Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 2 / b

Table of contents
- Solution b

Solution

Choose stopping times

T_{m} ↑ \infty

such that

X^{T_{m}}

is a bounded martingale. Part a constructs

[X^{T_{m}}]

. Uniqueness in the identity

(X^{T_{m}})^{2} - [X^{T_{m}}] is a local martingale

shows consistency on overlapping stopped intervals, so define

[X]_{t \land T_{m}} = [X^{T_{m}}]_{t}

. The stopped dyadic sums converge uniformly on every compact interval in probability, and

X^{2} - [X]

is a local martingale. This localization constructs the quadratic variation of every continuous local martingale.

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