In stochastic calculus, the Kunita–Watanabe inequality is a generalization of the

Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...

to integrals of

stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...

. It was first obtained by Hiroshi Kunita and

Shinzo Watanabe Shinzō Watanabe (渡辺信三 Watanabe Shinzō, 23 December 1935) is a Japanese mathematician, who has made fundamental contributions to probability theory, stochastic processes and stochastic differential equations. He is regarded and revere ...

and plays a fundamental role in their extension of Ito's stochastic integral to square-integrable martingales.The Kunita–Watanabe Extension
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Statement of the theorem

Let ''M'', ''N'' be continuous local martingales and ''H'', ''K'' measurable processes. Then :

\int_0^t \left,  H_s \ \left,  K_s \ \left,  \mathrm \langle M,N \rangle_s \ \leq  \sqrt \sqrt

where the angled brackets indicates the quadratic variation and quadratic covariation operators. The integrals are understood in the Lebesgue–Stieltjes sense.

References

* {{DEFAULTSORT:Kunita-Watanabe theorem Probability theorems Probabilistic inequalities