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.[1]

Statement of the theorem

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

∫ 0 t | H s | | K s | | d ⟨ M , N ⟩ s | ≤ ∫ 0 t H s 2 d ⟨ M ⟩ s ∫ 0 t K s 2 d ⟨ N ⟩ s {\displaystyle \int _{0}^{t}\left|H_{s}\right|\left|K_{s}\right|\left|\mathrm {d} \langle M,N\rangle _{s}\right|\leq {\sqrt {\int _{0}^{t}H_{s}^{2}\,\mathrm {d} \langle M\rangle _{s}}}{\sqrt {\int _{0}^{t}K_{s}^{2}\,\mathrm {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

  1. ^ The Kunita–Watanabe Extension
  • Rogers, L. C. G.; Williams, D. (1987). Diffusions, Markov Processes and Martingales. Vol. II, Itô, Calculus. Cambridge University Press. p. 50. doi:10.1017/CBO9780511805141. ISBN 0-521-77593-0.