AI Assistant
Transcript
00:01:400Paolo Guiotto: Okay, good morning. So, this morning…
00:06:240Paolo Guiotto: Before we introduce the last topic, I would like,
00:10:340Paolo Guiotto: To do one exercise, at least, on the application of the… Central limit theorem.
00:20:80Paolo Guiotto: So, I do the exercise, 9… 4… 7th.
00:29:120Paolo Guiotto: It says we have a sequence XN of Independent, identically distributed random variables.
00:38:470Paolo Guiotto: All of them are uniform in the interval 0, 1.
00:47:90Paolo Guiotto: And the exercise says, use a central limit theorem.
00:56:120Paolo Guiotto: to determine… The limit when… And goes to plus infinity of the probability.
09:35:770Paolo Guiotto: It is recording, but it is not sharing.
09:45:430Paolo Guiotto: So, let's see if we can continue.
09:50:30Paolo Guiotto: Okay, so we get minus 2 integrals 0 to 1 of log X dx.
09:57:20Paolo Guiotto: Again, this is a generalized integral. We can do the calculation the same way. We put the 1, so minus 2. We have X log of X between 0 and 1, minus integral 0, 1.
10:12:10Paolo Guiotto: The derivative moves on log, and this is 1 over X.
10:18:90Paolo Guiotto: So again, the evaluation here is 0, this is minus integral 0 to 1 of 1, which is equal to minus 1.
10:26:970Paolo Guiotto: So with the minus 2, it comes at plus 2.
10:30:250Paolo Guiotto: So this proves at once, first, that the variable is in L2, no?
10:37:690Paolo Guiotto: So this, proves this, but also… well, actually, we need to compute the… so, the YK is in L2.
10:47:580Paolo Guiotto: If it is in L2, it is in L1, because, four probability spaces.
10:53:310Paolo Guiotto: L2 is contented in L1, and therefore the mean value of YK also makes sense.
11:00:660Paolo Guiotto: And this is the mean value of log XK.
11:06:910Paolo Guiotto: which is just the quantity we computed here. It is the integral between 0, 1 of log X, DX,
11:15:50Paolo Guiotto: And this is equal to… after the calculation, it's equal to minus 1.
11:23:510Paolo Guiotto: Okay, so, we can say that. So, this, sequence, YK,
11:30:460Paolo Guiotto: made by the log of PXK is a sequence in L2,
11:36:550Paolo Guiotto: Such that the common mean value of the YK is minus 1, and the variance of the YK
11:47:160Paolo Guiotto: That can be also written as the expectation of YK squared.
11:52:890Paolo Guiotto: minus the square of the expectation of YK,
11:59:210Paolo Guiotto: Now, the expectation of YK squared has been computed here.
12:04:470Paolo Guiotto: And it comes equal plus 2.
12:08:280Paolo Guiotto: So, is equal to plus 2 minus the square of the mean value, so minus 1 squared, so 2 plus 1.
12:18:490Paolo Guiotto: So they have mean value equal to minus 1, and variance, common variance, equal to plus 1.
12:27:170Paolo Guiotto: According to the central limit theorem by…
12:32:420Paolo Guiotto: the central limit theorem, we can say that, let's give a look to the statement. Exactly, we have a 1 over sigma, which is the root of the variance.
12:44:810Paolo Guiotto: Ruth of Anna.
12:46:360Paolo Guiotto: sum, K from 1 to an XK minus M in distribution goes to a standard Gaussian.
12:53:400Paolo Guiotto: So we can say that, 1 over the sigma is 1, the sigma squared, this would be sigma squared, so sigma is 1. Root of n, sum for k going from 1 to n, we are applying to the YK.
13:10:740Paolo Guiotto: Minus the mean value, which is minus 1.
13:15:350Paolo Guiotto: This thing goes in distribution to a standard Gaussian
13:21:390Paolo Guiotto: So, in particular, this means that we have 1 over root of n.
13:27:270Paolo Guiotto: the sum for k going from 1 to N. The YK are the log of the XK. Minus minus 1 is plus 1.
13:37:600Paolo Guiotto: Now, if I can say that if I do the sum of plus 1 for cake that goes from 1 to N, I get plus n here.
13:47:150Paolo Guiotto: Which is exactly the quantity we have here, as you can see, no? Sum for k going from 1 to n, log XK plus n divided by a root of n.
13:58:100Paolo Guiotto: Okay, that's exactly, the quantity we have,
14:03:410Paolo Guiotto: over there. So, in particular, because of the convergence in distribution.
14:10:200Paolo Guiotto: we can say that the probability that… well, let's write the exact values, minus log B.
14:19:820Paolo Guiotto: So, minus log B.
14:22:910Paolo Guiotto: That should equal this thing, a sum for k going from 1 to NXK plus n divided root of n, which is the initial probability, minus log of n.
14:36:220Paolo Guiotto: This is equal to the probability that, minus, block B.
14:45:230Paolo Guiotto: This should equal 1 over root of n, sum for k going from 1 to N, EYK,
14:52:660Paolo Guiotto: Minus the common mean value.
14:55:700Paolo Guiotto: Which is minus 1. Well, let's say minus N.
15:01:740Paolo Guiotto: less or equal than minus log of A, Now, this is, So this is M.
15:11:190Paolo Guiotto: This probability… now, we can say that it goes to the integral from this to this, from minus log of B minus log of A.
15:25:860Paolo Guiotto: of the standard Gaussian, E minus X squared divided 2, dx divided root of 2 sigma.
15:32:920Paolo Guiotto: This because we know that the probability that the variable belongs to an interval goes to the same probability for the limit when
15:46:480Paolo Guiotto: this is always true when the limit variable is absolutely continuous. So this, is the conclusion.
15:56:890Paolo Guiotto: for this program.
16:00:760Paolo Guiotto: Okay, so now, we, we switch,
16:07:190Paolo Guiotto: quickly to the last part, which is somehow connected to the central limit theorem, in fact, so this is the Browning motion.
16:18:420Paolo Guiotto: Just limit, Brownian.
16:21:810Paolo Guiotto: Motion.
16:25:390Paolo Guiotto: Understand what is this.
16:27:750Paolo Guiotto: You see the definition?
16:31:490Paolo Guiotto: maybe we have an idea about how it is,
16:38:120Paolo Guiotto: How it works, the construction, and few properties of this.
16:44:560Paolo Guiotto: Now, the Browner motion is basically the starting point of modern probability, we may say. It was observed,
16:55:910Paolo Guiotto: long time before the modern probability was introduced. And it consists in the motion of small particles suspended in a fluid.
17:15:550Paolo Guiotto: Where the motion is characterized by continuous collisions between the particles you are observing and the particles
17:25:00Paolo Guiotto: of the fluid. Now, what happens is that if you observe this motion, you see that these parts are extremely irregular.
17:36:690Paolo Guiotto: paths. We may think in space, or in plane, if the fluid is two-dimensional, or in principle, even one dimension, if the fluid is just a one-dimensional fluid.
17:48:410Paolo Guiotto: But let's say that for geometrical intuition, we plot these trajectories in the plane. Now, empirically, it was observed the following, well, the empirical
18:07:130Paolo Guiotto: empirical.
18:11:00Paolo Guiotto: properties.
18:16:300Paolo Guiotto: of the… trajectories.
18:25:830Paolo Guiotto: Well, number one.
18:27:530Paolo Guiotto: there is a, say, a natural physical condition that says the trajectories are continuous. We cannot think that a particle
18:40:930Paolo Guiotto: disappear at certain locations and reappears at certain other locations. So, naturally, the trajectories, let's call, gamma, as function of T, function t is time, are continuous functions.
19:07:330Paolo Guiotto: Nonetheless, even if they are continuous, as you can imagine from this figure, they are extremely irregular. So, when you observe, you see just like if it is a crazy movement, changing direction, basically at
19:26:100Paolo Guiotto: at each moment.
19:27:910Paolo Guiotto: So this means that the trajectory is continuous, but it changes direction continuously, so it means that you are plenty of angles in this trajectory.
19:39:640Paolo Guiotto: spikes, things like this. So, if we think this as function of T, we think… we may think that they are continuous, so there are no jumps.
19:52:180Paolo Guiotto: But, at the same time, we may think that this function won't have
19:57:320Paolo Guiotto: a tangent, because it's like, if you think to the simple function, the absolute value, where you have just one angle, there is no tangent to that angle. Now, imagine that these angles are
20:10:660Paolo Guiotto: basically, at each moment, you have an angle, so you may imagine that this function won't have a tangent at every time. So this basically says that they are continuous function.
20:23:350Paolo Guiotto: But… not… differentiable.
20:33:20Paolo Guiotto: So this makes this motion very…
20:36:90Paolo Guiotto: particularly unusual in physics, because in physics, we are used to regular motions with the velocity. Velocity is what? It's the derivative of the position, so this is not going to be the kind of motion we have here.
