Wiener process

2020 Mathematics Subject Classification: Primary: 60J65 [MSN][ZBL]

A homogeneous Gaussian process $ X( t) $ with independent increments. A Wiener process serves as one of the models of Brownian motion. A simple transformation will convert a Wiener process into the "standard" Wiener process $ X( t) $, $ t \geq 0 $, for which

$$ X ( 0) = 0,\ \ {\mathsf E} ( X ( t) - X ( s)) = 0, $$

$$ {\mathsf D} [ X ( t) - X ( s)] = t - s,\ s \leq t. $$

For these average values and incremental variances, this is the only almost-surely continuous process with independent increments. In what follows, the Wiener process will be understood to be this process.

The Wiener process $ X( t) $, $ 0 \leq t \leq 1 $, can also be defined as the Gaussian process with zero expectation and covariance function

$$ B ( s, t) = \min ( s, t). $$

The Wiener process $ X= X( t) $, $ t \geq 0 $, may also be defined as the homogeneous Markov process with transition function

$$ P ( t, x, \Gamma ) = \int\limits _ \Gamma p ( t, x, y) dy, $$

where the transition density $ p( t, x, y) $ is the fundamental solution of the parabolic differential equation

$$ \frac{\partial p }{\partial t } = \ { \frac{1}{2} } \frac{\partial ^ {2} p }{\partial x ^ {2} } , $$

given by the formula

$$ p ( t, x, y) = \ \frac{1}{\sqrt {2 \pi t } } e ^ {-( y- x) ^ {2} / 2t } . $$

The transition function $ P( t, x, \Gamma ) $ is invariant with respect to translations in the phase space:

$$ P ( t, x + y, \Gamma ) = P ( t, x, \Gamma - y), $$

where $ \Gamma - y $ denotes the set $ \{ {z } : {z + y \in \Gamma } \} $.

The Wiener process is the continuous analogue of the random walk of a particle which, at discrete moments of time $ t= k \Delta t $( multiples of $ \Delta t $), is randomly displaced by a quantity $ \Delta X( t) $, independent of the past ( $ {\mathsf E} \Delta X( t) = 0 $, $ {\mathsf D} \Delta X( t) = \Delta t $); more precisely, if

$$ X ( t) = \sum _{k = 0 } ^ {m-1} \Delta X \left ( { \frac{k}{n} } \right ) + ( nt - m) \Delta X \left ( { \frac{m}{n} } \right ) ,\ \ 0 \leq t \leq 1, $$

is the random trajectory of the motion of such a particle on the interval $ [ 0, 1] $( where $ m= [ nt] $ is the integer part of $ nt $, $ X( t) = nt \Delta X ( 0) $ if $ 0 \leq t < 1/n $ and $ {\mathsf P} _ {n} $ is the corresponding probability distribution in the space of continuous functions $ x= x( t) $, $ 0 \leq t \leq 1 $), then the probability distribution $ {\mathsf P} $ of the trajectory of the Wiener process $ X( t) $, $ 0 \leq t \leq 1 $, is the limit (in the sense of weak convergence) of the distributions $ {\mathsf P} _ {n} $ as $ n \rightarrow \infty $.

As a function with values in the Hilbert space $ L _ {2} ( \Omega ) $ of all random variables $ X $ with $ {\mathsf E} X ^ {2} < \infty $, in which the scalar product is defined by the formula

$$ \langle X _ {1} , X _ {2} \rangle = {\mathsf E} X _ {1} X _ {2} , $$

the Wiener process $ X = X( t) $, $ 0 \leq t \leq 1 $, may be canonically represented as follows:

$$ X ( t) = \sum _ {k = 0 } ^ \infty z _ {k} \phi _ {k} ( t), $$

where $ z _ {k} $ are independent Gaussian variables:

$$ {\mathsf E} z _ {k} = 0,\ \ {\mathsf D} z _ {k} = \frac{1}{\left [ { \frac \pi {2} } ( 2k + 1) \right ] ^ {2} } , $$

and

$$ \phi _ {k} ( t) = \ \sin \left [ { \frac \pi {2} } ( 2k + 1) t \right ] ,\ \ k = 0, 1 \dots $$

are the eigenfunctions of the operator $ B $ defined by the formula

$$ B \phi ( t) = \int\limits _ { 0 } ^ { 1 } B ( s, t) \phi ( s) ds $$

in the Hilbert space $ L _ {2} [ 0, 1] $ of all square-integrable (with respect to Lebesgue measure) functions $ \phi $ on $ [ 0, 1] $.

