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''for a process  $  X=( X _ {t} ) _ {t\geq } 0 $
with respect to a [[Wiener process|Wiener process]]  $  W = ( W _ {t} ) _ {t\geq } 0 $''

{{MSC|60H10}}

[[Category:Stochastic analysis]]

An equation of the form

$$ \tag{1 }
dX _ {t}  =  a( t, X)  dt + b( t, X)  dW _ {t} ,\  X _ {0}  =  \xi ,
$$

where  $  a( t, X) $
and  $  b( t, X) $
are non-anticipative functionals, and the random variable  $  \xi $
plays the part of the initial value. There are two separate concepts for a solution of a stochastic differential equation — strong and weak.

Let  $  ( \Omega , {\mathcal F} , {\mathsf P}) $
be a probability space with an increasing family of  $  \sigma $-
algebras  $  \mathbf F = ( {\mathcal F} _ {t} ) _ {t\geq } 0 $, 
and let  $  W = ( W _ {t} , {\mathcal F} _ {t} ) _ {t\geq } 0 $
be a Wiener process. One says that a continuous stochastic process  $  X = ( X _ {t} , {\mathcal F} _ {t} ) _ {t\geq } 0 $
is a strong solution of the stochastic differential equation (1) with drift coefficient  $  a( t, X) $, 
diffusion coefficient  $  b( t, X) $
and initial value  $  \xi $, 
if for every  $  t > 0 $
with probability one:

$$ \tag{2 }
X _ {t}  =  \xi + \int\limits _ { 0 } ^ { t }  a( s, X)  ds + \int\limits _ { 0 } ^ { t }  b( s, X)  dW _ {s} ,
$$

where it is supposed that the integrals in (2) are defined.

The first general result on the existence and uniqueness of a strong solution of a stochastic differential equation of the form

$$ \tag{3 }
dX _ {t}  =  a( t, X _ {t} )  dt + b( t, X _ {t} )  dW _ {t}  $$

was obtained by K. Itô. He demonstrated that if for every  $  t > 0 $
the functions  $  a( t, x) $
and  $  b( t, x) $
satisfy a Lipschitz condition with respect to  $  x $
and increase not faster than linearly, then a continuous solution  $  X = ( X _ {t} , {\mathcal F} _ {t} ) _ {t\geq } 0 $
of the equation (3) exists, and this solution is unique in the sense that if  $  Y = ( Y _ {t} , {\mathcal F} _ {t} ) _ {t\geq } 0 $
is another continuous solution, then

$$ 
{\mathsf P} \left \{ \sup _ { s\leq  } t  | X _ {s} - Y _ {s} | > 0
\right \}  =  0,\  t \geq  0.
$$

If  $  b( t, x) \equiv \textrm{ const } $, 
the measurability and boundedness of the drift coefficient (vector)  $  a( t, x) $
guarantees the existence and uniqueness of a strong solution of (3). The equation  $  dX _ {t} = a( t, X)  dt+  dW _ {t} $, 
generally speaking, does not have a strong solution for any bounded non-anticipative functional  $  a( t, X) $.

When studying the concept of a weak solution of the stochastic differential equation (1), the probability space  $  ( \Omega , {\mathcal F} , {\mathsf P}) $
with the family of  $  \sigma $-
algebras  $  \mathbf F = ( {\mathcal F} _ {t} ) _ {t\geq } 0 $, 
the Wiener process  $  W = ( W _ {t} , {\mathcal F} _ {t} ) _ {t\geq } 0 $
and the random variable  $  \xi $
are not fixed in advance, but the non-anticipative functionals  $  a( t, X) $, 
$  b( t, X) $, 
defined for continuous functions  $  X = ( X _ {t} ) _ {t\geq } 0 $, 
and the distribution function  $  F( x) $(
so to speak, the initial value) are fixed. Then by a weak solution of the equation (1) with given  $  a( t, X) $, 
$  b( t, X) $
and  $  F( x) $
one understands a set of objects

$$ 
\widetilde {\mathcal A}    =  ( \widetilde \Omega   , \widetilde {\mathcal F}   , ( \widetilde {\mathcal F}   _ {t} ) _ {t\geq } 0 ,\ \ 
\widetilde{W}   =  ( \widetilde{W}  _ {t} ) _ {t\geq } 0 ,\ \ 
\widetilde{X}   =  ( \widetilde{X}  _ {t} ) _ {t\geq } 0 , {\mathsf P} ),
$$

where  $  \widetilde{W}  = ( \widetilde{W}  {} _ {t} ) _ {t\geq } 0 $
is a Wiener process relative to  $  (( {\mathcal F} _ {t} ) _ {t\geq } 0 , {\mathsf P}) $, 
and  $  \widetilde{W}  $
and  $  \widetilde{X}  $
are related by