20:50:210Paolo Guiotto: A second, important,
20:53:200Paolo Guiotto: observation is that if you measure a variation of position between two different times, so if you take an increment, gamma T minus gamma S,
21:06:610Paolo Guiotto: Well, observations suggest that this increment, if we would plot for several gammas, so we observe one gamma.
21:16:790Paolo Guiotto: we have a plot, we have a number there, no? You fix 2 times time t, which is, let's say, after time S, so S is now, t is some time in the future. You do several experiments, repeating the observation along different trajectories, okay?
21:35:780Paolo Guiotto: Now, what happens if you repeat for the same time frame this assessment of this quantity? You will see random values. You will see different values.
21:50:10Paolo Guiotto: So you may think that if you look at how these values are distributed, if you plot the numerical value, no, for different gamma, you would see that this value follows something like a Gaussian distribution. So,
22:07:890Paolo Guiotto: have… Shows, let's say.
22:15:950Paolo Guiotto: a Gaussian.
22:20:560Paolo Guiotto: Distribution.
22:24:550Paolo Guiotto: With the average equals zero, with… Zero. Mean.
22:34:810Paolo Guiotto: And the variance… Which is proportional.
22:40:720Paolo Guiotto: to what… well, the idea is that the longer is the time frame, the longer… the bigger will be the variance. So, you may expect that it is proportional to something that depends on
22:54:840Paolo Guiotto: T minus S. But the surprise, and this is coherent with 0.1, is that the variance is proportional Proportional.
23:05:690Paolo Guiotto: To the square root of the time increment.
23:10:190Paolo Guiotto: So, very, very roughly, we may say, like, if the increment, gamma T minus gamma S, isaac.
23:21:190Paolo Guiotto: well, the variance, sorry, the variance is the square, so it's proportional to T minus S. So, we are saying that the square of this is approximately T minus S, so it is like if we say that the increment, gamma T minus gamma S
23:40:800Paolo Guiotto: is proportional to the root of T minus S.
23:44:880Paolo Guiotto: So if you want, in particular, if we assume that, for example, the initial point where we observe the trajectory is conventionally the origin, we may think that this is saying that gamma of t is like the root of t.
24:01:330Paolo Guiotto: And you understand why this shouldn't be differentiable, because it goes like the root, and the root is not differentiable at zero, okay?
24:11:50Paolo Guiotto: Now, the surprising fact is not the Gaussian distribution.
24:15:270Paolo Guiotto: Because we may think, and in fact, now it's a long story, we don't have time here to do that, but a naive way to build a Brownian motion is the so-called random walk model. So, let's do very, very shortly in one dimension. Imagine that you start at point 0, and you are the
24:39:580Paolo Guiotto: particle.
24:40:670Paolo Guiotto: And you have just two options. Move, one unit left or one unit right, no? So you may think that you… you toss a coin, and you decide if you… if to go right or left, and to move off, for example, one unit. So this means that the position after, N,
25:04:790Paolo Guiotto: tosses can be represented as the value that you obtain by summing these plus 1, minus 1, so something like this.
25:16:690Paolo Guiotto: Where these variables, XJ, are variables that take plus 1 with probability 1 half and minus 1 with probability 1 half. So, they are basically Bernoulli, such that probability that XJ is plus 1 is 1 half.
25:36:680Paolo Guiotto: And probability that XJ is… Minus 1 is equal to 1 half.
25:43:580Paolo Guiotto: Moreover, we may assume that the decisions are totally independent, so the Kate step is independent of what are the previous steps. So this formally says that these are independent
26:04:80Paolo Guiotto: XJ Independent.
26:07:290Paolo Guiotto: random variables. So, if we call this PN, this would be the position after n steps. So, it will be distributed at most between minus N and N, because if you move,
26:22:420Paolo Guiotto: n times to the right, you go to plus N. Well, actually, here, we have n minus 1, it's a detail, because we have…
26:31:800Paolo Guiotto: only.
26:32:720Paolo Guiotto: We start from zero. So,
26:36:310Paolo Guiotto: But in any case, we can get one of these positions.
26:42:80Paolo Guiotto: Between minus N plus 1, minus n minus 1, and then minus 1.
26:47:370Paolo Guiotto: Okay, now, in average, if you compute the average here, the expected value, the expected position after n,
26:57:760Paolo Guiotto: steps is zero, because all these variables have mean zero, okay?
27:03:810Paolo Guiotto: Now…
27:07:30Paolo Guiotto: the variance of this variable here, if you compute the variance of this, you have to do the variance of PN, of the position after n steps. It is equal to the expectation of PN squared.
27:22:430Paolo Guiotto: minus the expected value of PN.
27:26:340Paolo Guiotto: squared.
27:27:660Paolo Guiotto: This is zero.
27:29:320Paolo Guiotto: And when you do this expectation of Pn squared, you have to do the expectation of the square of the sum
27:40:470Paolo Guiotto: K going from 1 to n of the XK.
27:45:210Paolo Guiotto: So when you do the square, you have the sum of the squares, so sum of… Squares.
27:53:210Paolo Guiotto: And then there are the mixed products, so XIXJ, where I is different from J.
28:00:840Paolo Guiotto: But when you do the expectation of these second terms, you get zero, because they are independent, so the expectation of Xi times XJ will be the product of the expectation Xi times expectation of XJ.
28:18:910Paolo Guiotto: And these are both mean zero variables, and so you get zero.
28:24:50Paolo Guiotto: But at the end, you get only the sum of these. And the sum of these, they have the same value, okay? So it doesn't matter, you can… we can compute, of course, what is the variance of this variable, but let's say… let's call it the common variance sigma squared, and then you get times n, because it's the same number summed.
28:46:110Paolo Guiotto: and times. So the variance of this is proportional to
28:55:330Paolo Guiotto: is proportional to N. So when n gets big, this becomes huge, no? So this means that the variance represents the mean quadratic displacement from the average
29:11:210Paolo Guiotto: this is going to be huge when n is big. Now, imagine that instead of thinking in terms of steps, we think in terms of time. So we want to say that this is the position at time t, and we divide time into n steps, or n minus… n steps, this is the position minus Y. However, each of these steps would represent
29:34:320Paolo Guiotto: A little, displacement in time.
29:37:850Paolo Guiotto: of, say, a time interval of length t over n, so very small time interval. So we may expect that to rescale this.
29:51:820Paolo Guiotto: We have that rescaling dividing by N, this… it means to carry in the variance N, it means to carry in the variance a root of n. So you have that the variance of the rescaled quantity, 1 over root of n, the position after n, is equal to sigma square.
30:11:180Paolo Guiotto: Now, this quantity is exactly what you have in the central limit theory, 1 over root of n, the sum for k going from 1 to n of DXK minus m. DM is 0 here. So this quantity is the quantity that goes in distribution to the standard Gaussian.
30:28:810Paolo Guiotto: So, it's normal to think that when n is bigger, that displacement, rescale displacement, will have a Gaussian natural jet, just because of the, of the central limit theorem.
30:44:410Paolo Guiotto: So, when we observe this property, we shouldn't be surprised if we think that this motion is like if at each moment you have a random, identically distributed and independent of the previous collisions, and a random change of direction.
31:04:540Paolo Guiotto: With the equal probability, so… it makes sense that it becomes quotient, okay?
31:10:690Paolo Guiotto: And the third remarkable observation is that
31:16:150Paolo Guiotto: When you consider different increments of the position relative to different, time intervals, so it says, suppose that you have, this is time, so we have a time, say…
31:34:380Paolo Guiotto: T1 times T2 times T3 times T4.
31:40:380Paolo Guiotto: It can be extended. And you look at the increments between time T1, T2, this is gamma T2 minus gamma T1, so the, the displacement
31:53:240Paolo Guiotto: from Gamma T1, And this is the second displacement, gamma T4.
32:01:230Paolo Guiotto: minus gamma T3. Now, if you repeat these observations over a large number of trajectories gamma, Ehhhh
32:12:790Paolo Guiotto: And you measure correlations between these quantities in a statistical way, you will discover that this correlation is very low, if not zero.
32:23:360Paolo Guiotto: So this is associated to the idea that correlation is zero when, for example, variables are independent. It's not an if and all if, but if the variables are Gaussians, it's actually an if and all if. So these increments, the third observation is that R
32:42:770Paolo Guiotto: Independent.
32:44:470Paolo Guiotto: Statistically independent.
32:50:640Paolo Guiotto: So, basically, these three main properties are the key properties of a Brownian motion.
32:57:200Paolo Guiotto: So, Brownian motion, now the point is, what kind of mathematical object should we use to describe something which is coherent with these three properties? So, we want continuous trajectories, but non-differentiable trajectories.
