A formula by which one can compute the stochastic differential of a function of an Itô process. Let a (random) function be defined for all real and , be twice continuously differentiable in and once continuously differentiable in , and suppose that a process has stochastic differential
Then the stochastic differential of the process has the form
This formula was obtained by K. Itô . An analogous formula holds for vectorial and . Itô's formula can be generalized to certain classes of non-smooth functions  and semi-martingales (cf. Semi-martingale).
|||K. Itô, "On a formula concerning stochastic integration" Nagoya Math. J. , 3 (1951) pp. 55–65|
|||N.N. Krylov, "On Itô's stochastic integral equation" Theor. Probab. Appl. , 14 : 2 (1969) pp. 330–336 Teor. Veroyatnost. i Primenen. , 14 : 2 (1969) pp. 340–348|
Nowadays, Itô's formula is more generally the usual name given to the change of variable formula in a stochastic integral with respect to a semi-martingale. Either in its narrow or enlarged meaning, Itô's formula is one of the cornerstones of modern stochastic integral and differential calculus.
|[a1]||K.L. Chung, R.J. Williams, "Introduction to stochastic integration" , Birkhäuser (1983)|
|[a2]||A. Freedman, "Stochastic differential equations and applications" , 1 , Acad. Press (1975)|
|[a3]||N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland (1981)|
|[a4]||K. Itô, H.P. McKean jr., "Diffusion processes and their sample paths" , Acad. Press (1964)|
|[a5]||H.P. McKean jr., "Stochastic integrals" , Acad. Press (1969)|
|[a6]||L.C.G. Rogers, D. Williams, "Diffusions, Markov processes, and martingales" , 2. Itô calculus , Wiley (1987)|
|[a7]||T.G. Kurtz, "Markov processes" , Wiley (1986)|
Itô formula. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=It%C3%B4_formula&oldid=23336