Difference between revisions of "Wiener process"
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{{MSC|60J65}} | {{MSC|60J65}} | ||
[[Category:Markov processes]] | [[Category:Markov processes]] | ||
− | A homogeneous [[Gaussian process|Gaussian process]] | + | A homogeneous [[Gaussian process|Gaussian process]] $ X( t) $ |
+ | with independent increments. A Wiener process serves as one of the models of [[Brownian motion|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. | 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 | + | 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|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 | given by the formula | ||
− | + | $$ | |
+ | p ( t, x, y) = \ | ||
+ | |||
+ | \frac{1}{\sqrt {2 \pi t } } | ||
+ | |||
+ | e ^ {-( y- x) ^ {2} / 2t } . | ||
+ | $$ | ||
− | The transition function | + | 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 | + | where $ \Gamma - y $ |
+ | denotes the set $ \{ {z } : {z + y \in \Gamma } \} $. | ||
− | The Wiener process is the continuous analogue of the [[Random walk|random walk]] of a particle which, at discrete moments of time | + | The Wiener process is the continuous analogue of the [[Random walk|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 | + | 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 | + | 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 | + | 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 | + | 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 | and | ||
− | + | $$ | |
+ | \phi _ {k} ( t) = \ | ||
+ | \sin \left [ { | ||
+ | \frac \pi {2} | ||
+ | } ( 2k + 1) t \right ] ,\ \ | ||
+ | k = 0, 1 \dots | ||
+ | $$ | ||
− | are the eigenfunctions of the operator | + | 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 | + | 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: | 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; | which is the law of the iterated logarithm; | ||
− | + | $$ | |
+ | \lim\limits _ {h \rightarrow + 0 } \ | ||
+ | \sup _ {0 \leq t \leq \delta - h } \ | ||
− | characterizing the modulus of continuity on | + | \frac{| X ( t+ h) - X ( t) | }{\sqrt {2 h \mathop{\rm ln} { |
+ | \frac \delta {h} | ||
+ | } } } | ||
+ | = 1, | ||
+ | $$ | ||
+ | |||
+ | 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 | + | 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 | + | 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 | + | 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 | + | (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 | + | 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|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: | 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 | + | as $ T \rightarrow \infty $, |
+ | with probability one for any bounded Borel sets $ A _ {1} $ | ||
+ | and $ A _ {2} $. | ||
− | Wiener random fields, introduced by P. Lévy {{Cite|L}}, are analogues of the Wiener process | + | Wiener random fields, introduced by P. Lévy {{Cite|L}}, are analogues of the Wiener process $ X = X( t) $ |
+ | for a vector parameter $ t = ( t _ {1} \dots t _ {n} ) $. | ||
====References==== | ====References==== | ||
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====Comments==== | ====Comments==== | ||
− | The Wiener process is more commonly referred to as [[Brownian motion|Brownian motion]] in the Western literature. It is by far the most important construct in stochastic analysis. See {{Cite|Du}}–{{Cite|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 | + | The Wiener process is more commonly referred to as [[Brownian motion|Brownian motion]] in the Western literature. It is by far the most important construct in stochastic analysis. See {{Cite|Du}}–{{Cite|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|random field]] | + | There exists an almost-surely jointly-continuous [[Random field|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|Markov process]]; [[Stochastic differential equation|Stochastic differential equation]]. | See also [[Markov process|Markov process]]; [[Stochastic differential equation|Stochastic differential equation]]. |
Latest revision as of 16:32, 14 January 2024
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 |
Wiener process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_process&oldid=26975