33:13:600Paolo Guiotto: Gaussian increments with that particular feature, so the increment… the Gaussian with the variance proportional to the increment of time. And third, we want also independent increments.
33:29:20Paolo Guiotto: Now, the first point is to understand what kind of mathematical objects should be used to model this thing, okay? Now, we are using, even if I'm not referring to any probabilistic
33:42:190Paolo Guiotto: set up, in particular to any probability space, we are using this language, in particular for property 2, saying that the increment is a Gaussian distribution. What do we mean, really?
33:55:540Paolo Guiotto: We… what we can observe, I repeat, is that increment over a large number of gamma.
34:03:280Paolo Guiotto: So we observe 1 million of trajectory sales, we measure the increment, we see different increments, so we cannot say that it is all always the same. Sometimes it will be small, sometimes it will be big. We put this on a histogram, and we observe the distribution of the values.
34:19:230Paolo Guiotto: And we see that most of the values are concentrated around the value 0, like the typical shape of a Gaussian distribution. So this yields the idea that we do not… we cannot say the increment is always this one.
34:35:130Paolo Guiotto: It is distributed, so the natural object we should think for that instrument, it's a random variable with the Gaussian distribution.
34:44:370Paolo Guiotto: Okay? And the same here. When we take two different increments, and we repeat the observation of a long num… a big number of trajectories, these two exhibit a very low correlation. If they are Gaussian random, by all, we may think they are independent.
35:02:330Paolo Guiotto: So, it seems natural to… to…
35:06:880Paolo Guiotto: To fit this with a probabilistic structure.
35:11:210Paolo Guiotto: And now, let's see what should be this probabilistic structure. So, we may think…
35:22:490Paolo Guiotto: to the following.
35:29:460Paolo Guiotto: probabilistic… structure.
35:40:100Paolo Guiotto: So, we have a probability space, so let's for the moment say that omega F B.
35:47:900Paolo Guiotto: is a generic,
35:53:940Paolo Guiotto: Probability.
35:57:100Paolo Guiotto: space, huh?
35:58:960Paolo Guiotto: Now, since,
36:02:700Paolo Guiotto: We want to think this as random variables. It means that they must be also function not only of time, but on this parameter omega. And in fact, we may think that omega is responsible to tell you which trajectory are you considering.
36:17:900Paolo Guiotto: So, let's imagine that we have a function that we will denote with the
36:22:850Paolo Guiotto: letter capital W, sometimes you will be… you will see the letter capital B, which is a function defined
36:31:990Paolo Guiotto: Well, let's emphasize first the role of time. Well, let's start from time t equals 0 to plus infinity.
36:40:440Paolo Guiotto: And a function also of the sample variable in the probability space omega, with the… now, it depends what we are modeling. If we are modeling the one-dimensional Brownian motion, the two-dimensional Brownian motion, three-dimensional, this deck…
36:56:610Paolo Guiotto: Dimension means, do we want to model a motion in one unique dimension, so on an infinite line, or on a plane, or in the space?
37:07:230Paolo Guiotto: Technically speaking, it is the same. So, for simplicity, I will stay on the real Brownian motion case, okay? Then, if you want to build a three-dimensional Brownian motion, so something that moves in space, you just have to take three copies of the Brownian motion, and coordinate one first copy, coordinate two second copy, coordinate with third copy.
37:31:940Paolo Guiotto: Okay.
37:32:820Paolo Guiotto: We could start directly from this, but it's preferable to start with the Scarlet case.
37:38:360Paolo Guiotto: for simplicity.
37:41:10Paolo Guiotto: So now, this W is a function of two letters, let's say two variables, T and omega, with entirely different meaning.
37:50:510Paolo Guiotto: Because T is the classical, real variable, so it is on the offline 0 plus infinity. The omega belongs to a not better specified probability space, capital omega.
38:03:00Paolo Guiotto: And, for example, this means that also we will have different structures. So, for example, here P is time, and when we say continuity, it makes sense, not because the standard real variable saying that the function is continuous in time is something that we more or less know.
38:21:650Paolo Guiotto: Respect to omega, the story is different. Continuity might not have any meaning, but this is not what we needed here, because omega plays the role just as randomness.
38:32:900Paolo Guiotto: So, we introduce some notation and names. So, when we look at this function as a function of T with the omega frizzed, so the function, say, let's say.
38:47:640Paolo Guiotto: the function that maps T into W, T.
38:53:800Paolo Guiotto: Omega.
38:55:30Paolo Guiotto: with omega fixed them, so this is now a function that has domain on 0 plus infinity, and it is real valued, so it's just the kind of function we know since first year, no? This is also denoted by this symbol.
39:11:500Paolo Guiotto: we will write W sub T, and we will almost never specify the omega. This is normal, because when we deal with random variables, we never express D variable. This is typical improbability, no?
39:27:130Paolo Guiotto: So we denote this by WT.
39:30:400Paolo Guiotto: And we call this, path.
39:36:390Paolo Guiotto: for… Beach.
39:41:00Paolo Guiotto: Omega in the sample space, fixed.
39:45:820Paolo Guiotto: Path or trajectory?
39:52:260Paolo Guiotto: So, saying that the trajectory is continuous means that this function, as a function of E is continuous.
40:01:140Paolo Guiotto: Now, of course, since this is very close on omega, you have to say, for which omega will be continuous?
40:07:510Paolo Guiotto: For example, we want that all the possible trajectories are continuous. We will have to say that this function is continuous for every omega.
40:16:930Paolo Guiotto: Now, we know that this omega is a probabilistic object, so statements like all the trajectories won't make particular sense, so we will always have a probabilistic structure, and…
40:32:370Paolo Guiotto: it makes sense to say that trajectory is continuous for, let's say, with probability 1, that means for a set of omegas, which probability is equal to 1? This makes more sense. I will be between a second on this. So this is the trajectory. While…
40:51:170Paolo Guiotto: Mmm… While… If, we… Pizza.
41:02:950Paolo Guiotto: T… This one, omega, goes to W sub T of omega. This is now a random variable.
41:12:880Paolo Guiotto: It's a function that, it's defined on the sample space and takes numerical values.
41:20:470Paolo Guiotto: Okay, so for this function, Properties like continuity won't simply make sense in general.
41:28:740Paolo Guiotto: But, since we have a probabilistic structure, we can say this is a random variable, so a measure of function. There is an expected value, there is a mean, there is a variance, things like that. We can talk about the distribution and things like this.
41:44:540Paolo Guiotto: So, this, WT is called the… It's called the position.
41:53:250Paolo Guiotto: At. Bye.
41:56:310Paolo Guiotto: G.
41:58:30Paolo Guiotto: So, we always use the same symbol, W sub T, but it depends on the context. We may think as this as a function of T, so this is a trajectory, omega pixel, or vice versa. We freeze T, and we look at this as a function of omega, that's a random variable.
42:14:720Paolo Guiotto: Now, as you can see, this is, so, a function of two variables. Normally, this type of functions, positionality, we require, of course, we require
42:31:20Paolo Guiotto: that WT be at least a measurable function, so… Random variable.
42:39:370Paolo Guiotto: On that space omit.
42:41:550Paolo Guiotto: Now, this type of functions, W, that depends on a parameter, which is a real parameter.
42:48:820Paolo Guiotto: And on a second variable, which is a sample variable, are called stochastic processes, okay? So, normally they are denoted in this way.
43:02:460Paolo Guiotto: This, or if you want, you can look this as a family of random variables, parameterized by this time t, in this case.
43:11:160Paolo Guiotto: is… What we call a stochastic
43:19:540Paolo Guiotto: process.
43:23:390Paolo Guiotto: So, stochastic is a random phenomena which is evolving in time. That's what… when it… literally, stochastic is a synonym.
43:35:100Paolo Guiotto: of random, okay? They mean the same thing. But in mathematics, normally, and no literature that has to deal, not only in mathematics, when we read stochastic, we normally think to a function, a random variable, depending on some
43:50:270Paolo Guiotto: Numerical parameter that could be even more complicated, but normally this is.
43:55:960Paolo Guiotto: Okay, so we can now… we are now ready to define what the Browning motion should be, so let's introduce this definition.
44:06:370Paolo Guiotto: So, stochastic.
44:10:590Paolo Guiotto: process.
44:14:760Paolo Guiotto: WT is greater or equal than zero.
44:18:970Paolo Guiotto: He is… older.
44:22:780Paolo Guiotto: Brownian motion.
44:32:350Paolo Guiotto: Now, the origin of this letter W is because it is also called Wiener, process.
44:44:10Paolo Guiotto: It's just as synonymous.
