[MUSIC] Now I am going to discuss various interesting properties of Brownian motion. And I would like to start with one property which follows from the so-called Kolmogorov Continuity Theorem. Before we'll formulate this theorem, let me give one definition. And what follows, I will say that processes xt and yt are stochastically equivalent, If the probability that xt = yt = 1 for any t larger than 0 or equal to 0. In other words, we can say that yt is a modification of xt. And sometimes, processes which are stochastically equivalent have completely different properties. I just would like to provide one very simple example. Let me consider a process, let x = 0 for any t from 0 to 1. And the process yt is a process which is equal to the indicator that some random variable having absolutely continuous distribution 2 is equal to t. For instance, tau can have uniform distribution on 0, 1. It's easy to see that these two processes are stochastically equivalent. In fact, the probability that xt = yt, this is the same as the probability of the process yt = 0. Or in other words, this is the probability that t is not equal to 2. So this probability is equal to 1 because 2 has an absolutely continuous distribution. But if I look at the trajectories of these two processes, xt and yt, we conclude that they are completely different. For instance, if we speak about xt, then [INAUDIBLE] to readjust equal to 0 for any t from 0 to 1. And if we consider trajectories of the second process, yt, they all are discontinuous. So at some point between 0 and 1, this process is equal to 1, and at all other points, it is equal to 0. So here, all trajectories are continuous, and here all trajectories are discontinuous, so they look very different. And it's very natural to ask whether we can modify our process so that the equivalent process will have continuous trajectories. And the answer of this question is given by the Kolmogorov Continuity Theorem. Let me at least formulate this theorem. So let xt be a stochastic processes, and assume that there exists a constant C, a constant alpha and beta, all of these constants will be positive. Such that mathematical expectation of Xt- Xs to the power alpha is less or equal than this constant C multiplied by t- s in the power 1- beta. This inequality shall be fulfilled for any t and s from a to b. So if this condition is fulfilled, then there exists a process yt which is stochastically equivalent to xt, such that yt has continuous, Trajectories. In this context, one can say that the process xt has a continuous modification. Let me show that the theorem can be applied to the Brownian motion. In fact, let Bt be a Brownian motion and then let me consider the mathematical expectation of Bt- Bs is a power of 4. You know that Bt- Bs has normal distribution with parameters 0 and t- s. Therefore, this difference, Bt- Bs, can be represented as square root of t- s multiplied with some standard normal random variable, xi. Of course, here, I can assume that t is larger then s, as well as formulation of the theorem. And what we'll have here is the fourth moment of this random variable. So if we now consider mathematical expectation of t- s squared, multiplied with xi to the power of 4, so, We'll conclude that this mathematical expectation is the fourth moment of the standard normal random variable, it is actually [INAUDIBLE] 3. And we get that this mathematical expectation is equal to 3 multiplied by t- s squared. Therefore, this condition is fulfilled because we can take alpha equal to 4, C equal to 3, and beta equal to 1. So we conclude that a Brownian motion has continuous modification. In fact, exactly this modification is called the Brownian motion. And if we would like to be precise, we should include this condition in all of our definitions, both definition one and two. When you think about Brownian motion, you should immediately realize that all trajectories of this process are continuous. And this is a kind of mathematical jargon, so you can omit this condition as a definition. But you will think about exactly this modification of Brownian motion. Starting from now, when I will say Brownian motion, I will mean exactly this continuous modification. [MUSIC]