Difference between revisions of "Diffusion process"
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====References==== | ====References==== | ||
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− | + | |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}} | ||
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====Comments==== | ====Comments==== | ||
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====References==== | ====References==== | ||
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+ | |valign="top"|{{Ref|IW}}|| N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland & 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}} | ||
+ | |} |
Revision as of 17:45, 11 May 2012
2020 Mathematics Subject Classification: Primary: 60J60 [MSN][ZBL]
A continuous Markov process with transition density
which satisfies the following condition: There exist functions
and
, known as the drift coefficient and the diffusion coefficient respectively, such that for any
,
![]() | (1) |
it being usually assumed that these limit relations are uniform with respect to in each finite interval
and with respect to
,
. 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 is continuous in
and
together with its derivatives
and
, it is the fundamental solution of the differential equation
![]() | (2) |
![]() |
which is known as the backward Kolmogorov equation (cf. also Kolmogorov equation).
In the homogeneous case, when the drift coefficient and the diffusion coefficient
are independent of the time
, the backward Kolmogorov equation for the respective transition density
has the form
![]() |
If the transition density has a continuous derivative
in
and
such that the functions
and
are continuous in
, it is the fundamental solution of the differential equation
![]() | (3) |
![]() |
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 as the solution of the Itô stochastic differential equation
![]() |
![]() |
where is the standard process of Brownian motion. Roughly speaking,
is considered to be connected with some Brownian motion process
in such a way that if
, then the increment
during the next period of time
is
![]() |
If this asymptotic relation is understood in the sense that
![]() |
![]() |
where are magnitudes of the same type as in equations (1), the
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 in an
-dimensional vector space
whose transition density
satisfies the following conditions: For any
,
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
The vector characterizes the local drift of the process
, and the matrix
,
, characterizes the mean square deviation of the random process
from the initial position
in a small period of time between
and
.
Subject to certain additional restrictions, the transition density of a multi-dimensional diffusion process satisfies the forward and backward Kolmogorov differential equations:
![]() |
![]() |
A multi-dimensional diffusion process may also be described with the aid of Itô's stochastic differential equations:
![]() |
where are mutually-independent Brownian motion processes, while
![]() |
are the eigen vectors of the matrix .
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 |
Diffusion process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diffusion_process&oldid=23600