44:45:810Paolo Guiotto: in honor of this mathematician, Norbert Wiener, who was the first… who really treated this from a mathematical
45:09:710Paolo Guiotto: We do not need to specify this, because this will be actually a consequence of what we… of the other… of these three properties combined. So the non-differentiability, it's…
45:22:710Paolo Guiotto: an outcome, it's not needed in the assumption. So, we first say that we want that trajectories are continuous. Trajectories are
45:33:190Paolo Guiotto: the function W as function of T. So let's be a bit, say, formally, 100% precise. So, if W, as function of T, omega freezed, okay, so the function W,
45:52:970Paolo Guiotto: Dr. Omega.
45:55:910Paolo Guiotto: If this function is a continuous function.
45:59:310Paolo Guiotto: For all times, from 0 to plus infinity.
46:03:450Paolo Guiotto: And this for which omega?
46:07:530Paolo Guiotto: Well, omega belongs to the probability space, so we could say for every omega, that's too much, and we say for almost
46:16:110Paolo Guiotto: every omega in the space capital omega, okay?
46:22:180Paolo Guiotto: Number two.
46:25:260Paolo Guiotto: Well, we want to say that the… increments… Where is it?
46:32:710Paolo Guiotto: The increments are Gaussian with mean 0 and variance proportional to T minus S.
46:39:820Paolo Guiotto: Now, a Gaussian distribution is entirely characterized by these two parameters, mean and variance.
46:45:810Paolo Guiotto: Okay, so the standard agreement is to take the constant of proportionality equal to 1, and say that
46:53:780Paolo Guiotto: the increment, so WT minus WS, now I'm thinking this as the difference between these two random variables. So here, T and S are freezed, so, well, this is 0, this is S, this is T.
47:09:630Paolo Guiotto: So, S will be less than T, so at the end we will have, for every
47:13:850Paolo Guiotto: S positive, less, well, greater or equal, less than T.
47:20:270Paolo Guiotto: This T is Gaussian, so S distribution, which is normal, with mean Z, and variance just T minus S.
47:30:880Paolo Guiotto: Okay?
47:32:780Paolo Guiotto: Number 3… The third condition was that the increments are independent.
47:42:330Paolo Guiotto: Now, we extend this because we have seen two increments, but it can be proved for any number of increments. The important point is that you don't take here… you have to observe that you don't take any two increments. Here, there is a particular feature.
47:59:120Paolo Guiotto: That the time it allows must be consecutive.
48:03:410Paolo Guiotto: The sense that this second time Itaba comes after the first one. So, you know that E3 here, for example.
48:10:160Paolo Guiotto: So, this is saying in this way. If we take n times.
48:15:60Paolo Guiotto: The consecutive increments are independent, and this is an extension of this one. So… The property is, if…
48:24:530Paolo Guiotto: We have, so this is the timeline, 0, T1, T2.
48:30:420Paolo Guiotto: etc.
48:32:620Paolo Guiotto: TN, huh?
48:34:370Paolo Guiotto: So, if 0, less or equal than T1, less than T2, less than T3, etc, less than Yeah, no?
48:49:450Paolo Guiotto: Then, the increments… D.
48:56:10Paolo Guiotto: increments.
49:02:340Paolo Guiotto: So the consecutive increments are WT2 minus… well, we use the notation with the sub, so WT2 minus WT1.
49:16:480Paolo Guiotto: WT3 minus WT2, and so on, until the last one, WTN minus WTN minus 1.
49:28:500Paolo Guiotto: Well, all this… R… independent.
49:35:510Paolo Guiotto: Random variable, of course.
49:39:240Paolo Guiotto: Well, we are the final, which is just conventional. We assume that all trajectories start at time zero from the origin, okay? So W is identical equal to 0. It's a trivial variable, constantly equal to 0.
49:58:370Paolo Guiotto: Now, this is the Brownian motion, or vena process, in one dimension.
50:04:760Paolo Guiotto: Basically, we may say that with this definition, you could do everything that concerns the Brownian motion. So, the study of properties of the Brownian motion.
50:15:600Paolo Guiotto: for example, the properties of the trajectories, and do all the constructions that are based on the Brownian motion. Like, there is… well, we cannot do differential calculus, because at the end, there are no derivatives for this.
50:32:60Paolo Guiotto: But we can do integral calculus. You will see these things if you do the course on stochastic differential equations. So there are particular differential equations based on the Brownian motion that are used to model certain random phenomena, exactly as we use differential equation as a model for deterministic
50:54:980Paolo Guiotto: phenomena, like trajectory of motion of a particle that moves according to the laws of classical physics. Newton law is a differential equation.
51:05:890Paolo Guiotto: You use that paradigm.
51:07:630Paolo Guiotto: To model that kind of phenomena, okay?
51:10:870Paolo Guiotto: Stochastic equations are used to model phenomena where, the variation… if you think about it, the differential equation says that the variation Y prime equal F of PY, no?
51:27:870Paolo Guiotto: Y prime is the Y over DT. So you are saying that the variation of Y, of position, is proportional to the variation of time.
51:37:460Paolo Guiotto: Which is not the case for this kind of phenomena, because we expect that the variation of position is proportional to the root
51:44:400Paolo Guiotto: Of the variation of time.
51:47:90Paolo Guiotto: And this makes your regular phenomenon, okay? So this is, the idea, to… well, it's not, something we are going to talk about here.
51:58:620Paolo Guiotto: So, however, with this definition, without knowing how this process is built, assuming that there is a process that verifies these conditions, you can do everything, okay?
52:12:560Paolo Guiotto: Well, what we do here is to do one step back and say, okay, but is there a process that verifies this condition? How this process can be built? This is also interesting because
52:28:800Paolo Guiotto: We can have an idea of a numerical way to build this process. That could be useful when you do modeling.
52:38:60Paolo Guiotto: Now, let's talk about the, construction construction.
52:49:970Paolo Guiotto: off.
52:51:650Paolo Guiotto: the Brownian motion.
52:56:460Paolo Guiotto: Now, there are several different ways to construct this process to show that there is a structure, a probability space, a function W, explicitly written that verifies this condition.
53:12:40Paolo Guiotto: Basically, there are two different main important approaches.
53:17:50Paolo Guiotto: One is the approach that starts from the space omega.
53:22:250Paolo Guiotto: And making trivial the function W.
53:26:750Paolo Guiotto: So, the function W is just,
53:32:610Paolo Guiotto: Well, let's say… I don't, I don't expand this one, but let's say there is a path… space.
53:46:200Paolo Guiotto: approach.
53:50:930Paolo Guiotto: that says, you know what we do? We take as a sample space omega.
53:57:630Paolo Guiotto: just the set of all possible continuous trajectories. So, you take just continuous functions on 0 plus infinity.
54:07:320Paolo Guiotto: real value.
54:09:430Paolo Guiotto: With this choice, the definition of W will be trivial, because the function W of T omega… now, omega, an element of this set, is itself a continuous function. So, it's… well, let's say that C0, where this means that omega of 0 is equal to 0. So, trajectories that
54:32:590Paolo Guiotto: start from the origin. Now, I do the figure as if we have R2 here, but because in R1, it doesn't count very nice. So, let's imagine that we have a trajectory that at time 0 is in the origin, and then it starts moving, a continuous trajectory.
54:51:120Paolo Guiotto: in the Cartesian plane.
54:53:240Paolo Guiotto: Now, continuous does not mean necessarily differentiable. Of course, we have nice trajectories like this one.
55:00:20Paolo Guiotto: Which is a perfectly smooth trajectory, we can do derivatives. But most of the functions will be irregular, no?
55:08:900Paolo Guiotto: Now, how do you define the W? Easy. This is, you take omega, which is the trajectory, and you evaluate at time t, the position at time t of the element omega. Now, the problem is to build on omega a probabilistic structure, so a probability measure.
55:28:10Paolo Guiotto: The main problem is, How?
55:33:990Paolo Guiotto: Due.
55:34:930Paolo Guiotto: Wheat.
55:36:50Paolo Guiotto: Define.
55:37:830Paolo Guiotto: a probability.
55:40:960Paolo Guiotto: a probability measure.
55:44:220Paolo Guiotto: P?
55:45:420Paolo Guiotto: on this omega.
55:47:910Paolo Guiotto: So, on the space of continuous functions, it's…
55:53:200Paolo Guiotto: really complicated, because we are not saying, take the interval 0, 1, and take the LeBag measure.
55:59:170Paolo Guiotto: We are saying, take this set, which is a set made of functions, so it's a huge set, we don't have any intuition.
56:07:760Paolo Guiotto: Realistically, on this, and we have to… the problem is, how can we build a measure, a probability P, on this omega in such a way that the least 1 to 4 is verified? Well, number 4 is already verified, because if you take t equals 0,
56:27:680Paolo Guiotto: it is a consequence of this condition. But the other three are a bit more complicated. In such a way.
