Before I give the formal definition of stochastic process, I would like to recall the most important notion in the probability theory namely probability space. If you're familiar with this notion, I'd advise you to skip this section. On the other side, if you're not confident that you know everything about this, please follow this lecture. So I would like to draw a following table. Under the first column, I will write the general theory, and in the second column, I will show what the general theory yield. If I speak about the well known Bernoulli scheme. Bernoulli scheme is exactly tossing a coin. So basically you have two outcomes, one and zero. For instance a head and a tail. It is very common to associate one with success and zero is failure. Afterwards, you could repeat this experiment many times and finally get the sequence A1 and so on, an, where all of those numbers are either zero or one. And finally, in the third column, I would like to write the results for those experiment, when you just throw a coin at an interval from zero to one. So, we have three columns: general theory, Bernoulli scheme, and throwing a coin to a point at an interval from zero to one. Okay, first of all, we have the element omega in the probability space. Omega is known as a sample space and this is basically a space of any nature which includes all possible outcomes of an experiment. The Bernoulli scheme, this omega consists on two points namely zero and 1, and if we repeat the experiment many times, then this is a set of all possible vectors of size M, where each element is either zero or one. In this case, the amount of elements in omega is equal to two power n. In this example, when the similar coin at point between zero and one, a set of possible outcomes is just an interval from zero to one. Okay everything is clear at the moment. Now a bit more complicated thing is F. F is sigma algebra. Sigma algebra is a set of subsets omega with the following properties. Let me show these properties here. First of all, omega should be included in F. Secondly, F should be close on the complement. This means that if subset A is an F, then the complement to A, that is omega without A should be also in F. And thirdly, F should be close under countable units. That is if we have A1 and so on, an and maybe, this infinite but countable amount of sets from F, then its unit should be also in F. Sigma algebra is used for answering all questions related to the experiment. For instance, here in the Bernoulli scheme, as well as in all other [inaudible] cases, we should include all possible subsets of omega into F. In this case, F is known as powerset, the amount of elements in F is equal to two raised to the power amount of elements in omega, and in this particular case, it is equal to two to the power of n. As far as second example, one might ask what is a probability that a point is in some close interval of a beta. This interval is a sub-interval of the interval zero, one, and therefore, it seems to be very logic to include into F all close intervals alpha beta. But if you look effectively, in the definition of sigma algebra, we realize that all intervals like alpha, beta. Alpha is not included, but beta is included. We should also include F, and therefore, all elements like alpha beta when both alpha and beta are not included in the interval for the F. And all elements which consist only on one point beta should be also in F. And all of these considerations leads us to the conclusion that all elements of this type and also all countable units of such elements should be included in F and the minimum F which involves all closed intervals that are the same or open intervals is known as Borel sigma algebra. I have called this sigma algebra minimal, in the sense that all other sigma algebras which include all closed intervals should include also the whole Borel sigma algebra. Okay, this is the second element of the sigma algebra, and therefore, the third one is P which is called the probability measure. The probability measure should satisfy the following two conditions. First of all, the probability measure of the whole sample space omega should be equal to one, and secondly, if you have a countable set or subsets, A1, A2, and so on from F, there is a probability of Z unit as the case when they are not intersectable , that is AI intersected with AJ is equal to the intersect if I is not equal to J, and should be equal to the sum of the probabilities A I. So generally speaking for a view to measure is a function from F into the interval from zero to one. In the first example, where we considered the Bernoulli scheme, we just introduce the probability of one and zero, for instance, probability of one is equal to P, the probability of zero is equal to one minus P. As for the second example, we can say that the probability of the interval alpha beta is equal to beta minus alpha. These three elements form a probability space and actually this is a very important notion, not only in the context of probability theory, but also for stochastic processes.
