Kunita-Watanabe inequality

In stochastics , the Kunita-Watanabe inequality denotes a generalization of the Cauchy-Schwarz inequality for integrals of stochastic processes . The inequality was proven in 1967 by Hiroshi Kunita and Shinzō Watanabe .

Statement of the inequality

Be and constant local martingales and , measurable processes . Then applies to ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle H}$ ${\displaystyle K}$ ${\displaystyle A\subseteq [0,\infty ]}$

${\displaystyle \int \limits _{A}\left|H_{s}\right|\left|K_{s}\right|\left|\mathrm {d} \langle M,N\rangle _{s}\right|\leq {\sqrt {\int \limits _{A}H_{s}^{2}\,\mathrm {d} \langle M\rangle _{s}}}{\sqrt {\int \limits _{A}K_{s}^{2}\,\mathrm {d} \langle N\rangle _{s}}}}$,

The angle brackets denote the quadratic variation and the integral is to be understood as a Stieltjes integral .

literature

• L. Rogers, David Williams: Diffusions, Markov Processes and Martingales, Volume 2: Ito Calculus, Cambridge UP 2000
• Richard Durrett : Stochastic Calculus. An Introduction, CRC Press 1996

Individual evidence

1. ↑ Kunito, Watanabe, On square integrable Martingales, Nagoya Math. J., Volume 30, 1967, pp. 209-245, Project Euclid