And now, I would like to introduce the notion of stochastic derivative, which says that a random process Xt is differentiable as a point t equal to t0 if the following inclement exists. So, we take Xt0 plus h minus Xt0 divide this difference by h, and then consider the limit of this expression as H goes to 0 as the means coerced sense, and if this limit exists and does equal to some random variable eta, so I would say is that eta is a derivative of the process Xt as a point t equal t0. And I will denote this derivative as Xt0 prime. Well, I guess that this definition is quite clear. Once more, this conversions means that mathematical expectation of Xt0 plus H minus Xt0 divided by H minus eta squared converges to 0 as H goes to 0. And the full length proposition completed, it characterizes this definition in terms of mathematical expectation as a covariance function. The following proposition holds. Let me assume that the process Xt has finite moment. This mathematical expectation of Xt squared is finite. Then, this process Xt is differentiable at the moment t equal to t0, if and only if, the following two conditions hold. The first condition that the medical expectation is differentiable as a moment t equal to t0, and second condition is that there exist a mixed derivative according to dt and ds of the covariance function. This mixed derivative should exist also at point t0, t0. Once more, the function is differentiable if and only if, both of these conditions are fulfilled. Let me provide a couple of examples. First of all, let me assume that the process Xt is a weekly stationary. This means that the mathematical expectation is a constant. And the covariance function is equal to the autocovariance function and the moment t minus s. The first condition from this proposition is definitely fulfilled because the constant is a differentiable function. As for the second one, it turns out that the second derivative of the function K has a moment t0, t0 is equal to minus second derivative of the function gamma as a point zero. So this condition can vary from related as follows. The weekly stationary process is differentiable if you only if, the second derivative of the autocovariance function exist as a point zero. In some cases, the second derivative exist, in some cases, not. Let me provide a couple of examples. For instance, there's a process Xt is such that it's autocovariance function is equal to exponent in the power minus alpha f so its the value of r. So the process Xt is not differentiable because the second derivative of this function as well as the first derivative is this function zero doesn't exist. So in this situation Xt is not differentiable. Another example, if gamma of r is equal to cosine of r. This function is differentiable at zero, and also as the second derivative exist. And therefore, in this case, the corresponding process Xt is differentiable. I mean, here is differentiable at any time t. This is the first example. Let me provide some further examples. The second example is the Brownian motion. It turns out that the Brownian motion is not differentiable at any time point. How to show this? As you know, the covariance function or the Brownian motion is equal to minimum between the arguments. And therefore, I should show that this mixed derivative doesn't exist. How to shows this? Well, this is simply a function of two arguments. And let me construct the following thing. I will take function K at t0 plus h, t0 minus K at t0, t0 divided by h. This object is equal to minimum between t0 and t0 plus H minus t0 divided by h. Next, this expression is equal to zero if h is positive. Because in this case, you have that minimum is equal to t0 and therefore, t0 minus t0 is equal to zero. And otherwise, if h is less than zero, then we have here t0 plus h minus t0 and get h and then you divide it by h and get one. So if h is turned into zero, this limit is equal to zero or one depending on zero h is smaller than zero. So the limit as h goes to zero doesn't exist at all. Okay. So we get that the function K is not differentiable at any point t0, t0, and therefore, the Brownian motion doesn't exist. Okay, this is a very typical example just because of the following situation. Actually, if Xt is any process with independent increments, and which takes the value zero at zero, then the covariance function of this process is equal to the variants of X at time moment minimum between t and s. Therefore, most properties of this type are not differentiable. Of course, one can construct a counter example when phrases variances as a constant. In this case, this process will be differentiable but this is like a degenerate case. The most typical situation when variants depends on minimum between t and s. And therefore, it's a covariance function is not differentiable. There is no mixed derivative of the covariance function and therefore according to this proposition, the correspondings to Xt process is not differentiable. Let me show why this formula holds. This is a very simple exercise because a covariance function is equal to covariance between Xt and Xs and defies human addition is that t is larger than s can represent this covariance as follows. This covariance between Xt minus Xs and Xs plus covariance between Xs and Xs. And since X0 is equal to zero almost surely, we can just imagine that here we have not Xs, but Xs minus X0. And therefore, what we basically have here are two increments or the process Xt with respect to non inter acceptable intervals. And therefore this covariance is equal to zero. And as for the second summant, this covariance between Xs and Xs, it is equal by definition to the variants of Xs. And if you now take into account that we have assumed that t is larger than s, then you can easily conclude that in general case, you will have here variants of X as a moment minimum between t and s. So this formula is actually true, and therefore, as I said, processes of this type are typically non differentiable. And this is exactly one of the essential differences between stochastic processes and real valued functions. Because most, let me say, well-used or most good functions are differentiable and it's a theory of stochastic processes, most processes as well as old label processes, at this point appears to be an essential difference between real valued functions and stochastic processes. Because as you know, most functions, [inaudible] functions are from differentiable. Let me say, most good or most well-used functions are differentiable. And in stochastic processes, most processes for instance, almost all label processes are not differentiable.