Namespaces
Variants
Actions

Difference between revisions of "Diffusion process"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
A continuous [[Markov process|Markov process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d0323801.png" /> with transition density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d0323802.png" /> which satisfies the following condition: There exist functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d0323803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d0323804.png" />, known as the drift coefficient and the diffusion coefficient respectively, such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d0323805.png" />,
+
<!--
 +
d0323801.png
 +
$#A+1 = 72 n = 0
 +
$#C+1 = 72 : ~/encyclopedia/old_files/data/D032/D.0302380 Diffusion process
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d0323806.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
it being usually assumed that these limit relations are uniform with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d0323807.png" /> in each finite interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d0323808.png" /> and with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d0323809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238010.png" />. An important representative of this class of processes is the process of [[Brownian motion|Brownian motion]], which was originally considered as a mathematical model of diffusion processes (hence the name  "diffusion process" ).
+
{{MSC|60J60}}
  
If the transition density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238011.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238013.png" /> together with its derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238015.png" />, it is the fundamental solution of the differential equation
+
[[Category:Markov processes]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
A continuous [[Markov process|Markov process]]  $  X = X ( t) $
 +
with transition density  $  p ( s , x , t , y ) $
 +
which satisfies the following condition: There exist functions  $  a ( t , y ) $
 +
and  $  \sigma  ^ {2} ( t , x ) $,
 +
known as the drift coefficient and the diffusion coefficient respectively, such that for any  $  \epsilon > 0 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238017.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
\left . \begin{array}{c}
 +
\int\limits _ {| y - x | = \epsilon }
 +
p ( t , x , t + \Delta t , y )  dy  = o ( \Delta t ) ,
 +
\\
 +
\int\limits _ {| y - x | \leq  \epsilon }
 +
( y - x ) p ( t , x , t + \Delta t , y )  dy  =
 +
\\
 +
= a ( t , x ) + o ( \Delta t ) ,
 +
\\
 +
\int\limits _ {| y - x | \leq  \epsilon }
 +
( y - x )  ^ {2} p ( t , x , t + \Delta t , y )  dy  =
 +
\\
 +
= \sigma  ^ {2} ( t , x ) + o ( \Delta t ),
 +
\end{array}
 +
 
 +
\right \}
 +
$$
 +
 
 +
it being usually assumed that these limit relations are uniform with respect to  $  t $
 +
in each finite interval  $  t _ {0} \leq  t \leq  t _ {1} $
 +
and with respect to  $  x $,
 +
$  - \infty < x < \infty $.
 +
An important representative of this class of processes is the process of [[Brownian motion|Brownian motion]], which was originally considered as a mathematical model of diffusion processes (hence the name "diffusion process" ).
 +
 
 +
If the transition density  $  p ( s , x , t , y ) $
 +
is continuous in  $  s $
 +
and  $  x $
 +
together with its derivatives  $  ( \partial  / \partial  x ) p ( s , x , t , y ) $
 +
and  $  ( \partial  ^ {2} / \partial  x  ^ {2} ) p ( s , x , t , y ) $,
 +
it is the fundamental solution of the differential equation
 +
 
 +
$$ \tag{2 }
 +
 
 +
\frac \partial {\partial  s }
 +
p ( s , x , t , y )  =  - a ( s , x )
 +
\frac \partial {\partial  x }
 +
p ( s , x , t , y ) +
 +
$$
 +
 
 +
$$
 +
-
 +
\frac{1}{2}
 +
\sigma  ^ {2} ( s , x )
 +
\frac{\partial  ^ {2} }{\partial  x  ^ {2} }
 +
p ( s , x , t , y ) ,
 +
$$
  
 
which is known as the backward Kolmogorov equation (cf. also [[Kolmogorov equation|Kolmogorov equation]]).
 
which is known as the backward Kolmogorov equation (cf. also [[Kolmogorov equation|Kolmogorov equation]]).
  
