3.1. Main result

Here is the main result of this note.

Theorem 3.1. There exists a unique function $${\cal H}_q^{(\omega )}{\kern 1pt} :{\mathbb R}{\kern 1pt} \to (0,\infty )$$ satisfying (the arbitrary normalization condition) $${\cal H}_q^{(\omega )}(0) = 1$$ such that

(3.1) $${\cal B}_q^{(\omega )}(x,c) = {{{\cal H}_q^{(\omega )}(x)} \over {{\cal H}_q^{(\omega )}(c)}}{\rm{for\,all\,real\,}}x \le c.$$

The function $${\cal H}_q^{(\omega )}$$ enjoys the following properties.

1. (I) It is non-decreasing (hence locally bounded), continuous, and it is strictly increasing provided ω> 0.

2. (II) For each c ∈ R the following holds: ω ₁ (⋅ ∧c) = ω ₂ (⋅ ∧c) implies $${{\cal H}^{({\omega _1})}}( \cdot \wedge {\kern 1pt} c) = \alpha {{\cal H}^{({\omega _2})}}( \cdot \wedge {\kern 1pt} c)$$ for some α ∈ (0, ∞), this α being 1 if c ≥ 0.

3. (III) If ω ₁ ,ω ₂ : ℝ → [0, ∞) are both locally bounded and Borel measurable with ω ₁ ≤ ω ₂ (resp. ω ₁ <ω ₂ ), then $${\cal H}_q^{({\omega _1})} \le {\cal H}_q^{({\omega _2})}$$ (resp. $${\cal H}_q^{({\omega _1})} \lt {\cal H}_q^{({\omega _2})}$$) on (0, ∞) and $${\cal H}_q^{({\omega _1})} \ge {\cal H}_q^{({\omega _2})}$$ (resp. $${\cal H}_q^{({\omega _1})} > {\cal H}_q^{({\omega _2})}$$) on (−∞, 0) ; of course, $${\cal H}_q^{({\omega _1})}(0) = 1 = {\cal H}_q^{({\omega _2})}(0)$$

4. (IV) For real x ≤ c $${\cal H}_q^{(\omega )}(x) \le {\cal H}_q^{(\omega )}(c){{\rm{e}}^{ - \Phi (q)(c - x)}}$$; in particular, $${\cal H}_q^{(\omega )}(x) \le {{\rm{e}}^{\Phi (q)x}}$$ for all $$x \in ( - \infty ,0]$$, so that $${\cal H}_q^{(\omega )}$$ has a left tail that is Ф(q) -subexponential.

Furthermore, if (ωe^{Ф(q + p)⋅)} ⋆W ^(q) is finite-valued for all p ∈ (0, ∞) , in particular if ω has a left tail that is bounded, then for some unique $$L_q^{(\omega )} \in [0,1]$$, $${\cal H}_q^{(\omega )}$$ satisfies the convolution equation

(3.2) $${\cal H}_q^{(\omega )} = L_q^{(\omega )}{{\rm{e}}^{\Phi (q) \cdot }} + (\omega {\cal H}_q^{(\omega )}) \star {W^{(q)}}.$$

More specifically:

1. (i) If, moreover, ω has a left tail that is bounded and bounded below away from zero, then $${\cal H}_q^{(\omega )}$$ satisfies the (homogeneous) convolution equation

   (3.3) $${\cal H}_q^{(\omega )} = (\omega {\cal H}_q^{(\omega )}) \star {W^{(q)}}.$$

2. (ii) If $$(\omega {{\rm{e}}^{\Phi (q) \cdot }}) \star {W^{(q)}}$$ is finite-valued, in particular if ω has a subexponential left tail, then $${\cal H}_q^{(\omega )}$$is the unique locally bounded Borel measurable function H : ℝ → ℝ admitting a left tail that is Ф(q) -subexponential and satisfying the (inhomogeneous) convolution equation

   (3.4) $$H = L_q^{(\omega )}{{\rm{e}}^{\Phi (q) \cdot }} + (\omega H) \star {W^{(q)}},$$

   where

   (3.5) $$\matrix{ {L_q^{(\omega )} = \mathop {\lim }\limits_{x \to - \infty } {\cal H}_q^{(\omega )}(x){{\rm{e}}^{ - \Phi (q)x}}\quad } \hfill \cr { = \mathop {\lim }\limits_{x \to - \infty } {{\rm{P}}_x}[\exp \{ - \int_0^{\tau _0^ + } \omega ({X_s}){\kern 1pt} {\rm{d}}s\} {\kern 1pt} |{\kern 1pt} \tau _0^ + \lt {e_q}] \in (0,1].} \hfill \cr } $$

   This function is given as $${\cal H}_q^{(\omega )} = \uparrow - \mathop {\lim }\nolimits_{n \to \infty } {H_n}$$, where $${H_0}L_q^{(\omega )}{{\rm{e}}^{\Phi (q) \cdot }}$$ and recursively $${H_{n + 1}}:=L_q^{(\omega )}{{\rm{e}}^{\Phi (q) \cdot }} + (\omega {H_n}) \star {W^{(q)}}$$ for n ∈ ℕ₀.

   After some remarks and examples we turn to the proof of this theorem on p. 480.

Example 3.2. When ω = γ e^α., with γ ∈ [0, ∞) and α ∈ (0, ∞), a situation that falls under case (ii ), using (2.2) we obtain

(3.6) $${\cal H}_q^{(\gamma {{\rm{e}}^{\alpha \cdot }})}(x) = \sum\limits_{k = 0}^\infty {{{\gamma ^k}{{\rm{e}}^{(\Phi (q) + \alpha k)x}}} \over {\prod\nolimits_{l = 1}^k (\psi (\Phi (q) + l\alpha ) - q)}}/\sum\limits_{k = 0}^\infty {{{\gamma ^k}} \over {\prod\nolimits_{l = 1}^k (\psi (\Phi (q) + l\alpha ) - q)}},\quad x \in {\mathbb R},$$

with the series converging to finite values. (As usual the empty product is interpreted as being equal to 1.) Of course, when γ > 0 then, from (3.1), by spatial homogeneity, $${\cal H}_q^{(\gamma {{\rm{e}}^{\alpha \cdot }})}(x) = {{{\cal H}_q^{({{\rm{e}}^{\alpha \cdot }})}(x + {1 \over \alpha }\log \gamma )} \over {{\cal H}_q^{({{\rm{e}}^{\alpha \cdot }})}({1 \over \alpha }\log \gamma )}},\,x \in {\mathbb R}$$. Note that this reproduces (up to trivial transformations) Patie’s [Reference Patie14] scale functions from the fluctuation theory of spectrally negative pssMp [Reference Kyprianou11, Section 13.7]. We can also identify the limit (3.5) as $$L_q^{(\gamma {{\rm{e}}^{\alpha \cdot }})} = (\sum\nolimits_{k = 0}^\infty {{{\gamma ^k}} \over {\prod\nolimits_{l = 1}^k (\psi (\Phi (q) + l\alpha ) - q)}}{)^{ - 1}}$$. For the special case when X is Brownian motion and when γ > 0 (γ = 0 being trivial), if q = 0 we have $${\cal H}_q^{(\gamma {{\rm{e}}^{\alpha \cdot }})}(x) = {\cal H}_0^{(\gamma {{\rm{e}}^{\alpha \cdot }})}(x) = {{{I_0}(2{{\rm{e}}^{\alpha x/2}}\sqrt \gamma /\alpha )} \over {{I_0}(2\sqrt \gamma /\alpha )}}$$, x ∈ ℝ, where I ₀ is the modified Bessel function of the first kind (of order 0); for q > 0 the function $${\cal H}_q^{(\gamma {{\rm{e}}^{\alpha \cdot }})}$$ evaluates in terms of a generalized hypergeometric function.

Remark 3.3. Let γ ∈[0,∞] and α∈(0, ∞). Suppose $$\omega (x) \le\gamma {{\rm{e}}^{\alpha x }}$$ for all x ∈ ℝ. Then, again via (2.2), we get the following a priori bound on the absolute error in case (ii) from computing only finitely many terms of the recursion for $${\cal H}_q^{(\omega )}$$:

$$\matrix{ {{\cal H}_q^{(\omega )} - {H_n} \le L_q^{(\omega )}\sum\limits_{k = n + 1}^\infty {{{\gamma ^k}} \over {\prod\nolimits_{l = 1}^k (\psi (\Phi (q) + \alpha l) - q)}}{{\rm{e}}^{(\Phi (q) + \alpha k) \cdot }}} \cr { \le \sum\limits_{k = n + 1}^\infty {{{\gamma ^k}} \over {\prod\nolimits_{l = 1}^k (\psi (\Phi (q) + \alpha l) - q)}}{{\rm{e}}^{(\Phi (q) + \alpha k) \cdot }}} \cr } $$

for all n ∈ ℕ₀. (In particular, $${\cal H}_q^{(\omega )} \le L_q^{(\omega )}\sum\nolimits_{k = 0}^\infty {{{\gamma ^k}} \over {\prod\nolimits_{l = 1}^k (\psi (\Phi (q) + \alpha l) - q)}}{{\rm{e}}^{(\Phi (q) + \alpha k) \cdot }} \le $$ $$\sum\nolimits_{k = 0}^\infty {{{\gamma ^k}} \over {\prod\nolimits_{l = 1}^k (\psi (\Phi (q) + \alpha l) - q)}}{{\rm{e}}^{(\Phi (q) + \alpha k) \cdot }}$$

In the following, an expression of the form $${\cal H}_q^{(\omega )}(x) \propto b(x)$$’ means that there is a C ∈ (0, ∞) such that $${\cal H}_q^{(\omega )}(x) = Cb(x)$$ for all x in the relevant domain; the constant C depends only on ω, q, and the characteristics of X, not on x (it is ultimately determined by the (arbitrary) normalization condition $${\cal H}_q^{(\omega )}(0) = 1$$).

