Kunita–Watanabe inequality

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.

Statement of the theorem

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

$\underset{0}{\overset{t}{\int }}\left|H_s\right|\left|K_s\right||\mathrm{d}⟨M,N{⟩}_s|\le \sqrt{\underset{0}{\overset{t}{\int }}{H}_s^2\mathrm{d}⟨M{⟩}_s}\sqrt{\underset{0}{\overset{t}{\int }}{K}_s^2\mathrm{d}⟨N{⟩}_s}$

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

References

• Rogers, L. C. G.; Williams, D. (1987). Diffusions, Markov Processes and Martingales. II, Itô; Calculus. Cambridge University Press. p. 50. doi:10.1017/CBO9780511805141. ISBN 0-521-77593-0. https://books.google.com/books?id=bDQy-zoHWfcC&pg=PA50.