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 .
|||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|
|||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)|
Stochastic differential. A.N. Shiryaev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Stochastic_differential&oldid=14242