Example 3.3. Let c ∈ (0, ∞), γ ∈ (0, ∞), and $$\omega (x) = {\gamma \over {|x|}}$$ for x ∈ (−∞, −c]. Using the result for csbp of [Reference Duhalde, Foucart and MA6, Theorem 1] we identify $${\cal H}_q^{(\omega )}$$ up to a multiplicative constant;

$${\cal H}_q^{(\omega )}(x) \propto \int_{\Phi (q)}^\infty {{{\kern 1pt} {\rm{d}}z} \over {\psi (z) - q}}\exp \,\{ xz + \int_\theta ^z {\gamma \over {\psi (u) - q}}{\kern 1pt} {\rm{d}}u\} ,\quad x \in ( - \infty , - c],$$

where θ ∈ (Ф(q), ∞) is arbitrary but fixed. Indeed, [Reference Duhalde, Foucart and MA6, Eq. (11)] identifies (by other means) the expression (1.3), and even the joint Laplace transform ‘$${{\rm{P}}_y}[{{\rm{e}}^{ - \delta T_d^ + - p\tau _d^ + }};{\kern 1pt} T_d^ + \lt \zeta ]$$’, from which the expression for $${\cal H}_q^{(\omega )}$$ can easily be recovered via (3.1). Note that this ω falls under neither case (i) nor case (ii ), but it does fall under (3.2). In fact, while it is not so obvious, an easy computation shows that (3.2) is verified in this case with $$L_q^{(\omega )} = 0$$. For the special case when X is Brownian motion and when γ > 0: if q = 0 we have $${\cal H}_q^{(\omega )}(x) \propto 2\sqrt { - 2\gamma x} {K_1}(2\sqrt { - 2\gamma x} )$$, $$x \in (\infty , - c]$$, where K ₁ is the modified Bessel function of the second kind (of order 1); for q > 0 the function $${\cal H}_q^{(\gamma {{\rm{e}}^{\alpha \cdot }})}$$ does not appear to have a closed-form expression in terms of elementary/special functions.

Example 3.4. Let n ∈ N_≥2, γ ∈ [0, ∞), c ∈ (0, ∞), and $$\omega (x) = {\gamma \over {|x{|^n}}}$$ for x ∈ (−∞, − c]. Except possibly when q = Ф(0) = ψ′ (0 +) = 0, we then automatically have, because of the asymptotic properties of W ^(q) (see (2.3)) that (ωe ^Ф(q).)* W ^(q) is finite-valued, and in any event we assume now that this is so. Then note, using (2.2), that for x ∈ (−∞, − c], v ∈ [Ф(q), ∞), and for α > 0,$${{{{\rm{d}}^n}} \over {{\rm{d}}{\alpha ^n}}}\int_0^\infty {{{{\rm{e}}^{(v + \alpha )(x - y)}}} \over {{{(x - y)}^n}}}{W^{(q)}}(y){\kern 1pt} {\rm{d}}y = {{{{\rm{e}}^{(v + \alpha )x}}} \over {\psi (v + \alpha ) - q}}$$, which implies that

$$\matrix{ {\int_0^\infty {{{{\rm{e}}^{(v + \alpha )(x - y)}}} \over {|x - y{|^n}}}{W^{(q)}}(y){\kern 1pt} {\rm{d}}y = \int_{v + \alpha }^\infty {\kern 1pt} {\rm{d}}{v_1}\int_{{v_1}}^\infty {\kern 1pt} {\rm{d}}{v_2} \cdots \int_{{v_{n - 1}}}^\infty {\kern 1pt} {\rm{d}}{v_n}{{{{\rm{e}}^{{v_n}x}}} \over {\psi ({v_n}) - q}}} \cr { = \int_{v + \alpha }^\infty {\kern 1pt} {\rm{d}}{v_n}{{{{\rm{e}}^{x{v_n}}}} \over {\psi ({v_n}) - q}}\int_{v + \alpha }^{{v_n}} {\kern 1pt} {\rm{d}}{v_{n - 1}} \cdots \int_{v + \alpha }^{{v_2}} {\kern 1pt} {\rm{d}}{v_1} = \int_{v + \alpha }^\infty {\kern 1pt} {\rm{d}}y{{{{\rm{e}}^{xy}}} \over {\psi (y) - q}}{{{{(y - v - \alpha )}^{n - 1}}} \over {(n - 1)!}}.} \cr } $$

Hence, letting α ↓ 0, by monotone convergence,

$$\matrix{ {((\omega {{\rm{e}}^{v \cdot }}) \star {W^{(q)}})(x) = \gamma \int_0^\infty {{{{\rm{e}}^{v(x - y)}}} \over {|x - y{|^n}}}{W^{(q)}}(y){\kern 1pt} {\rm{d}}y} \cr { = \gamma \int_v^\infty {\kern 1pt} {\rm{d}}y{{{{\rm{e}}^{xy}}} \over {\psi (y) - q}}{{{{(y - v)}^{n - 1}}} \over {(n - 1)!}} = \gamma \int_0^\infty {\kern 1pt} {\rm{d}}y{{{{\rm{e}}^{x(v + y)}}} \over {\psi (v + y) - q}}{{{y^{n - 1}}} \over {(n - 1)!}}.} \cr } $$

Thus, the recursion of case (ii) allows us to identify $${\cal H}_q^{(\omega )}$$, up to a proportionality constant, in terms of an infinite series of iterated integrals:

$$\matrix{ {{\cal H}_q^{(\omega )}(x) \propto {{\rm{e}}^{\Phi (q)x}}[1 + \sum\limits_{k = 1}^\infty {{({\gamma \over {(n - 1)!}})}^k}\int_{{{(0,\infty )}^k}} {{{{({x_1} \cdots {x_k})}^{n - 1}}{{\rm{e}}^{x({x_1} + \cdots + {x_k})}}{\kern 1pt} {\rm{d}}{x_1} \cdots {\kern 1pt} {\rm{d}}{x_k}} \over {(\psi (\Phi (q) + {x_1}) - q) \cdots (\psi (\Phi (q) + {x_1} + \cdots + {x_k}) - q)}}]} \cr { = {{\rm{e}}^{\Phi (q)x}}[1 + {\gamma \over {(n - 1)!}}\int_0^\infty {\kern 1pt} {\rm{d}}y{{{y^{n - 1}}{{\rm{e}}^{xy}}} \over {\psi (\Phi (q) + y) - q}}} \cr { + {\kern 1pt} {{({\gamma \over {(n - 1)!}})}^2}\int_0^\infty {\kern 1pt} {\rm{d}}y{{{y^{n - 1}}{{\rm{e}}^{xy}}} \over {\psi (\Phi (q) + y) - q}}\int_0^\infty {\kern 1pt} {\rm{d}}z{{{z^{n - 1}}{{\rm{e}}^{xz}}} \over {\psi (\Phi (q) + y + z) - q}} + \cdots ],\quad x \in ( - \infty , - c].} \cr } $$

For instance, when X is Brownian motion, q = 0, and n = 3, then $${\cal H}_q^{(\omega )}(x) \propto 1 + {\gamma \over { - x}} + {{{\gamma ^2}} \over {3{x^2}}} + \cdots $$, with the first three terms of the preceding series having been made explicit; already the fourth term does not (appear to) evaluate to any nice expression. It is clear from the representation of the series, however, that in this special case its kth term will be of the form c _k / | x | ^k, for some c _k ∈ (0, ∞), so that by adding the successive terms to the series we get an asymptotic expansion of $${\cal H}_q^{(\omega )}$$.

As previously indicated, we turn now to the proof of our main result. We resume providing further complements to Theorem 3.1 in Proposition 3.1 on p. 485. Below, as usual, the statement $$\tau _d^ + \downarrow \tau _c^ + as d \downarrow c'$$ means that the map $$(c,\infty )d \mapsto \tau _d^ + $$ is non-decreasing and that $$\mathop {\lim }\nolimits_{d \to c,d > c} \tau _d^ + = \tau _c^ + $$ (all pointwise). Other statements of this form are interpreted in an analogous manner.

Proof of Theorem 3.1. We separate the proof into several parts. The reader who wishes (at first) only to see the main idea without being preoccupied with the technical details may proceed (first) to (•₃) below.

(•₁) We begin by arguing that there exists a unique function $${\cal H}_q^{(\omega )}{\kern 1pt} : {\mathbb R}{\kern 1pt} \to (0,\infty )$$ satisfying $${\cal H}_q^{(\omega )}(0) = 1$$ and (3.1). This is basically a consequence of the strong Markov property and of the absence of positive jumps of X.

Let x ≤ y ≤ c be real numbers. Since X has no positive jumps, then P_x -a.s. $${X_{\tau _y^ + }} = y$$ on $$\{ \tau _y^ + \lt \infty \} $$, and it follows by the strong Markov property of X applied at the time $$\tau _y^ + $$, and by the memoryless property of the exponential distribution, that one has the multiplicative structure

$${\cal B}_q^{(\omega )}(x,c) = {\cal B}_q^{(\omega )}(x,y){\cal B}_q^{(\omega )}(y,c).$$

Furthermore, it is clear that $${\cal B}_q^{(\omega )}(x,c) > 0$$ for all real x ≤ c. As a consequence, we may unambiguously define (with a preemptive choice of notation) $${\cal H}_q^{(\omega )}(x):={{{\cal B}_q^{(\omega )}(x,c)} \over {{\cal B}_q^{(\omega )}(0,c)}}$$ for real x ≤ c, c ≥ 0. In short, then, $${\cal H}_q^{(\omega )}{\kern 1pt} : {\mathbb R}{\kern 1pt} \to (0,\infty )$$, $${\cal H}_q^{(\omega )}(0) = 1$$ and (3.1) holds. It is clear that $${\cal H}_q^{(\omega )}$$ is unique in having the preceding properties.

