2. Preliminaries

2.1. General settings

Let X={X _t ; t≥0} be a Markov process and the probability space be $$({\rm \Omega },{\scr F},{\Bbb F},{\Bbb P})$$ where $${\scr F}_{t} $$ is generated by {X _s }_0≤s≤t and $${\Bbb F}{\equals}\left\{ {{\cal F}_{t} ,t\geq 0} \right\}$$ . Denote the translated probability measure for X by $${\Bbb P}_{x} $$ if X ₀=x and $${\Bbb P}{\equals}{\Bbb P}_{0} $$ if X ₀=0. Correspondingly we use expectation E _x to denote the integration with respect to probability measure $${\Bbb P}_{x} $$ .

Definition 2.1 X is a Lévy process if X has a càdlàg sample path and its increment X _t+s −X _t is independent of $${\cal F}_{t} $$ and is identically distributed as X _s , for each s, t≥0.

Generally a Lévy process can make jumps in both positive and negative directions. If a Lévy process only makes non-positive jumps, it is referred to as a spectrally negative Lévy process. Suppose that X is a spectrally negative Lévy process. Because X only makes negative jumps, its moment generating function is well-defined at least for θ≥0. Specifically, $$E[\rm {\rm e}^{{\theta X_{t} }} ]{\equals}\rm {\rm e}^{{\phi (\theta )t}} $$ exists for t≥0, where ϕ(θ) is its Laplace exponent and it is given by

(2.1) $$\phi (\theta )\,{\equals}\,\gamma \theta {\plus}{1 \over 2}\sigma ^{2} \theta ^{2} {\minus}\mathop{\int}\limits_0^\infty (1{\minus}{\rm e}^{{{\minus}\theta x}} {\minus}\theta x{\bf{1}} _{{\left\{ {0\,\lt\,x\,\lt\,1} \right\}}} )\nu (dx)$$

In the above formula, γ, σ and μ characterise Lévy process X and they are called a Lévy triplet (γ, σ, ν) of X.

The Lévy triplet (γ, σ, ν) satisfies the following conditions:

1. (1) The Lévy measure ν is a positive Radon measure on (0, ∞).

2. (2) $$\mathop{\int}\limits_0^\infty \left( {1\wedge x\!^{2} } \right)\nu (dx)\,\,\lt\,\infty$$ .

Suppose in the following that the underlying capital process is a spectrally negative Lévy process X in the absence of dividend payments. We shall restrict the optimisation problem to Markov strategies. Denote the dividend strategy by $$\pi {\equals}\left\{ {L_{t}^{\pi } \,\\;\,t\geq 0} \right\}$$ , a left-continuous, non-decreasing, $${\cal F}_{t} $$ -adapted process starting at 0. L ^π is the accumulated dividend process under the dividend strategy π. By this, under the dividend strategy π, the dividend payment rate at time t can be denoted by $$dL_{t}^{\pi } $$ . Associated with the underlying capital process X and dividend process L ^π , the surplus process will be defined by $$U_{t}^{\pi } {\equals}X_{t} {\minus}L_{t}^{\pi } ,\,{\rm for}\, t\geq 0$$ . The ruin time, which is the moment that the surplus process goes below 0, is denoted by $$\tau _{0}^{{\minus}} {\equals}{\rm inf}\left\{ {t\geq 0{\rm \,\mid\,}U_{t}^{\pi } \,\lt\,0} \right\}$$ . A strategy π is called admissible if $$L_{{t{\plus}}}^{\pi } {\minus}L_{t}^{\pi } \leq U_{t}^{\pi } ,\,{\rm for}\,t\geq 0$$ . Denote the collection of all admissible dividend strategies by Π.

Suppose the interest rate r is a positive constant for which the accumulated interest rate is a linear deterministic function rt, for t≥0. The expected discounted dividend payment under strategy π is

$$v_{\pi } (x)\,{\equals}\,E_{x} \left[ {\mathop{\int}\limits_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}rs}} dL_{s}^{\pi } } \right]$$

which is a function dependent on the initial capital X ₀=x. The maximal expected discounted dividend payment is denoted by

$$v_{{\asterisk}} (x)\,{\equals}\,\mathop {{\rm sup}}\limits_{\pi \in{\rm \Pi }} E_{x} \left[ {\mathop{\int}\limits_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}rs}} dL_{s}^{\pi } } \right]$$

2.2. Exponential Lévy process as the discounting factor

The new change to the classical setting is that the interest rate r is replaced by a stochastic process whose accumulated process is modelled by a Lévy process R={R _t ; t≥0}. Accordingly the objective maximal expected discounted dividend payment can be given as

$$v_{{\asterisk}} (x,r)\,{\equals}\,\mathop {{\rm sup}}\limits_{\pi \in{\rm \Pi }} \;E_{{x,r}} \left[ {\mathop{\int}\limits_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}R_{{s^{{\minus}} }} }} dL_{s}^{\pi } } \right]$$

where the surplus process, dividend process and ruin time are not changed and r is the initial state of R, i.e., R ₀=r.

For convenience, v _*(x, r) is called the value function.

Remark 2.1 The stochastic integtal $$\mathop{\int}\nolimits_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}R_{{s^{{\minus}} }} }} dL_{s}^{\pi } $$ is defined in semimartingale theory where it is important to evaluate the integrand to the left of jumps; e.g., see chapter 2 of Protter (Reference Revuz and Yor2005).

2.3. Classical theorems and definitions

For the results of this paper we require the following notion of the extended infinitesimal generator for the general Markov processes.

Definition 2.2 (Extended infinitesimal generator) (see Revuz & Yor, Reference Sato2013) Suppose that X is a Markov process. A Borel-measurable function f is said to belong to the domain of the extended infinitesimal generator if there exists a Borel-measurable function g such that $$\mathop{\int}\nolimits_0^t \left| {g(X_{s} )} \right|ds\,\lt\,\infty$$ for each t, and $$f(X_{t} ){\minus}f(X_{0} ){\minus}\nomathop{\int}\limits_0^t g(X_{s} )ds$$ is a ( $${\cal F}_{t} ,{\Bbb P}_{x} $$ ) right-continuous martingale for every x. If we define Af:=g, then A is called the extended infinitesimal generator and its domain is denoted by $${\cal D}_{A} $$ .

