[SOUND] Now I would like to introduce a couple of stochastic models and show how the inter formula can be used for the analysis of these models. I would like to start with the well-known Black Scholes model. According to this model, the process Xt is a solution of the following stochastic differential equation dXt = Xt mu dt + Xt sigma dWt. Here, mu and sigma are just numbers, and we will assume that the parameter sigma is strongly larger than 0. Okay, this is a well-used formula. It is commonly used for describing stock prices, and the essence of this model called the number of pricing economics. Now, let me solve this stochastic differential equation using either formula. But before I do this, let me just shortly recall that this notation is by definition exactly the same as another one, as Xt = X0 + integral from 0 to t Xs mu ds + sigma integral from 0 to t Xs dWs. These two lines are exactly the same by definition. The context of the stochastic modeling, it's more common to work with differential form. And therefore I will mainly deal with this representation, and in order to do this I will also represent the formula in the differential form. Here is the exact formulation. Now I would like to use this formula with the function f(t,x) = log(x). What I have on the left-hand side is logarithm of Ht, and in this example, I will take Ht = Xt. Okay, we have, on the left-hand side, the differential of logarithm Xt is equal to. Now I would like to take the first derivative of the function f with respect to t, and it is equal to 0. Then the first derivative of the function f with respect to x is equal to 1 divided by x. Then I shall substitute Xt instead of X, and I will get 1 divided by Xt dXt. But this is not all because I should also take the second derivative with respect to X. And the second derivative is equal to 1 divided by X squared with sign minus. So it will be -1 divided by Xt squared multiplied by sigma t squared divided by 2. Sigma t squared is exactly the term which stands before Wt, this term. So I should consider the second power of this term, Xt squared sigma squared, and divide it by 2, dt. Okay, here you see that Xt squared vanishes, and as for the first term, I can do the following. I will just substitute the original stochastic differential equation instead of dXt. Therefore I will get d logarithm Xt = 1 divided by Xt (Xt mu dt + Xt sigma dWt ) and minus sigma square divided by 2 dt. Now, you see that Xt actually vanishes here, here, and here. And what we finally get is that d logarithmic Xt is equal to mu minus sigma squared divided by 2 dt, plus sigma multiplied by dWt. in other words, we can write the same formula in terms of integrals, and from there we will finally get that Xt = X0. Multiplied by the exponent (mu- sigma squared divided by 2)t + sigma Wt, and this is exactly the answer. So this is the solution of the following stochastic differential equation. So the inter formula helped us to solve this equation, and this formula basically can be used for modeling of stock prices. Let me finally mention that since Brownian motion has almost surely continuous trajectories, Xt has also almost surely continuous trajectories. And therefore, as this model can be used only for describing prices which are continuous in time. Nevertheless, this model is still widely used for describing stock prices. And I think that it's a very good example of the application of the inter formula. [MUSIC]