(•₂) We consider statements ( I ), (II ), (III ), and (IV ). Apart from the continuity of $${\cal H}_q^{(\omega )}$$, (I), ( II ), and (II I) follow immediately from (3.1) and (1.1) and simple comparison arguments. To prove the continuity of $${\cal H}_q^{(\omega )}$$, note that, for real c,

(r)$$\tau _d^ + \downarrow \tau _c^ + as d \downarrow c'$$ (this is simply directly from the definition of the first passage times $${(\tau _y^ + )_{y \in {\mathbb R}}}$$ involving the strict inequality >),

and further that for x ∈ (−∞, c), by quasi-left-continuity and regularity of 0 for (0, ∞), also

(l) $$\tau _d^ + \uparrow \tau _c^ + \,{P_x}$$ -a.s. (on $$\{ \mathop {\lim }\nolimits_{d \uparrow c} \tau _d^ + \lt \infty \} $$} and hence everywhere) as d ↑ c.

Then item (r), the fact that ω is locally bounded, and bounded convergence in (3.1) and (1.1), immediately yield the right-continuity of $${\cal H}_q^{(\omega )}$$. Left-continuity follows similarly from item (l) in lieu of (r), except that now one notices in addition that P_x -a.s. on $$\{ \tau _c^ + = \infty \} $$ also $$\tau _d^ + = \infty $$ for all d < c that are sufficiently close to c (this is needed to ensure, when q = 0, the P_x -a.s. convergence $${{\bf{1}}_{\{ \tau _d^ + \lt {e_q}\} }} \to {{\bf{1}}_{\{ \tau _c^ + \lt {e_q}\} }}$$ as d ↑ c also on the event$$\{ \tau _c^ + = \infty \} $$; the issue is moot when q > 0 because then automatically $${{\rm{P}}_x}(\tau _c^ + = {e_q}) = 0$$. The latter property in turn is a consequence of the fact that the law of the overall supremum, $${\overline X _\infty }$$, being exponential (2.1), has no finite atoms.

For (IV), notice that by (2.1), for real x ≤ c, $${\cal H}_q^{(\omega )}(x) \le {\cal H}_q^{(\omega )}(c){{\rm{P}}_x}(\tau _c^ + \lt {e_q}) = {\cal H}_q^{(\omega )}(c){\mathbb P}{_x}[{{\rm{e}}^{ - q\tau _c^ + }};\tau _c^ + \lt \infty ] = {\cal H}_q^{(\omega )}(c){{\rm{e}}^{ - \Phi (q)(c - x)}}$$.

(•₃) We now prove (i), exploiting the marked Poisson process technique of [12]. Suppose then that ω has a left tail that is bounded and bounded below away from zero.

As a preliminary observation, note that the homogeneous convolution equation (3.3) may be checked ‘locally’, separately on each (−∞, c] for c ∈ ℝ (this is trivial); then replacing ω with ω(⋅∧c) if necessary, we may assume – by (II), and because W ^(q) vanishes on (−∞, 0) – without loss of generality that ω is bounded by a λ ∈ (0, ∞). Also, by assumption, there is an x ₀ ∈ ℝ such that ω is bounded below away from zero on (−∞, x ₀ ] by some a > 0.

Next, again let x ≤ c be real numbers. Recall the expression for the resolvent [Reference Kuznetsov, Kyprianou and Rivero10, Theorem 2.7(ii)],

$$\int_0^\infty {{\rm{e}}^{ - \lambda t}}{{\rm{P}}_x}({X_t} \in {\rm{d}}y,t \lt \tau _c^ + \wedge {e_q}){\kern 1pt} {\rm{d}}t = ({{\rm{e}}^{ - \Phi (\lambda + q)(c - x)}}{W^{(\lambda + q)}}(c - y) - {W^{(\lambda + q)}}(x - y)){\kern 1pt} {\rm{d}}y$$

for y ∈ (−∞, c], and the classical identity (2.1) $${{\rm{P}}_x}[{{\rm{e}}^{ - \lambda \tau _c^ + }};\tau _c^ + \lt {e_q}] = {{\rm{e}}^{ - \Phi (\lambda + q)(c - x)}}$$. Furthermore, let (T _i)_{i ∈ℕ} be the arrival times of a homogeneous Poisson process of intensity λ, marked by an independent sequence (M _i)_i∈ℕ of independent, identically, uniformly on [0 ,λ], distributed random variables, all independent of X. Then, by properties of marked Poisson processes,

(3.7) $$\matrix{ {{\cal B}_q^{(\omega )}(x,c) = {{\rm{P}}_x}({M_i} > \omega ({X_{{T_i}}}){\rm{for\,all\,i}} \in {\mathbb N}{\rm{such\,that\,}}{T_i} \lt \tau _c^ + ,\tau _c^ + \lt {e_q})} \cr { = {{\rm{P}}_x}({T_1} > \tau _c^ + ,\tau _c^ + \lt {e_q}) + {{\rm{P}}_x}[{\cal B}_q^{(\omega )}({X_{{T_1}}},c);{T_1} \lt \tau _c^ + \wedge {e_q},{M_1} > \omega ({X_{{T_1}}})]} \cr { = {{\rm{e}}^{ - \Phi (\lambda + q)(c - x)}} + \int_{ - \infty }^c {\cal B}_q^{(\omega )}(y,c)({{\rm{e}}^{ - \Phi (\lambda + q)(c - x)}}{W^{(\lambda + q)}}(c - y) - {W^{(\lambda + q)}}(x - y))(\lambda - \omega (y)){\kern 1pt} {\rm{d}}y.} \cr } $$

Further, plugging (3.1) into (3.7), multiplying both sides by $${\cal H}_q^{(\omega )}(c)$$, and rearranging, we see that

(3.8) $$\matrix{ {0 \le {\cal H}_q^{(\omega )}(x) - \int_{ - \infty }^c {\cal H}_q^{(\omega )}(y)({{\rm{e}}^{ - \Phi (\lambda + q)(c - x)}}{W^{(\lambda + q)}}(c - y) - {W^{(\lambda + q)}}(x - y))(\lambda - \omega (y)){\kern 1pt} {\rm{d}}y} \cr { = {\cal H}_q^{(\omega )}(c){{\rm{e}}^{ - \Phi (\lambda + q)(c - x)}}.} \cr } $$

Letting c ↑∞ in (3.8), we then obtain, by monotone convergence and using (2.3), that

(3.9) $${\cal H}_q^{(\omega )} = {{\rm{e}}^{\Phi (\lambda + q) \cdot }}{h_\lambda } + ((\lambda - \omega ){\cal H}_q^{(\omega )}) \star \left({{{{\rm{e}}^{\Phi (\lambda + q) \cdot }}} \over {\psi '(\Phi (\lambda + q))}} - {W^{(\lambda + q)}}\right),$$

with $${h_\lambda }:=\mathop {\lim }\nolimits_{c \to \infty } {\cal H}_q^{(\omega )}(c){{\rm{e}}^{ - \Phi (\lambda + q)c}}$$; a priori this limit must exist in [0, ∞), because the limit of the left-hand side in (3.8) exists in [0, ∞) (by monotone convergence), hence so too must the limit of the right-hand side (so the limit defining h _λ is even monotone non-increasing, but we do not need this).

Note now the relations $${{\rm{e}}^{\Phi (\lambda + q) \cdot }} = \lambda {{\rm{e}}^{\Phi (\lambda + q) \cdot }} \star {W^{(q)}}$$ (which is a direct consequence of (2.2)) and W ^{(λ +q)} = W ^(q) + λ W ^{(λ +q)} ⋆ W ^(q) (which may be checked by taking Laplace transforms and again using (2.2)). They together imply that

$$\left({{{{\rm{e}}^{\Phi (\lambda + q) \cdot }}} \over {\psi '(\Phi (\lambda + q))}} - {W^{(\lambda + q)}}\right) \star \lambda {W^{(q)}} = \left({{{{\rm{e}}^{\Phi (\lambda + q) \cdot }}} \over {\psi '(\Phi (\lambda + q))}} - {W^{(\lambda + q)}}\right) + {W^{(q)}}.$$

Using this, we convolute both sides of (3.9) with λ W ^(q) to obtain

$$\matrix{ {\lambda {\cal H}_q^{(\omega )} \star {W^{(q)}} = {{\rm{e}}^{\Phi (\lambda + q) \cdot }}{h_\lambda } + ((\lambda - \omega ){\cal H}_q^{(\omega )}) \star \left({{{{\rm{e}}^{\Phi (\lambda ) \cdot }}} \over {\psi '(\Phi (q))}} - {W^{(\lambda + q)}} + {W^{(q)}}\right)} \cr { = {\cal H}_q^{(\omega )} + ((\lambda - \omega ){\cal H}_q^{(\omega )}) \star {W^{(q)}}.} \cr } $$

Then the estimate (IV) $${{{\cal H}_q^{(\omega )}(x)} \over {{\cal H}_q^{(\omega )}({x_0})}} = {\cal B}_q^{(\omega )}(x,{x_0}) = {\cal B}_{q + a}^{((\omega - a) \vee 0)}(x,{x_0}) \le {{\rm{e}}^{ - \Phi (q + a)({x_0} - x)}}$$ for x ∈ (−∞, x ₀] implies (via (2.2) and via the local boundedness of $${\cal H}_q^{(\omega )}$$ (see (I)) and of W ^(q)) that $${\cal H}_q^{(\omega )} \star {W^{(q)}}$$ is finite-valued, and upon subtracting finite quantities we obtain (3.3). This concludes the proof of (i).

