# Stochastic differential

A random interval function defined by the formula

for every process in the class of semi-martingales , with respect to a stochastic basis . In the family of stochastic differentials one introduces addition , multiplication by a process and the product operation according to the following formulas:

;

(a stochastic integral, being a locally bounded predictable process which is adapted to the filtration );

, where and are the left-continuous versions of and .

It then turns out that

where is an arbitrary decomposition of the interval , l.i.p. is the limit in probability, and .

In stochastic analysis, the principle of "differentiation" of random functions, or Itô formula, is of importance: If and the function , then

and

 (1)

where is the partial derivative with respect to the -th coordinate. In particular, it can be inferred from (1) that if , then

 (2)

where is the continuous martingale part of , .

Formula (2) can be given the following form:

where is the quadratic variation of .

#### 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)