In
stochastic calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
, the Kunita–Watanabe inequality is a generalization of the
Cauchy–Schwarz inequality to integrals of
stochastic processes.
It was first obtained by Hiroshi Kunita and
Shinzo Watanabe and plays a fundamental role in their extension of Ito's
stochastic integral
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
to square-integrable martingales.
The Kunita–Watanabe Extension
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Statement of the theorem
Let ''M'', ''N'' be continuous local martingale
In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local m ...
s and ''H'', ''K'' measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
processes. Then
:
where the angled brackets indicates the quadratic variation and quadratic covariation operators. The integrals are understood in the Lebesgue–Stieltjes sense.
References
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{{DEFAULTSORT:Kunita-Watanabe theorem
Probability theorems
Probabilistic inequalities