2.1 Some basic fluctuation theory facts

The following is standard. We provide some specific references for the reader’s benefit.

(i) Let $\psi$ be the Laplace exponent of X, $\psi(\lambda)\coloneqq \log \mathbb{P}[\text{e}^{\lambda X_1}]$ for $\lambda\in [0,\infty)$. It has the representation

(2.1) \begin{equation}\psi(\lambda)=\dfrac{\sigma^2}{2}\lambda^2+\mu\lambda+\int(\text{e}^{\lambda y}-\mathds{1}_{[-1,0)}(y)\lambda y-1)\nu(\text{d} y),\quad \lambda\in [0,\infty),\end{equation}

for some (unique) $\mu\in \mathbb{R}$, $\sigma^2\in [0,\infty)$, and measure $\nu$ on $\mathcal{B}_\mathbb{R}$, supported by $({-}\infty,0)$, and satisfying $\int (1\land y^2 )\nu(\text{d} y)<\infty$. When X has paths of finite variation, equivalently $\sigma^2=0$ and $\int (1\land \vert y\vert)\nu(\text{d} y)<\infty$, we set $\textsf d\coloneqq \mu+\int_{[-1,0)}\vert y\vert\nu(\text{d} y)$; in this case we must have $\textsf d\in (0,\infty)$ and $\nu$ non-zero. Differentiating under the integral sign in (2.1), $\psi$ is seen to be strictly convex, continuous, with $\lim_\infty\psi=\infty$, and indeed with $\lim_\infty \psi'=\infty$ provided X has paths of infinite variation.

We also let $\Phi$ be the right-continuous inverse of $\psi$, $\Phi(q)\coloneqq \inf\{\lambda\in [0,\infty)\colon \psi(\lambda)>q\}$ for $q\in [0,\infty)$, so that $\Phi(0)$ is the largest zero of $\psi$. Recall that X drifts to $\infty$, oscillates, or drifts to $-\infty$, respectively, as $\psi'(0+\!)>0$, $\psi'(0+\!)=0$, or $\psi'(0+\!)<0$ (the latter being equivalent to $\Phi(0)>0$).

(ii) For real $x\leq a$, $q\in [0,\infty)$, we have the classical identity [Reference Kyprianou14, equation (3.15)]

(2.2) \begin{equation}\mathbb{P}_x\big[\text{e}^{-q\tau_a^+};\, \tau_a^+<\infty\big]=\text{e}^{-\Phi(q)(a-x)},\end{equation}

where $\tau_a^+\coloneqq \inf\{t\in (0,\infty)\colon X_t>a\}$. Equation (2.2) causes $\overline{X}_\textsf{e}-X_0$ to have the exponential distribution of rate $\Phi(p)$.

(iii) Associated to the solution (2.2) of the first passage upwards problem is the family, in $\lambda\in[0,\infty)$, of exponential $\mathcal F$-martingales, $\mathcal{E}^\lambda=(\mathcal{E}^\lambda_t)_{t\in [0,\infty)}$,

(2.3) \begin{equation} \mathcal{E}^\lambda_t\coloneqq \text{e}^{\lambda(X_t-X_0)-\psi(\lambda)t},\quad t\in[0,\infty).\end{equation}

(iv) Regarding the position of X at first passage upwards, we have the following resolvent identity [Reference Kuznetsov, Kyprianou and Rivero13, Theorem 2.7(ii)]. For real $x\leq a$, $q\in [0,\infty)$, and $f\in \mathcal{B}_{\mathbb{R}}/\mathcal{B}_{[0,\infty]}$,

(2.4) \begin{equation}\int_0^\infty \,\text{e}^{-q t}\mathbb{P}_x[f(X_t);\, t<\tau_a^+] \,\text{d} t = \int_{ -\infty}^a f(y)\big(\text{e}^{-\Phi(q)(a-x)}W^{(q)}(a-y) - W^{(q)}(x-y) \big) \,\text{d} y,\end{equation}

where, for $q\in [0,\infty)$, $W^{(q)}\colon \mathbb{R}\to [0,\infty)$ is the q-scale function of X, characterized by being continuous on $[0,\infty)$, vanishing on $({-}\infty,0)$, and having Laplace transform

(2.5) \begin{equation}\int_0^\infty \,\text{e}^{-\theta x}W^{(q)}(x) \,\text{d} x=\dfrac{1}{\psi(\theta)-q},\quad \theta\in (\Phi(q),\infty).\end{equation}

The reader is referred to [Reference Kuznetsov, Kyprianou and Rivero13] for further background on scale functions of snLp. As usual we set $W^{(0)}=\!:\,W$, and recall that $W(0)>0$ or $W(0)=0$, respectively, as X has paths of finite or infinite variation [Reference Kuznetsov, Kyprianou and Rivero13, Lemma 3.1].

2.2 Wiener–Hopf factorization

The following falls under the umbrella of the Wiener–Hopf factorization.

(i) Because X is regular upwards [Reference Kyprianou14, page 232], the process $(X_t;\, t\in [0,G))$ is independent of the process $(X_{G+t}-X_{G-};\, t\in [0,\textsf{e}-G))$ [Reference Bertoin3, Lemma VI.6(ii)], and [Reference Bertoin3, comment following Lemma VI.6(ii)] $X_{G-}=X_G$ a.s. if and only if X is regular downwards, that is, if and only if X has paths of infinite variation [Reference Kyprianou14, page 232]. In particular, the Wiener–Hopf factors $(G,\overline{X}_\textsf{e})$ and $(\textsf{e}-G,\overline{X}_\textsf{e}-X_\textsf{e})$ are independent. When $p>0$, then their Laplace transforms are given, for $\{\gamma,\delta\}\subset [0,\infty)$, by [Reference Kyprianou14, Theorem 6.15(ii)]

\[\mathbb{P}\big[\text{e}^{-\gamma G-\delta \overline{X}_\textsf{e}}\big]=\dfrac{\kappa(p,0)}{\kappa(p+\gamma,\delta)}\quad\text{and}\quad\mathbb{P}\big[\text{e}^{-\gamma (\textsf{e}-G)+\delta (X_\textsf{e}-\overline{X}_\textsf{e})}\big]=\dfrac{\hat{\kappa}(p,0)}{\hat{\kappa}(p+\gamma,\delta)},\]

where $\kappa$ and $\hat{\kappa}$ are the Laplace exponents of the increasing and decreasing ladder heights processes, respectively. The latter are themselves in turn expressed explicitly as [Reference Kyprianou14, Section 6.5.2]

\[\kappa(\gamma,\delta)=\Phi(\gamma)+\delta\quad \text{and}\quad \hat{\kappa}(\gamma,\delta)=\dfrac{\gamma-\psi(\delta)}{\Phi(\gamma)-\delta},\quad\{\gamma,\delta\}\subset [0,\infty) \]

(the expression for $\hat{\kappa}$ being understood in the limiting sense when $\Phi(\gamma)=\delta$).

(ii) We have

\[ \mathbb{P}(\overline{X}_\textsf{e}=X_\textsf{e})=\lim_{\beta\to\infty}\mathbb{P} \big[\text{e}^{\beta(X_\textsf{e}-\overline{X}_\textsf{e})}\big]=\lim_{\beta\to\infty}\dfrac{p(\Phi(p)-\beta)}{\Phi(p)(p-\psi(\beta))}, \]

so that

(2.6) \begin{equation} \mathbb{P}(\overline{X}_\textsf{e} = X_\textsf{e}) = \begin{cases} \dfrac{p}{\Phi(p)\textsf d}& \text{if \textit{X} has paths of finite variation,} \\ \\[-9pt] 0 & \text{if \textit{X} has paths of infinite variation.} \end{cases}\end{equation}

For convenience we shall understand $\textsf d=\infty$, and (hence) ${{p}/{(\Phi(p)\textsf d)}}=0$, when X has paths of infinite variation. Note also that ${{p}/{(\Phi(p)\textsf d)}}<1$.

