Stochastic diffusion process in probability theory
In
probability theory , a McKean–Vlasov process is a
stochastic process described by a
stochastic differential equation where the coefficients of the
diffusion depend on the distribution of the solution itself.
[1]
[2] The equations are a model for
Vlasov equation and were first studied by
Henry McKean in 1966.
[3] It is an example of propagation of chaos , in that it can be obtained as a limit of a mean-field system of interacting particles: as the number of particles tends to infinity, the interactions between any single particle and the rest of the pool will only depend on the particle itself.
[4]
Definition
Consider a measurable function
σ
:
R
d
×
P
(
R
d
)
→
M
d
(
R
)
{\displaystyle \sigma :\mathbb {R} ^{d}\times {\mathcal {P}}(\mathbb {R} ^{d})\to {\mathcal {M}}_{d}(\mathbb {R} )}
where
P
(
R
d
)
{\displaystyle {\mathcal {P}}(\mathbb {R} ^{d})}
is the space of
probability distributions on
R
d
{\displaystyle \mathbb {R} ^{d}}
equipped with the
Wasserstein metric
W
2
{\displaystyle W_{2}}
and
M
d
(
R
)
{\displaystyle {\mathcal {M}}_{d}(\mathbb {R} )}
is the space of
square matrices of dimension
d
{\displaystyle d}
. Consider a measurable function
b
:
R
d
×
P
(
R
d
)
→
R
d
{\displaystyle b:\mathbb {R} ^{d}\times {\mathcal {P}}(\mathbb {R} ^{d})\to \mathbb {R} ^{d}}
. Define
a
(
x
,
μ
)
:=
σ
(
x
,
μ
)
σ
(
x
,
μ
)
T
{\displaystyle a(x,\mu ):=\sigma (x,\mu )\sigma (x,\mu )^{T}}
.
A stochastic process
(
X
t
)
t
≥
0
{\displaystyle (X_{t})_{t\geq 0}}
is a McKean–Vlasov process if it solves the following system:
[3]
[5]
X
0
{\displaystyle X_{0}}
has law
f
0
{\displaystyle f_{0}}
d
X
t
=
σ
(
X
t
,
μ
t
)
d
B
t
+
b
(
X
t
,
μ
t
)
d
t
{\displaystyle dX_{t}=\sigma (X_{t},\mu _{t})dB_{t}+b(X_{t},\mu _{t})dt}
where
μ
t
=
L
(
X
t
)
{\displaystyle \mu _{t}={\mathcal {L}}(X_{t})}
describes the
law of
X
{\displaystyle X}
and
B
t
{\displaystyle B_{t}}
denotes a
d
{\displaystyle d}
-dimensional
Wiener process . This process is non-linear, in the sense that the associated
Fokker-Planck equation for
μ
t
{\displaystyle \mu _{t}}
is a non-linear
partial differential equation .
[5]
[6]
Existence of a solution
The following Theorem can be found in.
[4]
Existence of a solution — Suppose
b
{\displaystyle b}
and
σ
{\displaystyle \sigma }
are
globally Lipschitz , that is, there exists a constant
C
>
0
{\displaystyle C>0}
such that:
|
b
(
x
,
μ
)
−
b
(
y
,
ν
)
|
+
|
σ
(
x
,
μ
)
−
σ
(
y
,
ν
)
|
≤
C
(
|
x
−
y
|
+
W
2
(
μ
,
ν
)
)
{\displaystyle |b(x,\mu )-b(y,\nu )|+|\sigma (x,\mu )-\sigma (y,\nu )|\leq C(|x-y|+W_{2}(\mu ,\nu ))}
where
W
2
{\displaystyle W_{2}}
is the
Wasserstein metric .
Suppose
f
0
{\displaystyle f_{0}}
has finite variance.