Theorem 2.1 Suppose a Lévy process X has the Lévy triplet (γ, σ, ν), where Lévy measure ν is defined on $${\Bbb R}$$ and satisfies $$\mathop{\int}_{\Bbb R} \left( {1\wedge x^{2} } \right)\nu (dx)\,\lt\,\infty$$ . Given a function $$f\in{\Bbb C}^{2} ({\Bbb R})$$ , its extended infinitesimal generatorFootnote ¹ is

$$Af(x)\,{\equals}\,\gamma f\prime(x){\plus}{1 \over 2}\sigma ^{2} f\prime\prime(x){\plus}\mathop{\int}_{\Bbb R} \left( {f(x{\plus}y){\minus}f(x){\minus}{y \over {1{\plus}y^{2} }}f\prime(x)} \right)\nu (dy)$$

3. Fluctuation Identities and Eigenfunction

3.1. Eigenfunction of Lévy process under constraints

Proposition 3.1 Suppose R={R _t } _t≥0 is a Lévy process starting at r and it satisfies the following conditions: Assuming $$a\in{\Bbb R}$$ and a≠0

(1) $$E\left[ {{\rm e}^{{aR_{t} }} } \right]$$ is continuous at t=0 (2) $$E\left[ {{\rm e}^{{aR_{1} }} } \right]\,\lt\,\infty$$

Then an eigenfunction of the extended infinitesimal generator L of R={R _t } _t≥0 is given by e ^ar and corresponding eigenvalue is ϕ(a), where ϕ(a) is the Laplace exponent of R.

Proof: Denote $$E\left[ {{\rm e}^{{aR_{t} }} } \right]$$ by f _a (t). Since $$E\left[ {{\rm e}^{{aR_{t} }} } \right]$$ satisfies the conditions in Proposition 3.1, f _a (t) is continuous at x=0 and f _a (1)<∞. Consider the case that {R _t } starts at R ₀=0. Then using stationary independent increments, we have $$E\left[ {{\rm e}^{{aR_{{t{\plus}s}} }} } \right]{\equals}E\left[ {{\rm e}^{{a\left( {R_{{t{\plus}s}} {\minus}R_{s} {\plus}R_{s} } \right)}} } \right]{\equals}E\left[ {{\rm e}^{{a\left( {R_{{t{\plus}s}} {\minus}R_{s} } \right)}} } \right]E\left[ {{\rm e}^{{aR_{s} }} } \right]{\equals}E\left[ {{\rm e}^{{aR_{t} }} } \right]E\left[ {{\rm e}^{{aR_{s} }} } \right]$$ , i.e., f _a (t+s)=f _a (t)f _a (s). Thus, $$f_{a} (t){\equals}{\rm e}^{{{\rm ln}\left( {f_{a} (1)} \right)t}} $$ . Since $$E\left[ {{\rm e}^{{aR_{t} }} } \right]$$ exists, $$E\left[ {{\rm e}^{{aR_{t} }} } \right]{\equals}{\rm e}^{{\phi (a)t}} $$ where ϕ(a) is the Laplace exponent with parameter a. Next let us show that e ^ar is an eigenfunction for the extended infinitesimal generator L of R starting at r. By the definition, Lf:=g is an extended infinitesimal generator if $$f(R_{t} ){\minus}f(r){\minus}\mathop{\int}\nolimits_0^t g(R_{s} )ds$$ is a martingale. As discussed above, $$E_{r} \left[ {{\rm e}^{{aR_{t} }} } \right]\,{\equals}\,E_{r} \left[ {{\rm e}^{{a\left( {R_{t} {\minus}r{\plus}r} \right)}} } \right]{\equals}{\rm e}^{{ar}} E_{r} \left[ {{\rm e}^{{a\left( {R_{t} {\minus}r} \right)}} } \right]$$ . It can be noticed that {R _t −r} _t≥0 is still a Lévy process which starts at 0, and hence $$E_{r} \left[ {{\rm e}^{{a\left( {R_{t} {\minus}r} \right)}} } \right]{\equals}{\rm e}^{{\phi (a)t}} $$ . For now, assume that $${\rm e}^{{aR_{t} }} {\minus}{\rm e}^{{ar}} {\minus}\mathop{\int}\nplimits_0^t L{\rm e}^{{aR_{s} }} ds$$ is a martingale for each r (we will prove this result below). By the definition of the martingale, $$E_{r} \big[ {{\rm e}^{{aR_{t} }} {\minus}{\rm e}^{{ar}} {\minus}\mathop{\int}\nolimits_0^t L{\rm e}^{{aR_{s} }} ds} \big]{\equals}{\rm e}^{{ar}} {\minus}{\rm e}^{{ar}} {\equals}0$$ . Thus, $$E_{r} \big[ {{\rm e}^{{aR_{t} }} {\minus}{\rm e}^{{ar}} {\minus}\mathop{\int}\nolimits_0^t L{\rm e}^{{aR_{s} }} ds} \right]\,{\equals}\,0\Rightarrow E_{r} \big[ {{\rm e}^{{aR_{t} }} {\minus}{\rm e}^{{ar}} } \big]\,{\equals}\,E_{r} \big[ {\mathop{\int}\nolimits_0^t L{\rm e}^{{aR_{s} }} ds} \big].$$ By stochastic Fubini theorem, $$E_{r} \big[ {\mathop{\int}\nolimits_0^t L{\rm e}^{{aR_{s} }} ds} \right]{\equals}\mathop{\int}\nolimits_0^t E_{r} \left[ {L{\rm e}^{{aR_{s} }} } \big]ds$$ . Then we have the following calculations:

$$\eqalignno{ &#x0026; {d \over {dt}}E_{r} \left[ {{\rm e}^{{aR_{t} }} {\minus}{\rm e}^{{ar}} } \right]\left| {_{{t{\equals}0}} \,{\equals}\,{d \over {dt}}E_{r} \left[ {\mathop{\int}\limits_0^t L{\rm e}^{{aR_{s} }} ds} \right]} \right|_{{t{\equals}0}} \cr &#x0026; {d \over {dt}}{\rm e}^{{ar}} \left( {{\rm e}^{{\phi (a)t}} {\minus}1} \right)\left| {_{{t{\equals}0}} \,{\equals}\,{d \over {dt}}\mathop{\int}\limits_0^t E_{r} \left[ {L{\rm e}^{{aR_{s} }} } \right]ds} \right|_{{t{\equals}0}} \cr &#x0026; \phi (a){\rm e}^{{ar}} {\equals}E_{r} [L{\rm e}^{{ar}} ]\,{\equals}\,L{\rm e}^{{ar}} $$