2.3 Excursions from the maximum

We gather here some facts concerning the Itô point process of excursions [Reference Blumenthal5, Reference Itô11] from the maximum of X. For what follows, as well as the general references given in the Introduction, the reader may also consult [Reference Greenwood and Pitman10], [Reference Rogers20], and [Reference Avram, Kyprianou and Pistorius1, passim].

(i) Under $\mathbb{P}$ the running supremum $\overline{X}$ serves as a continuous local time for X at the maximum. Its right-continuous inverse is the process of first passage times $\tau^+=(\tau_a^+\!)_{a\in [0,\infty)}$. The time axis $[0,\infty)$ is partitioned $\mathbb{P}$-a.s. into

\[\textsf M\coloneqq \overline{\{t\in [0,\infty)\colon \overline{X}_t=X_t\}}\allowbreak =\{t\in [0,\infty)\colon \overline{X}_t=X_t\text{ or }\overline{X}_{t}=X_{t-}\},\]

the closure of the random set of times when X is at its running supremum, and the open intervals $(\tau_{a-}^+,\tau_a^+\!)$, $a\in \textsf D\coloneqq \{b\in (0,\infty)\colon \tau_{b-}^+<\tau_b^+\}$; the visiting set $\textsf M$ has $\mathbb{P}$-a.s. no isolated points.

(ii) The process $\varepsilon=(\varepsilon_a)_{a\in (0,\infty)}$, defined for $a\in (0,\infty)$ by

\[\varepsilon_a(u)\coloneqq \begin{cases} X_{(\tau_{a-}^++u)\land \tau_a^+}-X_{\tau_{a-}^+-} & \text{if $a\in \textsf D$,}\\ \\[-9pt] \Delta & \text{otherwise,}\end{cases}\]

for $u\in [0,\infty)$ (of course $X_{\tau_{a-}^+-}=a$ off the event $\{X_0\ne 0\}$, which is $\mathbb{P}$-negligible), where $\Delta\notin \mathbb D$ is a cemetery state, is, under $\mathbb{P}$, a Poisson point process (Ppp) with values in $(\mathbb D,\mathcal D)$, in the filtration $\mathcal F_{\tau^+}=(\mathcal F_{\tau^+_a})_{a\in [0,\infty)}$, absorbed on first entry into a path $\omega\in \mathbb D$ for which $\zeta(\omega)=\infty$, where $\zeta(\omega)\coloneqq \inf\{t\in (0, \infty)\colon \omega(t)\geq 0\}$, and whose characteristic measure we will denote by $\textsf{n}$ (so the intensity measure of $\varepsilon$ is $\textsf{l}\times \textsf{n}$, where $\textsf{l}$ is Lebesgue measure on $\mathcal{B}_{(0,\infty)}$). Note that $\textsf{n}$ is carried by the set $\{\zeta>0\}$ and also by $\{\xi_0<0\}$ (resp. $\{\xi_0=0\}$) when X has paths of finite (resp. infinite) variation. Besides, $\mathbb{P}$-a.s., for all $a\in \textsf D$, $\zeta(\varepsilon_a)=\tau_{a}^+-\tau_{a-}^+$.

(iii) We have

\[\mathbb{P} \biggl[\sum_{a\in \textsf D}Z_{a}(\varepsilon_a)\biggr]=\mathbb{P}\biggl[\int_0^{\overline{X}_\infty}\int Z_a(\omega)\textsf{n}(\text{d}\omega) \,\text{d} a\biggr]\]

by the compensation formula for Ppp, whenever $Z\in \mathcal P_{\mathcal F_{\tau^+}}\otimes \mathcal D/\mathcal{B}_{[0,\infty]}$, with $ \mathcal P_{\mathcal F_{\tau^+}}$ the $\mathcal F_{\tau^+}$-predictable $\sigma$-field. In this regard note that if $R\in \mathcal P_\mathcal F\otimes \mathcal D/\mathcal{B}_{[0,\infty]}$, where $\mathcal P_\mathcal F$ is now the $\mathcal F$-predictable $\sigma$-field, then $(R_{\tau_{a-}^+})_{a\in (0,\infty)}\in \mathcal P_{\mathcal F_{\tau^+}}\otimes \mathcal D/\mathcal{B}_{[0,\infty]}$.

(iv) The measure $\textsf{n}$ has the following Markov property: for all $t\in [0,\infty)$,

\[\textsf{n}[G\cdot H\circ \theta_t;\, t<\zeta]=\textsf{n}[G\mathbb{P}_{\xi_t}[H];\, t<\zeta],\quad G\in \mathcal D_t/\mathcal{B}_{[0,\infty]},H\in \mathcal D/\mathcal{B}_{[0,\infty]}.\]

(v) If X has paths of infinite variation (equivalently, X is regular downwards) the result of [Reference Chaumont and Doney6, Corollary 1] applied to $-X$ (in conjunction with [Reference Chaumont and Doney6, equations (2.5) and (2.8)] therein, (2.2), and the fact that in this case $\textsf{n}[1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}}]=\Phi(p)$: see Remark 4.1), implies that for all $t\in (0,\infty)$ and then all $F\in \mathcal D_t/\mathcal{B}_{\mathbb{R}}$ bounded and continuous in the Skorokhod topology on $\mathbb D$,

(2.7) \begin{equation}\textsf{n}[F;\, t<\zeta]=\textsf{k}\lim_{x\uparrow 0}\dfrac{\mathbb{P}_x[F;\, t<\tau_0^+]}{\textsf{g}(x)},\end{equation}

where

\[\textsf{g}(x)\coloneqq \lim_{q\downarrow 0}\dfrac{1-\text{e}^{\Phi(q)x}}{\Phi(q)}=\begin{cases}\dfrac{1-\text{e}^{\Phi(0)x}}{\Phi(0)} & \Phi(0)>0{,}\\\\[-9pt] -x & \Phi(0)=0{,}\end{cases}\quad x\in ({-}\infty,0),\]

and with $\textsf{k}\in (0,\infty)$ depending on the characteristics of X only.

(vi) We have the following representation of the scale function W [Reference Kuznetsov, Kyprianou and Rivero13, equation (31)]:

(2.8) \begin{equation}\dfrac{W(x)}{W(a)}=\exp\biggl\{-\int_x^a\textsf{n}({-}\underline{\xi}_{\zeta}>y) \,\text{d} y\biggr\},\quad 0\leq x < a,\end{equation}

where $\underline{\xi}=(\underline{\xi}_t)_{t\in [0,\infty]}$ is the running infimum process of $\xi$.

2.4 Point process of jumps

Under $\mathbb{P}$ the process $\Delta X=(\Delta X_t)_{t\in (0,\infty)}$ of the jumps of X is a Ppp in the filtration $\mathcal F$ with values in $(\mathbb{R}\backslash \{0\},\mathcal{B}_{\mathbb{R}\backslash \{0\}})$ and 0 as cemetery state, whose characteristic measure is the restriction to $\mathcal{B}_{\mathbb{R}\backslash \{0\}}$ of $\nu$. In this case the compensation formula for Ppp states that

\[\mathbb{P}\biggl[\sum_{t\in\textsf J}Z_{t}(\Delta X_t)\biggr]=\mathbb{P}\biggl[\int_0^\infty \int Z_t(x)\textsf{n}( \text{d} x) \,\text{d} t\biggr]\quad\text{for $Z\in \mathcal P_\mathcal F\otimes \mathcal{B}_{\mathbb{R}\backslash \{0\}}/\mathcal{B}_{[0,\infty]}$,}\]

where $\textsf J\coloneqq \{t\in (0,\infty)\colon \Delta X_t\ne 0\}$ is the set of jump times of X. See for instance [Reference Bertoin3, Theorem I.1].