In the homogeneous case, when the drift coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238018.png" /> and the diffusion coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238019.png" /> are independent of the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238020.png" />, the backward Kolmogorov equation for the respective transition density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238021.png" /> has the form
+
In the homogeneous case, when the drift coefficient $  a ( t , x ) = a ( x) $
 +
and the diffusion coefficient $  \sigma  ^ {2} ( t , x ) = \sigma  ^ {2} ( x) $
 +
are independent of the time $  t $,  
 +
the backward Kolmogorov equation for the respective transition density $  p ( s , x , t , y ) = p ( t - s , x , y ) $
 +
has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238022.png" /></td> </tr></table>
+
$$
  
If the transition density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238023.png" /> has a continuous derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238026.png" /> such that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238028.png" /> are continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238029.png" />, it is the fundamental solution of the differential equation
+
\frac \partial {\partial  t }
 +
p ( t , x , y )  = a ( x)
 +
\frac \partial {\partial  x }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
p ( t , x , y ) +
 +
\frac{1}{2}
 +
\sigma  ^ {2} ( x)
 +
\frac{\partial  ^ {2} }{
 +
\partial  x  ^ {2} }
 +
p ( t , x , y ) .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238031.png" /></td> </tr></table>
+
If the transition density  $  p ( s , x , t , y ) $
 +
has a continuous derivative  $  ( \partial  / \partial  t ) p ( s , x , t , y ) $
 +
in  $  t $
 +
and  $  y $
 +
such that the functions  $  ( \partial  / \partial  y ) [ a ( t , y ) p ( s , x , t , y ) ] $
 +
and  $  ( \partial  ^ {2} / \partial  y  ^ {2} ) [ \sigma  ^ {2} ( t , y ) p ( s , x , t , y )] $
 +
are continuous in  $  y $,
 +
it is the fundamental solution of the differential equation
  
known as the [[Fokker–Planck equation|Fokker–Planck equation]], or the forward Kolmogorov equation. The differential equations (2) and (3) for the probability density are the fundamental analytic objects of study of diffusion processes. There is also another, purely  "probabilistic" , approach to diffusion processes, based on the representation of the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238032.png" /> as the solution of the Itô stochastic differential equation
+
$$ \tag{3 }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238033.png" /></td> </tr></table>
+
\frac \partial {\partial  t }
 +
p ( s , x , t , y )  = -  
 +
\frac \partial {\partial  y
 +
}
 +
[ a ( t , y ) p ( s , x , t , y ) ] +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238034.png" /></td> </tr></table>
+
$$
 +
+
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238035.png" /> is the standard process of Brownian motion. Roughly speaking, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238036.png" /> is considered to be connected with some Brownian motion process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238037.png" /> in such a way that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238038.png" />, then the increment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238039.png" /> during the next period of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238040.png" /> is
+
\frac{1}{2}
 +
 +
\frac{\partial  ^ {2} }{\partial  y  ^ {2} }
 +
[ \sigma
 +
^ {2} ( t , y ) p ( s , x , t , y )] ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238041.png" /></td> </tr></table>
+
known as the [[Fokker–Planck equation|Fokker–Planck equation]], or the forward Kolmogorov equation. The differential equations (2) and (3) for the probability density are the fundamental analytic objects of study of diffusion processes. There is also another, purely "probabilistic" , approach to diffusion processes, based on the representation of the process  $  X ( t) $
 +
as the solution of the Itô stochastic differential equation
 +
 
 +
$$
 +
d X ( t)  =  a ( t , X ( t) )  d t + \sigma ( t , X ( t) )  d Y ( t) ,
 +
$$
 +
 
 +
$$
 +
X ( t)  = X ( t _ {0} ) + \int\limits _ {t _ {0} } ^ { t }  a ( s , X ( s)
 +
)  d s + \int\limits _ {t _ {0} } ^ { t }  \sigma ( s , X ( s) )  d Y ( s) ,
 +
$$
 +
 
 +
where  $  Y ( t) $
 +
is the standard process of Brownian motion. Roughly speaking,  $  X ( t) $
 +
is considered to be connected with some Brownian motion process  $  Y ( t) $
 +
in such a way that if  $  X ( t) = x $,
 +
then the increment  $  \Delta X ( t) = X ( t + \Delta t ) - X ( t) $
 +
during the next period of time  $  \Delta t $
 +
is
 +
 
 +
$$
 +
\Delta X ( t)  \sim  a ( t , x ) \Delta t + \sigma ( t , x ) \Delta Y ( t) .
 +
$$
  
 
If this asymptotic relation is understood in the sense that
 
If this asymptotic relation is understood in the sense that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238042.png" /></td> </tr></table>
+
$$
 +
{\mathsf E} \{ \Delta X ( t) - ( a ( t , x ) \Delta t - \sigma ( t , x )
 +
\Delta Y ( t) ) \mid  X ( t) = x \}  = o ( \Delta t ) ,
 +
$$
 +
 