$$ 
\widetilde{X}  _ {t}  =  \widetilde{X}  _ {0} + \int\limits _ { 0 } ^ { t }  a( s, \widetilde{X}  )  ds + \int\limits _ { 0 } ^ { t }  b( s, \widetilde{X}  )  d \widetilde{W}  _ {s} ,
$$

and  $  \widetilde {\mathsf P}   \{ \widetilde{X}  _ {0} \leq  x \} = F( x) $. 
The term "weak solution" sometimes applies only to the process  $  \widetilde{X}  $
that appears in the set  $  \widetilde {\mathcal A}   $. 
A weak solution of equation (3) exists under weaker hypotheses. It is sufficient, for example, that  $  b  ^ {2} ( t, x) \geq  c > 0 $, 
and that  $  b  ^ {2} ( t, x) $
be continuous in  $  ( t, x) $, 
that  $  a( t, x) $
be measurable in  $  ( t, x) $, 
and that  $  | a | + | b | \leq  \textrm{ const } $.

The development of the theory of stochastic integration (see [[Stochastic integral|Stochastic integral]]) using semi-martingales (cf. [[Semi-martingale|Semi-martingale]]) and random measures has led to the study of more general stochastic differential equations, where semi-martingales and random measures are used as generators (along with a Wiener process). The following result is typical. Let  $  ( \Omega , {\mathcal F} , {\mathsf P}) $
be a probability space, let  $  \mathbf F = ( {\mathcal F} _ {t} ) _ {t\geq } 0 $
be an increasing family of  $  \sigma $-
algebras, let  $  Z = ( Z _ {t} , {\mathcal F} _ {t} ) _ {t\geq } 0 $
be an  $  m $-
dimensional semi-martingale, and let  $  G( t, X) = \| g  ^ {ij} ( t, X) \| _ {ij} $
be a matrix consisting of non-anticipative functionals  $  g  ^ {ij} ( t, X) $
such that

$$ 
| g  ^ {ij} ( t, X) - g  ^ {ij} ( t, Y) |  \leq   L _ {t}  ^ {ij}  \sup _ { s\leq  } t  | X _ {s} - Y _ {s} | ,
$$

where the  $  L _ {t}  ^ {ij} $
do not increase too rapidly (in  $  t $). 
Then the stochastic differential equation  $  dX _ {t} = G( t, X)  dZ _ {t} $, 
$  X _ {0} = 0 $, 
has a unique strong solution.

If the functions  $  a( t, x) $
and  $  b( t, x) $, 
$  t \geq  0 $, 
$  x \in \mathbf R $, 
satisfy a Lipschitz condition (in  $  x $) 
and do not increase faster than linearly, then the solution  $  X = ( X _ {t} ) _ {t\geq } 0 $
of equation (3) (unique up to [[Stochastic equivalence|stochastic equivalence]]) will be a Markov process. If, moreover,  $  a( t, x) $
and  $  b( t, x) $
are continuous in all variables, then this will be a diffusion process. Using stochastic differential equations, starting only from a Wiener process, it is thus possible to construct Markov and diffusion processes.

Given certain extra conditions of smoothness on the functions  $  a( t, x) $
and  $  b( t, x) $, 
the solution  $  ( X _ {t}  ^ {x} ) _ {t\geq } 0 $
of equation (3) with initial condition  $  X _ {0}  ^ {x} = x $
is such that the function  $  u( s, x) = {\mathsf E} f( X _ {s}  ^ {x} ) $, 
given a sufficiently smooth function  $  f( x) $, 
satisfies the backward Kolmogorov equation

$$ 

\frac{\partial  u( s, x) }{\partial  s }
 + a( s, x) 
\frac{\partial  u( s, x) }{\partial  x }

+ 
\frac{b  ^ {2} ( s, x) }{2}
 
\frac{\partial   ^ {2} u ( s, x) }{\partial  x  ^ {2} }
  = \ 
0,
$$

in the domain  $  s \in ( 0, t) $, 
$  x \in \mathbf R $, 
with the boundary condition

$$ 
\lim\limits _ { s\downarrow } t  u( s, x)  =  f( x).
$$

====References====
{|
|valign="top"|{{Ref|GS}}|| I.I. Gikhman, A.V. Skorokhod, "Stochastic differential equations and their applications" , Springer (1972) (Translated from Russian) {{MR|0678374}} {{ZBL|0557.60041}}
|-
|valign="top"|{{Ref|LS}}|| R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , '''1–2''' , Springer (1977–1978) (Translated from Russian) {{MR|1800858}} {{MR|1800857}} {{MR|0608221}} {{MR|0488267}} {{MR|0474486}} {{ZBL|1008.62073}} {{ZBL|1008.62072}} {{ZBL|0556.60003}} {{ZBL|0369.60001}} {{ZBL|0364.60004}}
|}

====Comments====

====References====
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|-
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|-
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|-
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|}