Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 1 / c / Solution

Apply part (a)'s square formula, extended by the Jordan decomposition of a function of bounded variation from nondecreasing functions to arbitrary càdlàg functions of bounded variation, to

f + g

,

f

, and

g

. Since

f g = \frac{1}{2} ((f + g)^{2} - f^{2} - g^{2}),

linearity of the Lebesgue-Stieltjes integral leaves

\int_{0}^{t} f d g + \int_{0}^{t} g df

. At each time

s

, polarization of the jump correction gives

\frac{1}{2} ((Δ f (s) + Δ g (s))^{2} - Δ f (s)^{2} - Δ g (s)^{2}) = Δ f (s) Δ g (s) .

Only common jump times contribute, and hence

f (t) g (t) = f (0) g (0) + \int_{0}^{t} f d g + \int_{0}^{t} g df - \sum_{s \in A_{f} \cap A_{g} \cap (0, t]} Δ f (s) Δ g (s) .

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