The last equality comes from the fact that the term Le ^ar inside the expectation is a function of r. Thus Le ^ar =ϕ(a)e ^ar . Note that $$\mathop{\int}\nolimits_0^t \left| {\phi (a){\rm e}^{{aR_{s} }} } \right|ds{\equals}\left| {\phi (a)} \right|\mathop{\int}\nolimits_0^t {\rm e}^{{aR_{s} }} ds$$ . We will show that $$\mathop{\int}\nolimits_0^t \left| {\phi (a){\rm e}^{{aR_{s} }} } \right|ds\,\lt\,\infty$$ a.s. for each t. It is assumed that $$E_{r} \big[ {\mathop{\int}\nolimits_0^t {\rm e}^{{aR_{s} }} ds} \big]\,\lt\,\infty$$ . Therefore, $$\mathop{\int}\nolimits_0^t {\rm e}^{{aR_{s} }} ds$$ is finite with probability one. Note that Laplace exponent is bounded as well for a constant a, i.e., |ϕ(a)|<∞. That is to say, $$\mathop{\int}\nolimits_0^t \left| {\phi (a){\rm e}^{{aR_{s} }} } \right|ds{\equals}\left| {\phi (a)} \right|\mathop{\int}\nolimits_0^t {\rm e}^{{aR_{s} }} ds\,\lt\,\infty$$ a.s. for each t. It remains to show that $${\rm e}^{{aR_{t} }} {\minus}{\rm e}^{{ar}} {\minus}\mathop{\int}\nolimits_0^t \phi (a){\rm e}^{{aR_{s} }} ds$$ is a martingale. Letting 0≤u<t and $$Y_{t} {\equals}{\rm e}^{{aR_{t} }} {\minus}{\rm e}^{{ar}} {\minus}\mathop{\int}\nolimits_0^t \phi (a){\rm e}^{{aR_{s} }} ds$$ ,

$$\matrix{ {E[Y_{t} {\rm \mid}{\cal F}_{u} ]} \hfill \hskip-9pt&#x0026; {\equals} &#x0026; \hskip-8pt{{\rm e}^{{aR_{u} }} E_{{R_{u} }} \left[ {{\rm e}^{{a\left( {R_{t} {\minus}R_{u} } \right)}} } \right]{\minus}{\rm e}^{{ar}} {\minus}\mathop{\int}\limits_0^u \phi (a){\rm e}^{{aR_{s} }} ds{\minus}E_{{R_{u} }} \left[ {\mathop{\int}\limits_u^t \phi (a){\rm e}^{{aR_{s} }} ds} \right]} \hfill \cr {} \hfill &#x0026; {\equals} \hfill &#x0026; \hskip-8pt{{\rm e}^{{aR_{u} }} {\rm e}^{{\phi (a)(t{\minus}u)}} {\minus}{\rm e}^{{ar}} {\minus}\mathop{\int}\limits_0^u \phi (a){\rm e}^{{aR_{s} }} ds{\minus}\mathop{\int}\limits_u^t E_{{R_{u} }} \left[ {\phi (a){\rm e}^{{aR_{s} }} ds} \right]} \hfill \cr {} \hfill &#x0026; {\equals} \hfill &#x0026; \hskip-8pt{{\rm e}^{{aR_{u} }} {\rm e}^{{\phi (a)(t{\minus}u)}} {\minus}{\rm e}^{{ar}} {\minus}\mathop{\int}\limits_0^u \phi (a){\rm e}^{{aR_{s} }} ds{\minus}\mathop{\int}\limits_u^t \phi (a){\rm e}^{{aR_{u} }} {\rm e}^{{\phi (a)(s{\minus}u)}} ds} \hfill \cr {} \hfill &#x0026; {\equals} \hfill &#x0026; \hskip-8pt{{\rm e}^{{aR_{u} }} {\rm e}^{{\phi (a)(t{\minus}u)}} {\minus}{\rm e}^{{ar}} {\minus}\mathop{\int}\limits_0^u \phi (a){\rm e}^{{aR_{s} }} ds{\minus}\left( {{\rm e}^{{aR_{u} }} {\rm e}^{{\phi (a)(t{\minus}u)}} {\minus}{\rm e}^{{aR_{u} }} } \right)} \hfill \cr {} \hfill &#x0026; {\equals} \hfill &#x0026; \hskip-8pt{{\rm e}^{{aR_{u} }} {\minus}{\rm e}^{{ar}} {\minus}\mathop{\int}\limits_0^u \phi (a){\rm e}^{{aR_{s} }} ds\,{\equals}\,Y_{u} } \hfill \cr } $$

where the first equality comes from the fact that {R _t −R _u } is a Lévy process starting at 0 and the second equality comes from the stochastic Fubini theorem. Thus, there exists a Borel-measurable function Le ^ar =ϕ(a)e ^ar such that $${\rm e}^{{aR_{t} }} {\minus}{\rm e}^{{ar}} {\minus}\mathop{\int}\nolimits_0^t L{\rm e}^{{aR_{s} }} ds$$ is a martingale. By the definition for the extended infinitesimal generator, the function e ^ar is inside the domain of L. Hence e ^ar is an eigenfunction of L and ϕ(a) is its eigenvalue.

3.2. Fluctuation identities

In this section, X _t is a spectrally negative Lévy process satisfying the general setting of section 2.1. In the next two propositions, two important identities are stated for ease of reference; see Bertoin (Reference Bayraktar, Kyprianou and Yamazaki1998: 195), Avram et al. (Reference Bertoin2004, theorem 1), Kyprianou (Reference Monique and Shiryaev2006, theorem 8.1) for the proof.

Definition 3.1 The right inverse of Laplace exponent of X is denoted as

$${\rm \Phi }(q){\equals}{\rm sup}\left\{ {a\geq 0\,\colon\,\phi (a){\equals}q} \right\}\;\;\;{\rm for}\;{\rm each}\;\;\;q\geq 0$$

Definition 3.2 (q-scale function) For q≥0, a function W ^(q)(x) is called a q-scale function of X if W ^(q)(x)=0 when x<0 and W ^(q)(x) is a strictly increasing and continuous function in (0, ∞) with Laplace transform satisfying:

$$\mathop{\int}\limits_0^\infty {\rm e}^{{{\minus}\theta x}} W^{{(q)}} (x)dx\,{\equals}\,{1 \over {\phi (\theta ){\minus}q}}\;\;\;{\rm for}\;\;\;\theta \,\gt\,{\rm \Phi }(q)$$

Proposition 3.2 (Two-sided exit identity). Suppose that X _t is a spectrally negative Lévy process defined in the above with the initial state X ₀=x∈[0, a] and q>0. Consider the stopping times $$\tau _{0}^{{\minus}} {\equals}{\rm inf}\left\{ {t\geq 0{\rm \,\mid\,}X_{t} 0} \right\}$$ and $$\tau _{a}^{{\plus}} {\equals}{\rm inf}\left\{ {t\geq 0{\rm \mid}X_{t}&#x003E; a} \right\}$$ . Then

$$E_{x} \left[ {{\rm e}^{{{\minus}q\tau _{a}^{{\plus}} }} I_{{\left\{ {\tau _{0}^{{\minus}} \,\gt\,\tau _{a}^{{\plus}} } \right\}}} } \right]{\equals}{{W^{{(q)}} (x)} \over {W^{{(q)}} (a)}}$$

