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
/ref>
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
:
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