By Luc Devroye (auth.), Luc Devroye, Bülent Karasözen, Michael Kohler, Ralf Korn (eds.)

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Infinitely divisible laws associated with hyperbolic functions. Can. J. Math. : On the height of trees. J. Aust. Math. Soc. : Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften, vol. 293. : The joint density of the maximum and its location for a Wiener process with drift. J. Appl. Probab. : Empirical Processes with Applications to Statistics. : A relation between Brownian bridge and Brownian excursion. Ann. Probab. : Various techniques used in connection with random digits.

Chap. 12 in Györfi et al. 2002) and the consistency for general continuation values and the rate of convergence of the estimate in case of smooth continuation values has been investigated. For smooth continuation values Egloff (2005) showed the usual optimal rate of convergence for estimation of smooth regression functions. , for sample size tending to infinity, as was shown in the theoretical results), but also for finite sample size. In Egloff et al. (2007) the error propagation was simplified by generating new data for each time point which was (conditioned on the data corresponding to time t) independent of all previously used data.

The decomposition into a product in the numerator follows from the Markovian nature of Brownian motion. Using scaling, we see that P √ √ min Y (s) ≥ 0 = 1 − exp(−2(a/ t)(x/ t)) = 1 − exp(−2ax/t), 0≤s≤t and similarly, P min Z(s) ≥ 0 = 1 − exp(−2bx/(1 − t)). t≤s≤1 Therefore, putting μ = a + t (b − a), 18 Luc Devroye P B me (t) ∈ dx = P a + t (b − a) + × t (1 − t)N ∈ dx (1 − exp(−2ax/t))(1 − exp(−2bx/(1 − t))) . 1 − exp(−2ab) The density of B me (t) is f (x) = g(x) × h(x), where, for x > 0, (x − μ)2 1 def exp − g(x) = √ , 2t (1 − t) 2πt (1 − t) ⎧ (1−exp(−2ax/t))(1−exp(−2bx/(1−t))) if ab > 0, ⎪ ⎪ 1−exp(−2ab) ⎪ ⎪ ⎪ ⎨ x (1 − exp(−2bx/(1 − t))) if a = 0, b > 0, bt def h(x) = x ⎪ if a > 0, b = 0, ⎪ a(1−t) (1 − exp(−2ax/t)) ⎪ ⎪ ⎪ 2 ⎩ 2x if a = b = 0.