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)