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

Table of contents
- Solution b

Solution

Apply the Itô product rule to the deterministic discount factor and

u (X_{t})

:

d (e^{- λ t} u (X_{t})) = e^{- λ t} (L u - λ u) (X_{t}) d t + e^{- λ t} \nabla u (X_{t})^{T} σ (X_{t}) d W_{t} .

The prescribed differential equation makes the drift vanish. Therefore

M_{t} = e^{- λ t} u (X_{t}) is a local martingale .

This is the discounted generator-eigenfunction martingale underlying the Feynman-Kac formula.

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