56:36:180Paolo Guiotto: Such.
56:38:200Paolo Guiotto: way.
56:40:720Paolo Guiotto: That, W.
56:45:820Paolo Guiotto: of T.
56:47:560Paolo Guiotto: verifies… properties.
56:57:370Paolo Guiotto: 1, 2, 4.
57:00:100Paolo Guiotto: off… B.
57:02:550Paolo Guiotto: definition of browning marsh.
57:07:530Paolo Guiotto: There is an answer to that, and this was done… was proven by Wiener, and the probability that comes out is the so-called Wiener
57:21:710Paolo Guiotto: measure.
57:24:250Paolo Guiotto: Which is a complicated way, so I don't know what we learned from this course, no? But one thing that you must have somehow clear is that it is not easy to define a measure.
57:38:30Paolo Guiotto: Because you have to deal with the sigma algebra of sets, which is a complete, because we're made of huge number of sets, and that they are hard to be characterized. You don't even know what are the elements of the sigma algebra, and you pretend to define a measure. And that's already a problem in dimension 1 with the Levesque measure, it's not easy.
57:56:520Paolo Guiotto: So it's… even the back measure that seems an ordinary tool, it's not at all an ordinary tool. Imagine now you have to do a measure on a space which is a space of functions. These are infinite dimensional things, so you don't see anything. You have no intuition.
58:14:480Paolo Guiotto: So it's complicated. So this way is definitely not the easiest way to do.
58:20:810Paolo Guiotto: The second way…
58:24:220Paolo Guiotto: As you will see, the structure of omega is irrelevant, but we try to define the function W, okay, in a suitable way. So, this is… the first one is path space approach, and this could be, say, the…
58:42:780Paolo Guiotto: as a, director… definition.
58:49:420Paolo Guiotto: off.
58:50:790Paolo Guiotto: W.
58:52:50Paolo Guiotto: Also here, the original idea was,
58:56:150Paolo Guiotto: was, given by Wiener. So, the, Wiener idea…
59:07:930Paolo Guiotto: was the following. We take our function W as a function of T. It's… the starting point is extremely informal.
59:18:110Paolo Guiotto: Now, the idea is we want to build something which is, at the end, continuous, but not differentiable.
59:26:730Paolo Guiotto: So it says, let's compute informally the derivative with respect to t of this.
59:32:750Paolo Guiotto: So let's do this, huh?
59:34:920Paolo Guiotto: and say that this is a very irregular thing. So, let's represent this as…
59:45:520Paolo Guiotto: So, what we expect is that this derivative, if you want to represent it, let's see, imagine that you want to represent a function like that, no? So, let's take a piecewise linear function.
00:02:470Paolo Guiotto: moving up and down. Now, if you look at the derivative, so this is WT, what would you see in the derivative? Derivative with respect to T,
00:11:850Paolo Guiotto: This is still differentiable, except for a few points where you have the angle. But you would see this. So, for example, this is, until to this time, you would see a function which is constant equal to this. Then you have a negative value, something like here, then a positive value. You see?
00:30:300Paolo Guiotto: So these are… The values where the derivative change.
00:44:560Paolo Guiotto: Imagine that that's… that's piecewise linear, so…
00:48:490Paolo Guiotto: It's not curved, so we would see something like this. This is positive, negative, then we see a small positive value, maybe a big negative value, then we have a positive value, then we get… go down negative, yet positive, I don't know, like that, negative here, and so on.
01:09:310Paolo Guiotto: Okay?
01:10:220Paolo Guiotto: So, the green, function is what we would see for the derivative of the black function.
01:18:290Paolo Guiotto: So, the idea is, the green… this function is not definitely an element of the space of a continuous function, because
01:27:670Paolo Guiotto: There are these junctions, and you may expect that if these, these, angles are more and more, these irregularities will be more and more.
01:39:680Paolo Guiotto: So, this green function is an element of what kind of space?
01:44:790Paolo Guiotto: Well, it won't be an element of the space of continuous functions, But the natural,
01:51:400Paolo Guiotto: The dimensional space for these objects is an L1, L2, etc. space, because they are functions of this type, not necessarily continuous, but hopefully
02:04:920Paolo Guiotto: Not extremely bad, huh?
02:07:430Paolo Guiotto: So the idea of Wiener is we think this as an element of the space L2 in time.
02:17:760Paolo Guiotto: Now, to fix ideas, instead of taking all possible times, we decided to take, to fix an horizon, so a final time.
02:29:550Paolo Guiotto: capital T, for example Okay?
02:32:700Paolo Guiotto: then we will remove this capital T sending to infinity. There is a reason to do that, because on L2,
02:39:800Paolo Guiotto: finite interval, I can think that elements can be represented in some orthonormal basis, like the Fourier, the trigonometric basis. So the idea was, write this as a Fourier series.
03:03:40Paolo Guiotto: So, to represent this as, well, in abstract form, whatever is the orthonormal basis, we say sum for n going from 1 to infinity, DTWT is color EN,
03:17:840Paolo Guiotto: dot EN.
03:20:630Paolo Guiotto: Well, this is, end off to normal.
03:28:960Paolo Guiotto: basis.
03:33:30Paolo Guiotto: off.
03:34:420Paolo Guiotto: this space.
03:36:560Paolo Guiotto: Now, what Wiener did was to try with the trigonometric, classical trigonometric basis, so this sine and cosine.
03:47:500Paolo Guiotto: And, well, that's quite hard.
03:50:850Paolo Guiotto: But he did this, I don't remind exactly which year, about, let's say, if I'm not wrong, in 90…
04:03:50Paolo Guiotto: 12, something like this, it did.
04:05:880Paolo Guiotto: Now, it took a lot of years to have an extreme simplification of this approach just by changing the basis. So, by using a basis that makes all the calculation much easier. And this is the Shaziel scheme.
04:25:470Paolo Guiotto: Lesquite.
04:32:820Paolo Guiotto: construction. This is 1962, if I'm not wrong.
04:38:750Paolo Guiotto: Which is based on the same idea.
04:41:490Paolo Guiotto: But, using a different basis, which is important in application, so, same… Same.
04:53:30Paolo Guiotto: idea.
04:56:280Paolo Guiotto: But… using… the…
05:04:790Paolo Guiotto: basis.
05:07:850Paolo Guiotto: Now, we have never talked about this basis, and this is a concrete, important basis, so maybe it's now the moment to tell something about this. So, let's start seeing what is the R basis. So, to write the R basis conveniently.
05:26:800Paolo Guiotto: we will just take T equals 1, okay? But we can do for every time T, okay? So, L2… you take L201,
05:39:90Paolo Guiotto: it is easy to describe this basis, then we need to do little calculations to write down the functions. So, we start saying the first element of this basis is the function E0,
05:56:510Paolo Guiotto: X con… well, we use letter T here, because the variable for these functions will be T. Is the function constantly equal to 1? So the first element is quite trivial, is the function constantly equal to 1 on the interval 0, 1.
06:13:650Paolo Guiotto: Then what we do?
06:15:300Paolo Guiotto: Well, we take the interval 0, 1, and we start dividing into two parts, so 1 half 1.
06:23:330Paolo Guiotto: And the second function is the function that is equal to something. This is equal to 1.
06:31:750Paolo Guiotto: Well, you can… you can check that the L2 norm of E0,
06:38:120Paolo Guiotto: since it is the integral from 0 to 1 of modulus E0 squared E0 is 1, squared is 1, it comes equal to 1, so it's also a unitary factor. Now, you take as function E2, now.
06:57:750Paolo Guiotto: the function which is plus something, I won't be precise for a moment on this value, let's say alpha here, and minus something, minus alpha.
07:11:100Paolo Guiotto: On the interval 1 half 1.
07:13:900Paolo Guiotto: Now, this is the function E1, and you choose alpha in such a way that this is a unit vector. You can check that. So, let's take alpha.
07:26:140Paolo Guiotto: Such that, huh?
07:28:240Paolo Guiotto: The norm of E1 is equal… the L2 norm is equal to 1. That's easy. We start computing E1, L2 norm square. This is the integral 0 to 1 of modulus E1 squared.
07:43:110Paolo Guiotto: Well, this function is equal to alpha on 0, 1 half, so 0, 1 half, you have alpha squared, plus from 1 half to 1, it is minus alpha. With the absolute value, you get still alpha squared.
07:58:390Paolo Guiotto: So, at the end, you get alpha squared times 1, and so if you wanted this be 1, you just have to take alpha equal 1. So that's easy, no?
08:09:100Paolo Guiotto: So he, it turns out that this is just one for this case. And you can…
08:18:810Paolo Guiotto: verify that if you do this color product between E0 and E1, you get 0.
08:27:270Paolo Guiotto: Because this is the integral from 0 to 1 of E0 times E1.
