# Stochastic differential

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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 .

How to Cite This Entry:
Stochastic differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_differential&oldid=14242
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article