Itô formula

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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 [1]. An analogous formula holds for vectorial and . Itô's formula can be generalized to certain classes of non-smooth functions [2] and semi-martingales (cf. Semi-martingale).


[1] K. Itô, "On a formula concerning stochastic integration" Nagoya Math. J. , 3 (1951) pp. 55–65
[2] 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)
How to Cite This Entry:
Itô formula. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.A. Novikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article