 +
$$
 +
{\mathsf E} \{ \Delta X ( t) - ( a ( t , x ) \Delta t + \sigma ( t , x
 +
) \Delta Y ( t) )  ^ {2} \mid  X ( t) = x \}  = o ( \Delta t ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238043.png" /></td> </tr></table>
+
where  $  o ( \Delta t ) $
 +
are magnitudes of the same type as in equations (1), the  $  X ( t) $
 +
under consideration will constitute a diffusion process in the sense of this definition as well.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238044.png" /> are magnitudes of the same type as in equations (1), the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238045.png" /> under consideration will constitute a diffusion process in the sense of this definition as well.
+
Multi-dimensional diffusion process is the name usually given to a continuous Markov process  $  X ( t) = \{ X _ {1} ( t) \dots X _ {n} ( t) \} $
 +
in an  $  n $-
 +
dimensional vector space  $  E  ^ {n} $
 +
whose transition density  $  p( s , x , y ) $
 +
satisfies the following conditions: For any  $  \epsilon > 0 $,
  
Multi-dimensional diffusion process is the name usually given to a continuous Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238046.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238047.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238048.png" /> whose transition density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238049.png" /> satisfies the following conditions: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238050.png" />,
+
$$
 +
\int\limits _ {| y - x | > \epsilon } p ( t , x , t + \Delta t , y )
 +
dy  = o ( \Delta t ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238051.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {| y - x | \leq  \epsilon } ( y _ {k} - x _ {k} ) p ( t , x , t + \Delta t , y )  d y =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238052.png" /></td> </tr></table>
+
$$
 +
= \
 +
a _ {k} ( t , x ) \Delta t + o ( \Delta t ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238053.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {| y - x | \leq  \epsilon } ( y _ {k} - x _ {k} ) ( y _ {j} - x _ {j} ) p ( t , x , t + \Delta t , y )  d y =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238054.png" /></td> </tr></table>
+
$$
 +
= \
 +
2 b _ {kj} ( t , x ) \Delta t + o ( \Delta t ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238055.png" /></td> </tr></table>
+
$$
 +
k , j  = 1 \dots n ,\  x  = ( x _ {1} \dots x _ {n} ) ,\  y  = ( y _ {1} \dots y _ {n} ) .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238056.png" /></td> </tr></table>
+
The vector  $  a = \{ a _ {1} ( t , x ) \dots a _ {n} ( t , x ) \} $
 +
characterizes the local drift of the process  $  \xi ( t) $,
 +
and the matrix  $  \sigma  ^ {2} = \| 2 b _ {kj} ( t , x ) \| $,
 +
$  k , j = 1 \dots n $,
 +
characterizes the mean square deviation of the random process  $  \xi ( t) $
 +
from the initial position  $  x $
 +
in a small period of time between  $  t $
 +
and  $  t + \Delta $.
  
The vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238057.png" /> characterizes the local drift of the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238058.png" />, and the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238060.png" />, characterizes the mean square deviation of the random process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238061.png" /> from the initial position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238062.png" /> in a small period of time between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238064.png" />.
+
Subject to certain additional restrictions, the transition density  $  p ( s , x , t , y ) $
 +
of a multi-dimensional diffusion process satisfies the forward and backward Kolmogorov differential equations:
  
Subject to certain additional restrictions, the transition density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238065.png" /> of a multi-dimensional diffusion process satisfies the forward and backward Kolmogorov differential equations:
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238066.png" /></td> </tr></table>
+
\frac{\partial  p }{\partial  s }
 +
  = - \sum _ {k = 1 } ^ { n }  a _ {k} ( s , x )
 +
\frac{\partial  p }{\partial  x _ {k} }
 +
- \sum _ {k , j = 1
 +
} ^ { n }  b _ {kj} ( s , x )
 +
\frac{\partial  ^ {2} p }{\partial  x _ {k} \partial  x _ {j} }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238067.png" /></td> </tr></table>
+
$$
  
