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

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Del curso dictado por National Research University Higher School of Economics

Stochastic processes

90 calificaciones

National Research University Higher School of Economics

Stochastic processes

90 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 define the integrals of the following type,

the integrals Xt dHt from a to b.

Here, H is some stochastic process.

Namely, I will consider H of t from the class of Ito processes.

Well, this means that this process can be represented in the following form,

H of t is equal to H0 plus integral from zero to

t bs ds plus

integral from zero to t sigma s dWs.

W is a Brownian motion.

And here, we assume that there is some filtration behind this integral.

So, F of t is a filtration.

Bs and sigma s are two processes which are adapted to this filtration.

So, processes adapted to

Ft. Brownian motion is Ft Brownian motion.

And H0 is a random variable which is measurable

with respect to F0.

So, here, Ht is a process of this type,

and X is some process adapted to the same filtration.

And if the process Xt is such that the integral from a to b absolute value of

X as bs plus Xs squared sigma s squared,

ds is finate, if X is such that this condition is fulfilled,

then we define the integral from a to

b Xt dHt is equal to the integral from a to b

bs Xsds plus integral

from a to b sigma s XsdWs.

Know that here, we'll have two integrals of the type which we considered before,

namely: the first integral is

the first type and take s as the basic process and integrate it with respect to ds.

And the second integral was also considered before because here,

we have an integral with respect to Brownian motion.

This is the first and the third type of the integrals which were considered before.

This is just a definition, nothing more.

And if the process H of t kind of represents this form,

one can also use an alternative notation for this class of processes,

namely: one can write that the process Ht is formed in the following ways.

So, one can write dHt is equal to btdt plus sigma tdt.

This is just another notation of the same object, nothing more.

By definition,

this Stochastic differential equation Stochastic representation

is the same as this Stochastic integral, nothing more.

The following theorem plays essential role in the theory of integrals of this type.

Let me assume that H of t is an Ito process,

and F is some function of two arguments t and x,

which is twice continuously differential.

Then, the following statement holds.

The process F (t,

H of t) is itself an Ito process.

And what is very important is that it is equal to F(0,

H0) plus the derivative of the function F with respect to the first variable.

I will denote it by F1 prime.

At the moment, S,

H of s. And I will integrate this function with respect to ds,

integral from zero to t. Plus integral from zero to t,

is the first derivative of the function F with respect to the second argument.

F2 prime s, H of s, dH of s,

this is just an integral of this type,

an integral with respect to the Ito process.

But this is not all.

Plus, and here I should write,

one half integral from zero to t. And now I

should take the second derivative of the function F with respect to the second argument.

So, F22 to prime is at s,

H of s multiplied by the coefficient which is here.

So this sigma s is the power two.

So, sigma s squared.

And now I should write ds.

This formula is known as the Ito formula.

At the first glance,

it looks like a magic.

So it's a very complicated thing which is not clear how to use.

But in the next subsection,

I will show you an example how this formula helps

to calculate the basic integrals of different types.

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