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Predicting the ultimate supremum of a stable Lévy process with no negative jumps. (English) Zbl 1235.60036

Let \(X= (X_t)_{0\leq t\leq T}\) be a stable Lévy process of index \(\alpha\in(1,2)\) with no negative jumps, and define \(S_t:= \sup_{s\in[0,t]}X_S\). Consider the optimal prediction problem \(V=\text{inf}_{\tau\in[0,T]}{\mathbf E}(S_T- X_\tau)^p\), where the infimum is taken over all stopping times \(\tau\) of \(X\), \(p\in(1,\alpha)\) fixed. The authors proceed via reducing the problem to that of solving a fractional free-boundary problem of Riemann-Liouville type. They show that there exist an \(\alpha^*\in(1,2)\) and a strictly increasing function \(p^*: (\alpha^*,2)\to(1, 2)\) satisfying \(p^*(\alpha*+)= 1\), \(p^*(2-)= 2\) and \(p^*(\alpha*)< \alpha\) for \(\alpha\in(\alpha^*,2)\) such that for any \(\alpha\in(\alpha^*,2)\) and \(p\in(1,p^*(\alpha))\) an optimal stopping time \(\tau^*\) exists, \(\tau^*= \text{inf}\{t[0,T]: S_t- X_t\geq z^*(T- t)^{1/\alpha}\}\), where \(z^*\) is the unique solution of a transcendental equation involving \(\alpha\) and \(p\). Furthermore, they prove that if either \(\alpha\in(1,\alpha^*)\) or \(p\in(p^*(\alpha), \alpha)\), it is not optimal to stop at \(t\in[0,T]\) when \(S_t-X_t\) is sufficiently large.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60J75 Jump processes (MSC2010)
45J05 Integro-ordinary differential equations
60G25 Prediction theory (aspects of stochastic processes)
47G20 Integro-differential operators
26A33 Fractional derivatives and integrals
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