(•₄) We now turn to (3.2). Suppose then that (ωe^{Ф(q +p)⋅}) ⋆ W ^(q) is finite-valued for all p ∈ (0, ∞) (on account of (2.2) and because ω is locally bounded, this condition is satisfied when ω has a bounded left tail). The basic idea here is to reduce this case to (i) via a suitable approximation. As the reader will see, some care is needed to pass to the relevant limits.

Indeed, from (i), for each p ∈ (0, ∞) and n ∈ ℕ, one has, for all x ∈ ℝ,

(3.10) $$\matrix{ {{\cal H}_q^{(p + \omega \wedge n)}(x) = [((\omega \wedge n + p){\cal H}_q^{(p + \omega \wedge n)}) \star {W^{(q)}}](x)} \cr { = [((\omega \wedge n){\cal H}_q^{(p + \omega \wedge n)}) \star {W^{(q)}}](x) + p[{\cal H}_q^{(p + \omega \wedge n)} \star {W^{(q)}}](x)} \cr { = [\int_{ - \infty }^{0 \wedge x} + \int_0^{0 \vee x} ](\omega (y) \wedge n){\cal H}_q^{(p + \omega \wedge n)}(y){W^{(q)}}(x - y){\kern 1pt} {\rm{d}}y} \cr {\quad + p[\int_{ - \infty }^{0 \wedge x} + \int_0^{x \vee 0} ]{\cal H}_q^{(p + \omega \wedge n)}(y){W^{(q)}}(x - y){\kern 1pt} {\rm{d}}y.} \cr } $$

We now first pass to the limit n ↦∞ as follows. In (3.1) and (1.1),

monotone (for the integral against the Lebesgue measure)

and

bounded (for the expectation)

convergence yield $${\cal H}_q^{(p + \omega \wedge n)} \to {\cal H}_q^{(p + \omega )}$$. Then, in (3.10),

monotone (for the integrals on [0, x∨0]; recall (III)) and

dominated (for the integrals on (−∞, 0 ∧ x], using the assumed finiteness condition and the estimate (IV) $${\cal H}_q^{(p + \omega \wedge n)}(y) \le {{\rm{e}}^{\Phi (q + p)y}}$$ for y ∈ (−∞, 0])

convergence produce

(3.11) $$\matrix{ {{\cal H}_q^{(p + \omega )}(x) = [(\omega {\cal H}_q^{(p + \omega )}) \star {W^{(q)}}](x) + p[{\cal H}_q^{(p + \omega )} \star {W^{(q)}}](x)} \cr { = [\int_{ - \infty }^{0 \wedge x} + \int_0^{0 \vee x} ]\omega (y){\cal H}_q^{(p + \omega )}(y){W^{(q)}}(x - y){\kern 1pt} {\rm{d}}y + p\int_{ - \infty }^x {\cal H}_q^{(p + \omega )}(y){W^{(q)}}(x - y){\kern 1pt} {\rm{d}}y.} \cr } $$

Let us next write, for the purposes of the remainder of this proof only, $${\cal H}_q^{(p + \omega )} = :{\kern 1pt} {H_p}$$ and $${\cal H}_q^{(\omega )} = :{\kern 1pt} H$$ for short. We proceed to pass to the limit p ↓ 0.

In (3.1) and (1.1), as before, by

bounded (for the integral against the Lebesgue measure; recall that ω is locally bounded)

and

monotone (for the expectation)

convergence, we obtain that H _p ↦ H as p ↓ 0.

Then, in (3.11), [(ω H _p) W ^(q) ](x) ↦ [(ω H) ⋆ W ^(q) ](x) as p ↓ 0, by

monotone (for the integral on (−∞, 0 ∧ x] ; recall (III))

and

bounded (for the integral on [0, x∨0]; use the facts that H _p is non-decreasing (see (I)), that H _p (c) is bounded in bounded p given a fixed c ∈ [0, ∞) (see (III)), and that W ^(q) and ω are locally bounded)

convergence.

Finally, we consider

$${L_x}:=\mathop {\lim }\limits_{p \downarrow 0} {{\rm{e}}^{ - \Phi (q)x}}p({H_p} \star {W^{(q)}})(x) = {{\rm{e}}^{ - \Phi (q)x}}\mathop {\lim }\limits_{p \downarrow 0} p\int_{ - \infty }^x {H_p}(y){W^{(q)}}(x - y){\kern 1pt} {\rm{d}}y;$$

a priori this limit must exist in [0, ∞), because the limits of $${\cal H}_q^{(p + \omega )}(x)$$ and $$[(\omega {\cal H}_q^{(p + \omega )}) \star {W^{(q)}}](x)$$ in (3.11) have already been found to exist (and are finite, because the limit of $${\cal H}_q^{(p + \omega )}(x)$$, being $${\cal H}_q^{(\omega )}(x)$$, is finite). We show that L _x does not depend on x, thus demonstrating that (3.2) is indeed satisfied for some, necessarily unique, $$L_q^{(\omega )} \in [0,1]$$.

Now, since W ^(q) is locally bounded, since H _p is non-decreasing for each p ∈ (0, ∞), and since H _p (c) is bounded in bounded p given a fixed real c, it is clear that for any choice of a ∈ (−∞, x],

$${L_x} = \mathop {\lim }\limits_{p \downarrow 0} p\int_{ - \infty }^a {H_p}(y){{\rm{e}}^{ - \Phi (q)y}}{W^{(q)}}(x - y){{\rm{e}}^{ - \Phi (q)(x - y)}}{\kern 1pt} {\rm{d}}y.$$

Suppose first that ψ′ (Ф(q) +) > 0. Then, given any ∈ > 0 we may (2.3) choose this a to be (for simplicity) ≤ 0 and such as to render |W ^(q) (x − y)e ^{−Ф (q)(x −y)} − 1 /ψ′ (Ф(q) +) | ≤ ∈ for all y ≤ a. Consequently, since (using the estimate H _p (y) ≤ e^{Ф(q +p)y} for y ≤ 0, p ∈ (0, ∞))

$$\mathop {\limsup }\limits_{p \downarrow 0} p\int_{ - \infty }^0 {H_p}(y){{\rm{e}}^{ - \Phi (q)y}}{\kern 1pt} {\rm{d}}y \le \mathop {\lim }\limits_{p \downarrow 0} {p \over {\Phi (q + p) - \Phi (q)}} = \psi '(\Phi (q) + ) \lt \infty ,$$

we conclude that L _x in fact does not depend on x.

For the case when ψ′ (Ф(q) +) = 0, i.e. the case q = Ф(q) = ψ′ (0 +) = 0, we have that $${L_x} = \mathop {\lim }\nolimits_{p \downarrow 0} p\int_{ - \infty }^a {H_p}(y)W(x - y){\kern 1pt} {\rm{d}}y$$ for any a ∈ (−∞, x]. We argue that

$$Q\mathop {\limsup }\limits_{p \downarrow 0} p\int_{ - \infty }^a {H_p}(y)(W(x - y) - W(a - y)){\kern 1pt} {\rm{d}}y = 0$$

for a that are (again for simplicity) ≤ 0 ∧ x, which will complete the verification that L _x does not depend on x. Indeed, since $${H_p}(y) \le {H_p}(a){{\rm{e}}^{ - \Phi (p)(a - y)}} \le {{\rm{e}}^{ - \Phi (p)(a - y)}}$$ for real y ≤ a ≤ 0 and p > 0, we have that

$$\matrix{ {Q \le \mathop {\lim }\limits_{p \downarrow 0} p\int_{ - a}^\infty {{\rm{e}}^{ - \Phi (p)(y + a)}}(W(x + y) - W(a + y)){\kern 1pt} {\rm{d}}y} \cr { = \mathop {\lim }\limits_{p \downarrow 0} p\left[{{\rm{e}}^{\Phi (p)(x - a)}}\left({p^{ - 1}} - \int_0^{x - a} {{\rm{e}}^{ - \Phi (p)z}}W(z){\kern 1pt} {\rm{d}}z\right) - {p^{ - 1}}\right] = 0.} \cr } $$

(•₅) The claims of (ii) follow at once from the above (in particular from (IV)) and from Lemma 3.1 below.

Regarding the uniqueness of the solutions to (3.4), we have the following lemma.

The proof of the lemma is technical; the reader may safely skip it without it affecting their understanding of the remainder of the text.

Proof. (i). We have $$|{{\rm{e}}^{ - \Phi (q)x}}((\omega G) \star {W^{(q)}})(x)| \le \int_{ - \infty }^\infty \omega (y)|G(y)|{{\rm{e}}^{ - \Phi (q)y}}{W^{(q)}}(x - y){{\rm{e}}^{ - \Phi (q)(x - y)}}{\kern 1pt} {\rm{d}}y$$ for all x ∈ ℝ. Since G has a left tail that is Ф (q)-subexponential and since it is locally bounded, it follows that there is a γ < ∞ such that | G (y) | e^{− Ф(q)y} ≤ γ for all y ∈ (−∞, 0] (say). Therefore, for x ∈ (−∞, 0],

$$|{{\rm{e}}^{ - \Phi (q)x}}((\omega G) \star {W^{(q)}})(x)| \le \gamma \int_{ - \infty }^\infty \omega (y){W^{(q)}}(x - y){{\rm{e}}^{ - \Phi (q)(x - y)}}{\kern 1pt} {\rm{d}}y,$$

which is < ∞ by assumption. Now, by (2.3), W ^(q) (x − y)e ^{−Ф(q)(x −y)} is non-increasing to 0 as x ↓−∞. Thus the conclusion follows by dominated convergence.

