[MUSIC] Thank you so much for your interest to attend this course, let us speak about the stochastic integration. In the most broad sense, stochastic integral is any integral which is constructed via a stochastic process or even via a set of numerals. With this course we speak about the following four types of stochastic integrals. The first type, when we have a stochastic process Xt and integrated with respect to dt, and we consider this integral over an integral from a to b. the second type, when we take the deterministic function f(t) and integrate it with respect of dVt where Vt is a Brownian motion, the integral from a to b. The third type, when we take the integral from a to b, then we consider a stochastic process Xt and integrate with respect to dWt. And finally the last type, the most general one when we also consider the stochastic process Xt, but integrated not with respect to Brownian motion, but with respect to more general process Ht, from a to b. Well, the first step of these integrals, namely this integral, was already considered in this course. And I would like to shortly recall the most important aspects related to this kind of objects. You know that any stochastic process Xt is basically a function, from omega times R+, where omega is the first element, is a probability space, and R+ is a space of time t. So Xt is a function from omega time R+ to R. And therefore, if you fix an element of omega, say small omega, then, for any small omega, this integral is just a kind Riemann integrals, where it simply considered t just as a variable of integration. Therefore, you can determine these kind of integrals, integral Xt of omega dt, let me say in classical sense, as a limit of the sums Xtk-1, multiplied by (tk -tk-1). K is from 0 to n, and the points t0, t1 and so on tn, are just division of the interval a b into n small sub-intervals. Nevertheless, we should pay attention to the fact that Xt is a stochastic process. And therefore this limit, of course with a maximum of tk- tk- 1 tends into 0, but this figment should be understood in a little bit an unusual way. Namely, you should think of this limit as a limit of a mean's squared sense. That is, you should say that this limit is understood as a full link sense. You consider this sum, and then the sum converges to the integral from a to b Xtdt, in the mean squared sense. And this means that the mathematical expectation of the square of the difference between these two objects, should converge to 0 as a maximum of tk- tk- 1 tends to 0. The main reason of the introduction of these kind of conversions is the full link. If we take these limit exactly in this sense, then the following theory holds. If the mathematical expectation of the process Xt is a continuous function as a function of t, and a covariance function of this process is continuous as a function of t and s, then the integral from a to b Xtdt exists. And this theorem is very useful because most processes Xt, posses these two properties. So mathematical expectations continuous, and it's for a covariance function, it is continuous or it is discontinuous only in finite amount of points. And if it is so, you can just present this integral as the sum of integrals over smaller intervals, such that the process Xt continues covariance function within each interval. And to know that if these conditions are fulfilled, then the further analysis of this integral is rather simple. For instance if you consider the mathematical expectation of the integral a, b Xt dt, you can just change the places of the mathematical expectation and the integral. And this change is due to the fact that mathematical expectation is also a kind of integral. It's an integral with respect to the probability measure. And therefore, you can use a so called Fubini's theorem and conclude that this mathematical expectation is equal to the integral from a to b, mathematical expectation of Xt dt. Similarly, one can show that the variance of this integral. Is equal to the double integral, both integrals from a to b, of the convergence function, as the point t,s, dtds. This is also a consequence of the fact that you can change the places of mathematical expectations and the integrals. So, I guess that the complete picture or the situation with these integrals is clear, basically everything is quite clear here, and basically you can think of these integrals in the following way. So you fix some omega, and for any omega you compute this integral as a Riemann integral. So one other thing which is important here, is that this limit should be understood as a mean squared sense. Nevertheless, if we now proceed to more difficult types of integrals, for instance to this integral, this situation drastically changes. And in the next subsection, I would like to give you an intuition how one should define these kind of integrals. [MUSIC]