# Itô formula

A formula by which one can compute the stochastic differential of a function of an Itô process. Let a (random) function $ f ( t , x ) $
be defined for all real $ x $
and $ t $,
be twice continuously differentiable in $ x $
and once continuously differentiable in $ t $,
and suppose that a process $ X _ {t} $
has stochastic differential

$$ d X _ {t} = a ( t) d t + \sigma ( t) d W _ {t} . $$

Then the stochastic differential of the process $ f ( t , X _ {t} ) $ has the form

$$ d f ( t , X _ {t} ) = \ [ f _ {t} ^ { \prime } ( t , X _ {t} ) + a ( t) f _ {x} ^ { \prime } ( t , X _ {t} ) + $$

$$ + {} \sigma ^ {2} ( t) f _ {xx} ^ { \prime\prime } ( t , X _ {t} ) /2 ] d t + \sigma ( t) f _ {x} ^ { \prime } ( t , X _ {t} ) d W _ {t} . $$

This formula was obtained by K. Itô [1]. An analogous formula holds for vectorial $ X _ {t} $ and $ f ( t , x ) $. Itô's formula can be generalized to certain classes of non-smooth functions [2] and semi-martingales (cf. Semi-martingale).

#### References

[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 |

#### Comments

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.

#### References

[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: http://encyclopediaofmath.org/index.php?title=It%C3%B4_formula&oldid=47451