2.5 Patie’s scale functions

Assuming $\Phi(p)\notin \alpha\mathbb{N}$, we set

\[\mathcal{J}^{p,\alpha}(y)\coloneqq \sum_{k=0}^{\infty}\textsf{a}^{p,\alpha}_ky^k,\quad y\in [0,\infty),\quad \text{where }\textsf{a}^{p,\alpha}_k\coloneqq \biggl(\prod_{l=1}^k(\psi(l\alpha)-p)\biggr)^{-1},\quad k\in \mathbb{N}_0\]

(the condition on $\Phi(p)$ ensures all the $\textsf{a}^{p,\alpha}_k$ are well-defined), and, whether or not $\Phi(p)\notin \alpha\mathbb{N}$,

\[\mathcal{I}^{p,\alpha}(y)\coloneqq \sum_{k=0}^{\infty}\textsf{b}^{p,\alpha}_k y^k,\quad y\in [0,\infty),\quad \text{where }\textsf{b}^{p,\alpha}_k\coloneqq \biggl(\prod_{l=1}^k(\psi(\Phi(p)+l\alpha)-p)\biggr)^{-1},\quad k\in \mathbb{N}_0,\]

with (as usual) the empty product being interpreted as $=1$. These power series converge absolutely, indeed

\[ \lim_{k\to\infty}\dfrac{\textsf{a}^{p,\alpha}_{k+1}}{\textsf{a}^{p,\alpha}_k}=\lim_{k\to\infty}\dfrac{\textsf{b}^{p,\alpha}_{k+1}}{\textsf{b}^{p,\alpha}_k}=0, \]

and the coefficients $\textsf{a}^{p,\alpha}_k$ are ultimately of the same sign (even all strictly positive when $\Phi(p)<\alpha$). Finally

(2.9) \begin{equation}\mathbb{P}_x \big[\text{e}^{-\gamma I_{\tau_a^+}};\, \tau_a^+<\textsf{e}\big]=\dfrac{\text{e}^{\Phi(p)x}\mathcal{I}^{p,\alpha}(\gamma\text{e}^{\alpha x})}{\text{e}^{\Phi(p)a}\mathcal{I}^{p,\alpha}(\gamma\text{e}^{\alpha a})},\quad x\leq a,\gamma\in[0,\infty){.}\end{equation}

In other words

(2.10) \begin{equation}\mathbb{Q}_y\big[\text{e}^{-\gamma T_d^+};\, T_d^+<\infty\big]=\biggl(\dfrac{y}{d}\biggr)^{\Phi(p)}\dfrac{\mathcal{I}^{p,\alpha}(\gamma y^\alpha)}{\mathcal{I}^{p,\alpha}(\gamma d^\alpha)},\quad 0<y\leq d,\gamma\in[0,\infty),\end{equation}

where

\[T_d^+\coloneqq \inf\{s\in(0,\infty)\colon Y_s>d\}\quad \text{for $d\in (0,\infty)$.}\]

See [Reference Patie17, Theorem 2.1] (or [Reference Kyprianou14, Section 13.7], [Reference Vidmar22, Example 6]).

4.1 Laplace transform of L given $\overline{Y}_\infty$

Proposition 4.1. Let $\gamma\in [0,\infty)$. For $x\in \mathbb{R}$, a.s.-$\mathbb{P}_x$,

\[\mathbb{P}_x\biggl[\exp\biggl\{-\gamma\int_0^G\,\text{e}^{\alpha X_u} \,\text{d} u\biggr\}\mid \overline{X}_\textsf{e}\biggr]=\dfrac{\mathcal{I}^{p,\alpha}(\gamma\text{e}^{\alpha x})}{\mathcal{I}^{p,\alpha}(\gamma\text{e}^{\alpha \overline{X}_\textsf{e}})}.\]

In other words, for $y\in (0,\infty)$, a.s.-$\mathbb{Q}_y$,

\[\mathbb{Q}_y[\text{e}^{-\gamma L}\mid \overline{Y}_\infty]=\dfrac{\mathcal{I}^{p,\alpha}(\gamma y^\alpha)}{\mathcal{I}^{p,\alpha}(\gamma {\overline{Y}_\infty}^\alpha)}.\]

Remark 4.1 In the proof we will, en passant, establish the identity

\[ \textsf{n}[1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}}]=\Phi(p)-\dfrac{p}{\textsf d} \]

(recall that we interpret $\textsf d=\infty$, hence ${{p}/{\textsf d}}=0$, when X has paths of infinite variation).

Proof. Without loss of generality we work under $\mathbb{P}$: for $x\in \mathbb{R}$, the law of $(L,\overline{X}_\textsf{e})$ under $\mathbb{P}_x$ is that of $(\text{e}^{\alpha x}L,x+\overline{X}_\textsf{e})$ under $\mathbb{P}$. Then we are to determine, for $f\in \mathcal{B}_\mathbb{R}/\mathcal{B}_{[0,\infty]}$,

\begin{align*} \mathcal A & \coloneqq \mathbb{P}\biggl[\exp\biggl\{-\gamma \int_0^G \,\text{e}^{\alpha X_u} \,\text{d} u\biggr\}f(\overline{X}_\textsf{e})\biggr]\\* & =\mathbb{P} \biggl[\sum_{a\in \textsf D}\exp\biggl\{-\gamma \int_0^{\tau_{a-}^+} \,\text{e}^{\alpha X_u} \,\text{d} u\biggr\}f\big(\overline{X}_{\tau_{a-}^+}\big)\mathds{1}_{\{\tau_{a-}^+<\textsf{e}\leq \tau_a^+\}}\biggr]\\* &\quad\, +\mathbb{P}\biggl[\exp\biggl\{-\gamma \int_0^G \,\text{e}^{\alpha X_u}\,\text{d} u\biggr\}f(\overline{X}_G);\, \overline{X}_\textsf{e}=X_\textsf{e}\biggr]. \end{align*}

Here the second term only appears when $p>0$. From the Wiener–Hopf factorization, the event $\{\overline{X}_\textsf{e}=X_\textsf{e}\}$ is independent of the process X on the time interval [0,G), and (2.6) $\mathbb{P}(\overline{X}_\textsf{e}=X_\textsf{e})={{p}/{(\Phi(p) \textsf d)}}$. In consequence

\begin{align*} \mathcal A\biggl(1-\dfrac{p}{\Phi(p)\textsf d}\biggr) & = \mathbb{P} \biggl[\sum_{a\in \textsf D}\exp\biggl\{-p \tau_{a-}^+-\gamma \int_0^{\tau_{a-}^+} \,\text{e}^{\alpha X_u} \,\text{d} u\biggr\}f\big(\overline{X}_{\tau_{a-}^+}\big) \big(1-\text{e}^{-p(\tau_a^+-\tau_{a-}^+\!)}\mathds{1}_{\{\tau_a^+<\infty\}}\big)\biggr]\\* &=\mathbb{P} \biggl[\sum_{a\in \textsf D}\exp\biggl\{-p \tau_{a-}^+-\gamma \int_0^{\tau_{a-}^+} \,\text{e}^{\alpha X_u} \,\text{d} u\biggr\}f\big(\overline{X}_{\tau_{a-}^+}\big)\big(1-\text{e}^{-p\zeta(\varepsilon_a)}\mathds{1}_{\{\zeta(\varepsilon_a)<\infty\}}\big)\biggr]. \end{align*}

Now, by the absence of positive jumps, $f(\overline{X}_{\tau_{a-}^+})=a$ for all $a\in (0,\infty)$ a.s.-$\mathbb{P}$. The compensation formula for $\varepsilon$ then entails that

\begin{align*} \mathcal A\biggl(1-\dfrac{p}{\Phi(p)\textsf d}\biggr) &=\mathbb{P}\biggl[\int_0^{\overline{X}_\infty} \,\text{e}^{-p\tau_{a-}^+-\gamma\int_0^{\tau_{a-}^+}\text{e}^{\alpha X_u} \,\text{d} u}f(a) \,\text{d} a\biggr]\textsf{n}[1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}}]\\*&=\textsf{n}[1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}}]\mathbb{P}\biggl[\int_0^{\overline{X}_\infty} \,\text{e}^{-p\tau_a^+-\gamma\int_0^{\tau_{a}^+}\,\text{e}^{\alpha X_u} \,\text{d} u}f(a) \,\text{d} a\biggr]\\* & =\textsf{n}[1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}}]\int_0^{\infty} \mathbb{P} \big[\text{e}^{-\gamma\int_0^{\tau_{a}^+}\,\text{e}^{\alpha X_u} \,\text{d} u};\, \tau_{a}^+<\textsf{e}\big]f(a) \,\text{d} a, \end{align*}

where the penultimate equality follows from the fact that $\tau^+$ has at most countably many jump times, the set of which therefore has zero Lebesgue measure. Using (2.9) this can be expressed as

\[\mathcal A\biggl(1-\dfrac{p}{\Phi(p)\textsf d}\biggr)=\textsf{n}[1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}}]\int_0^{\infty}\dfrac{\mathcal{I}^{p,\alpha}(\gamma)}{\mathcal{I}^{p,\alpha}(\gamma\text{e}^{\alpha a})}f(a)\,\text{e}^{-\Phi(p)a} \,\text{d} a.\]