A multi-dimensional diffusion process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238068.png" /> may also be described with the aid of Itô's stochastic differential equations:
+
\frac{\partial  p }{\partial  t }
 +
  =  - \sum _ {k = 1 } ^ { n }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238069.png" /></td> </tr></table>
+
\frac \partial {\partial  y _ {k} }
 +
[ a _ {k} ( t , y ) p ] + \sum
 +
_ {k , j = 1 } ^ { n } 
 +
\frac{\partial  ^ {2} }{\partial  y _ {k} \partial  y _ {j} }
 +
[ b _ {kj} ( t , y ) p ] .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238070.png" /> are mutually-independent Brownian motion processes, while
+
A multi-dimensional diffusion process  $  X ( t) $
 +
may also be described with the aid of Itô's stochastic differential equations:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238071.png" /></td> </tr></table>
+
$$
 +
d X _ {k} ( t)  = a _ {k} ( t , X ( t) )  dt + \sum _ {j = 1 } ^ { n }  \sigma _ {kj} ( t , X ( t) )  d Y _ {j} ( t) ,
 +
$$
  
are the eigen vectors of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032380/d03238072.png" />.
+
where  $  Y _ {1} ( t) \dots Y _ {n} ( t) $
 +
are mutually-independent Brownian motion processes, while
  
====References====
+
$$
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.I. Gikhman,   A.V. Skorokhod,   "Introduction to the theory of random processes" , Saunders (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. Gikhman,   A.V. Skorokhod,  "Stochastic differential equations and their applications" , Springer  (1972)  (Translated from Russian)</TD></TR></table>
+
\sigma _ {j}  \{ \sigma _ {1j} ( t , x ) \dots \sigma _ {nj} ( t ,\
 +
x ) \} ,\ j = 1 \dots n ,
 +
$$
  
 +
are the eigen vectors of the matrix  $  \sigma  ^ {2} = \| 2 b _ {kj} ( t , x ) \| $.
  
 +
====References====
 +
{|
 +
|valign="top"|{{Ref|GS}}|| I.I. Gikhman, A.V. Skorokhod, "Introduction to the theory of random processes" , Saunders (1969) (Translated from Russian) {{MR|0247660}} {{ZBL|0573.60003}} {{ZBL|0429.60002}} {{ZBL|0132.37902}}
 +
|-
 +
|valign="top"|{{Ref|GS2}}|| I.I. Gikhman, A.V. Skorokhod, "Stochastic differential equations and their applications" , Springer (1972) (Translated from Russian) {{MR|0678374}} {{ZBL|0557.60041}}
 +
|}
  
 
====Comments====
 
====Comments====
Line 82: Line 255:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Ikeda,   S. Watanabe,   "Stochastic differential equations and diffusion processes" , North-Holland &amp; Kodansha (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"D.W. Stroock,   S.R.S. Varadhan,   "Multidimensional diffusion processes" , Springer (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"L. Arnold,   "Stochastische Differentialgleichungen" , R. Oldenbourg (1973) (Translated from Russian)</TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|IW}}|| N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland &amp; Kodansha (1981) {{MR|0637061}} {{ZBL|0495.60005}}
 +
|-
 +
|valign="top"|{{Ref|SV}}|| D.W. Stroock, S.R.S. Varadhan, "Multidimensional diffusion processes" , Springer (1979) {{MR|0532498}} {{ZBL|0426.60069}}
 +
|-
 +
|valign="top"|{{Ref|A}}|| L. Arnold, "Stochastische Differentialgleichungen" , R. Oldenbourg (1973) (Translated from Russian) {{MR|0443082}} {{ZBL|0266.60039}}
 +
|}

Latest revision as of 19:35, 5 June 2020


2010 Mathematics Subject Classification: Primary: 60J60 [MSN][ZBL]

A continuous Markov process $ X = X ( t) $ with transition density $ p ( s , x , t , y ) $ which satisfies the following condition: There exist functions $ a ( t , y ) $ and $ \sigma ^ {2} ( t , x ) $, known as the drift coefficient and the diffusion coefficient respectively, such that for any $ \epsilon > 0 $,

$$ \tag{1 } \left . \begin{array}{c} \int\limits _ {| y - x | = \epsilon } p ( t , x , t + \Delta t , y ) dy = o ( \Delta t ) , \\ \int\limits _ {| y - x | \leq \epsilon } ( y - x ) p ( t , x , t + \Delta t , y ) dy = \\ = a ( t , x ) + o ( \Delta t ) , \\ \int\limits _ {| y - x | \leq \epsilon } ( y - x ) ^ {2} p ( t , x , t + \Delta t , y ) dy = \\ = \sigma ^ {2} ( t , x ) + o ( \Delta t ), \end{array} \right \} $$

it being usually assumed that these limit relations are uniform with respect to $ t $ in each finite interval $ t _ {0} \leq t \leq t _ {1} $ and with respect to $ x $, $ - \infty < x < \infty $. An important representative of this class of processes is the process of Brownian motion, which was originally considered as a mathematical model of diffusion processes (hence the name "diffusion process" ).

