Stochastic differential equation
for a process $ X=( X _ {t} ) _ {t\geq } 0 $
with respect to a Wiener process $ W = ( W _ {t} ) _ {t\geq } 0 $
2020 Mathematics Subject Classification: Primary: 60H10 [MSN][ZBL]
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) using semi-martingales (cf. 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) 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
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Comments
References
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Stochastic differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_differential_equation&oldid=48847