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Discounted symmetric random-walk exit transform

For simple symmetric random walk stopped on hitting

- a

or

b

and

0 < z < 1

, choose

w > 1

by

z = \frac{2}{w + w ^{- 1}}, w = \frac{1 + 1 - z ^{2}}{z} .

Then

z^{n} w^{\pm X_{n}}

are martingales. Solving the two boundary equations gives

E_{0} z^{T} = \frac{w ^{a} + w ^{b} - w ^{- a} - w ^{- b}}{w ^{a + b} - w ^{- (a + b)}} .

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