Almost-all trajectories of the Wiener process have the following properties:

$$ {\lim\limits \sup } _ {h \rightarrow \infty } \ \frac{X ( h) }{\sqrt {2 h \mathop{\rm ln} \mathop{\rm ln} { \frac{1}{h} } } } = 1,\ \ X ( 0) = 0, $$

which is the law of the iterated logarithm;

$$ \lim\limits _ {h \rightarrow + 0 } \ \sup _ {0 \leq t \leq \delta - h } \ \frac{| X ( t+ h) - X ( t) | }{\sqrt {2 h \mathop{\rm ln} { \frac \delta {h} } } } = 1, $$

characterizing the modulus of continuity on $ [ 0, h] $; and

$$ \lim\limits _ {h \rightarrow \infty } \sum_{k = 0 } ^ {n-1} | \Delta X ( kh) | ^ {2} = \delta , $$

$$ h = \frac \delta {n} ,\ \Delta X ( t) = X ( t+ h) - X ( t). $$

When applied to the Wiener process $ X _ {1} ( t) = tX( 1/t) $, $ 0 \leq t < \infty $, the law of the iterated logarithm reads:

$$ {\lim\limits \sup } _ {t \rightarrow \infty } \frac{X ( t) }{\sqrt {2 t \mathop{\rm ln} \mathop{\rm ln} t } } = 1. $$

The distributions of the maximum $ \max _ {0 \leq s \leq t } X( s) $, of the time $ r _ {x} $ at which the Brownian particle first reaches a fixed point $ x > 0 $ and of the location $ \tau $ of the maximum $ \max _ {0 \leq s \leq t } X( s) $ give insight in the nature of the movement of a Brownian particle; these distributions are given by the following formulas:

$$ {\mathsf P} \left \{ \max _ {0 \leq s \leq t } X( s) \right \} = \frac{2}{\sqrt {2 \pi t } } \int\limits _ { x } ^ \infty e ^ {- u ^ {2} /2t } du , $$

$$ P \{ \tau _ {x} \geq t \} = P \left \{ \max _ {0 \leq s \leq t } X ( s) \leq x \right \} = $$

$$ = \ \sqrt { \frac{2} \pi } \int\limits _ { 0 } ^ { {x/ } \sqrt t } e ^ {- u ^ {2} /2 } d u , $$

while the simultaneous density of the maximum $ \max _ {0 \leq s \leq t } X( s) $ and its location $ \tau $ is given by:

$$ {\mathsf P} \{ \tau _ {x} \geq t \} = \ \sqrt { \frac{2} \pi } \int\limits _ { 0 } ^ { {x } / \sqrt t } e ^ {- u ^ {2} /2t } d u , $$

$$ 0 < s < t ,\ 0 \leq x < \infty . $$

(The laws of the Wiener process remain unchanged on transforming the phase space $ x \rightarrow - x $.) The joint distribution of the maximum point $ \tau $, $ 0 \leq \tau \leq t $, and of the maximum $ \max _ {0\leq s \leq t } X( s) $ itself has the probability density

$$ p ( s, x) = \ { \frac{1}{\pi \sqrt {s ( t- s) } } } { \frac{x}{s} } e ^ {- x ^ {2} / 2s } , $$

$$ 0 \leq s \leq t,\ 0 \leq x < \infty , $$

while the point $ \tau $ by itself (with probability one there is only one maximum on the interval $ 0 \leq s \leq t $) is distributed according to the arcsine law:

$$ {\mathsf P} \{ \tau \leq s \} = \ { \frac{2} \pi } \mathop{\rm arc} \sin \sqrt { \frac{s}{t} } ,\ \ 0 \leq s \leq t, $$

with the probability density:

$$ p ( s) = { \frac{1}{\pi \sqrt s( t- s) } } ,\ \ 0 \leq s \leq t. $$

The following properties of the Wiener process are readily deduced from the formulas given above. The Brownian trajectory is nowhere differentiable; on starting from any point $ x $ this trajectory crosses the "level" $ x $( returns to its initial point) infinitely many times with probability one, however short the time $ \delta $; the Brownian trajectory passes through all points $ x $( more precisely, $ \tau _ {x} < \infty $) with probability one (the most probable value of $ \tau _ {x} $ is of the order $ x ^ {2} $ for large $ x $); this trajectory, if considered on a fixed interval $ [ 0, t] $, tends to attain the extremal values near the end-points $ s= 0 $ and $ s= t $.

