Abstract
For a spectrally positive and strictly stable process with index in (1, 2), a series representation is obtained for the joint distribution of the “first passage triple” that consists of the time of first passage and the undershoot and the overshoot at first passage. The result leads to several corollaries, including (1) the joint law of the first passage triple and the pre-passage running supremum, and (2) at a fixed time point, the joint law of the process’ value, running supremum, and the time of the running supremum. The representation can be decomposed as a sum of strictly positive functions that allow exact sampling of the first passage triple.
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The research is partially supported by NSF Grant DMS 1720218. The author would like to thank two referees for their careful reading of the manuscript and useful suggestions, in particular, one that significantly simplifies the proof of the main theorem.
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Appendix
Appendix
On the connection between (9) and (10) When \(\alpha =2\), \(\sin (\pi k/\alpha )\) is 0 if k is even and is \((-1)^j\) is \(k = 2j + 1\) for integer \(j\ge 0\). Then, the series in (9) can be written as
Write \(n=l+1\) and \(m=j+l\). Then, the series becomes
Since \((X_t)_{t\ge 0}\sim (W_{2t})_{t\ge 0}\), this is essentially the same result as (10). \(\square \)
Proof of Eq. (16) We need the following refined version of Lemma 16(a).
Lemma 21
There is a constant \(M>0\), such that for all \(0<x<1/2\) and all \(t>0\),
Assume the lemma is true for now. Then, given \(c>0\), by scaling, for all \(0<x<c/2\), \(h_c(c-x, t)\le M x(t+t^{1-{1/\alpha }})\) for some \(M = M(c)>0\). Then, by Lemma 16(a) and dominated convergence, for each \(q>0\),
However, by scaling (6) and Proposition 10, for \(0<x<c/2\),
As a result,
Provided that \(\beta > q^{1/\alpha }\), integration term by term of the r.h.s. yields
By analytic extension, the equality still holds for \(0\le \beta \le q^{1/\alpha }\). Then, by (15), the proof is complete. \(\square \)
Proof of Lemma 21
By (22) and integral by parts,
where \(\overline{F}_{-x}(t) = \int ^\infty _t f_{-x}(s)\,\mathrm {d}s = \mathbb {P}\{\tau _{-x} > t\}\). For \(0<x<1/2\), \(g_t(1-x) - g_t(1) = - g'_t(z) x\) for some \(z\in (1-x,1)\). Clearly, \(z>1/2\). It is not hard to show that \(M_1 := \sup _{y>0} [y^{\alpha + 2} |g'_1(y)|]<\infty \) ([18], p. 88). On the other hand, by \(g_t(z) = t^{-{1/\alpha }} g_1(t^{-{1/\alpha }} z)\), \(g'_t(z) = t^{-2/\alpha } g'_1(t^{-{1/\alpha }} z)\). Then,
Next, by \(g_s(1) = s^{-{1/\alpha }} g_1(s^{-{1/\alpha }})\), \(|\partial g_s(1)/\partial s| \le ({1/\alpha }) [s^{-{1/\alpha }-1} g_1(s^{-{1/\alpha }}) + s^{-2/\alpha -1} |g'_1(s^{-{1/\alpha }})|]\) is bounded. Then, for some \(M_2>0\),
where the equality is due to \(\overline{F}_{-x}(s) = \overline{F}_{-1}(x^{-\alpha } s)\) and change of variable. Because \(\overline{F}_{-1}(s)\) is decreasing with \(\overline{F}_{-1}(0)=1\) and is slowly varying at \(\infty \) with index \(-{1/\alpha }\), there is a constant \(M_3>0\) such that \(\int ^y_0 \overline{F}_{-1}(s)\,\mathrm {d}s \le M_3 y^{1-{1/\alpha }}\) for all \(y>0\). It follows that
Then, the proof is complete by combining (45)–(47). \(\square \)
Proof of Eq. (20) Denote the r.h.s. of (20) by \(v^q(x)\). The task is to show \(\widehat{v^q} = \widehat{u^q}\), where, for example, \(\widehat{v^q}(x) = {v^q}(-x)\). Since \(v^q\) is a version of the q-resolvent density, according to the proof of Proposition I.13 of [2], \((r-q) U^r \widehat{v^q}\uparrow \widehat{u^q}\) as \(r\rightarrow \infty \), where \(U^r\) is the r-resolvent operator. For \(r>q\),
Then, by monotone convergence, \((r-q) U^r \widehat{v^q}(x)\rightarrow \widehat{v^q}(x)\), giving \(\widehat{v^q}(x) = \widehat{u^q}(x)\). \(\square \)
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Chi, Z. Law of the First Passage Triple of a Spectrally Positive Strictly Stable Process. J Theor Probab 33, 715–748 (2020). https://doi.org/10.1007/s10959-019-00898-w
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DOI: https://doi.org/10.1007/s10959-019-00898-w