 Scale increments for a birth-death chain

ID: scale-increments-for-a-birth-death-chain

p_{k} (h (k + 1) - h (k)) = q_{k} (h (k) - h (k - 1))

. Positive scale increments satisfy

w_{k + 1} = (q_{k} / p_{k}) w_{k}

. For absorbing boundaries

L < U

, the lower exit probability from

x

\frac{\sum _{j = x + 1}^{U} w _{j}}{\sum _{j = L + 1}^{U} w _{j}} .

This satisfies the harmonic equation and the boundary values and therefore is the hitting probability. For parity-dependent probabilities,

w_{j + 2} = ρ w_{j}

with

ρ = q_{0} q_{1} / (p_{0} p_{1})

; geometric sums distinguish rightward, balanced and leftward escape.

// Target: markov-process.bigb

 New to topics? Read the docs here!