For the next proposition and the main theorem thereafter, we need to introduce constant barrier strategy and its related dividend process. Here is a simple definition for constant barrier strategy: a dividend strategy is called constant barrier strategy and denoted by π _a if any amount of X above a prefixed cash level a>0 is paid out as the dividends to its shareholders. Its related dividend process is denoted by $$\left\{ {L_{t}^{a} } \right\}_{{t\geq 0}} $$ . Moreover, the dividend process satisfies: $$L_{t}^{a} {\equals}(\overline{X} _{t} {\minus}a)\vee0$$ , where $$\overline{X} _{t} {\equals}{\rm sup}\left\{ {X_{s} {\rm \,\mid\,}0\leq s\leq t} \right\}$$ .

Proposition 3.3.

$$E_{x} \left[ {{\int}_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}qt}} dL_{t}^{a} } \right]\,{\equals}\,\left\{ {\matrix{ \hskip-28pt{{{W^{{(q)}} (x)} \over {W^{{(q)\prime}} (a)}},\;\,{\rm if}\;\,x\in[0,a].} \cr {x{\minus}a{\plus}{{W^{{(q)}} (a)} \over {W^{{(q)\prime}} (a)}},\;\,{\rm if}\;\,x\in(a,\infty)} \cr } } \right.$$

4. Main Theorem

In the previous section, we presented an eigenfunction and fluctuation identities for Lévy processes which are restricted by some specific conditions. But in this section, we will show the important role they play in the proof of the main result for this paper. In the following, ϕ _r and ϕ _x will be the Laplace exponents of R and X, respectively. Before the main theorem and its proof, by assuming that q=−ϕ _r (−1), a level c _* is introduced below:

$$c_{{\asterisk}} \,{\equals}\,{\rm sup}\left\{ {a\,\gt\,0{\rm \,\mid\,}W^{{(q)\prime}} (a)\leq W^{{(q)\prime}} (x),\;\,{\rm for}\;{\rm all}\;\,x\geq 0} \right\}$$

In the main theorem which is stated below, it is assumed that X is a Lévy process satisfying the setting specified in section 2.1 and R is a Lévy process under the conditions of Proposition 3.1 when a=−1.

Suppose that the underlying capital process is X and the accumulated interest rate process is R. Under the dividend strategy π _a , the expected discounted dividend payement will be denoted as: $$v_{{\pi _{a} }} (x,r){\equals}E_{{x,r}} \big[ {\mathop{\int}\nolimits_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}R_{s} }}{\!^\minus}dL_{s}^{a} } \big]$$ , where x, r are the initial states for X and R, respectively. In the previous sections, two extended generators A and L have been defined for processes X and R, respectively. Here we introduce another extended generator G for the bi-variate Markov process (X, R) for which A, L are marginal extended generators, respectively.

Proposition 4.1.

$$v_{{\pi _{a} }} (x,r)\,{\equals}\,E_{{x,r}} \left[ {{\int}_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}R_{s} }} dL_{s}^{a} } \right]{\equals}{\rm e}^{{{\minus}r}} v_{a} (x)$$

where $$v_{a} (x){\equals}E_{x} \left[ {{\int}_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}qs}} dL_{s}^{a} } \right]$$ for which q=−ϕ _r (−1).

Proof:

$$\eqalignno{ v_{{\pi _{a} }} (x,r)\,\,{\equals}\,\, &#x0026; E_{{x,r}} \left[ {{\int}_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}R_{s} }} dL_{s}^{a} } \right]\,{\equals}\,E_{x} \left[ {\mathop{\int}_{\rm \Omega } {\int}_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}R_{s} (\omega )}} dL_{s}^{a} {\Bbb P}_{r} (d\omega )} \right] \cr \,{\equals}\, &#x0026; E_{x} \left[ {{\int}_0^{\tau _{0}^{{\minus}} } \mathop{\int}_{\rm \Omega } {\rm e}^{{{\minus}(R_{s} {\minus}r){\minus}r}} {\Bbb P}_{r} (d\omega )dL_{s}^{a} } \right]\,{\equals}\,E_{x} \left[ {{\int}_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}r}} {\rm e}^{{\varphi_r} ({\minus}1)s}} dL_{s}^{a} } \right] \cr \,{\equals}\, &#x0026; {\rm e}^{{{\minus}r}} E_{x} \left[ {{\int}_0^{\tau _{0}^{{\minus}} } {\rm e}^{{\phi _{r} ({\minus}1)s}} dL_{s}^{a} } \right] $$

where the third equality holds by stochastic Fubini theorem since the ruin time $$\tau _{0}^{{\minus}} $$ does not depend on R _s under constant barrier strategy π _a , and the fourth equality holds by Proposition 3.1 and the fact that {R _s −r} _s≥0 is a Lévy process starting at 0.

Let q=−ϕ _r (−1). Then

$$v_{{\pi _{a} }} (x,r)\,{\equals}\,{\rm e}^{{{\minus}r}} E_{x} \left[ {{\int}_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}qs}} dL_{s}^{a} } \right]\,{\equals}\,{\rm e}^{{{\minus}r}} v_{a} (x)$$

$$v_{{\pi _{a} }} (x,r)\,{\equals}\,\left\{ {\matrix{ \hskip-42pt{{\rm e}^{{{\minus}r}} {{W^{{(q)}} (x)} \over {W^{{(q)\prime}} (a)}}\;\,{\rm if}\;\,x\in[0,a]} \cr {{\rm e}^{{{\minus}r}} \left( {x{\minus}a{\plus}{{W^{{(q)}} (a)} \over {W^{{(q)\prime}} (a)}}} \right)\;\:{\rm if}\;\,x\in(a,\infty)} \cr } } \right.$$

Proof of Lemma 4.1: By Proposition 4.1 and Proposition 3.3

$$v_{{\pi _{a} }} (x,r)\,{\equals}\,{\rm e}^{{{\minus}r}} E_{x} \left[ {{\int}_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}qs}} dL_{s}^{a} } \right]\,{\equals}\,\left\{ {\matrix{ \hskip-43pt{{\rm e}^{{{\minus}r}} {{W^{{(q)}} (x)} \over {W^{{(q)\prime}} (a)}},\,{\rm if}\;\,x\in[0,a]} \cr {{\rm e}^{{{\minus}r}} \left( {x{\minus}a{\plus}{{W^{{(q)}} (a)} \over {W^{{(q)\prime}} (a)}}} \right),\;\,{\rm if}\;\,x\in(a,\infty)} \cr } } \right.$$

$$v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)\geq v_{{\pi _{a} }} (x,r)$$

which means that the strategy $$\pi _{{c_{{\asterisk}} }} $$ dominates all the constant barrier strategies.

