# Stochastic differential

A random interval function $dX$ defined by the formula

$$( dX) I = X _ {t} - X _ {s} ,\ \ I = ( s, t],$$

for every process $X = ( X _ {t} , {\mathcal F} _ {t} , {\mathsf P})$ in the class of semi-martingales $S$, with respect to a stochastic basis $( \Omega , {\mathcal F} , ( {\mathcal F} _ {t} ) _ {t \geq 0 } , {\mathsf P})$. In the family of stochastic differentials $dS = \{ {dX } : {X \in S } \}$ one introduces addition $( A)$, multiplication by a process $( M)$ and the product operation $( P)$ according to the following formulas:

$( A)$ $dX + dY = d( X+ Y)$;

$( M)$ $( \Phi dX) ( s, t] = \int _ {s} ^ {t} \Phi dX$( a stochastic integral, $\Phi$ being a locally bounded predictable process which is adapted to the filtration $( {\mathcal F} _ {t} ) _ {t \geq 0 }$);

$( P)$ $dX \cdot dY = d( XY) - X _ {-} dY - Y _ {-} dX$, where $X _ {-}$ and $Y _ {-}$ are the left-continuous versions of $X$ and $Y$.

It then turns out that

$$( dX \cdot dY) ( s, t] = \mathop{\rm l}.i.p. _ {| \Delta | \rightarrow 0 } \sum _ { i= } 1 ^ { n } ( X _ {t _ {i} } - X _ {t _ {i-} 1 } )( Y _ {t _ {i} } - Y _ {t _ {i-} 1 } ),$$

where $\Delta = ( s= t _ {0} < t _ {1} < \dots < t _ {n} = t)$ is an arbitrary decomposition of the interval $( s, t]$, l.i.p. is the limit in probability, and $| \Delta | = \max | t _ {i} - t _ {i-} 1 |$.

In stochastic analysis, the principle of "differentiation" of random functions, or Itô formula, is of importance: If $X ^ {1} \dots X ^ {n} \in S$ and the function $f = f( x _ {1} \dots x _ {n} ) \in C ^ {2}$, then

$$Y = f( X ^ {1} \dots X ^ {n} ) \in S ,$$

and

$$\tag{1 } df( X ^ {1} \dots X ^ {n} ) = \sum _ { i= } 1 ^ { n } \partial _ {i} f \cdot dX ^ {i} + \frac{1}{2} \sum _ { i,j= } 1 ^ { n } \partial _ {i} \partial _ {j} f \cdot dX ^ {i} dX ^ {j} ,$$

where $\partial _ {i}$ is the partial derivative with respect to the $i$- th coordinate. In particular, it can be inferred from (1) that if $X \in S$, then

$$\tag{2 } f( X _ {t} ) = f( X _ {0} ) + \int\limits _ { 0 } ^ { t } f ^ { \prime } ( X _ {s - } ) dX _ {s} +$$

$$+ \frac{1}{2} \int\limits _ { 0 } ^ { t } f ^ { \prime\prime } ( X _ {s - } ) \ d\langle X\rangle ^ {c} + \sum _ {0< s\leq t } [ f( X _ {s} ) - f( X _ {s - } ) - f ^ { \prime } ( X _ {s - } ) \Delta X _ {s} ] ,$$

where $X ^ {c}$ is the continuous martingale part of $X$, $\Delta X _ {s} = X _ {s} - X _ {s - }$.

Formula (2) can be given the following form:

$$f( X _ {t} ) = f( X _ {0} ) + \int\limits _ { 0 } ^ { t } f ^ { \prime } ( X _ {s - } ) dX _ {s} + \frac{1}{2} \int\limits _ { 0 } ^ { t } f ^ { \prime\prime } ( X _ {s - } ) d[ X, X] _ {s} +$$

$$+ \sum _ { 0< } s\leq t \left [ f( X _ {s)} - f ( X _ {s - } ) - f ^ { \prime } ( X _ {s - } ) \Delta X _ {s} - \frac{1}{2} f ^ { \prime\prime } ( X _ {s - } ) ( \Delta X _ {s} ) ^ {2} \right ] ,$$

where $[ X, X]$ is the quadratic variation of $X$.

#### References

 [1] K. Itô, S. Watanabe, "Introduction to stochastic differential equations" K. Itô (ed.) , Proc. Int. Symp. Stochastic Differential Equations Kyoto, 1976 , Wiley (1978) pp. I-XXX [2] I.I. Gikhman, A.V. Skorokhod, "Stochastic differential equations and their applications" , Kiev (1982) (In Russian)

The product $dX \cdot dY$ is more often written as $d[ X, Y]$, where the so-called "square bracket" $[ X, Y]$ is the process with finite variation such that $[ X, Y] _ {t} = X _ {0} Y _ {0} + dX \cdot dY( 0, t]$. When $X= Y$, one obtains the quadratic variation $[ X, X]$ used at the end of the main article. Actually, it is a probabilistic quadratic variation: when $X$ is a standard Brownian motion, $d[ X, X]$ is the Lebesgue measure, but the true quadratic variation of the paths is almost surely infinite. See also Semi-martingale; Stochastic integral; Stochastic differential equation.