Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 3 / c

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- Solution c

Solution

Let

X^{x}

solve

d X_{s} = b (X_{s}) d s + a (X_{s}) d B_{s}, X_{0} = x .

The Feynman-Kac formula is

u (t, x) = E_{x} [f (X_{t}) exp (\int_{0}^{t} V (X_{r}) d r)] .

Fix

t

and apply the two-variable Itô formula to

F (s, y) = u (t - s, y)

and the semimartingale vector

(s, X_{s})

. Multiplying by

R_{s} = exp (\int_{0}^{s} V (X_{r}) d r)

and using the Itô product rule, the drift of

R_{s} F (s, X_{s})

is

R_{s} (- \partial_{t} u + Lu + V u) (t - s, X_{s}) d s = 0.

The remaining stochastic integral is a true martingale because the coefficients and derivatives are bounded. Taking expectations at

s = 0

and

s = t

gives the formula.

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