At a specific time point , is a random variable with a specific density function. Given a specific , is a realization or a path of the process.
The stochastic process is clearly identified, when the system of its density functions is known. If for any the joint distribution function is known, the underlying stochastic process is uniquely determined.
Next we will define moment functions of real stochastic process. Here we will assume that the moments exist. If this is not the case, then the corresponding function is not defined.
In general depends on time , as, for example, processes with a seasonal or periodical structure or processes with a deterministic trend.
The autocovariance function is symmetric, i.e., . For the special case the result is the variance function . In general is dependent on as well as on . In the following we define the important concept of stationarity, which will simplify the moment functions in many cases.
A stochastic process is strictly stationary if for any and for all it holds that
For covariance stationary the term weakly stationary is often used. One should notice, however, that a stochastic process can be strictly stationary without being covariance stationary, namely then, when the variance (or covariance) does not exist. If the first two moment functions exist, then covariance stationary follows from strictly stationary.
The ACF is normalized on the interval [-1,1] and thus simplifies the interpretation of the autocovariance structure from various stochastic processes. Since the process is required to be covariance stationary, the ACF depends only on one parameter, the lag . Often the ACF is plotted as a function of , the so called correlogram. This is an important graphical instrument to illustrate linear dependency structures of the process.
Next we define two important stochastic processes which build the foundation for further modelling.
If is a process from i.i.d. random values with expectation 0 and finite variance, then it is a white noise. This special case is called independent white noise. On the contrary the white noise could have dependent third or higher moments, and in this case it would not be independent.
If is not zero, then the variables have a non-zero mean. We call it a random walk with a drift (see Section 4.1). In contrast to Section 4.3 we do not require here that the variables are independent. The random walk defined here is the boundary case for an AR(1) process introduced in Example 10.1 as . When we require, as in Section 4.3, that is independent white noise, then we will call a random walk with independent increments. Historically the random walk plays a special role, since at the beginning of the last century it was the first stochastic model to represent the development of stock prices. Even today the random walk is often assumed as an underlying hypothesis. However the applications are rejected in its strongest form with independent increments.
In order to determine the moment functions of a random walk, we will simply assume that the constant and the initial value are set to zero. Then through recursive substitutions we will get the representation
and for the variance function, since there is no correlation of , we obtain
The variance of the random walk increases linearly with time. For
the autocovariance function the following holds for :
For the autocovariance is thus strictly positive. Since the covariance function depends on time (and not only on the lags ), the random walk is not covariance stationary. For the autocorrelation function we obtain
As further illustration we consider a simple, but important stochastic process.
If is given for a particular (for example, the initial value of the process), the characteristics of the process are obviously dependent on this value. This influence disappears, however, over time, since we have assumed that and thus for . For there exists a limit in the sense of squared deviation, thus we can write the process as
In other words, the conditional distribution of a Markov process at a specific point in time is completely determined by the condition of the system at the previous date. One can also define Markov processes of higher order, from which the conditional distribution only depends on the finite number of past levels. Two examples for the Markov process of first order are the above mentioned random walk with independent variables and the AR(1) process with independent white noise.
Sometimes a fair game is also called a martingale difference. If is namely a martingale, then is a fair game.
After these mathematical definitions we arrive at the more econometric definitions, and in particular, at the term return. We start with a time series of prices and are interested in calculating the return between two periods.
The log return is defined for the case of continuous compounding.
For the average return over several periods we have
For small price changes the difference of the simple return and
log return is negligible. According to the Taylor approximation it