(ii). Denote, for x ∈ ℝ, ||G||_x ≔ sup_{y∈(−∞, x]} |G (y) |e^−Ф(q)y. Note that this quantity is finite because G has a tail that is Ф(q)-subexponential and because it is locally bounded. Then G = (G ω) ⋆ W ^(q) implies that for all x ∈ ℝ we have

$$|G(x)|{{\rm{e}}^{ - \Phi (q)x}} \le \parallel G{\parallel _x}\int_{ - \infty }^\infty \omega (y){W^{(q)}}(x - y){{\rm{e}}^{ - \Phi (q)(x - y)}}{\kern 1pt} {\rm{d}}y = \parallel G{\parallel _x}(\omega \star ({{\rm{e}}^{ - \Phi (q) \cdot }}{W^{(q)}}))(x).$$

By (i), (ω ⋆ (e^−Ф(q)⋅ W ^(q)))(x) = e ^−Ф(q)x ((ωe^(q)⋅) ⋆ W ^(q))(x) → 0 as x ↓−∞, so there is an x ₀ ∈ ℝ such that $$(\omega \star ({{\rm{e}}^{ - \Phi (q) \cdot }}{W^{(q)}}))(x) \le {I_{{x_0}}}:=(\omega \star ({{\rm{e}}^{ - \Phi (q) \cdot }}{W^{(q)}}))({x_0}) \lt 1$$ for all x ∈ (−∞, x ₀]. At the same time, the above estimate implies $$\parallel G{\parallel _{{x_0}}} \le {I_{{x_0}}}\parallel G{\parallel _{{x_0}}}$$, hence $$\parallel G{\parallel _{{x_0}}} = 0$$, which forces G to vanish on (−∞, x ₀].

Let us now shift all the functions by x ₀ for (notational) convenience; to wit, F : = G (x ₀ +⋅) and θ : = ω(x ₀ +⋅) are Borel measurable and locally bounded, and F = (F ⋆ θ) W ^(q). From this we obtain finally that F = 0 by the following argument.

Fix y ₀ ∈ [0, ∞) and let θ ₀ be an upper bound for θ on [0, y ₀]. We may choose s ₀ ∈ (0, ∞) such that ψ (s ₀) − q >θ ₀. Denote ||F||≔ sup_{y∈[0, y 0]} |F (y) | e ^{−s ₀y}. Then, for each y ∈ [0, y ₀], F = (F θ) ⋆ W ^(q) and (2.2) imply that

$$|F(y)|{{\rm{e}}^{ - {s_0}y}} \le \parallel F\parallel {\theta _0}\int_0^y {{\rm{e}}^{ - {s_0}(y - z)}}{W^{(q)}}(y - z){\kern 1pt} {\rm{d}}z \le \parallel F\parallel {\theta _0}/(\psi ({s_0}) - q).$$

Again this renders ||F|| = 0, and completes the proof of (ii).

(iii). By induction we can prove that G _n ≤ G for all n ∈ ℕ₀. To see that the sequence (G _n)_{n ∈ℕ0} is pointwise non-decreasing, define the operator $$\cal L$$, acting on Borel measurable functions h : ℝ → [0, ∞], by ℒ h: = (ω h) ⋆ W ^(q), and argue by induction that, for n ∈ ℕ₀, $${G_n} = \sum\nolimits_{k = 0}^n {{\cal L}^k}g$$, where for k ∈ℕ₀, $${{\cal L}^k}:=\underbrace {{\cal L} \circ \cdots \circ {\cal L}}_{k - {\rm{times}}}$$ is the k-fold composition of the operator $$\cal L$$ with itself (of course, $$\cal L ^0$$ is just the identity operator). Then, passing to the limit in the recursion via monotone convergence, we find that G _∞ = g + (ω G _∞) ⋆ W ^(q). It follows that G − G _∞ has a Ф(q)-subexponential left tail, is locally bounded, Borel measurable, and satisfies G − G _∞ = ((G − G _∞) ω) ⋆ W ^(q). By (ii), this means that G = G _∞.

3.2. Further complements

As is to be expected, the solution to (1.1) is associated with a family of (local) martingales. Recall the process Y from (1.2).

Proposition 3.1. Let c ∈ℝ, γ ∈ (0, ∞). Define the processes Z = (Z _t)_t∈[0,∞) and V = (V _s)_s∈[0,∞) as follows:

$${Z_t}:=\exp \left\{ - \int_0^t \omega ({X_u}){\kern 1pt} {\rm{d}}u - qt\right\} {\cal H}_q^{(\omega )}({X_t}),\quad t \in [0,\infty ),$$

and

$${V_s}:={{\rm{e}}^{ - \gamma s}}{\cal H}_q^{(\gamma \omega )}({Y_s}){{\bf{1}}_{\{ s \lt \zeta \} }},\quad s \in [0,\infty ).$$

Further, let $$\cal F = {({\cal F_t})_{t \in [0,\infty )}}$$ be any right-continuous filtration relative to which X is adapted and has independent increments. Then:

1. (i) The stopped process $${Z^{\tau _c^ + }}$$ is a bounded cádlág martingale in the filtration $$\cal F$$ under P_x for each x ∈ ∝; for real x ≤ c the P_x -terminal value of this martingale is $${\cal H}_q^{(\omega )}(c)\exp \left\{ - \int_0^{\tau _c^ + } \omega ({X_s}){\kern 1pt} {\rm{d}}s - q\tau _c^ + \right\} {{\bf{1}}_{\{ \tau _c^ + \lt \infty \} }}$$

2. (ii) Assume ω is strictly positive and e _q is independent of $$\cal F _\infty$$. Then the stopped process $${V^{T_c^ + }}$$ is a cádlág bounded martingale in the filtration $${\cal G} = ({{\cal G}_s}{)_{s \in [0,\infty )}}:={({{\cal F}_{{\rho _s}}} \vee \sigma (\{ \{ {\rho _u} \lt {e_q}\} :u \in [0,s]\} ))_{s \in [0,\infty )}}$$ under P_x for each x ∈ ℝ.

Proof of Proposition 3.1. We may assume x ≤ c.

(i). Let t ∈ [0, ∞). Then in Markov process theory parlance, with (for the purposes of this proof, and for notational simplicity only) (θ _u)_u∈[0,∞) being the shift operators, we have $$\tau _c^ + \wedge t + \tau _c^ + ^\circ {\theta _{\tau _c^ + \wedge t}} = \tau _c^ + \,and\,{P_x} - as $$,

$$\matrix{ {{Z_{t \wedge \tau _c^ + }} = {\cal H}_q^{(\omega )}(c){{\rm{e}}^{ - \int_0^{t \wedge \tau _c^ + } \omega ({X_u}){\kern 1pt} {\rm{d}}u - q(t \wedge \tau _c^ + )}}{{\rm{P}}_{{X_{t \wedge \tau _c^ + }}}}[\exp \{ - \int_0^{\tau _c^ + } \omega ({X_u}){\kern 1pt} {\rm{d}}u - q\tau _c^ + \} ;\tau _c^ + \lt \infty ]} \cr { = {\cal H}_q^{(\omega )}(c){{\rm{P}}_x}[{{\rm{e}}^{ - \int_0^{t \wedge \tau _c^ + } \omega ({X_u}){\kern 1pt} {\rm{d}}u - q(t \wedge \tau _c^ + )}}(\exp \{ - \int_0^{\tau _c^ + } \omega ({X_u}){\kern 1pt} {\rm{d}}u - q\tau _c^ + \} {{\bf{1}}_{\{ \tau _c^ + \lt \infty \} }}) \circ {\theta _{\tau _c^ + \wedge t}}{\kern 1pt} |{\kern 1pt} {{\cal F}_{t \wedge \tau _c^ + }}]} \cr { = {\cal H}_q^{(\omega )}(c){{\rm{P}}_x}[\exp \{ - \int_0^{\tau _c^ + } \omega ({X_u}){\kern 1pt} {\rm{d}}u - q\tau _c^ + \} {{\bf{1}}_{\{ \tau _c^ + \lt \infty \} }}{\kern 1pt} |{\kern 1pt} {{\cal F}_{t \wedge \tau _c^ + }}]} \cr { = {\cal H}_q^{(\omega )}(c){{\rm{P}}_x}[\exp \{ - \int_0^{\tau _c^ + } \omega ({X_u}){\kern 1pt} {\rm{d}}u - q\tau _c^ + \} {{\bf{1}}_{\{ \tau _c^ + \lt \infty \} }}{\kern 1pt} |{\kern 1pt} {{\cal F}_t}],} \cr } $$

which establishes the first claim.