Proof of Proposition 4.2: We must show that $$v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)\geq v_{{\pi _{a} }} (x,r)$$ , for two cases a<c* and a>c _*. It suffices to show only one of them. Now suppose a<c _*.

For $$x\,\lt\,a,v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)\,{\equals}\,{\rm e}^{{{\minus}r}} {{W^{{(q)}} (x)} \over {W^{{(q)\prime}} (c_{{\asterisk}} )}}\geq {\rm e}^{{{\minus}r}} {{W^{{(q)}} (x)} \over {W^{{(q)\prime}} (a)}}\,{\equals}\,v_{{\pi _{a} }} (x,r)$$ by the definition of c _* and Lemma 4.1.

If a≤x≤c _*, by the definition of c _*,W ^(q)′(a)≥W ^(q)′(c _*). By Lemma 4.1

$$x{\minus}a{\plus}{{W^{{(q)}} (a)} \over {W^{{(q)\prime}} (a)}}\leq x{\minus}a{\plus}{{W^{{(q)}} (a)} \over {W^{{(q)\prime}} (c_{{\asterisk}} )}}\leq {{W^{{(q)}} (x)} \over {W^{{(q)\prime}} (c_{{\asterisk}} )}}$$

where the second inequality comes from the arguments used in proof of Proposition 3 in Avram et al. (Reference De Finetti2007). Therefore, $$v_{{\pi _{a} }} (x,r)\,{\equals}\,{\rm e}^{{{\minus}r}} \left( {x{\minus}a{\plus}{{W^{{(q)}} (a)} \over {W^{{(q)\prime}} (a)}}} \right)\leq {\rm e}^{{{\minus}r}} {{W^{{(q)}} (x)} \over {W^{{(q)\prime}} (c_{{\asterisk}} )}}\,{\equals}\,v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)$$ .

If x>c _*, similarly

$$\eqalignno{ v_{{\pi _{a} }} (x,r)\,{\equals}\, &#x0026; v_{{\pi _{a} }} (x,r)\,{\equals}\,{\rm e}^{{{\minus}r}} \left( {x{\minus}a{\plus}{{W^{{(q)}} (a)} \over {W^{{(q)\prime}} (a)}}} \right)\,{\equals}\,{\rm e}^{{{\minus}r}} \left( {x{\minus}c_{{\asterisk}} {\plus}c_{{\asterisk}} {\minus}a{\plus}{{W^{{(q)}} (a)} \over {W^{{(q)\prime}} (a)}}} \right) \cr &#x0026;\leq {\rm e}^{{{\minus}r}} \left( {x{\minus}c_{{\asterisk}} {\plus}c_{{\asterisk}} {\minus}a{\plus}{{W^{{(q)}} (a)} \over {W^{{(q)\prime}} (c_{{\asterisk}} )}}} \right)\leq {\rm e}^{{{\minus}r}} \left( {x{\minus}c_{{\asterisk}} {\plus}{{W^{{(q)}} (c_{{\asterisk}} )} \over {W^{{(q)\prime}} (c_{{\asterisk}} )}}} \right)\,{\equals}\,v_{{\pi _{{c_{{\asterisk}} }} }} (x,r) $$

Through these three cases under a<c _*, it shows that $$v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)\geq v_{{\pi _{a} }} (x,r)$$ . With similar discussions, we can show $$v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)\geq v_{{\pi _{a} }} (x,r)$$ when a>c _* .

(4.1) $${\rm max}\left\{ {Gw(x,r),{\rm e}^{{{\minus}r}} {\minus}w_{x} (x,r)} \right\}\,{\equals}\,0$$

then $$w(x,r)\geq {\rm sup}_{{\pi _{l} \in{\rm \Pi }_{{\leq l}} }} v_{{\pi _{l} }} (x,r)$$ , where Π_≤l is the collection of all dividend strategies that control the surplus process under level l.

Proof of Lemma 4.2: Suppose that π _l ∈Π_≤l and the controlled surplus process is defined by $$U_{t}^{{\pi _{l} }} \,{\equals}\,X_{t} {\minus}L_{t}^{{\pi _{l} }} $$ . Since $$w\in C^{{2,2}} ({\Bbb R}_{{\plus}} ,{\Bbb R})$$ , Itô’s Lemma can be applied to $$w\left( {U_{{t\wedge\tau _{0}^{{\minus}} }}^{{\pi _{l} }} ,\,R_{{t\wedge\tau _{0}^{{\minus}} }} } \right)$$ and

$$\eqalignno{ w\left( {U_{{t\wedge\tau _{0}^{{\minus}} }}^{{\pi _{l} }} ,\,R_{{t\wedge\tau _{0}^{{\minus}} }} } \right){\minus}w(x,r)\,{\equals}\, &#x0026; {\cal M}_{{t\wedge\tau _{0}^{{\minus}} }} {\minus}{\int}_0^{t\wedge\tau _{0}^{{\minus}} } w_{x} \left( {U_{s}^{{\pi _{l} }} ,\,R_{s} } \right)dL_{s}^{{\pi _{l} }} \cr&#x0026;{\plus}{\int}_0^{t\wedge\tau _{0}^{{\minus}} } Gw\left( {U_{s}^{{\pi _{l} }} ,\,R_{s} } \right)ds $$

where $$\tau _{0}^{{\minus}} $$ is the ruin time for $$U_{t}^{{\pi _{l} }} $$ and $${\cal M}$$ is a local martingale with $${\cal M}_{0} \,{\equals}\,0$$ .