If the transition density $ p ( s , x , t , y ) $ is continuous in $ s $ and $ x $ together with its derivatives $ ( \partial / \partial x ) p ( s , x , t , y ) $ and $ ( \partial ^ {2} / \partial x ^ {2} ) p ( s , x , t , y ) $, it is the fundamental solution of the differential equation

$$ \tag{2 } \frac \partial {\partial s } p ( s , x , t , y ) = - a ( s , x ) \frac \partial {\partial x } p ( s , x , t , y ) + $$

$$ - \frac{1}{2} \sigma ^ {2} ( s , x ) \frac{\partial ^ {2} }{\partial x ^ {2} } p ( s , x , t , y ) , $$

which is known as the backward Kolmogorov equation (cf. also Kolmogorov equation).

In the homogeneous case, when the drift coefficient $ a ( t , x ) = a ( x) $ and the diffusion coefficient $ \sigma ^ {2} ( t , x ) = \sigma ^ {2} ( x) $ are independent of the time $ t $, the backward Kolmogorov equation for the respective transition density $ p ( s , x , t , y ) = p ( t - s , x , y ) $ has the form

$$ \frac \partial {\partial t } p ( t , x , y ) = a ( x) \frac \partial {\partial x } p ( t , x , y ) + \frac{1}{2} \sigma ^ {2} ( x) \frac{\partial ^ {2} }{ \partial x ^ {2} } p ( t , x , y ) . $$

If the transition density $ p ( s , x , t , y ) $ has a continuous derivative $ ( \partial / \partial t ) p ( s , x , t , y ) $ in $ t $ and $ y $ such that the functions $ ( \partial / \partial y ) [ a ( t , y ) p ( s , x , t , y ) ] $ and $ ( \partial ^ {2} / \partial y ^ {2} ) [ \sigma ^ {2} ( t , y ) p ( s , x , t , y )] $ are continuous in $ y $, it is the fundamental solution of the differential equation

$$ \tag{3 } \frac \partial {\partial t } p ( s , x , t , y ) = - \frac \partial {\partial y } [ a ( t , y ) p ( s , x , t , y ) ] + $$

$$ + \frac{1}{2} \frac{\partial ^ {2} }{\partial y ^ {2} } [ \sigma ^ {2} ( t , y ) p ( s , x , t , y )] , $$

known as the Fokker–Planck equation, or the forward Kolmogorov equation. The differential equations (2) and (3) for the probability density are the fundamental analytic objects of study of diffusion processes. There is also another, purely "probabilistic" , approach to diffusion processes, based on the representation of the process $ X ( t) $ as the solution of the Itô stochastic differential equation

$$ d X ( t) = a ( t , X ( t) ) d t + \sigma ( t , X ( t) ) d Y ( t) , $$

$$ X ( t) = X ( t _ {0} ) + \int\limits _ {t _ {0} } ^ { t } a ( s , X ( s) ) d s + \int\limits _ {t _ {0} } ^ { t } \sigma ( s , X ( s) ) d Y ( s) , $$

where $ Y ( t) $ is the standard process of Brownian motion. Roughly speaking, $ X ( t) $ is considered to be connected with some Brownian motion process $ Y ( t) $ in such a way that if $ X ( t) = x $, then the increment $ \Delta X ( t) = X ( t + \Delta t ) - X ( t) $ during the next period of time $ \Delta t $ is

$$ \Delta X ( t) \sim a ( t , x ) \Delta t + \sigma ( t , x ) \Delta Y ( t) . $$

If this asymptotic relation is understood in the sense that

$$ {\mathsf E} \{ \Delta X ( t) - ( a ( t , x ) \Delta t - \sigma ( t , x ) \Delta Y ( t) ) \mid X ( t) = x \} = o ( \Delta t ) , $$

$$ {\mathsf E} \{ \Delta X ( t) - ( a ( t , x ) \Delta t + \sigma ( t , x ) \Delta Y ( t) ) ^ {2} \mid X ( t) = x \} = o ( \Delta t ) , $$

where $ o ( \Delta t ) $ are magnitudes of the same type as in equations (1), the $ X ( t) $ under consideration will constitute a diffusion process in the sense of this definition as well.

