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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
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<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>
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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
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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  $  f ( t , x ) $
 +
be defined for all real  $  x $
 +
and  $  t $,
 +
be twice continuously differentiable in  $  x $
 +
and once continuously differentiable in  $  t $,
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and suppose that a process  $  X _ {t} $
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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/i0530309.png" /></td> </tr></table>
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$$
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d X _ {t}  = a ( t)  d t + \sigma ( t)  d W _ {t} .
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$$
  
<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>
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Then the stochastic differential of the process  $  f ( t , X _ {t} ) $
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has the form
  
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]]).
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$$
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d f ( t , X _ {t} )  = \
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[ f _ {t} ^ { \prime } ( t , X _ {t} )
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+ a ( t) f _ {x} ^ { \prime } ( t , X _ {t} ) +
 +
$$
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$$
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+
 +
{} \sigma  ^ {2} ( t) f _ {xx} ^ { \prime\prime } ( t , X _ {t} ) /2  ]  d
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t + \sigma ( t) f _ {x} ^ { \prime } ( t , X _ {t} )  d W _ {t} .
 +
$$
 +
 
 +
This formula was obtained by K. Itô [[#References|[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 [[#References|[2]]] and semi-martingales (cf. [[Semi-martingale|Semi-martingale]]).
  
 
====References====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 22:13, 5 June 2020


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=16058
This article was adapted from an original article by A.A. Novikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article