2009/12/03 by Pieter C. Allaart, Allaart, Pieter C.
Mathematics · #60G25 #60G40 #60G50 #60G51 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60G25 #msc:60G40 #msc:60G50 #msc:60G51
paper · pdf · doi:10.48550/arxiv.0912.0615
20 pages; added a few specific examples and additional references
arxiv created 2012/03/20 · arxiv updated 2012/03/21
Let Xt, 0<=t<=T be a one-dimensional stochastic process with independent and stationary increments. This paper considers the problem of stopping the process Xt "as close as possible" to its eventual supremum MT:=supXt: 0<=t<=T, when the reward for stopping with a stopping time tau<=T is a nonincreasing convex function of MT-Xtau. Under fairly general conditions on the process Xt, it is shown that the optimal stopping time tau is of "bang-bang" form: it is either optimal to stop at time 0 or at time T. For the case of random walk, the rule tau=T is optimal if the steps of the walk stochastically dominate their opposites, and the rule tau=0 is optimal if the reverse relationship holds. For Le'vy processes Xt with finite Le'vy measure, an analogous result is proved assuming that the jumps of Xt satisfy the above condition, and the drift of Xt has the same sign as the mean jump. Finally, conditions are given under which the result can be extended to the case of nonfinite Le'vy measure.