Hello and welcome to the last lecture of this course duly devoted to the series of Levy processes. Levy processes are widely used in stochastic analysis for modelings that jump their dynamics. And before we will give the formal definition, I would like to compare this class of processes with two main examples of the Levy processes namely, with the Poisson process and the Brownian motion. Let me draw the following table. The first column, I will write the properties of the Poisson process, N_t. In the second column, let me write properties for Brownian motion, W_t. And in the third column, I will write properties of the general Levy process. Okay. Well, you know that Poisson process is 0, N_0 almost surely. The same property has a Brownian motion, so W at zero is also equal to zero almost surely. And for Levy process, L_t, the same property also holds, L_0 is equal to zero almost surely. Secondly, you know that Poisson process has independent increments. This means the following. That for any time moments, t_0, t_1, and so on, t_n. I mean the t_0 is less than t_1 and so on. t_n minus 1 is less than t_n. The set of random variables, X_tn minus X_tn minus 1 and so on, X_1 minus X_0 are jointly independent. In this case, we say that the process X_t has independent increments. And you know that Poisson process, N_t, has this property. The same is true for the Brownian motion, W_t, and basically the same is true also for Levy process, L_t. Certainly, we know that Poisson process has stationary increments. But mathematically, this means that for any time moments, t and s, and any h larger than zero, X_t plus h minus X_s plus h has the same distribution as X_t minus X_s. And you know that this property also holds for a Brownian motion, W_t, and it is also true for the Levy process. Okay. Moreover, it is known that Poisson process has the following property that N_t minus N_s has a Poisson distribution with parameter lambda multiplied by t minus s. Something similar holds also for Brownian motion. You know that W_t minus W_s has a normal distribution with mean zero and variance t minus s. This is true for any t larger than s. And as the case of a general Levy process, don't assume any specific distribution or the difference between L_t and L_s, but from the general theory, it will follow that this difference, as well as distribution of L_t, is from a class of probability distributions depending on the difference between t and s and basically, this clears the so-called the class of infinitely divisible distributions. I will discuss this issue in details a bit later. But let me now summarize the definition of Levy process. What was your claim here? So, the Levy process is a process which is 0 at 0, which has independent increments, also stationary increments, and we need one more property of Levy process to conclude that this process is the Levy process. And this property is the following one. We should assume that the process L_t is stochastically continuous. That is L_t plus h converges to L_t in probability when h goes to zero. Mathematically, this means that the probability that absolute value of the difference, L_t plus h minus L_t, is larger than epsilon for any epsilon larger than zero, this probability should tend to zero, when h goes to zero. Okay, this is the mathematical formulation of stochastic continuity. What does this property mean if we translate this property from mathematical language to, let me say, normal language? Well, assume that the process L_t is used for modeling the log return of a stock price. Let's say it is equal to log S_t divided by S_0, when S_t is a stock price. Okay, this process is starting from zero and now it has somehow changed. And in some points, this process may have jumps. For instance, the point t jumps, the price rapidly changes and so we observe a jump in this process. And basically, this property of stochastic continuity means that we do not allow calendar ethics. That is, if we know as the process L_t jumps as the time moment t equals, for instance, the first of May, at for instance value 2, then this process cannot be modeled by a Levy process. Why? Well, let me consider this property was t equal to the first of May and epsilon equal to one. What we'll have then? So, we'll have the probability that L_t plus h minus L_t larger than 1 is equal to 1, because you know that this difference when h goes to infinity, is equal to 2 and therefore, this probability is equal to 1, and 1 doesn't converge to 0. So, this property means that no calendar ethics can be incorporated in the process L_t. Nevertheless, this class of properties is very useful and before we go further, let me say a couple of laws about the trajectories of a Levy process. The trajectories of a Levy process are the so-called Cadlag functions. Cadlag is abbreviation from French words continue a droite, limite a gauche. Well there, right continuous, with left limits, and f is a Cadlag function if there exists a limit from the left side and there exists a limit from the right side, this limit f of s when s goes to t, but s is less than t. I will denote this limit by f(t-), it exists, and also limit f of s when s goes to t, but s is larger than t. This limit also exists and I will denote it by f(t+). And we will assume that f(t+) is equal to f(t). So, the plot of a Cadlag function is the following one. So, this function is continuous until the first jump occurs. So, this jump at time moment t, there exists a left limit, there is a right limit, and the value at time moment t is equal to the value of the right limit. Okay, and what one can show for a Levy process is that there exists a Cadlag modification of a Levy process. That is, there is another process which is equivalent to the process L_t and all trajectories of this process are almost surely Cadlag functions. The situation is very similar to the Brownian motion. You know that there is a continuous modification of a Brownian motion. That is a modification with almost surely continuous trajectories. And when you think about Brownian motion, you think only about this modification. You can forget all the story about modifications. You can just think that W_t has almost surely continuous trajectories. And here is exactly the same. You can forget about this notion of modification and just think that Levy process has exactly these trajectories and you see the stochastic processes with these trajectories can be well used for modeling jumps. And these jumps occur everywhere, in economics, in techniques, in all fields of human activity. And this is exactly the reason why Levy processes are so well used and well understood in the context of many applied fields. Okay, this was an introduction to the theory of Levy processes namely, I gave you the definition of a Levy process. In the next subsection, I would like to discuss in details the class of infinitely divisible distributions, which is very closely related to Levy processes.
