Gambler's ruin with random stopping

Let {Xj, j = 0} denote a Markov process on [-N - 1, N + 1]. {c}. Suppose P(Xj+ 1 = m + 1|Xj = m) = ph, P(Xj+ 1 = m 1|Xj = m) = (1 - p)h, all j = 1 and |m| = N, where p = 12 + bN and h = 1 - cN for cN = 12 a2/N2. Define P(Xj+ 1 = c|Xj = m) = cN, j = 0, |m| = N. {Xj} terminates at the first j such that Xj. {-N - 1, N + 1, c}. Let L = max{j = 0 : Xj = 0}. On . = {Xj terminates at c}, denote by R. and L., respectively, as the numbers of runs and steps from L until termination. Denote . = L. - 2R.. Then limN.8 E{e it N. | .} = Ca, b v c2+t2 cosh v c2+t2-cosh(2b) (a2+t2) sinh v c2+t2, where c2 = a2 + 4b2.