Predicting the ultimate supremum of a stable Lévy process with no negative jumps

Abstract

Given a stable Lévy process $X=(X_t{)}_{0\le t\le T}$ of index $\alpha \in (1,2)$ with no negative jumps, and letting $S_t={\sup }_{0\le s\le t}{X}_s$ denote its running supremum for $t\in [0,T]$, we consider the optimal prediction problem \[V=\inf_{0\leτ\le T}\mathsf{E}(S_T-X_τ)^p,\] where the infimum is taken over all stopping times $\tau$ of $X$, and the error parameter $p\in (1,\alpha )$ is given and fixed. Reducing the optimal prediction problem to a fractional free-boundary problem of Riemann--Liouville type, and finding an explicit solution to the latter, we show that there exists ${\alpha }_{\ast }\in (1,2)$ (equal to 1.57 approximately) and a strictly increasing function $p_{\ast }:({\alpha }_{\ast },2)\to (1,2)$ satisfying $p_{\ast }({\alpha }_{\ast }+)=1$, $p_{\ast }(2-)=2$ and $p_{\ast }(\alpha )<\alpha$ for $\alpha \in ({\alpha }_{\ast },2)$ such that for every $\alpha \in ({\alpha }_{\ast },2)$ and $p\in (1,p_{\ast }(\alpha ))$ the following stopping time is optimal \[τ_*=\inf\{t\in[0,T]:S_t-X_t\ge z_*(T-t)^{1/α}\},\] where $z_{\ast }\in (0,\infty )$ is the unique root to a transcendental equation (with parameters $\alpha$ and $p$). Moreover, if either $\alpha \in (1,{\alpha }_{\ast })$ or $p\in (p_{\ast }(\alpha ),\alpha )$ then it is not optimal to stop at $t\in [0,T)$ when $S_t-X_t$ is sufficiently large. The existence of the breakdown points ${\alpha }_{\ast }$ and $p_{\ast }(\alpha )$ stands in sharp contrast with the Brownian motion case (formally corresponding to $\alpha =2$), and the phenomenon itself may be attributed to the interplay between the jump structure (admitting a transition from lighter to heavier tails) and the individual preferences (represented by the error parameter $p$).

Published in at http://dx.doi.org/10.1214/10-AOP598 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• The trap of complacency in predicting the maximum
• The law of the supremum of a stable Lévy process with no negative jumps

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