Diffusion process

2020 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