Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 203 / 3 / c

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- Solution c

Solution

The Strong Markov property and part (b) show that, before

τ = τ_{x} \land τ_{y}

,

F (V_{t}) = P (τ_{x} > τ_{y} ∣ F_{t}) .

Thus

F (V_{t \land τ})

is a bounded martingale. From the given stochastic differential equation,

d [V]_{t} = \frac{( 1 - V _{t} ) ^{2}}{Y _{t}^{2}} d t .

The Itô formula says that the drift of

F (V_{t})

is

\frac{1}{2 Y _{t}^{2}} [(1 - v)^{2} F^{''} (v) + (\frac{d - 1}{v} + (3 - d) v - 2) F^{'} (v)]_{v = V_{t}} d t .

It must vanish. Dividing by

(1 - v)^{2} /2

and using the algebraic identity supplied in the question gives

F^{''} (v) + (\frac{d - 1}{v} + \frac{2 d - 4}{1 - v}) F^{'} (v) = 0, v < 0.

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