08:35:270Paolo Guiotto: Here we are in the real Scala product.
08:38:350Paolo Guiotto: So we have from 0 to 1 half, E0 is 1, E1 is 1.
08:43:830Paolo Guiotto: from 1 half to 1, E0 is still 1, and E1 is minus 1. So it clearly, you get 0 from this. So the two functions are orthogonal.
08:57:29Paolo Guiotto: Okay, now we divide, as you understand, we will divide, now, the individual 01 in 4 equal parts, then in 8 equal parts, then in 16. So, each time, we double the number of elements of the subdivision.
09:14:510Paolo Guiotto: Here, when we divide in four equal parts, we have two functions.
09:20:510Paolo Guiotto: So… This is the first one.
09:24:750Paolo Guiotto: The first one is still, let's put plus alpha on 0, 1 fourth, and minus alpha on 1 fourth to one alpha.
09:36:270Paolo Guiotto: And then 0 from 1 half 1.
09:40:520Paolo Guiotto: And the next one will be similar.
09:44:359Paolo Guiotto: So we take the interval 0, 1. We divide in four equal parts, but now we have… the function is 0 from 0 to 1 half, it is plus alpha from…
09:57:390Paolo Guiotto: One half, two, three.
09:59:300Paolo Guiotto: 4th, and minus alpha, From, 3 over 4 to 1.
10:06:160Paolo Guiotto: Okay, let's give names to these, so we said E0, E1,
10:12:840Paolo Guiotto: it is not convenient to proceed with one single index, because at this point, I have two functions. At the next step, I will have 4, then I will have 8, then 16, so it is better to say that
10:28:390Paolo Guiotto: Now we… we use a double index, which is, E…
10:36:710Paolo Guiotto: Let's say that here we… we use the 2 because this is the exponent of, you know, we are dividing in 4 parts. 4 is equal to 2 squared, so the 2 is the 2 that you see here.
10:51:240Paolo Guiotto: And then we put a comma, let's say there is a first function, which is,
10:57:150Paolo Guiotto: maybe 0, and this second, E2, 1.
11:02:730Paolo Guiotto: Let's see if it is the good choice.
11:05:700Paolo Guiotto: Now, first of all, the choice of alpha. If you redo the calculation of the L2 norm, it's the same for the two cases, so we do for the first, E20,
11:16:960Paolo Guiotto: the alto norm square.
11:18:880Paolo Guiotto: We now have the integral from 0 to 1 half of alpha squared, because from 0 to 1 fourth, you have alpha squared, then from 1 fourth to 1 half, you have minus alpha squared, so again, alpha squared. And so this time we get… and then 0 for the integral from 1 half to 1, so we get alpha squared divided 2.
11:43:220Paolo Guiotto: So this must be 1. It means that alpha squared is equal to 2, so alpha is now root of 2, so it is better to write 2-1 half.
11:56:60Paolo Guiotto: Okay? So, this is the same for the two. So, the E20 is 2 to 1 half, if you want the indicator of 0, 1 half.
12:12:460Paolo Guiotto: Minus the indicator of 1 half
12:16:730Paolo Guiotto: Well, it is better to… to write this,
12:25:510Paolo Guiotto: I don't know, I don't know if I want to write analytically these functions, maybe later we will do, I don't know. So, 1 fourth, sorry, 1 fourth, one fourth to one half.
12:36:920Paolo Guiotto: and the E21 Is 2 to 1 half… Indicator of,
12:45:820Paolo Guiotto: You see, if you want to have a unified way to write this, it would be better to write this 4 as a power of 2, 22 square, so this is, now it is,
12:59:320Paolo Guiotto: We are from one half, so it is 2 to 3 over 2 square, and then we have minus 1,
13:06:830Paolo Guiotto: 3 over 2 square, which is 3 fourths.
13:11:200Paolo Guiotto: 2, 1, which is 4… over 2 square.
13:18:790Paolo Guiotto: Now, it is easy that if you do the square product between these two, when you just multiply the two into the integral and get zero, because you see, this is different from zero, where this is 0. When you multiply these two, you get 0. So the integral will be zero, they actually 0.
13:38:400Paolo Guiotto: But if you do the multiplication with one of the previous, for example, with E1, D1 is constantly equal to 1, so when you multiply,
13:49:140Paolo Guiotto: to do the Scala product, you get one of these two. And if you do, the integral of this between 0, 1, you get 0, the integral.
13:58:970Paolo Guiotto: And the same for the product with the E1. So.
14:05:370Paolo Guiotto: These functions, E0, E1, E2, E2, 1, are perpendicular.
14:15:420Paolo Guiotto: Okay? And you can continue like that.
14:19:750Paolo Guiotto: So, let's say that, at… step… N.
14:26:120Paolo Guiotto: What do we have? We take our interval, 0, 1.
14:30:970Paolo Guiotto: We divide into 2 to the n parts, hmm?
14:36:50Paolo Guiotto: So, we divide.
14:41:420Paolo Guiotto: 01, huh?
14:44:70Paolo Guiotto: in two to the end, parts.
14:49:250Paolo Guiotto: So, they are described in this way. As you can see, this is 1 over 2 to the n, this is 2 over 2 to the n, this is 3 over 2 to the n, and so on.
15:02:920Paolo Guiotto: Now, what we need is to work in this way. You take the first two, the second two, then the next two, and so on.
15:14:290Paolo Guiotto: How these are described.
15:16:780Paolo Guiotto: So these are two consecutive intervals, where you see that the first is 0 over 2 to the n, the first k is 0 to the
15:27:710Paolo Guiotto: the next one will be 4 over 2 to the n, then there will be 6 over 2 to the n. So you understand that the first K is even, so it will be something like 2H,
15:41:550Paolo Guiotto: over 2 to the n, then there will be 2H plus 1 over 2 to the n, and then there will be 2H plus 2 over 2 to the n.
15:52:660Paolo Guiotto: Now, what is the function?
15:54:940Paolo Guiotto: The function, here will be 0 everywhere.
16:00:590Paolo Guiotto: Except in these two intervals, where you get the positive value in the first half and the negative value
16:07:800Paolo Guiotto: Opposite in the second half.
16:10:360Paolo Guiotto: Now, we call this function, since, we still use this notation. So this, the first index is the exponent we have here, so EN.
16:21:490Paolo Guiotto: Okay? And the second index we use here could be, this. Actually, this number here.
16:31:290Paolo Guiotto: H, the H, because when H is 0, it's 0.
16:35:90Paolo Guiotto: And, for example, in this example, when H is 1, it's just the 2 to 2 other force, so it's one.
16:44:920Paolo Guiotto: Now, you can check that this function, the value of the alpha that you have to choose here to have, so alpha and minus alpha, to have a norm 1
17:00:390Paolo Guiotto: This is norm of ENH.
17:05:200Paolo Guiotto: in L2 is equal to 1 if and only if… so in the previous case, we have seen that the value is this one, 2 to 1 half.
17:16:110Paolo Guiotto: Now, what is the next value? Is 2 to 1 third, 2 to 1 fourth is 2 to 1 over n.
17:22:860Paolo Guiotto: So it is alpha equal to 2 1 over n. This can be simply, found.
17:31:980Paolo Guiotto: And so, the function ENH of T is defined as, to 1 over n.
17:42:790Paolo Guiotto: times the indicator of the first interval, which is the 2H.
17:50:40Paolo Guiotto: over 2 to the n.
17:52:210Paolo Guiotto: 2H plus 1 over 2 to the n, huh?
17:57:410Paolo Guiotto: of T, minus the indicator of the second interval, 2H plus 1.
18:02:770Paolo Guiotto: It doesn't matter if you have the repetition at the point, at the midpoint, because this won't have any weight at the end.
18:12:670Paolo Guiotto: So this is analytically what these functions are.
18:17:330Paolo Guiotto: Now, you can easily understand that, with a little bit of tedious work, that these functions are also perpendicular, okay?
18:28:540Paolo Guiotto: So, the functions E0, E1, and all these ones, ENH, or N greater or equal than, 2,
18:42:390Paolo Guiotto: And, H, H depends on N, because, at, step N, you have, exactly one half of, these two to the n part, so it's,
18:55:140Paolo Guiotto: This is, 2 to N minus H ranges from…
19:01:570Paolo Guiotto: Sorry, from 0 to exactly 2 to n minus 1.
19:08:40Paolo Guiotto: So it can be proved, now we accept this, it can be proved.
19:13:820Paolo Guiotto: So let's state as theorem.
19:16:300Paolo Guiotto: If you are curious, the proof is on the chapter.
19:20:540Paolo Guiotto: where we talk about autonomal basis, there is this proof. That this… there is directly the proof, okay? E0, E1,
19:29:500Paolo Guiotto: Enh… We then greater or equal than 2H from 0 to 2DN plus minus 1 is N.
