Week 8.1: Definition of a Lévy process. Stochastic continuity and càdlàg paths.

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En provenance du cours de National Research University Higher School of Economics

Stochastic processes

101 notes

National Research University Higher School of Economics

Stochastic processes

101 notes

The purpose of this course is to equip students with theoretical knowledge and practical skills, which are necessary for the analysis of stochastic dynamical systems in economics, engineering and other fields. More precisely, the objectives are 1. study of the basic concepts of the theory of stochastic processes; 2. introduction of the most important types of stochastic processes; 3. study of various properties and characteristics of processes; 4. study of the methods for describing and analyzing complex stochastic models. Practical skills, acquired during the study process: 1. understanding the most important types of stochastic processes (Poisson, Markov, Gaussian, Wiener processes and others) and ability of finding the most appropriate process for modelling in particular situations arising in economics, engineering and other fields; 2. understanding the notions of ergodicity, stationarity, stochastic integration; application of these terms in context of financial mathematics; It is assumed that the students are familiar with the basics of probability theory. Knowledge of the basics of mathematical statistics is not required, but it simplifies the understanding of this course. The course provides a necessary theoretical basis for studying other courses in stochastics, such as financial mathematics, quantitative finance, stochastic modeling and the theory of jump - type processes.

À partir de la leçon

Week 8: Lévy processes

Upon completing this week, the learner will be able to understand the main properties of Lévy processes; construct a Lévy process from an infinitely-divisible distribution; characterize the activity of jumps of a given Lévy process; apply the Lévy-Khintchine representation for a particular Lévy process and understand the time change techniques, stochastic volatility approach are other ideas for construction of Lévy-based models.

Week 8.1: Definition of a Lévy process. Stochastic continuity and càdlàg paths.12:36

Week 8.2: Examples of Lévy processes. Calculation of the characteristic function in particular cases17:37

Week 8.3: Relation to the infinitely divisible distributions7:24

Week 8.4: Characteristic exponent8:45

Week 8.5: Properties of a Lévy process, which directly follow from the existence of characteristic exponent7:45

Week 8.6: Lévy-Khintchine representation and Lévy-Khintchine triplet-17:03

Week 8.7: Lévy-Khintchine representation and Lévy-Khintchine triplet-27:33

Week 8.8: Lévy-Khintchine representation and Lévy-Khintchine triplet-38:20

Week 8.9: Modelling of jump-type dynamics. Lévy-based models7:28

Week 8.10: Time-changed stochastic processes. Monroe theorem9:07

Rencontrer les enseignants

Vladimir Panov
Assistant Professor
Faculty of economic sciences, HSE

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.

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