# Stochastic differential equation

for a process $X=( X _ {t} ) _ {t\geq } 0$ with respect to a Wiener process $W = ( W _ {t} ) _ {t\geq } 0$

2010 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

 [GS] I.I. Gikhman, A.V. Skorokhod, "Stochastic differential equations and their applications" , Springer (1972) (Translated from Russian) MR0678374 Zbl 0557.60041 [LS] R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , 1–2 , Springer (1977–1978) (Translated from Russian) MR1800858 MR1800857 MR0608221 MR0488267 MR0474486 Zbl 1008.62073 Zbl 1008.62072 Zbl 0556.60003 Zbl 0369.60001 Zbl 0364.60004