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.