Considering that we do not know about the boundedness of the integrals and the local martingale in the above equation, define a sequence of stopping times $$\left\{ {T_{n} ,\,n\in{\Bbb Z}^{{\plus}} } \right\}$$ of which for each n, T _n is the first time when absolute value of any of the terms of RHS of the above equation exceeds n. $$\left\{ {T_{n} ,\,n\in{\cal Z}^{{\plus}} } \right\}$$ is a localising sequence for $${\cal M}$$ , and T _n →∞ a.s. Then let t=T _n , the equation above is changed to

$$\eqalignno{ w\left( {U_{{T_{n} \wedge\tau _{0}^{{\minus}} }}^{{\pi _{l} }} ,\,R_{{T_{n} \wedge\tau _{0}^{{\minus}} }} } \right){\minus}w(x,r)\,{\equals}\, &#x0026; {\cal M} _{{T_{n} \wedge\tau _{0}^{{\minus}} }} {\minus}{\int}_0^{T_{n} \wedge\tau _{0}^{{\minus}} } w_{x} \left( {U_{s}^{{\pi _{l} }} ,\,R_{s} } \right)dL_{s}^{{\pi _{l} }} \cr&#x0026;{\plus}{\int}_0^{T_{n} \wedge\tau _{0}^{{\minus}} } Gw\left( {U_{x}^{{\pi _{l} }} ,\,R_{s} } \right)ds $$

Note that $${\cal M}_{{T_{n} \wedge\tau _{0}^{{\minus}} }} $$ is a local martingale. Taking the expectations on both sides and letting n→∞,

$$\eqalignno{ w\left( {U_{{\tau _{0}^{{\minus}} }}^{{\pi _{l} }} ,\,R_{{\tau _{0}^{{\minus}} }} } \right){\minus}w(x,r)\,{\equals}\, &#x0026; {\cal M}_{{\tau _{0}^{{\minus}} }} {\minus}{\int}_0^{\tau _{0}^{{\minus}} } w_{x} \left( {U_{s}^{{\pi _{l} }} ,\,R_{s} } \right)dL_{s}^{{\pi _{l} }} \cr&#x0026;{\plus}{\int}_0^{\tau _{0}^{{\minus}} } Gw\left( {U_{s}^{{\pi _{l} }} ,\,R_{s} } \right)ds $$

Since $$U_{{\tau _{0}^{{\minus}} }}^{{\pi _{l} }} \leq 0$$ , the first term on LHS will be 0. However, since the strategy π _l controls the process $$U_{t}^{{\pi _{l} }} $$ under the level l, in the two integrals on RHS, the function w will satisfy the variational inequality (4.1). Therefore, by the assumption in this lemma, $$Gw\left( {U_{s}^{{\pi _{l} }} ,\,R_{s} } \right)\leq 0$$ and $$w_{x} \left( {U_{s}^{{\pi _{l} }} ,\,R_{s} } \right)\geq {\rm e}^{{{\minus}R_{s} }} $$ for time $$s\in\left[ {0,\,\tau _{0}^{{\minus}} } \right]$$ . Thus after taking the expectations on both sides

$${\minus}w(x,r)\leq {\minus}E_{{x,r}} \left[ {{\int}_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}R_{{s^{{\minus}} }} }} dL_{s}^{{\pi _{l} }} } \right]\Rightarrow E_{{x,r}} \left[ {{\int}_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}R_{{s^{{\minus}} }} }} dL_{s}^{{\pi _{l} }} } \right]\leq w(x,r)$$

The last inequality is true ∀π _l ∈Π_≤l . Hence $${\rm sup}_{{\pi _{l} \in{\rm \Pi }_{{\leq l}} }} E_{{x,r}} \left[ {{\int}_0^{\tau _{0}^{{\minus}} } {\rm e}^{{{\minus}R_{{s^{{\minus}} }} }} dL_{s}^{{\pi _{l} }} } \right]\leq w(x,r)$$ .

In other words, $$w(x,r)\geq {\rm sup}_{{\pi _{l} \in{\rm \Pi }_{{\leq l}} }} v_{{\pi _{l} }} (x,r)$$ .

Next we will see a stopping time T _0,a , which is the first-exit time of an interval [0, a] defined as: T _0,a =inf{t≥0|X _t ∉[0, a]} for some constant a>0.

Proof of Lemma 4.3: Given that t>s,

$$\eqalignno{ E\left[ {\left. {{\rm e}^{{{\minus}R_{{t{\wedge} T{{0,a}} }} }} W^{{(q)}} \left( {U_{{t\wedge T{{0,a}} }}^{{\pi _{a} }} } \right)} \right|{\cal F}_{s} } \right]\, &#x0026; {\equals}\, I_{{\left\{ {s\,\lt\,T_{{0,a}} } \right\}}} E_{{X_{s} ,R_{s} }} \left[ {{\rm e}^{{{\minus}R_{{t\wedge T_{{0,a}} }} }} W^{{(q)}} \left( {U_{{t\wedge T_{{0,a}} }}^{{\pi _{a} }} } \right)} \right] \cr &#x0026; \;\,\;{\plus}I_{{\left\{ {s\geq T_{{0,a}} } \right\}}} E_{{X_{s} ,R_{s} }} \left[ {{\rm e}^{{{\minus}R_{{T_{{0,a}} }} }} W^{{(q)}} \left( {U_{{T_{{0,a}} }}^{{\pi _{a} }} } \right)} \right] \cr &#x0026; \,{\equals}\,I_{{\left\{ {s\,\lt\,T_{{0,a}} } \right\}}} {\rm e}^{{{\minus}R_{s} }} E_{{X_{s} }} \left[ {{\rm e}^{{{\minus}q(t^\wedge T_{{0,a}} {\minus}s)}} W^{{(q)}} \left( {U_{{t\wedge T_{{0,a}} }}^{{\pi _{a} }} } \right)} \right] \cr &#x0026; \;\,\;{\plus}I_{{\left\{ {s\geq T_{{0,a}} } \right\}}} {\rm e}^{{{\minus}R_{{T_{{0,a}} }} }} W^{{(q)}} \left( {U_{{T_{{0,a}} }}^{{\pi _{a} }} } \right) \cr &#x0026; \,{\equals}\,I_{{\left\{ {s\,\lt\,T_{{0,a}} } \right\}}} {\rm e}^{{{\minus}R_{s} }} E_{{X_{s} }} \left[ {{\rm e}^{{{\minus}q(T_{{0,a}} {\minus}s)}} W^{{(q)}} \left( {U_{{T_{{0,a}} }}^{{\pi _{a} }} } \right)} \right] $$

