Kunita–Watanabe theorem

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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[edit]

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.


  • Rogers, L. C. G.; Williams, D. (1987). "Diffusions, Markov Processes and Martingales, Itô Calculus". Cambridge University Press. p. 50.