Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 5 / c

Table of contents
- Solution c

Solution

Let

h (y) = g^{'} (g^{- 1} (y)) .

Differentiating and using the scale equation gives

h^{'} (y) = \frac{g ^{''} ( x )}{g ^{'} ( x )} = - 2 b (x), x = g^{- 1} (y) .

Since

b

is bounded,

h

has global Lipschitz continuity. Thus

d Y_{t} = h (Y_{t}) d W_{t}

has a pathwise unique strong solution by the standard Lipschitz existence-and-uniqueness theorem for a stochastic differential equation. Applying the deterministic inverse

g^{- 1}

gives a strong solution

X

, and uniqueness of

Y

gives pathwise uniqueness of

X

.

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