$$\eqalignno{ &#x0026;\quad{\plus}I_{{\left\{ {s\geq T_{{0,a}} } \right\}}} {\rm e}^{{{\minus}R_{{T_{{0,a}} }} }} W^{{(q)}} \left( {U_{{T_{{0,a}} }}^{{\pi _{a} }} } \right) \cr &#x0026; \,{\equals}\,I_{{\left\{ {s\,\lt\,T_{{0,a}} } \right\}}} {\rm e}^{{{\minus}R_{s} }} E_{{X_{s} }} \left[ {{\rm e}^{{{\minus}q(\tau _{a}^{{\plus}} {\minus}s)}} W^{{(q)}} (a)I_{{\left\{ {\tau _{0}^{{\minus}} \gt\, \tau _{a}^{{\plus}} } \right\}}} } \right] \cr &#x0026; \;\,\;{\plus}I_{{\left\{ {s\geq T_{{0,a}} } \right\}}} {\rm e}^{{{\minus}R_{{T_{{0,a}} }} }} W^{{(q)}} \left( {U_{{T_{{0,a}} }}^{{\pi _{a} }} } \right) \cr &#x0026; \,{\equals}\,I_{{\left\{ {s\,\lt\,T_{{0,a}} } \right\}}} {\rm e}^{{{\minus}R_{s} }} W^{{(q)}} (X_{s} ){\plus}I_{{\left\{ {s\geq T_{{0,a}} } \right\}}} {\rm e}^{{{\minus}R_{{T_{{0,a}} }} }} W^{{(q)}} \left( {U_{{T_{{0,a}} }}^{{\pi _{a} }} } \right) \cr &#x0026; \,{\equals}\,{\rm e}^{{{\minus}R_{{s\wedge T_{{0,a}} }} }} W^{{(q)}} \left( {U_{{s\wedge T_{{0,a}} }}^{{\pi _{a} }} } \right) $$

where the first equality holds since $$\left\{ {s \lt T_{{0,a}} } \right\}\in{\cal F}_{s} $$ . It can be seen that $${\rm e}^{{{\minus}q\left( {t\wedge T_{{0,a}} } \right)}} W^{{(q)}} \left( {U_{{t\wedge T_{{0,a}} }}^{{\pi _{a} }} } \right)$$ is a martingale by Avram et al. (Reference Bertoin2004, proposition 3):

$$\eqalignno{ E_{{X_{s} }} \left[ {\left. {{\rm e}^{{{\minus}q\left( {T_{{0,a}} {\minus}s} \right)}} W^{{(q)}} \left( {U_{{T_{{0,a}} }}^{{\pi _{a} }} } \right)} \right|{\cal F}_{t} } \right]\,{\equals}\, &#x0026; \mathop {lim}\limits_{m\to\infty} E_{{X_{s} }} \left[ {\left. {{\rm e}^{{{\minus}q(m\wedge T_{{0,a}} {\minus}s)}} W^{{(q)}} \left( {U_{{m\wedge T_{{0,a}} }}^{{\pi _{a} }} } \right)} \right|{\cal F}_{t} } \right] \cr \,{\equals}\, &#x0026; E_{{X_{s} }} \left[ {I_{{\left\{ {t\,\gt\,T_{{0,a}} } \right\}}} {\rm e}^{{{\minus}q\left( {T_{{0,a}} {\minus}s} \right)}} W^{{(q)}} \left( {U_{{T_{{0,a}} }}^{{\pi _{a} }} } \right){\plus}I_{{\left\{ {t\leq T_{{0,a}} } \right\}}} {\rm e}^{{{\minus}q(t{\minus}s)}} W^{{(q)}} \left( {U_{t}^{{\pi _{a} }} } \right)} \right] \cr \,{\equals}\, &#x0026; E_{{X_{s} }} \left[ {{\rm e}^{{{\minus}q(t\wedge T_{{0,a}} {\minus}s)}} W^{{(q)}} \left( {U_{{t\wedge T_{{0,a}} }}^{{\pi _{a} }} } \right)} \right] $$

The third equality follows from the above equality. The fourth equation is due to the decomposition of the expectation

$$E_{{X_{s} }} \left[ {{\rm e}^{{{\minus}q\left( {\tau _{a}^{{\plus}} {\minus}s} \right)}} W^{{(q)}} (a)} \right]\,{\equals}\,E_{{X_{s} }} \left[ {{\rm e}^{{{\minus}q\left( {T_{{0,a}} {\minus}s} \right)}} W^{{(q)}} \left( {U_{{T_{{0,a}} }}^{{\pi _{a} }} } \right)I_{{\left\{ {\tau _{0}^{{\minus}} \gt \tau _{a}^{{\plus}} } \right\}}} } \right]{\plus}E_{{X_{s} }} \left[ {{\rm e}^{{{\minus}q\left( {T_{{0,a}} {\minus}s} \right)}} W^{{(q)}} \left( {U_{{T_{{0,a}} }}^{{\pi _{a} }} } \right)I_{{\left\{ {\tau _{0}^{{\minus}} \leq \tau _{a}^{{\plus}} } \right\}}} } \right]$$

and the fact that W ^(q)(x)=0 for x<0. The fifth equality comes from Proposition 3.2.

Therefore it justifies that $$\left\{ {{\rm e}^{{{\minus}R_{{t\wedge T_{{0,a}} }} }} W^{{(q)}} \left( {U_{{t\wedge T_{{0,a}} }}^{{\pi _{a} }} } \right)} \right\}_{{t\geq 0}} $$ is a martingale.

Proof of Lemma 4.4: By Lemma 4.3, we know that $$\left\{ {{\rm e}^{{{\minus}R_{{t\wedge T_{{0,c_{{\asterisk}} }} }} }} W^{{(q)}} \left( {X_{{t\wedge T_{{0,c_{{\asterisk}} }} }} } \right)} \right\}_{{t\geq 0}} $$ is a martingale, i.e., $$E_{{x,r}} \left[ {{\rm e}^{{{\minus}R_{{t\wedge T_{{0,c_{{\asterisk}} }} }} }} W^{{(q)}} \left( {X_{{t\wedge T_{{0,c_{{\asterisk}} }} }} } \right)} \right]\,{\equals}\,{\rm e}^{{{\minus}r}} W^{{(q)}} (x)$$ . Let $$\tilde{X}\,{\equals}\,\left\{ {X_{{t\wedge T_{{0,c_{{\asterisk}} }} }} } \right\}_{{t\geq 0}} $$ . Then,

$$\eqalignno{ 0\,{\equals}\, &#x0026; \mathop {lim}\limits_{t\to0^{{\plus}} } {{E_{{x,r}} \left[ {{\rm e}^{{{\minus}R_{{t\wedge T_{{0,c_{{\asterisk}} }} }} }} W^{{(q)}} \left( {X_{{t\wedge T_{{0,c_{{\asterisk}} }} }} } \right)} \right]{\minus}{\rm e}^{{{\minus}r}} W^{{(q)}} (x)} \over t} \cr \,{\equals}\, &#x0026; \mathop {lim}\limits_{t\to0^{{\plus}} } {{E_{{x,r}} \left[ {{\rm e}^{{{\minus}R_{t} }} W^{{(q)}} (X_{t} )} \right]{\minus}{\rm e}^{{{\minus}r}} W^{{(q)}} (x)} \over t} \cr \,{\equals}\, &#x0026; \mathop {lim}\limits_{t\to0^{{\plus}} } {{E_{x} \left[ {{\rm e}^{{{\minus}r{\plus}\phi _{r} ({\minus}1)t}} W^{{(q)}} (X_{t} )} \right]{\minus}{\rm e}^{{{\minus}r}} W^{{(q)}} (x)} \over t} $$

