Well, this is a very interesting property of the covariance function.

And of course not all functions depending on two variables has

this property and this is a very special thing which

is true only for covariance functions.

And the origins of this statement is as the following: So,

actually if the covariance function is continuous at some point t_0 t_0,

then the process X_t

is continuous at t_0 is a mean square sense,

that is mathematical expectation of X_t minus X_t_0

converges to zero as t tends to t_0.

On the other side,

if the process X_t is continuous as

the stochastic process in points t_0 and s_0,

then, the function K at the point t_0 and s_0 is also continuous.

So, we conclude the following: if

the process K_t_s is continuous at any point t_0 t_0,

then from the first statement we get that X_t is

continuous in the mean square sense at any point t_0,

and from the second part we conclude that

the function K is continuous at any point t_0 s_0.

So, this statement is correct.

This means that we can check the second condition of the theorem only on the diagonals,

that is, on all points such as the first and second coordinate coincide.

Let me now discuss the main properties of the Stochastic integral,

so from now we'll assume that the integral exist,

and for most cases this is exactly so because almost all

of our examples are such that both of these conditions are satisfied.

The first property, let me

consider the mathematical expectation of the Stochastic integral.

It turns out that the signs of

integral and mathematical expectation can be changed in the places,

so it's an integral expectation of X_t dt.

The origins of this statement is a well known Fubini theorem.

Mathematical expectations, also integral and

actually what we have here is a doubled integral,

one of which is integral with respect to some probabilistic measure,

and the second one is the Stochastic integral.

So, if you will apply for Fubini's theorem carefully,

you will get that this change is always possible.

But there's some logic if we consider mathematical expectation of

the integral X_t dt squared.

We can also conclude that the signs of expectation integral can be changed

to the following form: let me first represent this square as a double integral.

It's double integral X_t X_s dt ds.

And now we have exactly the same situation,

we have some integral and expectation and definitely we can

change the places and get this as a doubled integral,

mathematical expectation of X_t X_s dt ds.

Third property which a direct corollary from the first two is

that if you consider the variance of this integral,

of course you can represent variance as the mathematical expectation of the square of

this integral minus mathematical expectation

of the integral squared and if you combine this to formulas,

you will immediately get that this is a doubled integral of

the covariance function t_s dt ds.

Here I didn't write any bounds of this integral,

but I assumed that they were taken by

some closed interval a b and actually since the function K is symmetric,

we can also represent this integral as two integrals from a

to b and from a to

s K_t_s dt ds.

Well, these are the basic properties of Stochastic integrals

and as I said before it will be a separate lecture about this topic.

But to continue our studies,

to continue with the introduction of various properties,

we already need the notion of this integral.