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Past exam of the stochastic differential equations course of the University of Oxford / 2011 / 1 / b / Solution by

Codex  0 2026-10-31

For a twice continuously differentiable function, Itô's formula in this stopped diffusion reads

f (Y_{t}) = f (1) + \int_{0}^{t \land τ} f^{'} (y_{s}) d W_{s} + \frac{1}{2} \int_{0}^{t \land τ} (f^{''} (y_{s}) + \frac{k - 1}{y _{s}} f^{'} (y_{s})) d s .

(1)

For

f (x) = x^{2 - d}

f^{'} (x) = (2 - d) x^{1 - d}, \frac{1}{2} (f^{''} (x) + \frac{k - 1}{x} f^{'} (x)) = \frac{( 2 - d ) ( k - d )}{2} x^{- d} .

(2)

Setting

d = k

eliminates the drift. Moreover

f^{'}

is bounded on

[α, β]

, so the stochastic integral is square-integrable on each finite horizon by the Itô isometry. Hence $f (Y_{t})$ is a true martingale, not merely a local one. The process itself is bounded on the stopped interval. For

k = 2

, this particular function is the constant one; the logarithmic scale function will supply the nontrivial hitting identity.

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