On the optimality of double barrier strategies for L\'evy processes

TL;DR

This paper proves the optimality of double barrier strategies for dividend problems in Le9vy risk models with both positive and negative jumps, extending previous spectrally one-sided results.

Contribution

It generalizes existing theorems to Le9vy processes with two-sided jumps, demonstrating the optimality of double barrier strategies beyond spectrally one-sided cases.

Findings

Double barrier strategies are optimal for Le9vy processes with two-sided jumps.

Scale functions are insufficient for analyzing these strategies in the two-sided case.

Sample path analysis is used to establish properties of expected NPVs.

Abstract

This paper studies de Finetti's optimal dividend problem with capital injection. We confirm the optimality of a double barrier strategy when the underlying risk model follows a L\'evy process that may have positive and negative jumps. The main result in this paper is a generalization of Theorem 3 in Avram et al.(2007), which is the spectrally negative case, and Theorem 3.1 in Bayraktar et al.(2013), which is the spectrally positive case. In contrast with the spectrally one-sided cases, double barrier strategies cannot be handled by using scale functions to obtain some properties of the expected net present values (NPVs) of dividends and capital injections. Instead, to obtain these properties, we observe changes in the sample path (and the associated NPV) when there is a slight change to the initial value or the barrier value.

Equations

$$e^{-t\Psi (\lambda )}={\mathbb{E}}_0\left[e^{i\lambda X_t}\right],\lambda \in \mathbb{R},t\ge 0.$$

$$\Psi (\lambda )=-i\gamma \lambda +\frac{1}{2}{\sigma }^2{\lambda }^2+{\int }_{\mathbb{R}}(1-e^{i\lambda x}+i\lambda x{1}_{\{|x|<1\}})\Pi (dx),\lambda \in \mathbb{R}.$$

$${\int }_{\mathbb{R}\backslash \{0\}}(1\wedge x^2)\Pi (dx).$$

$$\Psi (\lambda )=-i\delta +{\int }_{\mathbb{R}}(1-e^{i\lambda x})\Pi (dx),$$

$$\delta =\gamma -{\int }_{|x|<1}x\Pi (dx).$$

$${\tau }_x^{+}=\inf \{t>0:X_t>x\}\text{and}{\tau }_x^{-}=\inf \{t>0:X_t<x\}.$$

$${\overline{\phi }}_{a,0}={\mathbb{E}}_x\left[e^{-q{\tau }_a^{+}};{\tau }_a^{+}<{\tau }_0^{-}\right],{\underline{\phi }}_{0,a}(x)={\mathbb{E}}_x\left[e^{-q{\tau }_0^{-}};{\tau }_0^{-}<{\tau }_a^{+}\right].$$

$$Y_t^a=X_t-\left((\underset{s\in [0,t]}{\sup }{X}_s-a)\vee 0\right),t\ge 0.$$

$${\kappa }_x^{a,-}=\inf \{t>0:Y_t^a<x\}.$$

$${\mathbb{E}}_0\left[\left|X_1\right|\right]<\infty .$$

$${\int }_{|x|\ge 1}\left|x\right|\Pi (dx)<\infty .$$

$$E_a^{(1)}=\{x\in [0,a]:{\overline{\phi }}_{a,0}\text{ or }{\underline{\phi }}_{0,a}\text{ are discontinuous at }x\}.$$

$$E_a^{(2)}=\{x\in (0,a):{\overline{\phi }}_{a,0}^{\prime }\text{ or }{\underline{\phi }}_{0,a}^{\prime }\text{ are discontinuous at }x\}.$$

$$\left|f(x)\right|<b_1\left|x\right|+b_2,x\in \mathbb{R},$$

$$\mathcal{L}f(x)=\gamma f^{\prime }(x)+\frac{1}{2}{\sigma }^2{f}^{\prime \prime }(x)+{\int }_{\mathbb{R}\backslash \{0\}}(f(x+z)-f(x)-f^{\prime }(x)z{1}_{\{|z|<1\}})\Pi (dz),x\in (0,\infty ),$$

$${\int }_{\mathbb{R}\backslash \{0\}}\left|f(x+z)-f(x)-f^{\prime }(x)z{1}_{\{|z|<1\}}\right|\Pi (dz)$$

$$\le {\int }_{(-\infty ,-1\vee (\frac{-x}{2})]\cup [1,\infty )}\left|f(x+z)-f(x)-f^{\prime }(x)z{1}_{\{|z|<1\}}\right|\Pi (dz)$$

$$+{\int }_{(-1\vee (\frac{-x}{2}),1)\backslash \{0\}}\left|f(x+z)-f(x)-f^{\prime }(x)z\right|\Pi (dz).$$

