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Brief Introduction to Probability
Prof. Kuo works on three areas in stochastic analysis:
[1] White noise analysis, (2) Stochastic differential equations, and
(3) Probability theory on infinite dimensional spaces.
For the third area, he wrote a very popular book in 1975
``Gaussian measures in Banach spaces'' (Lecture Notes in Math.,
volume 463, Springer-Verlag.) This book has influenced much of
the research in infinite dimensional analysis. For the second
area, he is currently writing a book with M. de Faria and L. Streit
``Stochastic integration with applications.'' The book is expected
to be published in the year 2000. For the first area, he has made
fundamental contributions. He has coauthored with T. Hida, J.
Potthoff, and L. Streit the book ``White Noise: An Infinite
Dimensional Calculus'' (Kluwer Academic Publishers, 1993.) More
recently he wrote a book in 1996 ``White Noise Distribution
Theory'' (CRC Press, Boca Raton) to give a friendly presentation of
white noise analysis. Kuo's recent work is concentrated in the
theory of white noise. This theory was initiated by T. Hida in
1975. It is nowadays regarded as an infinite dimensional
distribution theory and has applications to physics, quantum
probability, stochastic integration, biology, control theory, among
others.
Below is a brief introduction to white noise theory and some
crucial ideas to give its mathematical meaning.
1. What is white noise?
White noise is a sound with equal intensity at all frequencies
within a broad band. Rock music, the roar of a jet engine, and
the noise at a stock market are examples of white noise. We
use the word ``white'' to describe this kind of noise because
of its similarity to ``white light'' which is made up of all
different colors (frequencies) of light combined together. In
applied science white noise is often taken as a mathematical
idealization of phenomena involving sudden and extremely large
fluctuations.
2. White noise as the derivative of a Brownian motion
White noise can be thought of as the derivative of a Brownian
motion. But what is a Brownian motion? As is well-known,
Robert Brown made microscopic observations in 1827 that small
particles contained in the pollen of plants, when immersed in
a liquid, exhibit highly irregular motions. This highly
irregular motion is called Brownian motion. Mathematically,
a Brownian motion is a continuous stationary stochastic process
B(t) having independent increments and for each t, B(t) is a
Gaussian random variable with mean 0 and variance t. It can be
shown that B(t) is nowhere differentiable, a mathematical fact
explaining the highly irregular motions that Robert Brown
observed. This means that white noise, being thought of as the
derivative dB(t)/dt of B(t), does not exist in the ordinary
sense.
3. White noise as a generalized stochastic process.
In order to motivate a mathematical definition of white noise,
we need a comparison between functions and stochastic processes.
An (ordinary) function is a function f(t) of a real number t. A
generalized function is a function f[u] of a test function u.
An (ordinary) stochastic process is a function X(t) of t such
that for each t, X(t) is a random variable. Thus a generalized
stochastic process is a function X[u] of u such that for each
u, X[u] is a random variable. White noise is defined as a
generalized stochastic process X[u] such that for each u, the
random variable X[u] is Gaussian with mean 0 and variance the
integral of u-square.
4. White noise as an infinite dimensional generalized
function.
Defining white noise as a generalized stochastic process is not
so satisfactory because its sample path property is lost and
nonlinear functionals of white noise cannot be defined in a
unified way. To overcome these difficulties, T. Hida introduced
in 1975 the theory of white noise. In this theory, for each t,
the white noise dB(t)/dt is a generalized function on an infinite
dimensional space. Not only dB(t)/dt, but also all derivatives
of Brownian motion are generalized functions on the same space.
5. An application of white noise to stochastic integration.
Informal integrals involving white noise had already appeared
before Ito introduced the stochastic integral (called Ito's
integral nowadays) in 1944. Ito combined dB(t)/dt with dt to
get dB(t) as an integrator and defined stochastic integrals
of nonanticipating stochastic processes with respect to B(t).
The celebrated Ito's formula is the ordinary chain rule plus
a correction term. It has applications to almost all applied
sciences involving sudden and extremely large fluctuations.
For example, P. Samuelson in 1970 and R. Merton and M. Scholes
in 1997 won Nobel Prizes in economics for their works where
Ito's theory plays an essential role.
From the white noise point of view, we do not need to combine
dB(t)/dt and dt together. In fact, recent applications of white
noise to quantum probability show that it is better not to do
so. An important fact for white noise integration is that we do
not need to assume the nonanticipating property for integrands.
For more information, see Kuo's 1996 CRC Press book.
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