Taking $\gamma=0$ and $f=1$ and plugging back in concludes the proof, since, under $\mathbb{P}$, the law of $\overline{X}_\textsf{e}$ is exponential of rate $\Phi(p)$. □

4.2 Law of J

Proposition 4.2. Assume X has paths of finite variation. For $h\in \mathcal{B}_{({-}\infty,0]}/\mathcal{B}_{[0,\infty]}$,

\[\mathbb{P}[h(X_G-\overline{X}_\textsf{e})]=\dfrac{1}{\Phi(p)\textsf d}\biggl[ph(0)+\int h(z)(1-\text{e}^{\Phi(p)z})\nu(\text{d} z)\biggr].\]

In other words, for $g\in \mathcal{B}_{(0,1]}/\mathcal{B}_{[0,\infty]}$,

(4.1) \begin{equation}\mathbb{Q}[g(J)]=\dfrac{1}{\Phi(p)\textsf d}\biggl[pg(1)+\int g(\text{e}^z)(1-\text{e}^{\Phi(p)z})\nu(\text{d} z)\biggr].\end{equation}

Proof. Let $f\in \mathcal{B}_\mathbb{R}/\mathcal{B}_{[0,\infty]}$. By the compensation formulas for $\varepsilon$ and $\Delta X$, for any (arbitrary) $q\in (0,\infty)$, we have

\begin{align*} q \textsf{n}[f(\xi_0)]\Phi(q)^{-1} & =q\mathbb{P}\biggl[\sum_{a\in \textsf D}\,\text{e}^{-q\tau_{a-}^+}f(\varepsilon_a(0))\biggr]\\[2pt] & =q\mathbb{P}\biggl[\sum_{t\in \textsf J}\,\text{e}^{-q t}\mathds{1}_{\{\overline{X}_{t-}=X_{t-}\}}f(\Delta X_t)\biggr]\\[2pt] &=\mathbb{P}\biggl[\int_0^\infty \mathds{1}_{\{\overline{X}_{t-}=X_{t-}\}}q \,\text{e}^{-q t} \,\text{d} t\biggr]\nu[\kern1.5ptf]\\[2pt] &=\mathbb{P}\biggl[\int_0^\infty \mathds{1}_{\{\overline{X}_{t}=X_{t}\}}q \,\text{e}^{-q t} \,\text{d} t\biggr]\nu[\kern1.5ptf]\\[2pt] &=\dfrac{q}{\Phi(q) \textsf d}\nu[\kern1.5ptf]. \end{align*}

Let h be bounded. As in the proof of Proposition 4.1, we compute

\begin{align*} \mathbb{P}[h(X_G-\overline{X}_\textsf{e})]&=\mathbb{P} \biggl[\sum_{a\in \textsf D}h(\varepsilon_a(0))\mathds{1}_{\{\tau_{a-}^+<\textsf{e}\leq \tau_a^+\}}\biggr]+h(0)\mathbb{P}(\overline{X}_\textsf{e}=X_\textsf{e})\\[2pt] &=\mathbb{P}\biggl[\sum_{a\in \textsf D}\,\text{e}^{-p\tau_{a-}^+}h(\varepsilon_a(0))(1-\text{e}^{-p\zeta(\varepsilon_a)}\mathds{1}_{\{\zeta(\varepsilon_a)<\infty\}})\biggr]+h(0)\dfrac{p}{\Phi(p)\textsf d}\\[2pt] &=\mathbb{P}\biggl[\int_0^{\overline{X}_a}\,\text{e}^{-p\tau_{a-}^+} \,\text{d} a\biggr]\textsf{n} [h(\xi_0)(1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}})]+h(0)\dfrac{p}{\Phi(p)\textsf d}\\[2pt] &=\Phi(p)^{-1}\lim_{t\downarrow 0}\textsf{n}[h(\xi_0)(1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}});\, t<\zeta]+h(0)\dfrac{p}{\Phi(p)\textsf d}\\[2pt] &=\Phi(p)^{-1}\lim_{t\downarrow 0}\textsf{n}[h(\xi_0)\,\text{e}^{-pt}(1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}})\circ \theta_t;\, t<\zeta]+h(0)\dfrac{p}{\Phi(p)\textsf d}\\[2pt] &=\Phi(p)^{-1}\lim_{t\downarrow 0}\textsf{n}[h(\xi_0)\mathbb{P}_{\xi_t}[1-\text{e}^{-p\tau_0^+}\mathds{1}_{\{\tau_0^+<\infty\}})];\, t<\zeta]+h(0)\dfrac{p}{\Phi(p)\textsf d}\\[2pt] &=\Phi(p)^{-1}\lim_{t\downarrow 0}\textsf{n}[h(\xi_0)(1-\text{e}^{\Phi(p)\xi_t});\, t<\zeta]+h(0)\dfrac{p}{\Phi(p)\textsf d}\\[2pt] &=\Phi(p)^{-1}\textsf{n}[h(\xi_0)(1-\text{e}^{\Phi(p)\xi_0})]+h(0)\dfrac{p}{\Phi(p)\textsf d}{.} \end{align*}

Here the fifth equality follows by dominated convergence since

\[(1-\text{e}^{-pt})\mathds{1}_{\{t<\zeta\}}\leq 1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}}\quad\text{and}\quad\textsf{n}[1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}}]<\infty{,}\]

the antepenultimate equality is by the Markov property of $\textsf{n}$, and the final equality is by dominated convergence since

\[(1-\text{e}^{\Phi(p)\xi_t})\mathds{1}_{\{t<\zeta\}}\leq (1-\text{e}^{\Phi(p)\underline{\xi}_{\zeta}})\]

and

\[\textsf{n}[1-\text{e}^{\Phi(p)\underline{\xi}_{\zeta}}]=\int_0^\infty \Phi(p)\,\text{e}^{-\Phi(p)a}\textsf{n}\Big({-}\underline{\xi}_{\zeta}>a\Big) \,\text{d} a,\]

which quantity is finite when X has paths of finite variation, as follows from (2.8) and the fact that in this case $W(0)>0$. By the first part of this proof the claims follow. □

4.3 Laplace transform of $T_0-L$ given $\overline{Y}_\infty$ and J

The following definition will allow us to state our next result more succinctly. Recall the notation of Section 2.5.

Definition 4.1. For $\beta\in[0,\infty)$ and $ y\in (0,\infty)$, set

\[\mathcal{N}^{p,\alpha}_\beta(y)\coloneqq \mathcal{J}^{p,\alpha}(\beta y^\alpha)+\dfrac{\alpha}{\Phi(p)}\biggl[\dfrac{\mathcal{J}^{p,\alpha}(\beta y^\alpha)}{\mathcal{I}^{p,\alpha}(\beta y^\alpha)}\sum_{k=1}^\infty k\textsf{b}^{p,\alpha}_k(\beta y^\alpha)^k-\sum_{k=1}^\infty k\textsf{a}^{p,\alpha}_k(\beta y^\alpha)^k\biggr],\]

where the expression must be understood in the limiting sense (as $\alpha\to \Phi(p)/m$), when $\Phi(p)=\alpha m$ for some $m\in \mathbb{N}$: the limit is seen to exist and identified in Remark 4.3 to follow.