Multi-dimensional diffusion process is the name usually given to a continuous Markov process $ X ( t) = \{ X _ {1} ( t) \dots X _ {n} ( t) \} $ in an $ n $- dimensional vector space $ E ^ {n} $ whose transition density $ p( s , x , y ) $ satisfies the following conditions: For any $ \epsilon > 0 $,

$$ \int\limits _ {| y - x | > \epsilon } p ( t , x , t + \Delta t , y ) dy = o ( \Delta t ) , $$

$$ \int\limits _ {| y - x | \leq \epsilon } ( y _ {k} - x _ {k} ) p ( t , x , t + \Delta t , y ) d y = $$

$$ = \ a _ {k} ( t , x ) \Delta t + o ( \Delta t ) , $$

$$ \int\limits _ {| y - x | \leq \epsilon } ( y _ {k} - x _ {k} ) ( y _ {j} - x _ {j} ) p ( t , x , t + \Delta t , y ) d y = $$

$$ = \ 2 b _ {kj} ( t , x ) \Delta t + o ( \Delta t ) , $$

$$ k , j = 1 \dots n ,\ x = ( x _ {1} \dots x _ {n} ) ,\ y = ( y _ {1} \dots y _ {n} ) . $$

The vector $ a = \{ a _ {1} ( t , x ) \dots a _ {n} ( t , x ) \} $ characterizes the local drift of the process $ \xi ( t) $, and the matrix $ \sigma ^ {2} = \| 2 b _ {kj} ( t , x ) \| $, $ k , j = 1 \dots n $, characterizes the mean square deviation of the random process $ \xi ( t) $ from the initial position $ x $ in a small period of time between $ t $ and $ t + \Delta $.

Subject to certain additional restrictions, the transition density $ p ( s , x , t , y ) $ of a multi-dimensional diffusion process satisfies the forward and backward Kolmogorov differential equations:

$$ \frac{\partial p }{\partial s } = - \sum _ {k = 1 } ^ { n } a _ {k} ( s , x ) \frac{\partial p }{\partial x _ {k} } - \sum _ {k , j = 1 } ^ { n } b _ {kj} ( s , x ) \frac{\partial ^ {2} p }{\partial x _ {k} \partial x _ {j} } , $$

$$ \frac{\partial p }{\partial t } = - \sum _ {k = 1 } ^ { n } \frac \partial {\partial y _ {k} } [ a _ {k} ( t , y ) p ] + \sum _ {k , j = 1 } ^ { n } \frac{\partial ^ {2} }{\partial y _ {k} \partial y _ {j} } [ b _ {kj} ( t , y ) p ] . $$

A multi-dimensional diffusion process $ X ( t) $ may also be described with the aid of Itô's stochastic differential equations:

$$ d X _ {k} ( t) = a _ {k} ( t , X ( t) ) dt + \sum _ {j = 1 } ^ { n } \sigma _ {kj} ( t , X ( t) ) d Y _ {j} ( t) , $$

where $ Y _ {1} ( t) \dots Y _ {n} ( t) $ are mutually-independent Brownian motion processes, while

$$ \sigma _ {j} = \{ \sigma _ {1j} ( t , x ) \dots \sigma _ {nj} ( t ,\ x ) \} ,\ j = 1 \dots n , $$

are the eigen vectors of the matrix $ \sigma ^ {2} = \| 2 b _ {kj} ( t , x ) \| $.

References

[GS] I.I. Gikhman, A.V. Skorokhod, "Introduction to the theory of random processes" , Saunders (1969) (Translated from Russian) MR0247660 Zbl 0573.60003 Zbl 0429.60002 Zbl 0132.37902
[GS2] I.I. Gikhman, A.V. Skorokhod, "Stochastic differential equations and their applications" , Springer (1972) (Translated from Russian) MR0678374 Zbl 0557.60041

Comments

Instead of backward Kolmogorov equation and forward Kolmogorov equation are also finds simply backward equation and forward equation.

References

[IW] N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland & Kodansha (1981) MR0637061 Zbl 0495.60005
[SV] D.W. Stroock, S.R.S. Varadhan, "Multidimensional diffusion processes" , Springer (1979) MR0532498 Zbl 0426.60069
[A] L. Arnold, "Stochastische Differentialgleichungen" , R. Oldenbourg (1973) (Translated from Russian) MR0443082 Zbl 0266.60039
How to Cite This Entry:
Diffusion process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diffusion_process&oldid=12304
This article was adapted from an original article by Yu.A. Rozanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article