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

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From the course by National Research University Higher School of Economics

Stochastic processes

70 ratings

National Research University Higher School of Economics

Stochastic processes

70 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

[MUSIC]

Let me consider the full length particle test.

Assume that the iterations of some Levy process Xt will enable at time moment,

delta, 2 delta and so on, n delta.

For instance, you have some data about local talks of a stock price.

And this data is given as equidistant to great, from here to n delta.

The question is,

how to estimate that the Levy measure of a Levy process from this data?

Of course, this question is highly important from practical point of view

because a Levy measure completely characterizes the structure of jobs.

So returning to our example about stocker terms, actually estimation of mu allows to

make a lot of conclusions about the job activity of a given asset.

Okay, I would like to provide an idea of how I can solve it.

And to construct an algorithm,

let me assume that Xt is a process of bounded variation.

As we have discussed before, from here,

it follows as a heuristic function of X delta

of u is equal to the exponent in the power,

delta multiplied by iu mu plus integral

over R exponent in the power iux -1.

And here, I would like to assume that the measure mu has a density,

a Levy density which I will denote by S.

So S of (x) dx.

Good.

The general idea is to compute derivatives of the function phi.

Let me first take the first derivative with respect to U.

And since derivative is equal to phi x delta of u,

multiplied by the derivative of this function.

Actually, it is equal to

delta multiplied by i,

and multiply it by mu plus

integral over R, exponent and

the power iux, x s of x dx.

By the way here, I used mu for the same element as was previously mu tilde.

Let me make that correction.

It isn’t very important for this kind of theory but

we should specify which mu was here.

Well, this is the derivative of the function phi, and

now I'd like to take the second derivative of phi.

This is a derivative of the product of the functions,

so it's the derivative of the phi function,

phi prime x delta of u, multiplied by the second function.

But we know that the second function is equal to phi prime divided by phi, and

I will substitute exactly this expression.

And afterwards, I shall sum this with the function phi,

Multiplied by the derivative of this function.

And the derivative is actually equal

to minus delta multiplied by integral over R,

exponent in the power iux, x-squared, s of x dx.

And this last function is exactly the fourier transform

of the function x-squared s of x, connected at the point U.

So from here, it follows that this fourier transform can be represented to me

as a function phi and is derivative, the first one and the second one.

Namely, we have the following formula.

So fourier transform on the function

x-squared, s of x as a point u,

is equal to 1 divided by minus delta,

multiply with phi 2 prime x delta of u,

divided by phi x delta of u, and

minus phi prime x delta of u,

divided by phi x delta of u.

And then we should take the second power of this expression.

And this formula gives rise for the following methods.

Actually, if you have delta X, delta X, delta X and delta,

we can estimate, The characteristic function of phi,

as well as the first and the second derivative.

For instance, the function phi can be

estimated by its natural non-parametric estimators.

So phi caret of u is equal to 1 divided

by n sum k from 1 to n exponent in the power

iuXk delta minus Xk minus 1 delta.

You know that these differences form a sample that is independent and

identically distributed.

And therefore,

this estimator is a consistent estimator of the heuristic function.

So you can calculate the estimator phi, phi prime, phi 2 prime.

And from that estimate, using this formula,

you can get the fourier transform of x-squared s of x.

Now from the estimator of this fourier transform, you can use the inverse fourier

techniques and to estimate the function x-squared s of x and the function s.

Okay, this is the general idea, once more, looking at the data,

you can estimate all derivatives as a function phi, and the function phi itself.

Then from this formula,

which was constructed mainly due to the Levy continuity theorem,

you can get the estimate of the fourier transform and then you can use

the inverse Fourier transform methods and get the Levy density s.

So this is the general idea,

details can be found in any manual about Levy processes, and but

I guess that this example helps a lot to understand the logic of this lecture.

Basically, using the Levy-Khintchine formula,

we can conclude a lot about the structure of jobs, and

in practical situations, we can estimate it by the data.

[SOUND]

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