Remark 4.3. Suppose $\Phi(p)=m\alpha$ for an $m\in \mathbb{N}$. Then, for

\[ \alpha'\in \biggl(\dfrac{\Phi(p)}{m+1},\dfrac{\Phi(p)}{m-1}\biggr)\bigg\backslash \biggl\{\dfrac{\Phi(p)}{m}\biggr\}, \]

provisionally setting

\[ \widetilde{\textsf{b}^{p,\alpha'}_k}\coloneqq \biggl(\prod_{l=1}^k(\psi(m\alpha'+l\alpha')-p)\biggr)^{-1}\quad {\text{for $k\in \mathbb{N}_0$,}} \]

we may write

\begin{align*} & \mathcal{N}^{p,\alpha'}_\beta(y)\\[2pt] &\quad =\sum_{k=0}^{m-1}\textsf{a}^{p,\alpha'}_k (\beta y^{\alpha'}\!)^k+\dfrac{\alpha'}{\Phi(p)}\biggl[\dfrac{\sum_{k=0}^{m-1}\textsf{a}^{p,\alpha'}_k (\beta y^{\alpha'}\!)^k}{\mathcal{I}^{p,\alpha}(\beta y^{\alpha'}\!)}\sum_{k=1}^\infty k\textsf{b}^{p,\alpha'}_k(\beta y^{\alpha'}\!)^k - \sum_{k=1}^{m-1}k\textsf{a}^{p,\alpha'}_k(\beta y^{\alpha'}\!)^k\biggr] \\* &\quad\quad\, + \dfrac{\textsf{a}^{p,\alpha'}_{m-1}(\beta y^{\alpha'}\!)^m}{\psi(\alpha' m)-p} \biggl[ \sum_{k=0}^\infty \widetilde{\textsf{b}^{p,\alpha'}_k} (\beta y^{\alpha'}\!)^k +\dfrac{\alpha'}{\Phi(p)} \\* &\quad \quad\, \times \biggl(\dfrac{\sum_{k=0}^\infty \widetilde{\textsf{b}^{p,\alpha'}_k} (\beta y^{\alpha'}\!)^k}{\mathcal{I}^{p,\alpha}(\beta y^{\alpha'}\!)} \sum_{k=1}^\infty k\textsf{b}^{p,\alpha'}_k (\beta y^{\alpha'}\!)^k- \sum_{k=0}^\infty \widetilde{\textsf{b}^{p,\alpha'}_k} (\beta y^{\alpha'}\!)^k(k+m) \biggr) \biggr]\\&\quad \xrightarrow{\alpha'\to \alpha}\sum_{k=0}^{m-1}\textsf{a}^{p,\alpha}_k (\beta y^{\alpha})^k+\dfrac{\alpha}{\Phi(p)}\biggl[\dfrac{\sum_{k=0}^{m-1}\textsf{a}^{p,\alpha}_k (\beta y^{\alpha})^k}{\mathcal{I}^{p,\alpha}(\beta y^{\alpha})}\sum_{k=1}^\infty k\textsf{b}^{p,\alpha}_k(\beta y^{\alpha})^k-\sum_{k=1}^{m-1}k\textsf{a}^{p,\alpha}_k(\beta y^{\alpha})^k\biggr]\\ &\quad\quad\, +\dfrac{\textsf{a}^{p,\alpha}_{m-1}(\beta y)^{\alpha m}}{\psi'(\Phi(p))\Phi(p)}\biggl[\alpha\biggl(1-\dfrac{\sum_{k=1}^\infty k \textsf{b}^{p,\alpha}_k(\beta y^\alpha)^k}{\mathcal{I}^{p,\alpha}(\beta y^\alpha)}\biggr)\sum_{k=1}^\infty\textsf{c}^{p,\alpha}_k(\beta y^\alpha)^k-\mathcal{I}^{p,\alpha}(\beta y^\alpha)\biggr]\\* &\quad=\mathcal{N}^{p,\alpha}_\beta(y) \end{align*}

(recall the $\textsf{c}^{p,\alpha}_k$ from Remark 3.3).

Proof. Again we may work without loss of generality under $\mathbb{P}$. Let f and h be bounded functions from $\mathcal{B}_{\mathbb{R}}/\mathcal{B}_{[0,\infty)}$. We compute

\begin{align*} & \mathbb{P}[\text{e}^{-\beta\int_G^\textsf{e} \,\text{e}^{\alpha X_u} \,\text{d} u}h(X_G-\overline{X}_\textsf{e})f(\overline{X}_\textsf{e})]\\* &\quad =\mathbb{P} \biggl[ \sum_{a\in \textsf D}\exp \biggl\{-\beta \,\text{e}^{\alpha \overline{X}_{\tau_{a-}^+}} \int_0^{\textsf{e}-\tau_{a-}^+} \,\text{e}^{\alpha \varepsilon_a(u)} \,\text{d} u\biggr\} h(\varepsilon_a(0))f(\overline{X}_{\tau_{a-}^+})\mathds{1}_{\{\tau_{a-}^+ < \textsf{e}\leq \tau_a^+\}} \biggr] \\* &\quad\quad\, +\mathbb{P} [h(0)f(\overline{X}_\textsf{e});\, \overline{X}_\textsf{e}=X_\textsf{e}]\\ &\quad =\mathbb{P}\biggl[\sum_{a\in \textsf D}\,\text{e}^{-p\tau_{a-}^+}\exp\biggl\{-\beta \,\text{e}^{\alpha \overline{X}_{\tau_{a-}^+}}\int_0^{\textsf{e}} \,\text{e}^{\alpha \varepsilon_a(u)} \,\text{d} u\biggr\}h(\varepsilon_a(0))f(a)\mathds{1}_{\{\textsf{e}\leq \zeta(\varepsilon_a)\}}\biggr]\\* &\quad\quad\, +h(0)\mathbb{P}[f(\overline{X}_\textsf{e})]\mathbb{P}(\overline{X}_\textsf{e}=X_\textsf{e})\\ &\quad =\int_0^\infty \,\text{e}^{-\Phi(p)a}f(a)\int \textsf{n}[\text{e}^{-\beta \,\text{e}^{\alpha a}\text{I}_u}h(\xi_0);\, u\leq \zeta]\,\text{Exp}_p(\text{d} u)\,\text{d} a \\* &\quad\quad\, +h(0)\int_0^\infty\Phi(p) \,\text{e}^{-\Phi(p)a}f(a) \,\text{d} a \dfrac{p}{\Phi(p)\textsf d}.\end{align*}

Here the second term in the second line (and subsequent lines) appears only if $p>0$, the penultimate equality is by the memoryless property of the exponential distribution and because X is independent of $\textsf{e}$, and $\text{Exp}_p$ is the exponential law of rate p on $\mathcal{B}_{(0,\infty]}$. Thus it remains to determine, for $a\in [0,\infty)$,

