[MUSIC] At the end of the lecture, I would like to discuss some properties of Brownian motion. All of these properties, we will formulate it now and I do not provide any proof. But you can definitely find the proof in any manual on stochastic processes. Let me start with a quadratic variation of Brownian motion, so let me start this quadratic variation. So if you take an interval between 0 and t, and consider a partition of this interval. So t0 is the first point in this partition, is equal to 0, then we have t1, t2, and so on, and tn = t, we can construct the following sum. We take Brownian motion as a moment Btk- Bt(k- 1). We can take square, and then sum over all k from 1 to n. Now we can consider as a limit as n tends to infinity. By this limit, we mean that the diameter of this partition is tending to 0. That this maximum over ti- t(i- 1) is tending to 0 as n tends to infinity. So if we take this limit, we get that it is equal to t. Actually, this limit is called quadratic variation, and this statement means that in the case of Brownian motion, quadratic variation is equal to t. This limit is considered the sense of l to 0, the piece if I will denote this sum by Sn. This statement means that mathematical expectation of Sn- t squared tends to 0 as n tends to infinity. Or in other words, if diameter of a partition is tending to 0. So this is what's about quadratic variation. Now we'll have a very similar object, which is called variation. Which will suggest absolute value of the difference, instead of the square function. And it turns out that this limit is infinite. An interesting point is that the proof of this fact follows from the fact that the quadratic variation is finite. Okay, this is the first property, and it will be very useful In for some of our further lectures. Second property is that Brownian motion is everywhere continuous, but nowhere differentiable. We will discuss differentiability in one of our next lectures. And about continuity, I would just like to mention that here, it means that it's stochastically continuous. That this B(t + h) tends to beta t in probability when h is tending to 0 for any non-negative t. And the third thing which will be rather helpful to understand Brownian motion is the question of how fast Bt is tending to infinity. In fact, this is not difficult to show that the limit of Bt divided by t as t tends to infinity is equal to 0. This equality should be considered almost surely. And also, it is not a difficult task to show that the limit of Bt divided by square root of t when t is tending to infinity, the upper limit is equal to infinity almost surely. The question is, what is the correct rate of convergence of Bt to infinity, and the answer is given by the so-called law of iterated logarithm. This law states that the limit of Bt divided by square root of 2t log (log t) so logarithm of the logarithm of t, when t is tending to infinity. That means the upper logarithm, this limit is exactly equal to 1. This is a very interesting law, and actually, later, we will discuss that similar laws are also fulfilled for more general classes, for limit processes, for instance. But nevertheless, this is one of the beautifullest result in the theory of Brownian motion. This is all for today, I would like to invite you to attend our next lecture. [SOUND]