Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / A / 3 / c

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- Solution c

Solution

Put

q = 1 - ϵ < 1

. The hypothesis implies

P (τ > n + N ∣ F_{n}) \leq q

. Since

{τ > n} \in F_{n}

and survival to

n + N

implies survival to

n

, the tower property of conditional expectation gives

P (τ > n + N) = E [1_{{τ > n}} P (τ > n + N ∣ F_{n})] \leq q P (τ > n) .

Apply this at

n = 0, N, 2 N, \dots

and use

P (τ > 0) \leq 1

to obtain

P (τ > k N) \leq (1 - ϵ)^{k} .

This is a geometric tail bound from a uniform escape probability. The tail-sum formula for a nonnegative integer-valued variable now gives

E τ = m \geq 0 \sum P (τ > m) \leq N k \geq 0 \sum (1 - ϵ)^{k} = \frac{N}{ϵ} < \infty.

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