\begin{align*} \mathfrak{O} & \coloneqq \int \textsf{n}[\text{e}^{-\beta \,\text{e}^{\alpha a}\text{I}_u}h(\xi_0);\, u\leq \zeta]\,\text{Exp}_p(\text{d} u) \\* &= \lim_{t\downarrow 0}\int \textsf{n}[\text{e}^{-\beta \,\text{e}^{\alpha a}\text{I}_u}h(\xi_0);\, t \lt u\leq \zeta] \,\text{Exp}_p(\text{d} u)\\* & =\lim_{t\downarrow 0}\int_{(t,\infty]} \textsf{n}[\text{e}^{-\beta \,\text{e}^{\alpha a}\text{I}_t}h(\xi_0)(\text{e}^{-\beta \,\text{e}^{\alpha a}\text{I}_{u-t}}\mathds{1}_{\{\zeta\geq u-t\}})\circ\theta_t;\, t<\zeta]\,\text{Exp}_p(\text{d} u)\\& =\lim_{t\downarrow 0}\int_{(t,\infty]} \textsf{n}[h(\xi_0)(\text{e}^{-\beta \,\text{e}^{\alpha a}\text{I}_{u-t}}\mathds{1}_{\{\zeta\geq u-t\}})\circ\theta_t;\, t<\zeta]\,\text{Exp}_p(\text{d} u)\\& =\lim_{t\downarrow 0}\int_{(t,\infty]} \textsf{n}[h(\xi_0)(\text{e}^{-\beta \,\text{e}^{\alpha a}\text{I}_{u-t}}\mathds{1}_{\{\zeta\geq u-t\}})\circ\theta_t;\, t<\zeta]\,\text{e}^{pt}\,\text{Exp}_p(\text{d} u)\\& =\lim_{t\downarrow 0}\int \textsf{n}[h(\xi_0)(\text{e}^{-\beta \,\text{e}^{\alpha a}\text{I}_{v}}\mathds{1}_{\{\zeta\geq v\}})\circ\theta_t;\, t<\zeta]\,\text{Exp}_p( \text{d} v)\\& =\lim_{t\downarrow 0}\int \textsf{n} [h(\xi_0)\mathbb{P}_{\xi_t}[\text{e}^{-\beta \,\text{e}^{\alpha a}\text{I}_v};\, \tau_0^+\geq v];\, t<\zeta]\,\text{Exp}_p( \text{d} v)\\& =\lim_{t\downarrow 0} \textsf{n} [h(\xi_0)\mathbb{P}_{\xi_t}[\text{e}^{-\beta \,\text{e}^{\alpha a}\text{I}_\textsf{e}};\, \tau_0^+\geq \textsf{e}];\, t<\zeta]\\*&=\lim_{t\downarrow 0}\textsf{n} [h(\xi_0)\mathcal{M}^{p,\alpha}_\beta( \text{e}^{a+\xi_t}, \text{e}^{ a});\, t<\zeta]{.} \end{align*}

Here the second line follows by monotone convergence, the fourth line is obtained by dominated convergence, since $\textsf{n}(\zeta\geq u)<\infty$ for each $u\in (0,\infty]$ and moreover $\textsf{n}[1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}}]<\infty$, and we used (3.1) in the last equality.

Suppose first that X is of finite variation. We know already from the proof of Proposition 4.2 that $\textsf{n}[1-\text{e}^{\Phi(p)\underline{\xi}_{\zeta}}]<\infty$. Since, for $t\in (0,\infty)$,

\[ \mathcal{M}^{p,\alpha}_\beta(\text{e}^{a+\xi_t},\text{e}^{ a})\leq 1-\text{e}^{\Phi(p)\xi_t}\leq 1-\text{e}^{\Phi(p)\underline{\xi}_{\zeta}}\quad \text{on $\{t<\zeta\}$,} \]

it follows by dominated convergence that

\[\mathfrak{O}=\textsf{n}[h(\xi_0)\mathcal{M}^{p,\alpha}_\beta( \text{e}^{a+\xi_0}, \text{e}^{ a})].\]

Hence, using the fact that under $\mathbb{P}$ the random variable ${\overline{X}_\textsf{e}}$ is exponential of rate $\Phi(p)$, Proposition 4.2, Remark 4.2, and the independence of $\overline{X}_\textsf{e}$ from $X_G-\overline{X}_\textsf{e}$, it follows that

\begin{align*} &\mathbb{P}[\text{e}^{-\beta\int_G^\infty \,\text{e}^{\alpha X_u} \,\text{d} u}h(X_G-\overline{X}_\textsf{e})f(\overline{X}_\infty)]\\[3pt] &\quad =\mathbb{P}\biggl[f(\overline{X}_\infty)h(X_G-\overline{X}_\infty)\biggl(\dfrac{\mathcal{M}^{p,\alpha}_\beta( \text{e}^{ X_G}, \text{e}^{ \overline{X}_\textsf{e}})}{1-\text{e}^{\Phi(p)(X_G-\overline{X}_\textsf{e})}}\mathds{1}_{({-}\infty,0)}(X_G-\overline{X}_\textsf{e})+\mathds{1}_{\{0\}}(X_G-\overline{X}_\textsf{e})\biggr)\biggr]\end{align*}

and (i) is proved.

Now let X be of infinite variation; take $h=1$. Because the coordinate projection $(\mathbb D\ni \xi\mapsto \xi_t)$ is continuous in the Skorokhod topology at all paths for which $t\in (0,\infty)$ is a continuity point, and in particular (by the Markov property of $\textsf{n}$ and since X has no fixed points of discontinuity a.s.) $\textsf{n}$-a.e., it follows from (2.7) that

\[\mathfrak{O}=\textsf{k}\lim_{t\downarrow 0}\lim_{x\uparrow 0} \dfrac{\mathbb{P}_x[\mathcal{M}^{p,\alpha}_\beta( \text{e}^{a+X_t}, \text{e}^{ a});\, t<\tau_0^+]}{\textsf{g}(x)}.\]

Hence the expression inside the limit $\lim_{t\downarrow 0}$, say r(t), is bounded by $\textsf{n}[1-\text{e}^{-p\zeta}\mathds{1}_{\{\zeta<\infty\}}]\,\text{e}^{pt}$, and the limit $\lim_{t\downarrow 0}r(t)$ exists a priori. Then

\[\lim_{q\to\infty}\int_0^\infty \,\text{e}^{-q t}q r(t) \,\text{d} t=\lim_{q\to\infty}\dfrac{q}{q-p}\int_0^\infty \,\text{e}^{-(q-p) t}(q-p) \,\text{e}^{-pt}r(t) \,\text{d} t=r(0+\!),\]

and since $\lim_{x\downarrow 0}{{\textsf{g}(x)}/{({-}x)}}=1$, we obtain

\[\mathfrak{O}=\textsf{k} \lim_{q \to\infty}\int_0^\infty q \,\text{e}^{-q t}\lim_{x\uparrow 0} \dfrac{\mathbb{P}_x[\mathcal{M}^{p,\alpha}_\beta( \text{e}^{a+X_t}, \text{e}^{ a});\, t<\tau_0^+]}{-x} \,\text{d} t.\]

Moreover, for all $t\in (0,\infty)$,

\[\mathcal{M}^{p,\alpha}_\beta( \text{e}^{a+X_t}, \text{e}^{ a})\leq 1-\text{e}^{\Phi(p)X_t}\quad \text{on $\{t<\tau_0^+\}$,}\]

and since

\begin{align*}\mathbb{P}_x[1-\text{e}^{\Phi(p)X_t};\, t<\tau_0^+] = \mathbb{P}_x[(1-\text{e}^{-p(\tau_0^+-t)}\mathds{1}_{\{\tau_0^+<\infty\}});\, t<\tau_0^+] \leq \mathbb{P}_x(\textsf{e}\leq \tau_0^+\!)=1-\text{e}^{\Phi(p)x},\end{align*}

dominated convergence yields

(4.2) \begin{align} \mathfrak{O}& =\textsf{k} \lim_{q \to\infty}\lim_{x\uparrow 0} \dfrac{\int_0^\infty q \,\text{e}^{-q t}\mathbb{P}_x [\mathcal{M}^{p,\alpha}_\beta( \text{e}^{a+X_t}, \text{e}^{ a});\, t<\tau_0^+] \,\text{d} t}{-x} \notag \\[3pt] &=\textsf{k} \lim_{q \to\infty}q \lim_{x\downarrow 0}\int_{0}^\infty\mathcal{M}^{p,\alpha}_\beta( \text{e}^{a-y}, \text{e}^{ a})\dfrac{\text{e}^{-\Phi(q)x}W^{(q)}(y)-W^{(q)}(y-x)}{x} \,\text{d} y,\end{align}

by (2.4).