(ii). For all real 0 ≤ s ≤ t, $$A\in \cal F _\rho s$$, applying the optional sampling theorem to the process $$Z ^{{\tau {{_c^ +}}}}$$ at the times ρ _s and ρ _t, we obtain

$$\matrix{ {{{\rm{P}}_x}[\exp \{ - \int_0^{{\rho _t} \wedge \tau _c^ + } \omega ({X_u}){\kern 1pt} {\rm{d}}u - q({\rho _t} \wedge \tau _c^ + )\} {\cal H}_q^{(\omega )}({X_{{\rho _t} \wedge \tau _c^ + }});{\kern 1pt} A \cap \{ {\rho _t} \wedge \tau _c^ + \lt \infty \} ]} \cr { = {{\rm{P}}_x}[\exp \{ - \int_0^{{\rho _s} \wedge \tau _c^ + } \omega ({X_u}){\kern 1pt} {\rm{d}}u - q({\rho _s} \wedge \tau _c^ + )\} {\cal H}_q^{(\omega )}({X_{{\rho _s} \wedge \tau _c^ + }});{\kern 1pt} A \cap \{ {\rho _s} \wedge \tau _c^ + \lt \infty \} ];} \cr } $$

that is because $${\rho _t} \wedge \tau _c^ + = {\rho _t} \wedge {\rho _{T_c^ + }} = {\rho _{t \wedge T_c^ + }}$$ on $$\{ {\rho _t} \wedge \tau _c^ + \lt {e_q}\} = \{ t \wedge T_c^ + \lt \zeta \} $$ since ρ is the inverse of $$\int_0^ \cdot \omega ({X_u}){\kern 1pt} {\rm{d}}u$$, and by the independence of e _q from $$\cal F _\infty$$,

$$\matrix{ {{{\rm{P}}_x}[\exp \{ - t \wedge T_c^ + \} {\cal H}_q^{(\omega )}({Y_{t \wedge T_c^ + }});{\kern 1pt} A \cap \{ t \wedge T_c^ + \lt \zeta \} ]} \cr { = {{\rm{P}}_x}[\exp \{ - s \wedge T_c^ + \} {\cal H}_q^{(\omega )}({Y_{s \wedge T_c^ + }});{\kern 1pt} A \cap \{ s \wedge T_c^ + \lt \zeta \} ].} \cr } $$

This implies that $${({{\rm{e}}^{ - s}}{\cal H}_q^{(\omega )}({Y_s}){{\bf{1}}_{\{ s \lt \zeta \} }})_{s \in [0,\infty )}}$$ stopped at $$T_c^ + $$ is a martingale in the filtration $${\cal G}$$ under P_x, because this process is constant on [ζ ∞), and since for s ∈ [0, ∞), { s < ζ}= {ρ _s < e _q } with the equality of the trace σ fields $${{\cal G}_s}{|_{\{ {\rho _s} \lt {e_q}\} }} = {{\cal F}_{{\rho _s}}}{|_{\{ {\rho _s} \lt {e_q}\} }}$$ holding true. Replacing ω with γω shows that the same is true of the process $${({{\rm{e}}^{ - \gamma s}}{\cal H}_q^{(\gamma \omega )}({Y_s}){{\bf{1}}_{\{ s \lt \zeta \} }})_{s \in [0,\infty )}}$$ stopped at $$T_c^ + $$.

Apart from the solutions presented in Examples 3.1–3.4, it seems difficult to come up with ‘nice’ ω for which $${\cal H}_q^{(\omega )}$$ is given explicitly (in terms of ψ and Ф, say), at least for a general W ^(q). However, based on Example 3.2, we can ‘reverse engineer’ a class of ω for which $${\cal H}_q^{(\omega )}$$ is (semi-)explicit, in the precise sense of the following proposition.

Proposition 3.2. Let ν be a probability measure on the Borel sets of (0, ∞) , whose support is compactly contained in (0, ∞) ;q ∈ [0, ∞) is still fixed. Set

$$w(x):={{\int {\cal H}_q^{({{\rm{e}}^{\alpha \cdot }})}(x){{\rm{e}}^{\alpha x}}\nu ({\rm{d}}\alpha )} \over {\int {\cal H}_q^{({{\rm{e}}^{\alpha \cdot }})}(x)\nu ({\rm{d}}\alpha )}},\quad x \in {\mathbb R}$$

(see Example 3.2 for an explicit expression for $${\cal H}_q^{({{\rm{e}}^{\alpha \cdot }})}$$. Then w : ℝ → ℝ is well defined, Borel measurable, and locally bounded, $$L_q^{(w)} = \int L_q^{({{\rm{e}}^{\alpha \cdot }})}\nu ({\rm{d}}\alpha )$$, and, for x ∈ Ф$${\cal H}_q^{(w)}(x) = \int {\cal H}_q^{({{\rm{e}}^{\alpha \cdot }})}(x)\nu ({\rm{d}}\alpha )$$.

Proof of Proposition 3.2. To ease the notation, set: for α ∈ (0, ∞), $${\cal H}_q^\alpha :={\cal H}_q^{({{\rm{e}}^{\alpha \cdot }})}$$ and $$L_q^\alpha :=L_q^{({{\rm{e}}^{\alpha \cdot }})}$$; $$L:=\int L_q^\alpha \nu ({\rm{d}}\alpha )$$ and for x ∈ ℝ, $$H(x):=\int {\cal H}_q^\alpha (x)\nu ({\rm{d}}\alpha )$$. From the explicit form of $${\cal H}_q^\alpha $$ given in Example 3.2 we see that H : ℝ → (0, ∞] is well defined and Borel measurable. Also, the fact that ν has a support bounded from above ensures, via Theorem 3.1, items (I) and (II I), that H is locally bounded, and in particular it is finite; then, w : ℝ → (0, ∞) is well defined, Borel measurable, and locally bounded (the latter again because the support of ν is bounded from above).

Further, we know that $${\cal H}_q^\alpha = L_q^\alpha {{\rm{e}}^{\Phi (q) \cdot }} + ({{\rm{e}}^{\alpha \cdot }}{\cal H}_q^\alpha ) \star {W^{(q)}}$$ for each α ∈ (0, ∞). Integrating both sides against ν (dα) we obtain, via Tonelli’s theorem, $$H = L{{\rm{e}}^{\Phi (q) \cdot }} + (Hw) \star {W^{(q)}}$$. At the same time, for x ∈(−∞, 0] $${{\rm{e}}^{ - \Phi (q)x}}H(x) = \int {\cal H}_q^\alpha (x){{\rm{e}}^{ - \Phi (q)x}}\nu ({\rm{d}}\alpha ) \le 1$$, so H has a left tail that is Ф(q)-subexponential. Next, because the support of ν is bounded from below away from zero, w has a subexponential left tail. By Lemma 3.1, we obtain $$H = {L \over {L_q^{(w)}}}{\cal H}_q^{(w)}$$. But we also have $$H(1) = 1 = {\cal H}_q^{(w)}(1)$$, hence $$L = L_q^{(w)}$$, and the proof is complete.

Another fairly general class of ω that can be handled with some success is considered in the following remark.

4.1. The model

Assume ω> 0. We consider the process S defined by

$${S_s}:={{\rm{e}}^{{Y_s}}}{{\bf{1}}_{\{ s \lt \zeta \} }},\quad s \in [0,\infty ),$$

as a model for the price of a (speculative) financial asset (here, Y is the process from (1.2)). When ω = 1 then ζ = e _q, Y = X on [0, e _q), and S is nothing but the classical exponential Lévy model for the price of a risky asset (defaulted at ζ); see [Reference Tankov19] for a recent review. The idea in allowing a more general ω is that the asset price may ‘move faster or slower along its trajectory’, depending on the price level, destroying the stationary independent increments property of the log-returns but preserving their Markovian character. Using time-changed Lévy processes to model financial assets is not new, of course – see, e.g., [Reference Carr and WU2, Reference Klingler, Kim, Rachev and Fabozzi9]. We set Q_z ≔ P_{log z}, z ∈ (0, ∞), for convenience.

4.2. The optimization problem

Suppose in the above setting that we are interested in the simple problem of the determination of the optimal level b at which to sell the asset, having bought it at the level z ∈ (0, ∞), under an inflation/impatience rate γ ∈ (0, ∞). In other words, if we let, for b ∈ (0, ∞), $$R_b^ + :=\inf \{ s \in (0,\infty ){\kern 1pt} :{\kern 1pt} {S_s} > b\} $$ denote the first hitting time of the set (b, ∞) by the process S, then we would like a solution to the problem

(4.1) $$\mathop {\max }\limits_{b \in [z,\infty )} {\cal A}(b){\rm{\,with\,}}{\cal A}(b):={{\rm{Q}}_z}[{S_{R_b^ + }}\exp \{ - \gamma R_b^ + \} ;{\kern 1pt} R_b^ + \lt \infty ]{\rm{\,for\,}}b \in [z,\infty ).$$

(More generally, we may simply be interested in $$\cal A{(b)}$$ itself if we are predetermined to sell at the level b ∈ [ z, ∞).)

But, for b ∈ [ z, ∞), since $${S_{R_b^ + }} = b$$ on $$\{ R_b^ + \lt \infty \} $$,

$${\cal A}(b) = b{{\rm{Q}}_z}[\exp \{ - \gamma R_b^ + \} ;{\kern 1pt} R_b^ + \lt \infty ] = b{{\rm{P}}_{\log z}}[\exp \{ - \gamma T_{\log b}^ + \} ;{\kern 1pt} T_{\log b}^ + \lt \zeta ]$$

(4.2) $$ = b{\cal B}_q^{(\gamma \omega )}(\log z,\log b) = b{{{\cal H}_q^{(\gamma \omega )}(\log z)} \over {{\cal H}_q^{(\gamma \omega )}(\log b)}}.$$

Hence, (4.1) is intimately related to the determination of the quantity (1.1).

