Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 202 / 4 / 1

Table of contents
- Solution 1

Solution

For every real

λ

, the quadratic variation of

λ (M - M_{s}) + (N - N_{s})

on

[s, t]

is nonnegative:

λ^{2} Δ ⟨ M ⟩ + 2 λ Δ ⟨ M, N ⟩ + Δ ⟨ N ⟩ \geq 0.

Its discriminant is therefore nonpositive. This gives the pathwise Kunita-Watanabe inequality

∣ ⟨ M, N ⟩_{t} - ⟨ M, N ⟩_{s} ∣ \leq ⟨ M ⟩_{t} - ⟨ M ⟩_{s} ⟨ N ⟩_{t} - ⟨ N ⟩_{s} .

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