19:41:250Paolo Guiotto: orthonormal.
19:48:400Paolo Guiotto: basis.
19:52:90Paolo Guiotto: for L201.
19:56:170Paolo Guiotto: And this basis is named the Di Khar.
20:00:690Paolo Guiotto: Phases.
20:06:640Paolo Guiotto: For example, this basis, it's interesting for applications, because the classical trigonometric basis is made of regular functions, in fact, you know?
20:19:750Paolo Guiotto: So, it doesn't look to be a very good basis to model a phenomena where there are irregularities. They both are basis of L2, but this one, it's better for that purpose.
20:32:760Paolo Guiotto: Okay, now let's assume that we have this basis, we know this basis. So, let's explore what happens to the venal idea. So, vener says…
20:43:960Paolo Guiotto: Idea…
20:45:880Paolo Guiotto: is to say that we take the Brownian motion, we do the derivative, and we expand in a Fourier series. We think that this will be, hopefully, an element of L2. We don't know yet if this is true, but this is just to get a formula that at the end will be the definition of the Brownian motion.
21:05:630Paolo Guiotto: So we do the Fourier series with respect to this basis.
21:09:710Paolo Guiotto: Now, the problem is that this basis is written through a double index, so, we should say that, we should write this, like, in this way, sorry, DW…
21:24:530Paolo Guiotto: DTWT is 0, scalar is 0, times E0, plus DTWTE1
21:34:190Paolo Guiotto: times C1, and then we have the sum of what? We take our function, DTWT, we do the coordinate in the element ENH,
21:47:240Paolo Guiotto: times ENH.
21:49:910Paolo Guiotto: So now we have to sum. First, over the H, h goes from 0 to 2 to the n minus 1, and then we sum over the n that goes from 2 to class infinity.
22:01:690Paolo Guiotto: So we have… we get this formula. This is the Fourier series of this function, the PWT, written on this basis.
22:12:280Paolo Guiotto: Now, what is interesting about this series is the interpretation of these coefficients.
22:19:340Paolo Guiotto: Because, well, the first one…
22:22:160Paolo Guiotto: is we are proceeding informally. We don't know yet if this W exists, and we know at the end that there is no derivative, so the calculation I'm doing is completely out of any formal basis.
22:36:60Paolo Guiotto: But at the end, it will suggest a formula that turns out to be correct, okay? So that's why we are doing,
22:43:690Paolo Guiotto: So, if you look at this one, let's take the first one to understand what is this coefficient.
22:49:890Paolo Guiotto: This is the scalar product, so integral between 01 of DTWT times E0T. But E0T is the function
23:02:30Paolo Guiotto: the first function of this list is the function constantly equal to 1, that is 0. So if we plug the 0 there.
23:14:180Paolo Guiotto: So we get integral from 0 to 1 of the derivative of WT. We know that this is not defined, but if it is defined, integral of derivative is easy, because it is final value minus initial value, so 2 should be W1
23:30:690Paolo Guiotto: minus W0. W0 is supposed to be 0, because the Browner motion starts from 0, so we get W1.
23:40:820Paolo Guiotto: Okay? Now, what this says, it says that this coefficient
23:47:240Paolo Guiotto: is W1, but W1 is what?
23:50:60Paolo Guiotto: at the end of the story, W1,
23:52:810Paolo Guiotto: is, if you go back, and we have the functional group here.
23:58:570Paolo Guiotto: in the… in what is written here, in particular to number 2, take S equals 0. W0 is 0. So this says WT is quotient…
24:10:850Paolo Guiotto: That is normal means zero for RSD.
24:14:680Paolo Guiotto: So W1 is mean zero variance 1.
24:19:720Paolo Guiotto: So it means that the first coefficient of this Fourier series is a, let's write here, a normal variable, mean 0, and value 1.
24:34:610Paolo Guiotto: Now, let's take the second here.
24:38:800Paolo Guiotto: And let's see what is it.
24:40:710Paolo Guiotto: So if we repeat the calculation, now it's a bit more complicated.
24:46:630Paolo Guiotto: And, however, it's the same that you will have inside all this.
24:51:620Paolo Guiotto: Because this time, E1 is what? E1 is a function which is not constant, is plus 1 and minus 1.
25:01:400Paolo Guiotto: So it takes two values. And that's what actually happens with all the other functions, because they take two values and zero everywhere else. Okay, so we can get an idea of what is the general formula here.
25:14:130Paolo Guiotto: So, when we do this product, we have integral 0, 1, dtwt.
25:19:540Paolo Guiotto: times E1, but E1 is plus 1, indicator of 0, 1 half.
25:27:510Paolo Guiotto: Minus 1 indicator 0, sorry, 1 half 1.
25:32:200Paolo Guiotto: That was E1. So this means that we have, we split this into two parts. Because of the indicator, we have integral 0 to 1 half of DTWT minus integral from 1 half to 1 dtwt.
25:50:110Paolo Guiotto: So if we do now the same calculation, assuming that we could do this by the fundamental formula of integral calculus, this would be W1 half minus W0 minus W1 minus W1 half.
26:10:70Paolo Guiotto: Okay?
26:11:770Paolo Guiotto: Now, it is convenient to do not do the algebra, but keep it this way, because here we have two increments relative to two consecutive intervals. Because you see, here this is the increment from 0 to 1 half, and this is the increment from 1 half to 1.
26:30:50Paolo Guiotto: And at the end of the story, if W is a Brownian motion, these two variables are both Gaussian.
26:37:620Paolo Guiotto: This is normal, mean zero, and variance, you remind that variance is easy, because it's just the difference of times, so it's 1 half. This is another Gaussian, mean zero, variance, still one half.
26:53:200Paolo Guiotto: And moreover, because the consecutive increments are independent, they are independent.
27:02:180Paolo Guiotto: But now, if we do the sum difference of two normal, mean zero and variance, one half, and they are independent, you know, what we get?
27:14:440Paolo Guiotto: All this is…
27:18:350Paolo Guiotto: Exactly. So, because if you think to the characteristic functions, they split into the product because of the independence, no? So they are both equal E minus 1 half the variance, 1 half axi squared.
27:34:730Paolo Guiotto: When you multiply back the two 1 half coefficient sums, and that's exactly… we get a normal N01. So, also, this coefficient is a normal, mean zero variance 1.
27:47:60Paolo Guiotto: And you expect that the same will happen for all these coefficients. All these are normal, standard normal, mean 0, and variance 1. It can be verified.
27:58:260Paolo Guiotto: So, this Fourier series is the Fourier series where we have these elements, which are the elements of the hard basis, we have formulas, we have explicitly, and these are random objects that have all the same distribution. Normal, standard normal, mini zero by S1.
28:17:580Paolo Guiotto: What can be observed, moreover, is that they are also independent.
28:23:260Paolo Guiotto: No? Because,
28:30:760Paolo Guiotto: Because,
28:35:670Paolo Guiotto: DEA… so, if we, soap.
28:41:860Paolo Guiotto: Let's give a name. Capital X, say N…
28:47:140Paolo Guiotto: H equal to the scalar product DTWT.
28:54:140Paolo Guiotto: ENH, and of course, we will set X0 equal DTWT
29:03:550Paolo Guiotto: E0, scalar is 0, and X1, DT, WT… You won.
29:12:20Paolo Guiotto: are all normal, means zero variance 1, and as you can easily see, And… independent.
29:23:660Paolo Guiotto: So we can now give a new look to our series. So, the series, informally, because it's
29:32:140Paolo Guiotto: for the moment, it's still informal, is X0E0 plus X1E1,
29:38:850Paolo Guiotto: Plus, we have this double sum that, however, we keep the notation of double sums, some for n equal 2 to infinity, some for H going from 0 to 2 to the n minus 1 of X, what is it, NH, E, and H.
29:57:510Paolo Guiotto: Where all these coefficients, no, are random variables, independent random variables with the same distribution, normal means 0 variance 1.
30:07:400Paolo Guiotto: Okay, now we can get a formula for W. All this may look as sorcery, because we know at the end that the derivative does not make sense, but if we integrate this thing, so we get integrating
30:23:490Paolo Guiotto: Now, we integrate from 0 to T in order to have W of T, okay? So imagine that you put an S here. So when you integrate from 0 to T, you get W .
30:35:980Paolo Guiotto: minus W of 0 equal… so here we should say integral 0 to t of X0E0 of, say, SDS. Notice that X0
30:50:420Paolo Guiotto: is a coefficient independent of time, because that's the Fourier coefficient, no? Time is washed out by the scarlet product. So that's a constant we can write here. And the same for all terms. So we can say X1 integral 0 to t of E1SDS,
31:10:610Paolo Guiotto: Placer.