Next, as the integral of a resolvent density, for all $q\in(0,\infty)$ and all $x\in (0,\infty)$, we have that

\[\int_{0}^\infty \,\text{e}^{-\Phi(q)x}W^{(q)}(y)-W^{(q)}(y-x) \,\text{d} y\]

is finite. At the same time we know from (2.5) that

(4.3) \begin{equation}(\psi(\lambda)-q)\int_0^\infty \,\text{e}^{-\lambda y}(\text{e}^{-\Phi(q)x}W^{(q)}(y)-W^{(q)}(y-x)) \,\text{d} y=\text{e}^{-\Phi(q)x}-\text{e}^{-\lambda x}\end{equation}

at least for $\lambda\in (\Phi(q),\infty)$. But by the theorems of Cauchy, Morera, and Fubini, the left-hand side and clearly the right-hand side are analytic, or can be extended to analytic functions in $\lambda\in \{z\in \mathbb{C}\colon \Re z>0\}$. Hence, by the principle of permanence for analytic function, the equality (4.3) prevails for $\lambda\in (0,\infty)$. Taking limits, by monotone convergence and continuity, we conclude that

\[\int_0^\infty \,\text{e}^{-\lambda y}(\text{e}^{-\Phi(q)x}W^{(q)}(y)-W^{(q)}(y-x)) \,\text{d} y=\dfrac{\text{e}^{-\Phi(q)x}-\text{e}^{-\lambda x}}{\psi(\lambda)-q}\]

for all $\lambda\in [0,\infty)$, provided the right-hand side is interpreted in the limiting sense at $\lambda=\Phi(q)$.

Consequently, integrating term by term (via linearity and monotone or dominated convergence) in (4.2), we obtain, assuming $\Phi(p)\notin\alpha\mathbb{N}$,

\begin{align*} \mathfrak{O} & = \textsf{k} \lim_{q \to\infty}^{*} q \lim_{x\downarrow 0} \, x^{-1}\biggl( \sum_{k=0}^\infty \textsf{a}^{p,\alpha}_k (\beta \,\text{e}^{\alpha a})^k \,\dfrac{\text{e}^{-\Phi(q)x} -\text{e}^{-\alpha k x}}{\psi(\alpha k)-q} \\* &\quad\, - \dfrac{\mathcal{J}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}{\mathcal{I}^{p,\alpha}(\beta \,\text{e}^{\alpha a})} \sum_{k=0}^\infty \textsf{b}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a}) ^k\, \dfrac{\text{e}^{-\Phi(q)x} -\text{e}^{-(\Phi(p)+k\alpha) x}}{\psi(\Phi(p)+k\alpha)-q}\biggr), \end{align*}

where $\overset{*}{\lim}$ indicates that we take the limit along a sequence $(q_k)_{k\in \mathbb{N}_0}$ that uniformly (we use this for convenience later on when arguing dominated convergence) avoids the grid $\psi(\alpha\mathbb{N}_0\cup (\Phi(p)+\alpha\mathbb{N}_0))$.

Then, by dominated convergence (recall the elementary estimate $1-\text{e}^{-u}\leq u$, $u\in [0,\infty)$),

\begin{align*} \mathfrak{O} & =\textsf{k} \lim_{q \to\infty}^{*}q \biggl[\sum_{k=0}^\infty \textsf{a}^{p,\alpha}_k (\beta \,\text{e}^{\alpha a})^k\dfrac{\alpha k-\Phi(q)}{\psi(\alpha k)-q}-\dfrac{\mathcal{J}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}{\mathcal{I}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}\sum_{k=0}^\infty \textsf{b}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a}) ^k\dfrac{\Phi(p)+k\alpha-\Phi(q)}{\psi(\Phi(p)+k\alpha)-q}\biggr]\\* & =\textsf{k} \lim_{q \to\infty}^*q\biggl[\sum_{k=0}^\infty \textsf{a}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a})^k\biggl(\dfrac{\alpha k-\Phi(q)}{\psi(\alpha k)-q}-\dfrac{\Phi(q)}{q}\biggr)\\* &\quad\, -\dfrac{\mathcal{J}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}{\mathcal{I}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}\sum_{k=0}^\infty \textsf{b}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a})^k\biggl(\dfrac{\Phi(p)+\alpha k-\Phi(q)}{\psi(\Phi(p)+k\alpha)-q}-\dfrac{\Phi(q)}{q}\biggr)\bigg]\\ &=\textsf{k} \lim_{q \to\infty}^{*}\biggl[\sum_{k=0}^\infty \textsf{a}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a})^k\dfrac{q\alpha k}{\psi(\alpha k)-q}-\dfrac{\mathcal{J}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}{\mathcal{I}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}\sum_{k=0}^\infty \textsf{b}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a})^k\dfrac{q(\Phi(p)+\alpha k)}{\psi(\Phi(p)+k\alpha)-q}\biggr]\\&\quad\, +\textsf{k} \lim_{q \to\infty}^{*}\sum_{k=0}^\infty \textsf{a}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a})^k\biggl(\dfrac{(\alpha k-\Phi(q))\psi(\alpha k)}{\psi(\alpha k)-q}-\dfrac{\alpha k\psi(\alpha k)}{\psi(\alpha k)-q}\biggr)\\* &\quad\, -\textsf{k}\dfrac{\mathcal{J}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}{\mathcal{I}^{p,\alpha}(\beta \,\text{e}^{\alpha a})} \lim_{q \to\infty}^{*} \sum_{k=0}^\infty \textsf{b}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a})^k \\* &\quad\, \times \biggl(\dfrac{(\alpha k+\Phi(p)-\Phi(q))\psi(\alpha k+\Phi(p))}{\psi(\alpha k+\Phi(p))-q}-\dfrac{(\alpha k+\Phi(p))\psi(\alpha k+\Phi(p))}{\psi(\alpha k+\Phi(p))-q} \biggr)\\&=\textsf{k} \lim_{q \to\infty}^{*}\biggl[\sum_{k=0}^\infty \textsf{a}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a})^k\dfrac{q\alpha k}{\psi(\alpha k)-q}-\dfrac{\mathcal{J}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}{\mathcal{I}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}\sum_{k=0}^\infty \textsf{b}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a})^k\dfrac{q(\Phi(p)+\alpha k)}{\psi(\Phi(p)+k\alpha)-q}\biggr]\\&=\textsf{k} \lim_{q \to\infty}^{*}\sum_{k=0}^\infty \textsf{a}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a})^k\biggl(\dfrac{q\alpha k}{\psi(\alpha k)-q}+\alpha k-\alpha k\biggr)\\* &\quad\, -\textsf{k} \dfrac{\mathcal{J}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}{\mathcal{I}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}\lim_{q \to\infty}^{*}\sum_{k=0}^\infty \textsf{b}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a})^k\\* &\quad\, \times \biggl(\dfrac{q(\Phi(p)+\alpha k)}{\psi(\Phi(p)+k\alpha)-q}+\Phi(p)+\alpha k-(\Phi(p)+\alpha k)\biggr)\\* &=\textsf{k}\biggl[\dfrac{\mathcal{J}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}{\mathcal{I}^{p,\alpha}(\beta \,\text{e}^{\alpha a})}\sum_{k=0}^\infty \textsf{b}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a})^k(\Phi(p)+\alpha k)-\sum_{k=1}^\infty \textsf{a}^{p,\alpha}_k(\beta \,\text{e}^{\alpha a})^k\alpha k\biggr], \end{align*}

where the antepenultimate equality is by dominated convergence, noting that

\[ \dfrac{z-w}{\psi(z)-\psi(w)} \]

is bounded in $z\ne w$, $\{z,w\}\subset [\Phi(p),\infty)$ by the strict convexity of $\psi$, and that

\[ \lim_{q\to\infty}\dfrac{\Phi(q)}{q}=0 \]

because $\lim_\infty\psi'= \infty$. The last equality is again by dominated convergence. Plugging in $f=1$, $\beta=0$, we identify $\textsf{k}=1$. The case $\Phi(p)\in \alpha\mathbb{N}$ follows by taking limits. □

4.4 Conditional temporal splitting at the maximum

Combining our results, we arrive at the following (recall the notation of Definitions 3.1 and 4.1).