Remark 4.2. More difficult is the optimal stopping problem

$$(*){\kern 1pt} :\quad \mathop {\max }\limits_T {{\rm{Q}}_z}[{S_T}\exp \{ - \gamma T\} ;{\kern 1pt} T \lt \infty ],$$

where T ranges over all stopping times (of the completed natural filtration) of S, i.e. we maximize the optimal selling time over all stopping rules of S, not just the first upcrossing times $${(R_b^ + )_{b \in [z,\infty )}}$$ as in (4.1). On the level of the process X it corresponds to

$$(**){\kern 1pt} :\quad \mathop {\max }\limits_\tau {{\rm{P}}_{\log z}}[{{\rm{e}}^{{X_\tau }}}\exp \{ - \gamma \int_0^\tau \omega ({X_t}){\kern 1pt} {\rm{d}}t\} ;{\kern 1pt} \tau \lt {e_q}],$$

where τ ranges over all stopping times (of the completed natural filtration) of X (progressively enlarged by e _q).

Now, for the case q = 0 (without loss of generality), such problems have recently been studied (in even greater generality: allowing an additive functional A of X in place of $$\int_0^ \cdot \gamma \omega ({X_u}){\kern 1pt} {\rm{d}}u$$; a general Lévy process for X; f (X _τ) in place of e^X τ for a lower semi-continuous map f) in [13]. In the latter paper, sufficient conditions for the optimality of a first upcrossing time were provided [Reference Long and Zhang13, Theorem 2.1], these being further strengthened in the spectrally negative case [Reference Long and Zhang13, Theorem 2.2]. In our present case, [Reference Long and Zhang13, Assumption 2.1] requires that

(C ₁): either X drifts to −∞, or else $${{\rm{P}}_x}(\int_0^\infty \omega ({X_u}){\kern 1pt} {\rm{d}}u = \infty ) = 1$$ for all x ∈ R (in the case when X drifts to ∞ the latter is equivalent to $$\int^\infty \omega (y){\kern 1pt} {\rm{d}}y = \infty $$ [Reference DöRing and Kyprianou5, Theorem 1]).

When this is so, [Reference Long and Zhang13, Theorem 2.2] provides relatively explicit sufficient conditions under which a first upcrossing time from the family $${(\tau _c^ + )_{c \in {\mathbb R}}}$$ is optimal in (**). Indeed, let e be an exponential random time of mean 1, independent of X, and set $$\tilde \eta :={\rho _{{\rm{e}}/\gamma }}$$. We then have, for real x < z,

$$\matrix{ {{{\rm{P}}_x}({{\overline X }_{\tilde \eta }} > z) = {{\rm{P}}_x}({{\overline X }_{{\rho _{{\rm{e/}}\gamma }}}} > z) = {{\rm{P}}_x}(\tau _z^ + \lt {\rho _{{\rm{e/}}\gamma }}) = {{\rm{P}}_x}(\int_0^{\tau _z^ + } \gamma \omega ({X_u}){\kern 1pt} {\rm{d}}u \lt {\rm{e}},\tau _z^ + \lt \infty )} \cr { = {{\rm{P}}_x}[\exp \{ - \int_0^{\tau _z^ + } \gamma \omega ({X_u}){\kern 1pt} {\rm{d}}u\} ;{\kern 1pt} \tau _z^ + \lt \infty ] = {\cal B}_0^{(\gamma \omega )}(x,z) = {{{\cal H}_0^{(\gamma \omega )}(x)} \over {{\cal H}_0^{(\gamma \omega )}(z)}}} \cr } $$

(here, $$\overline X $$ is the running supremum process of X), i.e. the ‘hazard rate’ of $${\overline X _{\tilde \eta }}$$ is equal to (cf. [Reference Long and Zhang13, Assumption 2.2])

$${\Lambda ^{(\gamma \omega )}}(z) :=- {{\rm{d}} \over {{\rm{d}}z}}\log {{\rm{P}}_x}({\overline X _{\tilde \eta }} > z) = {{\rm{d}} \over {{\rm{d}}z}}\log ({\cal H}_0^{(\gamma \omega )}(z)) = {{{\cal H}_0^{(\gamma \omega )'}(z)} \over {{\cal H}_0^{(\gamma \omega )}(z)}},\quad z \in {\mathbb R},$$

when the derivative of $${\cal H}_0^{(\gamma \omega )}$$ exists, which we now assume it does. If, further, $${\cal H}_0^{(\gamma \omega )}$$ is absolutely continuous, which occurs if $${\cal H}_0^{(\gamma \omega )'}$$ is locally bounded or even continuous [Reference Yeh21, Theorem 13.18], then it follows by the fundamental theorem of calculus that $${\Lambda ^{(\gamma \omega )}} \in L_{{\rm{loc}}}^1({\mathbb R})$$, while $$\int^\infty {\Lambda ^{(\gamma \omega )}} = \infty $$ (recall (C ₁)). Suppose for simplicity (for absolutely minimal conditions that apply in the context of [Reference Long and Zhang13] the reader is asked to consult the latter reference directly) that

$$({C_2}):{\cal H}_0^{(\gamma \omega )}$$ is continuously differentiable with a derivative that has no zeros.

Set $${{\cal K}^{(\gamma \omega )}}(x):={{\rm{e}}^x}(1 - {{{\cal H}_0^{(\gamma \omega )}(x)} \over {{\cal H}_0^{(\gamma \omega )'}(x)}})$$ for x ∈ ℝ. Then, according to [Reference Long and Zhang13, Theorem 2.2, Assumption 2.3, Lemma 2.2], if

(C ₃): there exists a (necessarily unique) x ^* ∈ [−∞, ∞ ] such that the function $${{\cal K}^{(\gamma \omega )}}{\kern 1pt} : {\mathbb R}{\kern 1pt} \to {\mathbb R}$$ is > 0 and non-decreasing on (x ^*, ∞), and is ≤ 0 on (−∞, x ^*),

then 0 is optimal, $$\tau _{{x^*}}^ + $$ is optimal, or $${(\tau _n^ + )_{n \in {\mathbb N}}}$$ is an optimizing sequence in (**) according as x ^* = −∞, x ^* ∈ ℝ, or x ^* = ∞.

Generally speaking, it appears to be a non-trivial matter to ascertain whether and when the conditions (C ₁), (C ₂), and (C ₃) are met. We leave investigating this for future research. In particular, without further qualifications, we make no claim that an optimizer in (4.1) is also a general optimal selling rule (it is merely optimal in the class of first upcrossing passage times). We can, however, make the following immediate observations concerning (C ₂) and (C ₃) (leaving aside (C ₁), whose validity or non-validity is not difficult to check, except possibly when X oscillates).

Let γ = 1 without loss of generality. By standard excursion-theoretic arguments (see, e.g., [Reference Vidmar20, p. 13, proof of Lemma 3.6] and [Reference Kuznetsov, Kyprianou and Rivero10, Section 2.3]) we obtain the following representation of the quantity $${\cal B}_0^{(\omega )}(x,c)$$ for real x ≤ c (incidentally, it is true even if ω is not strictly positive):

(4.3) $${{{\cal H}_0^{(\omega )}(x)} \over {{\cal H}_0^{(\omega )}(c)}} = {\cal B}_0^{(\omega )}(x,c) = {{\rm{e}}^{ - \int_x^c \omega (y){\kern 1pt} {\rm{d}}y/\delta }}\exp \{ - \int_x^c {\rm{n}}[1 - {{\rm{e}}^{ - \int_0^\chi \omega (g + {\xi _u}){\kern 1pt} {\rm{d}}u}}{{\bf{1}}_{\{ \chi \lt \infty \} }}]{\kern 1pt} {\rm{d}}g\} ,$$

where the first factor appears only when X has paths of finite variation, in which case δ is the drift of X, and where n is the (suitably normalized) excursion measure for excursions away from the maximum of X, ξ is the canonical process, and χ ≔ inf{ u ∈ (0, ∞): ξ _u ≥ 0 } is the lifetime of the excursions. Now, if ω is continuous and has a bounded left tail then, by dominated convergence, the map $${\mathbb R} \ni g \mapsto {\rm{n}}[1 - {{\rm{e}}^{ - \int_0^\chi \omega (g + {\xi _u}){\kern 1pt} {\rm{d}}u}}{{\bf{1}}_{\{ \chi \lt \infty \} }}]$$ is continuous so that (4.3) implies, via the fundamental theorem of calculus, that

$${{{\cal H}_0^{(\omega )'}(x)} \over {{\cal H}_0^{(\omega )}(x)}} = {{\omega (x)} \over \delta } + {\rm{n}}[1 - {{\rm{e}}^{ - \int_0^\chi \omega (x + {\xi _u}){\kern 1pt} {\rm{d}}u}}{{\bf{1}}_{\{ \chi \lt \infty \} }}],\quad x \in {\mathbb R},$$

(again, the first term appears only when X has paths of finite variation); $${\cal H}_0^{(\omega )'}$$ therefore exists, is continuous, and has no zeros (the latter because ω> 0), i.e. (C ₂) obtains. Besides, if in addition ω is non-decreasing (resp. non-increasing, strictly increasing on a neighborhood of −∞ and non-decreasing, strictly decreasing on a neighborhood of −∞ and non-increasing), then $${\mathbb R} \ni x \mapsto 1 - {{{\cal H}_0^{(\omega )}(x)} \over {{\cal H}_0^{(\omega )'}(x)}}$$ is non-decreasing (resp. non-increasing, strictly increasing, strictly decreasing). Hence, if ω is strictly increasing and bounded on a neighborhood of −∞, continuous, and non-decreasing, then there exists a (necessarily unique) x ^* ∈ [−∞, ∞ ] such that the function $${\mathbb R} \ni x \mapsto 1 - {{{\cal H}_0^{(\omega )}(x)} \over {{\cal H}_0^{(\omega )'}(x)}}$$ is > 0 on (x ^*, ∞) and is < 0 on (−∞, x ^*); with this x ^*, condition (C ₃) is met.

