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By Eugene Lukacs

This quantity experiences attribute functions--which play a vital function in likelihood and statistics-- for his or her intrinsic, mathematical curiosity.

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LNM 1857, pp. 30–41, 2005. c Springer-Verlag Berlin Heidelberg 2005 Exit problems of L´evy processes 31 and write Y = X − I for X reflected at its past infimum I. By Ta− , Ta+ we denote Ta− = inf{t Ta+ = inf{t 0 : Xt < a}, 0 : Xt > a}, the first passage times of X into the sets (−∞, a) and (a, ∞), respectively. Similarly, we write τa+ = inf{t 0 : Yt > a} for the first passage time of Y into the set (a, ∞). The following theorem gives the form of the Laplace transforms of these passage times: Theorem 1.

Specifically, assume that f (t, Xt ) is a martingale; then t f (t, Xt − It ) + f (s, ∆Is ) − f (s, 0) 1∆Is <0 fx (s, 0) dIsc + 0 (4) s t is a martingale. The fact that fx is evaluated at 0 in the integrals with respect to dS c and dI c is intuitively clear from the fact that S c and I c are local times at 0 for the processes S − X and X − I respectively (see Section 4 for an explanation of this fact). The measures dS c and dI c are then respectively carried by {S = X} and {I = X}. 4. Here we have treated the special case with S or I as the bounded variation process which perturbs X, but it is clear from the proof that one could work as well with a general bounded variation process Y (in the spirit of [13]).

Let f : R+ × R → R be a C 1,1 , function and for 0 < a < b, set T = inf{t : Xt ∈ / (a, b)}. Assume that (f (t ∧ T, Xt∧T ), t 0) is a Px martingale for all x ∈ (a, b). Set t Mtf = f (t, Xt − St ) + f (s, ∆Ss ) − f (s, 0) 1∆Ss>0 (3) fx (s, 0) dSsc + 0 s t f Then for all x ∈ (a, b), (Mt∧τ ,t where τ = inf{t : Rt ∈ / (a, b)}. 0) is a martingale under P[ . | R0 = x], Remarks 1. 1. , will be denoted whenever convenient either by ft or by ∂t f and so on. . 2. If f ∈ C 1,2 (R+ × R, R), Prop. 1 is an immediate consequence of Itˆ o’s formula.

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