# 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