View source for Itô formula

A formula by which one can compute the [[Stochastic differential|stochastic differential]] of a function of an [[Itô process|Itô process]]. Let a (random) function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053030/i0530301.png" /> be defined for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053030/i0530302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053030/i0530303.png" />, be twice continuously differentiable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053030/i0530304.png" /> and once continuously differentiable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053030/i0530305.png" />, and suppose that a process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053030/i0530306.png" /> has stochastic differential

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053030/i0530307.png" /></td> </tr></table>

Then the stochastic differential of the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053030/i0530308.png" /> has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053030/i0530309.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053030/i05303010.png" /></td> </tr></table>

This formula was obtained by K. Itô [[#References|[1]]]. An analogous formula holds for vectorial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053030/i05303011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053030/i05303012.png" />. Itô's formula can be generalized to certain classes of non-smooth functions [[#References|[2]]] and semi-martingales (cf. [[Semi-martingale|Semi-martingale]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Itô,   "On a formula concerning stochastic integration"  ''Nagoya Math. J.'' , '''3'''  (1951)  pp. 55–65</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR></table>



====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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.L. Chung,   R.J. Williams,   "Introduction to stochastic integration" , Birkhäuser  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Freedman,   "Stochastic differential equations and applications" , '''1''' , Acad. Press  (1975)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Ikeda,   S. Watanabe,   "Stochastic differential equations and diffusion processes" , North-Holland  (1981)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Itô,   H.P. McKean jr.,   "Diffusion processes and their sample paths" , Acad. Press  (1964)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H.P. McKean jr.,   "Stochastic integrals" , Acad. Press  (1969)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L.C.G. Rogers,   D. Williams,   "Diffusions, Markov processes, and martingales" , '''2. Itô calculus''' , Wiley  (1987)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  T.G. Kurtz,   "Markov processes" , Wiley  (1986)</TD></TR></table>