Predicting the supremum: optimality of "stop at once or not at all"

Details Predicting the supremum: optimality of "stop at once or not at all" Predicting the supremum: optimality of "stop at once or not at all"

2009-12-03

arxiv.org/abs/0912.0615v3

Mathematics

Probability

20 pages; added a few specific examples and additional references

Scientific paper

Let X_t, 0<=t<=T be a one-dimensional stochastic process with independent and stationary increments. This paper considers the problem of stopping the process X_t "as close as possible" to its eventual supremum M_T:=sup{X_t: 0<=t<=T}, when the reward for stopping with a stopping time tau<=T is a nonincreasing convex function of M_T-X_tau. Under fairly general conditions on the process X_t, 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 X_t with finite Le'vy measure, an analogous result is proved assuming that the jumps of X_t satisfy the above condition, and the drift of X_t 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.

Affiliated with

Also associated with

No associations

Rating

Predicting the supremum: optimality of "stop at once or not at all" does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Predicting the supremum: optimality of "stop at once or not at all", we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Predicting the supremum: optimality of "stop at once or not at all" will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-517203