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



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


where is the continuous martingale part of , .

Formula (2) can be given the following form:

where is the quadratic variation of .


[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 is more often written as , where the so-called "square bracket" is the process with finite variation such that . When , one obtains the quadratic variation used at the end of the main article. Actually, it is a probabilistic quadratic variation: when is a standard Brownian motion, 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.

For the study of continuous-path processes evolving on non-flat manifolds the Itô stochastic differential is inconvenient, because the Itô formula (2) is incompatible with the ordinary rules of calculus relating different coordinate systems. A coordinate-free description can be obtained using the Stratonovich differential; see [a1], [a2], Chapt. 5, [a3], and Stratonovich integral.


[a1] K.D. Elworthy, "Stochastic differential equations on manifolds" , Cambridge Univ. Press (1982)
[a2] N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland (1989) pp. 97ff
[a3] P.A. Meyer, "Geometrie stochastiques sans larmes" J. Azéma (ed.) M. Yor (ed.) , Sem. Probab. Strassbourg XV , Lect. notes in math. , 850 , Springer (1981) pp. 44–102
[a4] I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus" , Springer (1988)
[a5] L.C.G. Rogers, D. Williams, "Diffusion, Markov processes, and martingales" , 2. Itô calculus , Wiley (1987)
[a6] S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrøm, "Nonstandard methods in stochastic analysis and mathematical physics" , Acad. Press (1986)
How to Cite This Entry:
Stochastic differential. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article