Week 7.7: Itô’s formula

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Del curso dictado por National Research University Higher School of Economics

Stochastic processes

39 calificaciones

National Research University Higher School of Economics

Stochastic processes

39 calificaciones

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.

De la lección

Week 7: Stochastic integration & Itô formula

Upon completing this week, the learner will be able to calculate stochastic integrals of various types and apply Itô’s formula for calculation of stochastic integrals as well as for construction of various stochastic models.

Week 7.6: Integrals of the type ∫ X_t dY_t, where Y_t is an Itô process6:54

Week 7.7: Itô’s formula8:02

Week 7.8: Calculation of stochastic integrals using the Itô formula. Black-Scholes model6:18

Week 7.9: Vasicek model. Application of the Itô formula to stochastic modelling5:05

Week 7.10: Ornstein-Uhlenbeck process. Application of the Itô formula to stochastic modelling.4:06

Conoce a los instructores

Vladimir Panov
Assistant Professor
Faculty of economic sciences, HSE

Now I would like to show how's the inter formula can be

used for calculating the stochastic integrals of the following type.

Namely, I would like to calculate integrals from zero to T,

sub function small g a

the point SW of SDES.

Well we assume that G is a function which is good

enough that this twice continues to differentiable,

and the question is how to calculate this integral?

Well, you have already one example of this integrals,

integrals from zero to TW of SDVS.

And the calculation of each integral via

the definition of such integrals is rather complicated.

And moreover if it makes this function just a little bit more difficult for instance,

we consider WS squared DVS,

this integral can be also calculated but the calculation is much more difficult.

Okay. How to use the inter formula to calculate this integral.

That implies the inter formula for the function F equal to

the anti derivative of the function G with respect to the second argument.

That is F 2 prime is equal to G.

The process H of T will be equal to the Brownian motion.

Of course a Brownian motion is an inter process.

So the left hand side is clear.

It should be F D W of T. The right hand side we have F 0 W,

at 0 that's if F 00,

because Brownian motion is zero and zero.

So look how this integral does not seem more into than the integral

from 0 to T F1

prime S W of s d s. Next.

We shall take the derivative of a function F with respect to the second argument.

But, this is the same as a function G. So,

we have integral from zero to T G S W of

S D W of S. And now we have one half the last term Integral

from zero to T. And here I should take

the second derivative of the function F with respect to a second argument.

This is the same as the first derivative of

the function G with respect to the second argument.

And here we have S W of S. As for the seeking which is equal to

one because in this case the integral process of T is equal to the Brownian motion.

So, and we have here DS.

Let us look more precise to the last formula let inside to simple.

So, we see we have also just a number.

Here we have an integral of the Simplest type.

So, we also can calculate everything from this representation.

So, basically this is already a very good object.

As for the next object it is exactly equal to the integral which we should compute.

Just compare this to expressions,

say exactly the same.

And this was the last thing to grow.

It is also the simplest type.

It looks similar to this integral and it's

also not a question how to deal with this object?

So, from this formula we concluded that the integral which we

should calculate this integral from zero to T H,

excuse me G S W of S DVS is equal to

FT W of T minus F 0 0 minus.

Integral from 0 to T. And here I can combine these two integrals.

What I have here is f one prime S W of S plus

one half G two

prime S W of S. And here I will write the S. So,

basically this formula can be used for calculation

of any integral of this type and let me provide one example.

Let me show how this formula can be used for calculation of the integral which was

considered before Integral from zero to T W of SGW of S. Well,

first of all, let me compare this integral with this general form.

So, the function G of the arguments T and X in our case is equal to x.

The anti derivative of this function with respect to the second argument is

equal to 1/2 X squared.

Here we can at any function say is a function H depending on T. But,

if you look attentively to this formula you immediately realize that

this additional tone changes nothing in this formula.

Because, here we will have plus H of T,

here we will have minus H of zero,

and here we will have integral from zero to TH prime off SDS,

and this terms vanish.

So basically.

We can add a function but it makes no sense.

So we just can take any anti derivative with respect to the second argument.

Okay. If it will apply this function F in

this formula we will get that this integral is equal to,

okay, we shall take function f as a point t w of t. That is

one 1/2 V T squared Then we should consider F zero zero,

so our function F at zero zero is equal to zero.

Now we shall take the first derivative

of this function with respect to the first argument.

This derivative is equal to zero.

And now we should take minus 1/2 as a derivative of this function with respect to x.

This derivative is equal to one.

If we will integrate these from zero to t to S we will get t.

So we'll get the exact same answer as in one of our previous examples.

So, we'll get the same answer but using the inter formula.

Basically this type of ideas can be used for calculating any integral of this type.

And this is the first application of the inter formula.

In the next subsection.

I will show how the inter formula can be used for constructing Stochastic models.

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