Since a Wiener process is a homogeneous Markov process, there exists an invariant measure $ Q( d x) $ for it, namely:

$$ Q ( A) \equiv \int\limits Q ( dx) P ( t, x, A), $$

which, since the transition function $ P( t, x, A) $ has been seen to be invariant, coincides with the Lebesgue measure on the real line: $ Q( d x) = d x $. The time $ T( A) $ which a Brownian particle spends in $ A $ between the times 0 and $ T $ is such that

$$ \frac{T ( A _ {1} ) }{T ( A _ {2} ) } \rightarrow \ \frac{Q ( A _ {1} ) }{Q ( A _ {2} ) } $$

as $ T \rightarrow \infty $, with probability one for any bounded Borel sets $ A _ {1} $ and $ A _ {2} $.

Wiener random fields, introduced by P. Lévy [L], are analogues of the Wiener process $ X = X( t) $ for a vector parameter $ t = ( t _ {1} \dots t _ {n} ) $.

References

[IM]	K. Itô, H.P. McKean jr., "Diffusion processes and their sample paths" , Springer (1974) MR0345224 Zbl 0285.60063
[PR]	Yu.V. Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) MR0251754
[L]	P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) MR0190953 Zbl 0137.11602
[P]	V.P. Pavlov, "Brownian motion" , Large Soviet Encyclopaedia , 4 (In Russian)

Comments

The Wiener process is more commonly referred to as Brownian motion in the Western literature. It is by far the most important construct in stochastic analysis. See [Du]–[RY] for up-to-date accounts of its properties. Of particular importance is the theory of local time. The occupation time of a Borel set $ B \subset \mathbf R $ on the interval $ [ 0, t] $ is:

$$ \Gamma _ {t} ( B) = \int\limits _ { 0 } ^ { t } I _ {B} ( X( s)) ds . $$

There exists an almost-surely jointly-continuous random field $ L ( t, x) $ for $ ( t, x) \in \mathbf R _ {+} \times \mathbf R $ such that

$$ \Gamma _ {t} ( B) = 2 \int\limits _ { B } L( t, x) dx; $$

$ L( t, x) $ is the local time at $ x $. For fixed $ x \in \mathbf R $, sample paths of the process $ t \mapsto L( t, x) $ are increasing and continuous but singular with respect to Lebesgue measure.

See also Markov process; Stochastic differential equation.

References

[Du]	R. Durrett, "Brownian motion and martingales in analysis", Wadsworth (1984) MR0750829 Zbl 0554.60075
[KS]	I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus", Springer (1988) MR0917065 Zbl 0638.60065
[RY]	D. Revuz, M. Yor, "Continuous martingales and Brownian motion", Springer (1990) MR1725357 MR1303781 MR1083357 Zbl 1087.60040 Zbl 0917.60006 Zbl 0804.60001 Zbl 0731.60002
[Dy]	E.B. Dynkin, "Markov processes", 1, Springer (1965) (Translated from Russian) MR0193671 Zbl 0132.37901
[F]	W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1968–1971)
[GS]	I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes", III, Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027
[H]	T. Hida, "Brownian motion", Springer (1980) MR0566333 MR0562914 Zbl 0432.60002 Zbl 0423.60063
[S]	F. Spitzer, "Principles of random walk", v. Nostrand (1964) MR0171290 Zbl 0119.34304
[Y]	J. Yeh, "Stochastic processes and the Wiener integral", M. Dekker (1973) MR0474528 Zbl 0277.60018
[Do]	J.L. Doob, "Classical potential theory and its probabilistic counterpart", Springer (1984) MR0731258 Zbl 0549.31001