31:12:240Paolo Guiotto: sum, n going from 2 to infinity, sum, H going from 0 to infinity of this coefficient, X and H, and then we have the integral 0 to t of the hard function, ENHS dS.
31:34:500Paolo Guiotto: Now, if we give a look at these functions, we can… we compute… let's say we need just to compute these two, the first two, because they… they are very similar, because of this feature.
31:47:10Paolo Guiotto: So if I do the integral from 0 to t of E0S, that's easy, because E0 is constantly equal to 1. So I am doing the integral from 0 to t of 1DS, that's T.
32:01:350Paolo Guiotto: Then, if I do the integral from 0 to t of e1, S,
32:08:340Paolo Guiotto: Now, I don't really need to do the calculation. I can do, analytically, because the one is plus 1 and minus 1 stuff. That's not important, but…
32:21:770Paolo Guiotto: Let's do graphically to understand what is the result. So we have a function made like this.
32:29:420Paolo Guiotto: Which is plus 1.
32:31:610Paolo Guiotto: Here, and minus 1 here.
32:35:130Paolo Guiotto: So this is the function that you see in red, is the E1 of S.
32:40:270Paolo Guiotto: Now, the integral from 0 to t is, if you want, a function whose derivative is this one.
32:47:670Paolo Guiotto: So what should be the integral from 0 to t?
32:54:880Paolo Guiotto: So what you are watching here is,
32:58:320Paolo Guiotto: It's exactly the same process we did at the beginning with that,
33:03:980Paolo Guiotto: path, no? Saying… we start from the path, assuming that it is piecewise linear, we differentiate, we get a function which is piecewise constant, no? Now, imagine that we have to do the vice versa. So we start from a function which is piecewise constant, and we have to turn that into a function.
33:23:280Paolo Guiotto: Use a derivative is this one.
33:26:760Paolo Guiotto: And that at t equals 0 is 0, because integral from 0 to 0 is 0. So, you see, the slope is 1, the function is doing like that, up to 1 half.
33:37:230Paolo Guiotto: Then the slope is minus 1, it goes down like that.
33:41:520Paolo Guiotto: So this is the function that you see here.
33:45:860Paolo Guiotto: Now, we give a name to this function, we call S0 of T.
33:51:790Paolo Guiotto: These functions are called… will be called niche outer… Shall there.
33:59:130Paolo Guiotto: functions.
34:02:510Paolo Guiotto: And when you will integrate from 0 to t, the generic ENH…
34:12:320Paolo Guiotto: So the genetic DNAH is a function made like that. So, it is a 0,
34:19:610Paolo Guiotto: Until a little interval where we have positive value, negative value, and then again, 0 to 1.
34:26:220Paolo Guiotto: This is the interval 2H over 2 to the n, and this is 2H plus 1, plus 2 over 2 to the n.
34:35:510Paolo Guiotto: What happens when you integrate?
34:42:540Paolo Guiotto: Well, when you integrate 0, you will have zero.
34:46:120Paolo Guiotto: until the function is zero, you will get zero. So the integral will be like this.
34:53:770Paolo Guiotto: When you arrive at this point, you are integrating a function which is plus something.
34:59:960Paolo Guiotto: So, the integrals start to increase linearly with this slope, So it grows like that.
35:08:880Paolo Guiotto: up to the point where the derivative changes sign. The derivative is this function inside.
35:15:720Paolo Guiotto: So then, you start decreased down to the axis.
35:25:410Paolo Guiotto: and you get this, then you continue integrating, 0 gets 0. So this is the function that we call SNH of T, and it is a Schauder function.
35:37:770Paolo Guiotto: So you see, it's a function that can be also written explicitly, it's done on the notes. And so, at the end, we finish with this formula. So WT…
35:48:920Paolo Guiotto: apparently seems to be X0 times S0T, which is easy, it's just T, plus X1S1T, It is this function.
36:03:960Paolo Guiotto: BSwatna.
36:06:190Paolo Guiotto: Sorry, there is S1 here.
36:09:650Paolo Guiotto: And then we have plus the double sum, some for n going from 2 to infinity, sum for H going from 0 to 2 to the n minus 1, of the coefficient Xnh, S, and H of T.
36:28:800Paolo Guiotto: So we got this formula.
36:31:870Paolo Guiotto: So, if the Brownian motion exists.
36:35:70Paolo Guiotto: Basically, the idea is that it should be made of this sum, where the coefficients, X0, X1, X and H are all.
36:46:70Paolo Guiotto: standard Gaussian, random variables, independent standard Gaussian random variables.
36:53:230Paolo Guiotto: Now, the theorem is… We can prove that the right-hand side is convergent
37:01:610Paolo Guiotto: And moreover, it defines the Browning motion. So this quantity is well-defined, and it fulfills all the properties of the Browning motion. This is thersky theorem.
37:13:880Paolo Guiotto: So, if, X0. X1.
37:19:740Paolo Guiotto: XNH.
37:22:480Paolo Guiotto: And, from 2 to infinity.
37:25:200Paolo Guiotto: H from 0 to 2 to the n minus 1.
37:31:320Paolo Guiotto: Independent.
37:34:440Paolo Guiotto: standard.
37:38:930Paolo Guiotto: Goshen.
37:44:290Paolo Guiotto: random variable.
37:46:380Paolo Guiotto: Then… Let's give a name to this formula.
37:52:40Paolo Guiotto: star. Well, you see, there is a non-trivial problem. At the end, you have an infinite sum, it's not a finite sum that you can say. It is always defined, so it's a series, it must converge. Well, it turns out that then the
38:09:50Paolo Guiotto: serious.
38:12:40Paolo Guiotto: At Ryan and Side.
38:15:200Paolo Guiotto: of star.
38:18:280Paolo Guiotto: converges.
38:22:610Paolo Guiotto: uniformly, uniformly, means in L infinity.
38:30:540Paolo Guiotto: Well, be careful, because here, we are talking about convergence as a function of T in L infinity, 0, 1.
38:43:200Paolo Guiotto: with the… probability… 1. So it means that for almost every omega.
38:52:160Paolo Guiotto: the omega R in the value of capital X, so fixed and omega. That's a series in P. It's a series of functions, it's a sum of functions of P,
39:04:560Paolo Guiotto: it converges in the uniform norm to something. This convergence is important because
39:11:160Paolo Guiotto: Maybe you remember from the first part, of course, that the space of function, of continuous functions is complete. So this means that when you have a convergence of a sequence of continuous function to something, also the something must be continuous.
39:28:620Paolo Guiotto: And this is, important because these, functions are triggered continuous. This is T, this is the function, like the models. Let me play that. These are functions also continuous. All of them are continuous functions. So, when you have the uniform convergence, it means that almost the limit is continuous.
39:50:500Paolo Guiotto: If this happens with probability 1, it means that the first requirement of the definition of Brownian motion
39:58:780Paolo Guiotto: is fulfilled.
40:00:610Paolo Guiotto: The trajectories of this W are continuous.
40:04:720Paolo Guiotto: And to finish the statement, this function W of P, verifies
40:17:480Paolo Guiotto: the definition… off.
40:21:380Paolo Guiotto: Brownian. Motion.
40:24:690Paolo Guiotto: You can, see in notes if you, we have been long a bit.
40:29:240Paolo Guiotto: If you look at notes, there should be a figure, there is the proof of this theorem, I don't think we have time to do.
40:38:540Paolo Guiotto: Because I want to do at least some calculation next time. But you can see in the… there are three figures that represent 3 applications with common software of this sum, of course, of a finite sum.
40:56:830Paolo Guiotto: So I truncated the sum to a certain order, and as you can see, the third one, which is made, if I'm not wrong, it should be made of 16 terms in that series, so it's that one with n up to 4.
41:14:800Paolo Guiotto: When… to… to… to be clear.
41:17:500Paolo Guiotto: So, we stop index little n to n equal 4. We get the figure at the right-hand side, and as you can see, it's a pretty nice and precise representation. So, you don't need an inconcrete application to rightly find some that you will never
41:36:80Paolo Guiotto: There is not human software that can compute that sum, but you can truncate the sum, because I say, because this is… it has also very good rate of convergence, so you can start with a few terms and get a very nice representation of the Browning motion.
41:53:500Paolo Guiotto: Okay.
41:55:130Paolo Guiotto: Let's stop here.
41:58:810Paolo Guiotto: Let's see if you can try to…
42:01:260Paolo Guiotto: Well, you can try to do, because these are the exercises on the definition, tomorrow we see the solution, do, 11…
42:13:560Paolo Guiotto: 31… Maybe 2, maybe 3…
42:20:260Paolo Guiotto: The number 4 is made on the definition of marting, and maybe we will learn.
42:24:870Paolo Guiotto: Just say what is it tomorrow or not, I don't know. Okay, let's stop here.