Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 2 / b / ii

Table of contents
- Solution ii

Solution

With the notation from part (i), the total-variation process of

[M, N]

is

V_{t} = \int_{0}^{t} ∣ c ∣ d C .

For the centered bivariate normal distribution

(U, V)

used there,

c = E [U V]

. The integral triangle inequality gives

∣ c ∣ = ∣ E [U V] ∣ \leq E ∣ U V ∣.

Integrating this pointwise inequality against

d C

proves

V_{t} \leq V_{t}

for every

t \geq 0

.

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