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

${\displaystyle \int _{0}^{t}|H_{s}||K_{s}|d\langle M,N\rangle _{s}\leq {\sqrt {\int _{0}^{t}H_{s}^{2}d\langle M\rangle _{s}}}{\sqrt {\int _{0}^{t}K_{s}^{2}d\langle N\rangle _{s}}}}$

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.