Kunita–Watanabe inequality explained

In stochastic calculus, 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 to square-integrable martingales.^[1]

Statement of the theorem

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

\int ^{t} _{0} \left| H _{s} \right| \left| K _{s} \right| \left| d \langle M,N \rangle _{s} \right| \leq \sqrt{\int ^{t} _{0} H ^{2} _{s} d \langle M \rangle _{s}} \sqrt{\int ^{t} _{0} K ^{2} _{s} d \langle N \rangle _{s}}

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

References

• Book: Diffusions, Markov Processes and Martingales . II, Itô; Calculus . 50 . L. C. G. . Rogers . D. . Williams . David Williams (mathematician) . Chris Rogers (mathematician) . 1987 . Cambridge University Press . 0-521-77593-0 . 10.1017/CBO9780511805141 .

Notes and References

1. http://www-math.mit.edu/~dws/ito/ito7.pdf The Kunita–Watanabe Extension