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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2013

Paper 25

Apply the <Itô product rule> to the deterministic discount factor and $u(X_t)$:
$$
d(e^{-\lambda t}u(X_t))=e^{-\lambda t}(\mathcal Lu-\lambda u)(X_t)\,dt+e^{-\lambda t}\nabla u(X_t)^{\mathsf T}\sigma(X_t)\,dW_t.
$$
The prescribed differential equation makes the drift vanish. Therefore
$$
\boxed{M_t=e^{-\lambda t}u(X_t)\text{ is a local martingale}.}
$$
This is the discounted generator-eigenfunction martingale underlying the <Feynman-Kac formula>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 25 / 6 / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 25 / 6 / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 25 / 6 / b / Solution

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