Kunita–Watanabe theorem

In stochastic calculus, the Kunita–Watanabe theorem or Kunita-Watanabe inequality is a generalization of the Cauchy Schwarz inequality to integrals of stochastic processes.

Statement of the Theorem

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

$\int_0^t | H_s | | K_s | d \langle M,N \rangle_s \leq \sqrt{\int_0^t H_s^2 d \langle M \rangle_s} \sqrt{\int_0^t K_s^2 d \langle N \rangle_s}$

Where the 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, Itô Calculus". Cambridge University Press: 50.