Kunita–Watanabe Inequality
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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 The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...
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 in a probability space, where the index of the family often has the interpretation of time. Stoc ...
. 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 martingales 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 magnitude, mass, and probability of events. These seemingly distinct concepts hav ...
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 Theorems in probability theory Probabilistic inequalities