Proof. Items (1) and (2) follow from Propositions 4.2, 4.1, and 4.3, and from the comments concerning the (conditional) independences in $(L,\overline{Y}_\infty,J,T_0-L)$ made in the Introduction (as a consequence of the independence statement of the Wiener–Hopf factorization for X). The final statement is simply a consequence of $\log(\overline{Y}_\infty/Y_0)=\overline{X}_\textsf{e}-X_0$. □

Remark 4.6. It is clear from their defining representation as power series in Section 2.5, that the functions $\mathcal{I}^{p,\alpha}$ and $\mathcal{J}^{p,\alpha}$ as well as their first derivatives will be given in terms of generalized hypergeometric functions whenever $\psi$ is rational. Recalling Definitions 3.1 and 4.1, these are then, in turn, trivially combined into $\mathcal{M}^{p,\alpha}_\beta$ and $\mathcal{N}^{p,\alpha}_\beta$, except in ‘degenerate’ cases corresponding to $\Phi(p)\in \alpha\mathbb{N}$, when another limit intervenes. See [Reference Kuznetsov, Kyprianou and Rivero13, Section 5.4] for X with jumps of rational transform (which is equivalent to $\psi$ being rational); one such process is illustrated in Figure 1. In special cases the hypergeometric functions may themselves be expressed in terms of more elementary functions.

Figure 1: The function $(0,1)\ni j\mapsto {{\mathcal{M}^{p,\alpha}_\beta(j,1)}/{(1-j^{\Phi(p)}})}$ for $\alpha=2$, $p=1$, and $\psi(\lambda)=\lambda-{{\lambda}/{(\lambda+1)}}$, $\lambda\in [0,\infty)$, corresponding to X being the difference of a unit drift and of a homogeneous Poisson process of unit intensity.

Example 4.1. Let us take an instance of the situation described in the above remark: $\psi(\lambda)={{\lambda^2}/{2}}-{{\lambda}/{2}}$ for $\lambda\in [0,\infty)$, $\alpha=2$, and $p=0$. It is well known that it corresponds to Y being Brownian motion stopped on hitting zero. Note that $\Phi(p)=\Phi(0)=1\notin 2\mathbb{N}=\alpha\mathbb{N}$. For this example – admittedly the simplest one could think of – the expressions for the functions are compact enough to warrant an explicit presentation. For $y\in (0,\infty)$ and $\beta\in [0,\infty)$ we indeed obtain

\[\mathcal{J}^{p,\alpha}(y)=\cosh(\sqrt{2y}),\quad \mathcal{I}^{p,\alpha}(y)=\dfrac{\sinh(\sqrt{2y})}{\sqrt{2y}},\quad \text{and}\quad\mathcal{N}^{p,\alpha}_\beta(y)=\dfrac{\sqrt{2\beta} y}{\sinh(\sqrt{2\beta} y)}{,}\]

and therefore

\[\mathbb{Q}_y[\text{e}^{-\beta T_0}\mid \overline{Y}_\infty]=\dfrac{\overline{Y}_\infty}{y}\dfrac{\sinh(y\sqrt{2\beta} )}{\sinh(\overline{Y}_\infty\sqrt{2\beta} )}\times \dfrac{\overline{Y}_\infty\sqrt{2\beta}}{\sinh(\overline{Y}_\infty\sqrt{2\beta} )}\quad \text{a.s.-$\mathbb{Q}_y$,}\]

where the two factors either side of $\times$ correspond to the conditional expectations $\mathbb{Q}_y[\text{e}^{-\beta L}\mid \overline{Y}_\infty]$ and $\mathbb{Q}_y[\text{e}^{-\beta (T_0-L)}\mid \overline{Y}_\infty]$, respectively. As a check, using that $\log(\overline{Y}_\infty/Y_0)$ is exponential of rate $\Phi(p)$, we compute the $\mathbb{Q}_y$ integral of the right-hand side of the above identity to get $\text{e}^{-\sqrt{2\beta} y}$, which is the well-known Laplace transform of $\mathbb{Q}_y[\text{e}^{-\beta T_0}]$ (as it should be). Incidentally, it means that the constant $C^{p,\alpha}$ in Remark 3.5 is equal to $\sqrt{2}$ in this case.

5.1 Expected discounted payoff of a ‘regret’ lookback option

One immediate application of the above that springs to mind is computation of the expected discounted payoff (under the ‘physical’ measure) of a lookback option on the stock of a company that one views as eventually going bankrupt, and whose price is modeled by the process Y. The idea is that one holds equity in the company until termination, say for dividends, but at the same time one wants an option to provide some hedge against not selling the stock sooner (or indeed at its maximum). It is a ‘buy-and-hold’ strategy recognizing that the company will eventually terminate.

Specifically, we may imagine that the stock price is given by the process Y under the ‘physical’ measure $\mathbb{Q}_y$ for an initial price $y\in (0,\infty)$. Assumption 1.1 then means that eventually the price will hit zero a.s., either abruptly, say as a result of some one-off adverse event, when $p>0$, or continuously, say as a result of gradually deteriorating business conditions, when $p=0$. At the same time we have an option written on the stock that will pay some (non-decreasing) function $f\colon (0,\infty)\to [0,\infty)$ of the overall maximum $\overline{Y}_\infty$ at termination $T_0$, thus providing some measure of compensation for the ‘regret’ of not having sold the stock optimally.

Assuming a constant risk-free force of interest $r\in [0,\infty)$, then the expected discounted payoff of such an option is simply

\[\mathbb{Q}_y[\text{e}^{-rT_0}f(\overline{Y}_\infty)]=\mathbb{Q}_y[f(\overline{Y}_\infty)\mathbb{Q}_y[\text{e}^{-r L}\mid \overline{Y}_\infty]\mathbb{Q}_y[\text{e}^{-r(T_0-L)}\mid \overline{Y}_\infty,J]],\]

which may easily be expressed using the results of Theorem 4.1 in terms of (at most) a two-dimensional integral. Such an expectation provides some information on the ‘value’ of the option for the investor, though of course it does not correspond to a risk-neutral valuation thereof.

5.2 Properties of the joint law of $(L,\overline{Y}_\infty,J,T_0-L)$

It is immediate from the definition of $\mathcal{I}^{p,\alpha}$ and from Proposition 4.1 (but not obvious a priori) that, for given $\beta\in[0,\infty)$, $y\in (0,\infty)$, the conditional Laplace transform

\[ \mathbb{Q}_y[\text{e}^{-\beta L}\mid \overline{Y}_\infty]\overset{\text{a.s.-$\mathbb{Q}_y$}}{=}\dfrac{\mathcal{I}^{p,\alpha}(\beta y^\alpha)}{\mathcal{I}^{p,\alpha}(\beta{\overline{Y}_\infty}^\alpha)} \]

is decreasing as a function of $\overline{Y}_\infty$. On the other hand it is clear from Definition 4.1 that $\mathcal{N}^{p,\alpha}_\beta(y)$ depends on $\beta$ and y only through $\beta y^\alpha$. Hence it follows from Proposition 4.3 (ii) that

\[ \mathbb{Q}_y[\text{e}^{-\beta (T_0-L)}\mid \overline{Y}_\infty]\overset{\text{a.s.-$\mathbb{Q}_y$}}{=}\mathcal{N}^{p,\alpha}_\beta\big(\beta {\overline{Y}_\infty}^\alpha\big) \]

is decreasing as a function of $\overline{Y}_\infty$ when X has paths of infinite variation. In the opposite case, it follows similarly from Definition 3.1, Corollary 3.1 and from Proposition 4.3 (i) that

\[ \mathbb{Q}_y[\text{e}^{-\beta (T_0-L)}\mid \overline{Y}_\infty,J]\overset{\text{a.s.-$\mathbb{Q}_y$}}{=}\dfrac{\mathcal{M}^{p,\alpha}_\beta(\overline{Y}_\infty J,\overline{Y}_\infty)}{1-J^{\Phi(p)}}\mathds{1}_{(0,1)}(J)+\mathds{1}_{\{1\}}(J) \]

is also decreasing in $\overline{Y}_\infty$. However, its dependence on J is non-trivial; see Figure 1.

More generally one would be interested in the following question.

We have given a flavor of this in the above, but do not pursue the problem any further here.