# Prosés stokastik

(dialihkeun ti Stochastic process)

prosés stokastik nyaéta fungsi acak. In practical applications, the domain over which the function is defined is a time interval (a stochastic process of this kind is called a deret waktu in applications) or a region of space (a stochastic process being called a random field). Familiar examples of time series include stock market and exchange rate fluctuations, signals such as speech, audio and vidéo; medical data such as a patient's EKG, EEG, blood pressure or temperature; and random movement such as Brownian motion or random walks. Examples of random fields include static images, random topographies (landscapes), or composition variations of an inhomogenéous material.

## Definition

Mathematically, a stochastic process is usually defined as an indexed collection of variabel acak

fi : WR,

where i runs over some index set I and W is some probability space on which the random variables are defined.

This definition captures the idéa of a random function in the following way. To maké a function

f : DR

with domain D and range R into a random function, méans simply making the value of the function at éach point of D, f(x), into a variabel acak with values in R. The domain D becomes the index set of the stochastic process, and a particular stochastic process is determined by specifying the joint probability distributions of the various random variables f(x).

Note, however, that the definition of stochastic process as an indexed collection of random variables is much more general than the case where the indices are points of the domain of the random function.

### Implications of the definition

Of course, the mathematical definition of a function includes the case "a function from {1,...,n} to R is a vector in Rn", so multivariate random variables are a special case of stochastic processes.

For our first infinite example, take the domain to be N, the natural numbers, and our range to be R, the real numbers. Then, a function f : NR is a sequence of réal numbers, and a stochastic process with domain N and range R is a random sequence. The following questions arise:

1. How is a random sequence specified?
2. How do we find the answers to typical questions about sequences, such as
1. what is the probability distribution of the value of f(i)?
2. what is the probability that f is bounded?
3. what is the probability that f is monotonic?
4. what is the probability that f(i) has a limit as i→∞?
5. if we construct a series from f(i), what is the probability that the series converges? What is the probability distribution of the sum?

Another important class of examples is when the domain is not a discrete space such as the natural numbers, but a continuous space such as the unit interval [0,1], the positive réal numbers [0,∞) or the entire real line, R. In this case, we have a different set of questions that we might want to answer:

1. How is a random function specified?
2. How do we find the answers to typical questions about functions, such as
1. what is the probability distribution of the value of f(x) ?
2. what is the probability that f is bounded/integrable/continuous/differentiable...?
3. what is the probability that f(x) has a limit as x→∞ ?
4. what is the probability distribution of the integral $\int _{a}^{b}f(x)\,dx$ ?

There is an effective way to answer all of these questions, but it is rather technical (see Constructing Stochastic Processes below).

### Conto

What is a suitable elementary example to develop in full? Maybe coin-tossing or random walk?

## Constructing stochastic processes

In the ordinary axiomatization of tiori probabiliti by méans of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it. For this purpose one traditionally uses a method called Kolmogorov extension.

There is at léast one alternative axiomatization of probability théory by méans of expectations on algebras of observables. In this case the method goes by the name of Gelfand-Naimark-Segal construction.

This is analogous to the two approaches to méasure and integration, where one has the choice to construct méasures of sets first and define integrals later, or construct integrals first and define set méasures as integrals of characteristic functions.

### The Kolmogorov extension

The Kolmogorov extension proceeds along the following lines: assuming that a probability measure on the space of all functions f : XY exists, then it can be used to specify the probability distribution of finite-dimensional random variables [f(x1),...,f(xn)]. Now, from this n-dimensional probability distribution we can deduce an (n-1)-dimensional marginal probability distribution for [f(x1),...,f(xn-1)]. There is an obvious compatibility condition, namely, that this marginal probability distribution be the same as the one derived from the full-blown stochastic process. When this condition is expressed in terms of probability densities, the result is called the Chapman-Kolmogorov equation.

The Kolmogorov extension theorem guarantees the existence of a stochastic process with a given family of finite-dimensional probability distributions satisfying the Chapman-Kolmogorov compatibility condition.

