Kunita-Watanabe inequality

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Problem 10.7 from Bass R.F. "Stochastic Processes" [Bas11]More problems in Chapter 10 "Stochastic integrals"

Show that if MM and NN are square integrable martingales with continuous paths, then

⟨M,N⟩t≤(⟨M⟩t)1/2(⟨N⟩t)1/2 \langle M, N\rangle _{t} \leq \left(\langle M\rangle _{t}\right)^{1 / 2}\left(\langle N\rangle _{t}\right)^{1 / 2}

Hint: Imitate an appropriate proof of the Cauchy-Schwarz inequality. This result is a special case of the inequality of Kunita-Watanabe.

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Problem 15.2 from Schilling R.L. "Brownian Motion: A Guide to Random Processes and Stochastic Calculus" [Sch21]More problems in Chapter 15 "Stochastic integrals: L²-Theory"

Let (Mn,Fn)n⩾0\left(M_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0} and (Nn,Fn)n⩾0\left(N_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0} be L2L^{2} martingales. Show that ∣⟨M,N⟩n∣⩽⟨M⟩n⟨N⟩n\left|\langle M, N\rangle_{n}\right| \leqslant \sqrt{\left\langle M\right\rangle_{n}} \sqrt{\left\langle N\right\rangle_{n}}.

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References

[Bas11] Bass R.F. Stochastic Processes. Cambridge University Press, 2011.
[Sch21] Schilling R.L. Brownian Motion: A Guide to Random Processes and Stochastic Calculus. 3rd ed., De Gruyter, 2021.