is 100 times
some probability plus 1 minus p times minus 50. There's some probability
that you stop at 100. With all the rest, you're
going to stop at minus 50. You know it's set. It's equal to 0. What it gives is– I hope it
gives me the right thing I'm thinking about.

p, 100, yes. It's 150p minus 50 equals 0. p is 1/3. And if you remember, that was
exactly the computation we got. So that's just a
neat application. But the content of this,
it's really interesting. So try to contemplate about it,
something very philosophically. 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. So that's it for today. And next week, Peter will
give wonderful lectures. See you next week..

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