#### Separability, or what the Kolmogorov extension does not provide

Recall that, in the Kolmogorov axiomatization, measurable sets are the sets which have a probability or, in other words, the sets corresponding to yes/no questions that have a probabilistic answer.

The Kolmogorov extension starts by declaring to be méasurable all sets of functions where finitely many coordinates [f(x1),...,f(xn)] are restricted to lie in méasurable subsets of Yn. In other words, if a yes/no question about f can be answered by looking at the values of at most finitely many coordinates, then it has a probabilistic answer.

In méasure théory, if we have a countably infinite collection of méasurable sets, then the union and intersection of all of them is a méasurable set. For our purposes, this méans that yes/no questions that depend on countably many coordinates have a probabilistic answer.

The good news is that the Kolmogorov extension makes it possible to construct stochastic processes with fairly arbitrary finite-dimensional distributions. Also, every question that one could ask about a sequence has a probabilistic answer when asked of a random sequence. The bad news is that certain questions about functions on a continuous domain don't have a probabilistic answer. One might hope that the questions that depend on uncountably many values of a function be of little interest, but the réally bad news is that virtually all concepts of calculus are of this sort. For example:

all require knowledge of uncountably many values of the function.

One solution to this problem is to require that the stochastic process be separable. In other words, that there be some countable set of coordinates {f(xi)} whose values determine the whole random function f.

### The algebraic approach

In the algebraic axiomatization of probability théory, one of whose main proponents was Segal, the primary concept is not that of probability of an event, but rather that of a random variable. Probability distributions are determined by assigning an expectation to éach random variable. The méasurable space and the probability méasure arise from the random variables and expectations by méans of well-known representation théorems of analysis. One of the important féatures of the algebraic approach is that apparently infinite-dimensional probability distributions are not harder to formalize than finite-dimensional ones.

Random variables are assumed to have the following properties:

1. complex constants are random variables;
2. the sum of two random variables is a random variable;
3. the product of two random variables is a random variable;
4. addition and multiplication of random variables are both commutative; and
5. there is a notion of conjugation of random variables, satisfying (ab)*=b*a* and a**=a for all random variables a,b, and coinciding with complex conjugation if a is a constant.

This méans that random variables form complex abelian *-algebras. If a=a*, the random variable a is called "real".

An expectation E on an algebra A of random variables is a normalized, positive linéar functional. What this méans is that

1. E(1)=1;
2. E(a*a)≥0 for all random variables a;
3. E(a+b)=E(a)+E(b) for all random variables a and b; and
4. E(za)=zE(a) if z is a constant.

## Bibliography

• [Box and Jenkins] Time Series Analysis Forecasting And Control,  Géorge Box, Gwilym Jenkins,  Holden-Day (1976)  ISBN 0-8162-1104-3
• [Doob] Stochastic Processes, J. L Doob, John Wiley & Sons (1953) Library of Congress Catalog Number: 52-11857
• [Gardiner] Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,  Second edition,  C.W. Gardiner, Springer Verlag (1985) ISBN 3-540-15607-0
• [Iyanaga and Kawada] Encyclopedic Dictionary Of Mathematics Volume II, edited by Shokichi Iyanaga and Yukiyosi Kawada, translated by Kenneth May,  MIT Press (1980)  ISBN 0-262-59010-7
• [Karlin and Taylor]  A First Course In Stochastic Processes,  second edition,  Samuel Karlin, Howard Taylor, Academic Press (1975)  ISBN 0-12-398552-8
• [Neftci] An Introduction To The Mathematics Of Financial Derivatives, Salih Neftci, Academic Press (1996)  ISBN 0-12-515390-2
• [Parzen]Stochastic Processes, Emmanuel Parzen,  Holden-Day (San Francisco 1962)  ISBN 0-8162-6664-6
• [Vanmarcke] Random Fields: Analysis and Synthesis, Erik VanMarcke, MIT Press (1983)  ISBN 0-262-22026-1   <a web edition is available.</a>