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Abstract : |
Let fS n: n 0g be a random walk having normally distributed increments with mean ` and variance 1, and let be the time at which the random walk first takes a positive value, so that S is the first ladder height. Then the expected value E ` S, originally defined for positive `, may be extended to be an analytic function of the complex variable ` throughout the entire complex plane, with the exception of certain branch point singularities. In particular, the coefficients in a Taylor expansion about ` = 0 may be written explicitly as simple expressions involving the Riemann zeta function. Previously only the first coefficient of the series developed here was known; this term has been used extensively in developing approximations for boundary crossing problems for Gaussian random walks. Knowledge of the complete series makes more refined results possible; we apply it to derive asymptotics for boundary crossing probabilities and the limiting expected overshoot., |