4.3. Analysis in some special cases

If ω = 1 then, because of the martingale property of (exp{X _t − Ψ (1) t})_{t∈[0, ∞)} and on account of the optional sampling theorem, $${\cal A} (b) $$ is monotone in b ∈ [ z, ∞) and the problem (4.1) is trivial. However, for a general ω this is no longer true, as we will see in an example shortly. To this end we specialize now to the case when

(4.4) $${\rm{for\,some }}\,\alpha \in (0,\infty ){\rm\,{we\,have }}\,\omega = {{\rm{e}}^{\alpha \cdot }} \wedge 1.$$

As a possible rationale for such a choice we can imagine that the asset moves faster along its trajectory at smaller price levels, reflecting that the investors are then more jittery, increasing the velocity of the trades.

(4.5) $$P(x) = L{{\rm{e}}^{\Phi (q)x}}\sum\limits_{k = 0}^\infty {a_k}{(\gamma {{\rm{e}}^{\alpha x}})^k},\quad x \in {\mathbb R}, $$

where we have set $${a_k} : = {(\prod\nolimits_{l = 1}^k (\psi (\Phi (q) + l\alpha ) - q))^{ - 1}}$$ for k ∈ ℕ₀. Then, by Theorem 3.1(II), H = P on (−∞, 0], while by Theorem 3.1(ii), for x∈[0,∞),

(4.6) $$H(x) = h(x) + \gamma \int_0^x H(y){W^{(q)}}(x - y){\kern 1pt} {\rm{d}}y,$$

where

(4.7) $$h(x) := L{{\rm{e}}^{\Phi (q)x}} + \gamma \int_{ - \infty }^0 {{\rm{e}}^{\alpha y}}P(y){W^{(q)}}(x - y){\kern 1pt} {\rm{d}}y,\quad x \in [0,\infty ).$$

As in the proof of Lemma 3.1 we see that, on [0,∞), H = ↑− lim_{n →∞} H _n where H ₀ := h, and then recursively H _n+1 = h + γ H _n ⋆ W ^(q) for n ∈ ℕ₀, the latter convolution being now on [0, ∞). Taking Laplace transforms on [0, ∞) in (4.6) (denoting them by a hat) and using (2.2), it is also true that $$\hat H(s) = \hat h(s) + \gamma \hat H(s)/(\psi (s) - q)$$, and hence

(4.8) $$\hat H = \hat h{{\psi - q} \over {\psi - \gamma - q}}{\rm{on}}\,(\Phi (q + \gamma ),\infty )$$

(using Theorem 3.1(III) and the known solution for ω =1, it is easily checked that $$\hat H(s) \lt \infty $$ for S ∈(Ф(q + γ), ∞)).

Now, if q + γ <ψ (1) then a comparison argument (with ω = 1) shows that $${\cal A} (b) \to \infty $$ as b → ∞. However, it does not in general seem obvious how to analytically determine an optimal b in (4.1) (or, which amounts to the same thing, the x ^* of (C ₃) of Remark 4.2), nor is it our intent to pursue this further here. We content ourselves with demonstrating (numerically, on a concrete example) how 𝒜(b) may fail to be monotone in b, and how a non-trivial b may be optimal, i.e. how the x ^* from (C ₃) of Remark 4.2 can fall into (z, ∞).

In fact, this transpires already in the (presumably simplest) case when

1. 1. ψ (s) = s ², s ∈ [0, ∞), corresponding to X − X ₀ being a multiple (by the factor $$\sqrt 2 $$) of Brownian motion, and

2. 2. q = 0, α = 1,

to which we specialize all discussion henceforth.

Continuing with our analysis, under the above specifications Φ(0) = 0 and ψ(1) = 1, while W (x) = x and (via (4.7)) h (x) = 1 + b _γx, x ∈ [0, ∞), where

$${b_\gamma } := \gamma \sum\limits_{k = 0}^\infty {{{\gamma ^k}} \over {k{!^2}(k + 1)}}/\sum\limits_{k = 0}^\infty {{{\gamma ^k}} \over {k{!^2}}}.$$

Furthermore, inverting the Laplace transform (4.8) for H we obtain

(4.9) $$H(x) = {{\sqrt \gamma - {b_\gamma }} \over {2\sqrt \gamma }}{{\rm{e}}^{ - \sqrt \gamma x}} + {{\sqrt \gamma + {b_\gamma }} \over {2\sqrt \gamma }}{{\rm{e}}^{\sqrt \gamma x}},\quad x \in [0,\infty )$$

(H |_[0,∞) is uniquely determined by its Laplace transform, because by Theorem 3.1 (I) this function is continuous).

With (4.9) having been established, Figure 1 depicts the case z = 1of (4.1) for three different values of γ (0.9, 1, and 1.1) corresponding to three fundamentally different behaviors of the objective function $${\cal A} $$. When γ = 1.1 (still z = 1), then numerically the optimal b in (4.1) is approximately b ^* ≔ 2.533 42. We reinforce this using Remark 4.2. Indeed, we have already observed that conditions (C ₁), (C ₂), and (C ₃), for a suitable x ^*, of Remark 4.2 hold in this case. Let us further set $${K_\gamma }\,: = \,{{\cal K}^{(\gamma \omega)}} $$ for short. Then we can compute

Figure 1: The function A of (4.1)and (4.2) on the interval [Reference Bertoin1, 30] in the case when q = 0, α = z = 1, and ψ(s) = s ², s ∈ [0, ∞), for three values of the inflation/impatience parameter γ – from top to bottom, γ = 0.9 (blue, dotted), γ = 1 (red, dashed), and γ = 1.1 (black, full). The case γ = 1.1 exhibits nontrivial behavior. Unlike when ω = 1, for which the optimal b would be 1, now the optimal b is strictly greater than 1. Intuitively this is due to the fact that the clock ‘runs faster’ when the price level is small, thus ‘buying’ us some time in terms of the inflation depreciation as we wait for a higher price level.

(4.10) $${K_\gamma }(x) = \left( {\matrix{{{{\rm{e}}^x}\left( {\matrix{{1 - {1 \over {\sqrt \gamma }}{{1 + {{1 - {{{b_\gamma }} \over {\sqrt \gamma }}} \over {1 + {{{b_\gamma }} \over {\sqrt \gamma }}}}{{\rm{e}}^{ - 2\sqrt \gamma x}}} \over {1 - {{1 - {{{b_\gamma }} \over {\sqrt \gamma }}} \over {1 + {{{b_\gamma }} \over {\sqrt \gamma }}}}{{\rm{e}}^{ - 2\sqrt \gamma x}}}}} \cr} } \right)} \:{{\rm{for}}\,x \in {\rm{[0,}}\infty {\rm{),while}}} \cr {{{\rm{e}}^x}\left( {\matrix{{1 - {{\sum\nolimits_{k = 0}^\infty {{{{(\gamma {{\rm{e}}^x})}^k}} \over {k{!^2}}}} \over {\gamma {{\rm{e}}^x}\sum\nolimits_{k = 0}^\infty {{{{(\gamma {{\rm{e}}^x})}^k}} \over {k{!^2}(k + 1)}}}}} \cr} } \right)}\:\: {{\rm{for}}\,x \in {\rm{(}} - \infty {\rm{,0)}}{\rm{.}}} \cr} } \right.$$

This function is seen numerically, when γ = 1.1, to satisfy condition (C ₃) of Remark 4.2 with (to within numerical precision) x ^* ≔ log (b ^*) – see Figure 2. On the other hand, when, ceteris paribus, γ is 0.9 or 1, then numerical experiments also show (or at least suggest) that K _γ is bounded away from zero (resp. is tending to 0 at ∞) and that it is strictly negative when γ = 0.9 (resp. γ = 1). (Of course, given the explicit nature of the above expressions one could in principle also verify all the preceding numerical claims analytically, but it would be an extremely tedious exercise with little apparent gain, so we refrain from doing so.)

Figure 2: Plot of K _γ from (4.10) on the interval [− 2, 3] with γ = 1.1.

Finally, let us introduce the value function V :(0, ∞) → (0, ∞ ](cf.( 4.1) and (4.2)):

(4.11) $$V(z)\mathop {\sup }\limits_{b \in [z,\infty )} {{\rm{Q}}_z}[{S_{R_b^ + }}{{\rm{e}}^{ - \gamma R_b^ + }};{\kern 1pt} R_b^ + \lt \infty ] = \mathop {\sup }\limits_{b \in [z,\infty )} b{{H(\log {\kern 1pt} z)} \over {H(\log {\kern 1pt} b)}},\quad z \in (0,\infty ).$$

It then follows analytically from the explicit form (4.9) of H obtained above that V ≡ ∞ when γ = 0.9, while $$V(z) = z + {{1 - {b_1}} \over {1 + {b_1}}}{z^{ - 1}} \in (z,\infty )$$ for z ∈ [1, ∞) when γ = 1; for the latter case we focus (for simplicity) only on the interval [1, ∞), so that the relevant values of H are all given by (4.9) rather than being a mixture of these and the values of P from (4.5) (recall that H |_(−∞,0] = P |_(−∞,0]). The case γ = 1.1 is handled numerically and reported in Figure 3.

Figure 3: Plot of V from (4.11) on the interval [Reference Bertoin1, Reference Carr and WU2] with γ = 1.1 (top, black, full) together with the identity function (bottom, red, dashed) that corresponds to immediate selling (and must therefore everywhere minorize the value function).