$$\eqalignno{ {\equals}\, &#x0026; \mathop {lim}\limits_{t\to0^{{\plus}} } {{{\rm e}^{{{\minus}r}} \left( {{\rm e}^{{\phi _{r} ({\minus}1)t}} E_{x} \left[ {W^{{(q)}} (X_{t} )} \right]{\minus}E_{x} \left[ {W^{{(q)}} (X_{t} )} \right]{\plus}E_{x} \left[ {W^{{(q)}} (X_{t} )} \right]{\minus}W^{{(q)}} (x)} \right)} \over t} \cr \,{\equals}\, &#x0026; {\rm e}^{{{\minus}r}} \left( {\phi _{r} ({\minus}1)W^{{(q)}} (x){\plus}AW^{{(q)}} (x)} \right) $$

On the other hand, by Lemma 4.1,

$$v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)\,{\equals}\,{\rm e}^{{{\minus}r}} {{W^{{(q)}} (x)} \over {W^{{(q)\prime}} (c_{{\asterisk}} )}}$$

for x∈(0, c _*). In the following, denote $$\left\{ {U_{{t\wedge T_{{0,c_{{\asterisk}} }} }}^{{\asterisk}} } \right\}_{{t\geq 0}} $$ and $$\left\{ {R_{{t\wedge T_{{0,c_{{\asterisk}} }} }} } \right\}_{{t\geq 0}} $$ by $$\tilde{U}^{{\asterisk}} $$ and $$\tilde{r}$$ , respectively. Let the initial state for $$U_{t}^{{\asterisk}} $$ be x∈(0,c _*). Observe that starting from x∈(0,c _*), $$\tilde{X}\,{\equals}\,\tilde{U}^{{\asterisk}} $$ , i.e., $$X_{{t\wedge T_{{0,c_{{\asterisk}} }} }} \,{\equals}\,U_{{t\wedge T_{{0,c_{{\asterisk}} }} }}^{{\asterisk}} $$ , before the stopping time $$T_{{0,c_{{\asterisk}} }} $$ . Suppose that the generator for $$\tilde{U}^{{\asterisk}} $$ is denoted by J. Then $$(J{\plus}L)v_{{c_{{\asterisk}} }} \left( {U_{0}^{{\asterisk}} ,r} \right)\,{\equals}\,Gv_{{c_{{\asterisk}} }} (X_{0} ,r)\,{\equals}\,Gv_{{c_{{\asterisk}} }} (x,r)$$ which implies that the generators for $$v_{{c_{{\asterisk}} }} \left( {\tilde{U}^{{\asterisk}} ,\tilde{r}} \right)$$ and $$v_{{c_{{\asterisk}} }} (\tilde{X},\tilde{r})$$ are the same around x∈(0, c _*):

$$\eqalignno{ Gv_{{\pi _{{c_{{\asterisk}} }} }} (x,r)\,{\equals}\, &#x0026; G{\rm e}^{{{\minus}r}} {{W^{{(q)}} (x)} \over {W^{{(q)}} (c_{{\asterisk}} )}} \cr \,{\equals}\, &#x0026; {{{\rm e}^{{{\minus}r}} AW^{{(q)}} (x){\plus}W^{{(q)}} (x)L{\rm e}^{r} } \over {W^{{(q)}} (c_{{\asterisk}} )}} \cr \,{\equals}\, &#x0026; {{{\rm e}^{{{\minus}r}} AW^{{(q)}} (x){\plus}W^{{(q)}} (x){\rm e}^{{{\minus}r}} \phi _{r} ({\minus}1)} \over {W^{{(q)}} (c_{{\asterisk}} )}}\,{\equals}\,0 $$

where the last equality but one comes from Proposition 3.1. The last equality follows simply from the previous result.

Proof of main theorem 4.1: Since it is assumed that q=−ϕ(−1)>0, c _*<∞ by Lemma 2 in Avram et al. (Reference De Finetti2007). We need to prove the claim that $$v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)$$ is the solution of the variational inequality (4.1) for x∈(0,∞). By Lemma 4.4 above, it shows that $$v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)$$ is the solution for x∈(0,c _*). However observe that for $$x\in(c_{{\asterisk}} ,\infty),\,v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)\,{\equals}\,{\rm e}^{{{\minus}r}} \left( {x{\minus}c_{{\asterisk}} {\plus}{{W^{{(q)}} (c_{{\asterisk}} )} \over {W^{{(q)\prime}} (c_{{\asterisk}} )}}} \right)$$ . Then $${\partial \over {\partial x}}v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)\,{\equals}\,{\rm e}^{{{\minus}r}} ,\,\forall \,x\in(c_{{\asterisk}} ,\infty)$$ . In other words $$v_{{\pi _{{c_{{{\asterisk}_{x} }} }} }} (x,r)\,{\equals}\,{\rm e}^{{{\minus}r}} ,\,\forall \,x\in(c_{{\asterisk}} ,\infty)$$ . On the other hand, by the assumption of main Theorem 4.1, $$Gv_{{\pi _{{c_{{\asterisk}} }} }} (x,r)\leq 0\,{\rm for}\,x\in(c_{{\asterisk}} ,\infty)$$ . Thus, the claim that $$v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)$$ is the solution of variational inequality (4.1) for x∈(0, ∞) is justified. By Verification Lemma 4.2, $$v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)\geq {\rm sup}_{{\pi \in\Pi _{{\leq \infty}} }} v_{\pi } (x,r)$$ . By the definition of Π_≤∞, it can be seen that the optimal strategy $$\pi _{{\asterisk}} \in{\rm \Pi }_{{\leq \infty}} ,\,{\rm i}.{\rm e}.,\,v_{{\asterisk}} (x,r)\,{\equals}\,{\rm sup}_{{\pi \in\Pi _{{\leq \infty}} }} v_{\pi } (x,r)$$ . So we have $$v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)\geq v_{{\asterisk}} (x,r)$$ . By the definition of value function, $$v_{{\asterisk}} (x,r)\geq v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)$$ . In the end $$v_{{\asterisk}} (x,r)\,{\equals}\,v_{{\pi _{{c_{{\asterisk}} }} }} (x,r)$$ .