Dynkin's Formula - Example

Example

Dynkin's formula can be used to find the expected first exit time τK of Brownian motion B from the closed ball

which, when B starts at a point a in the interior of K, is given by

Choose an integer k. The strategy is to apply Dynkin's formula with X = B, τ = σk = min(k, τK), and a compactly-supported C2 f with f(x) = |x|2 on K. The generator of Brownian motion is Δ/2, where Δ denotes the Laplacian operator. Therefore, by Dynkin's formula,

Hence, for any k,

Now let k → +∞ to conclude that τK = limk→+∞σk < +∞ almost surely and

as claimed.

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