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Ito's Lemma

Let W(u) be a Wiener process. Then

V_t-V_0=int_0^tf_x(W(u),u)dW(u)-int_0^tf_tau(W(u),u)du+1/2int_0^tf_(xx)(W(u),u)du,

where V_t=f(W(t),tau) for 0<=tau=T-t<=T, and f in C^(2,1)((0,infty)×[0,T]).

Note that while Ito's lemma was proved by Kiyoshi Ito (also spelled Itô), Ito's theorem is due to Noboru Itô.

See also

Wiener Process

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References

Karatsas, I. and Shreve, S. Brownian Motion and Stochastic Calculus, 2nd ed. New York: Springer-Verlag, 1997.Kendall, W. S. "Stochastic Integrals and Their Expectations." Mathematica J. 9, 757-767, 2005.Price, J. F. "Optional Mathematics is Not Optional." Not. Amer. Math. Soc. 43, 964-971, 1996.

Referenced on Wolfram|Alpha

Ito's Lemma

Cite this as:

Weisstein, Eric W. "Ito's Lemma." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ItosLemma.html

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