Then for any
T
>
0
{\displaystyle T>0}
there is a unique strong solution to the McKean-Vlasov system of equations on
0
,
T
{\displaystyle [0,T]}
. Furthermore, its law is the unique solution to the non-linear
Fokker–Planck equation :
∂
t
μ
t
(
x
)
=
−
∇
⋅
{
b
(
x
,
μ
t
)
μ
t
}
+
1
2
∑
i
,
j
=
1
d
∂
x
i
∂
x
j
{
a
i
j
(
x
,
μ
t
)
μ
t
}
{\displaystyle \partial _{t}\mu _{t}(x)=-\nabla \cdot \{b(x,\mu _{t})\mu _{t}\}+{\frac {1}{2}}\sum \limits _{i,j=1}^{d}\partial _{x_{i}}\partial _{x_{j}}\{a_{ij}(x,\mu _{t})\mu _{t}\}}
Propagation of chaos
The McKean-Vlasov process is an example of propagation of chaos .
[4] What this means is that many McKean-Vlasov process can be obtained as the limit of discrete systems of stochastic differential equations
(
X
t
i
)
1
≤
i
≤
N
{\displaystyle (X_{t}^{i})_{1\leq i\leq N}}
.
Formally, define
(
X
i
)
1
≤
i
≤
N
{\displaystyle (X^{i})_{1\leq i\leq N}}
to be the
d
{\displaystyle d}
-dimensional solutions to:
(
X
0
i
)
1
≤
i
≤
N
{\displaystyle (X_{0}^{i})_{1\leq i\leq N}}
are i.i.d with law
f
0
{\displaystyle f_{0}}
d
X
t
i
=
σ
(
X
t
i
,
μ
X
t
)
d
B
t
i
+
b
(
X
t
i
,
μ
X
t
)
d
t
{\displaystyle dX_{t}^{i}=\sigma (X_{t}^{i},\mu _{X_{t}})dB_{t}^{i}+b(X_{t}^{i},\mu _{X_{t}})dt}
where the
(
B
i
)
1
≤
i
≤
N
{\displaystyle (B^{i})_{1\leq i\leq N}}
are i.i.d
Brownian motion , and
μ
X
t
{\displaystyle \mu _{X_{t}}}
is the
empirical measure associated with
X
t
{\displaystyle X_{t}}
defined by
μ
X
t
:=
1
N
∑
1
≤
i
≤
N
δ
X
t
i
{\displaystyle \mu _{X_{t}}:={\frac {1}{N}}\sum \limits _{1\leq i\leq N}\delta _{X_{t}^{i}}}
where
δ
{\displaystyle \delta }
is the
Dirac measure .
Propagation of chaos is the property that, as the number of particles
N
→
+
∞
{\displaystyle N\to +\infty }
, the interaction between any two particles vanishes, and the random empirical measure
μ
X
t
{\displaystyle \mu _{X_{t}}}
is replaced by the deterministic distribution
μ
t
{\displaystyle \mu _{t}}
.
Under some regularity conditions,
[4] the mean-field process just defined will converge to the corresponding McKean-Vlasov process.
Applications
References
^ Des Combes, Rémi Tachet (2011).
Non-parametric model calibration in finance: Calibration non paramétrique de modèles en finance (PDF) (Doctoral dissertation). Archived from
the original (PDF) on 2012-05-11.
^ Funaki, T. (1984).
"A certain class of diffusion processes associated with nonlinear parabolic equations" . Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete . 67 (3): 331–348.
doi :
10.1007/BF00535008 .
S2CID
121117634 .
^
a
b
McKean, H. P. (1966).
"A Class of Markov Processes Associated with Nonlinear Parabolic Equations" .
Proc. Natl. Acad. Sci. USA . 56 (6): 1907–1911.
Bibcode :
1966PNAS...56.1907M .
doi :
10.1073/pnas.56.6.1907 .
PMC
220210 .
PMID
16591437 .
^
a
b
c
d Chaintron, Louis-Pierre; Diez, Antoine (2022).
"Propagation of chaos: A review of models, methods and applications. I. Models and methods" . Kinetic and Related Models . 15 (6): 895.
arXiv :
2203.00446 .
doi :
10.3934/krm.2022017 .
ISSN
1937-5093 .
^
a
b
c Carmona, Rene; Delarue, Francois; Lachapelle, Aime.
"Control of McKean-Vlasov Dynamics versus Mean Field Games" (PDF) . Princeton University .
^
a
b Chan, Terence (January 1994).
"Dynamics of the McKean-Vlasov Equation" . The Annals of Probability . 22 (1): 431–441.
doi :
10.1214/aop/1176988866 .
ISSN
0091-1798 .