Week 8.6: Lévy-Khintchine representation and Lévy-Khintchine triplet-1

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From the course by National Research University Higher School of Economics

Stochastic processes

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National Research University Higher School of Economics

Stochastic processes

69 ratings

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.

From the lesson

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

Meet the Instructors

Vladimir Panov
Assistant Professor
Faculty of economic sciences, HSE

Now, let me formulate the Levy-Khintchine theorem.

The theorem says, that the characteristic exponent of any Levy process lt,

can be represented as follows.

So, psi of u is equal to iumu minus one-half u squared sigma squared

and plus integral over R exponent in the power iux minus

one minus iux multiplied with

the indicator that absolute value of x is less than one nu dx.

This is the most important formula

in the theory of Levy processes and maybe this is a process of the whole course.

Here, mu is a real number,

sigma is a non-negative real number,

and nu is a levy measure.

So, these three elements mu,

sigma and nu form the so-called levy triplet.

And these three objects determine completely

the distribution of the levy processes and any time point.

And therefore, basically they can characterize any Levy process.

Okay. You see that this formula looks similar to the formula from our first example.

In fact, what we have here is a cost of multiplied by Iu,

it was the same as for the process nu t. Here,

will have something very similar to a Brownian motion multiplied by sigma.

And here, to look at this integral without the last two,

then it looks very similar to the compound Poisson process.

And, actually the origins of this formula as the following.

So, are going to proof,

that any Levy process X t. Camber present presented as the sum of three elements.

First element is mu multiplied by t.

Second element is a Brownian motion W t multiplied by sigma,

and the third element is a jump process.

These two theorems, form the continuous part of a Levy process X t. And,

from this representation directly follows,

that the only continuous Levy process is a Brownian motion.

As for J t, this is a jump process and it determines the jump of a process X t,

this is jump part.

How one can represent as jump part?

Basically, this can be done by the full length thing.

We can say that J t is equal to

the sum of all jumps delta X s which occur before t. And,

which have sizes larger than suffix to L,

say larger than one.

Plus, limit as Epsilon turns to zero of the sum delta X

s. Where s has between zero and t and

the sizes of those jumps are between one and epsilon.

This is a formal equality.

But it turns out that the first sum is basically a compound Poisson process.

And this is exactly the reason why this part of

the K formula looks very similar to the compound Poisson process.

And, as for the limit,

it turns out that any element of this sum is also compound Poisson process.

But if it takes a limit,

then what we have as that the missing value is not necessary compound Poisson,

and actually it's equality should be understood only approximately.

By the way, one can easily imagine the practical sense of this composition.

If I draw a trajectory of a Levy process.

For instance a trajectory of the full length types,

we have here some process starting from zero,

it will goes up, down and here,

there is a point that is continuity.

And afterwards, runs farther and here one more point of this continuity and so on.

One can imagine the following thing.

So, we have this continuous process.

So, in some sense we glues these points of this continuity.

So, I take this point and puts it down.

So, what I finally get is this curved.

And here, I also glue this part and finally get a continuous trajectory.

This trajectory is exactly mu t plus sigma W t. And,

to make this operation more or less legal,

I should sum all jumps,

all values of these jumps to this continuous process.

And, this continuous part is represent by J of t. So,

what we have here,

is a sum of continuous part,

which is Brownian motion was a drift and to the discontinuous part, the jump part.

Okay. If you look more attentively,

there's a Levy-Khinchine theorem,

you get the full length oppression's that without

this element this Iux multiplied by indicator,

the formula will be much more beautifuler.

And this is exactly the point.

And in the next subsection,

I would like to present some sufficient conditions,

which allow us to provide

much more simple analog of the Levy-Khinchine theorem as an analog,

where this is turn come down with it.

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