Optimally stopping at a given distance from the ultimate supremum of a spectrally negative Lévy process

2019/04/26 by Pinto, Mónica B. Carvajal, van Schaik, Kees
#60G40 #62M20 #FOS: Economics and business #FOS: Mathematics #General Finance (q-fin.GN) #Optimization and Control (math.OC) #Probability (math.PR)

paper · doi:10.48550/arxiv.1904.11911

Abstract

We consider the optimal prediction problem of stopping a spectrally negative Lévy process as close as possible to a given distance b ≥ 0 from its ultimate supremum, under a squared error penalty function. Under some mild conditions, the solution is fully and explicitly characterised in terms of scale functions. We find that the solution has an interesting non-trivial structure: if b is larger than a certain threshold then it is optimal to stop as soon as the difference between the running supremum and the position of the process exceeds a certain level (less than b), while if b is smaller than this threshold then it is optimal to stop immediately (independent of the running supremum and position of the process). We also present some examples.

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