$$\text{eqref}214=$$

$$\text{eqref}216\le 2b_3{\int }_{(-1\vee (\frac{-x}{2}),1)\backslash \{0\}}\left|z\right|\Pi (dz),$$

$$\left|f^{\prime \prime }(y)\right|<b_4,y\in (x+(-1\vee (\frac{-x}{2})),x+1)$$

$$\text{eqref}216=$$

$$\le$$

$$=$$

$$x$$

$$\displaystyle\bigg{(}\text{resp., }x$$

$$U_t^{\pi }=X_t-L_t^{\pi }+R_t^{\pi },t\ge 0.$$

$$R_t^{\pi }\ge -(X_t-L_t^{\pi }),t\ge 0.$$

$$v_{\pi }^L(x)={\mathbb{E}}_x\left[{\int }_{[0,\infty )}{e}^{-qt}d{L}_t^{\pi }\right],v_{\pi }^R(x)={\mathbb{E}}_x\left[{\int }_{[0,\infty )}{e}^{-qt}d{R}_t^{\pi }\right].$$

$$v_{\pi }(x)=v_{\pi }^L(x)-\beta v_{\pi }^R(x),x\in \mathbb{R}$$

$$v(x):=\underset{\pi \in \mathcal{A}}{\sup }{v}_{\pi }(x)=v_{{\pi }^{\ast }}(x),x\in \mathbb{R}.$$

$$L_t^{\prime }=\eta -a+((X_t-X_{T_a})\vee 0),U_t^{\prime }=a+X_t-X_{T_a}-((X_t-X_{T_a})\vee 0).$$

$$R_t^{\prime }=-\eta -((X_t-X_{T_0})\wedge 0),U_t^{\prime }=X_t-X_{T_0}-((X_t-X_{T_0})\wedge 0).$$

Full text

1 Introduction

In this paper, we study the optimal dividend problem with capital injection. A Lévy process $X$ describes the risk process of an insurance company. The company pays dividends from the risk process $X$ and capital is injected into $X$ to avoid ruin. This paper aims to show a joint strategy for dividend payout and capital injection that maximizes the total expected dividend payments minus the cost of capital injections.

Here, we focus on the double barrier strategies. The double barrier strategy at $a\geq 0$ is the strategy in which (a) when the risk process exceeds the barrier $a$, the company pays a dividend determined by the excess over $a$, and (b) when the risk process falls below [math], capital is injected accordingly to avoid ruin. The controlled process that results from applying a double barrier strategy behaves as a doubly reflected Lévy process.

In previous studies of de Finetti’s optimal dividend problem for spectrally negative Lévy processes, the optimality of double barrier strategies has been proven. For example, Avram et al.[2] proved optimality for general spectrally negative Lévy processes. Furthermore, in expanded situations that deal with two-sided singular control problems, Baurdoux–Yamazaki[3] and Yamazaki[17] proved optimality for general spectrally negative Lévy processes. As with the spectrally negative case, some previous studies have considered spectrally positive Lévy processes. Avanzi et al.[1] proved the optimality of a double barrier strategy for spectrally positive compound Poisson processes. Bayraktar et al.[4] generalized the result to general spectrally positive Lévy processes. In addition, many papers have considered other optimal dividend problems with capital injections for spectrally one-sided Lévy processes ([22], [15], [23], [21], [13], [14], [12], [7], [10]).

Recently, de Finetti’s optimal dividend problem for Lévy processes with two-sided jumps has been studied. Some previous studies have considered de Finetti’s optimal dividend problem without bail-outs. In particular, double or mixed-exponential jump diffusion processes have been discussed. For example, Bo et al.[6] computed the expected net present values (NPVs) of dividends of barrier strategies and gave numerical results for double exponential jump diffusion processes. Yin et al.[18] computed the expected NPVs of dividends of barrier strategies for mixed-exponential jump diffusion processes. Yuen–Yin[20] and Yin et al.[19] claim to have proven the optimality of the barrier strategy for more general Lévy processes with two-sided jumps, but their proofs seem to have some flaws. In addition to these studies, Li et al.[11] gave some computational results that seem to provide the expected NPVs of dividends and capital injections of double barrier strategies for double-exponential jump diffusion processes. Overall, though, no existing paper seems to prove the optimality of any strategy.

The objective of this paper is to show the optimality of a double barrier strategy for Lévy processes that may have two-sided jumps. The class of Lévy processes that we consider contains Lévy processes with bounded variation paths and positive drifts, mixed-exponential jump diffusion processes, spectrally one-sided Lévy processes, and others.

Our proof the optimality of a double barrier strategy has two broad steps.

1. (i)

We select a candidate barrier $a^{\ast}\geq 0$ for double barrier strategies. In this step, we compute the derivative of $v_{\pi^{a}}(x)$ for $a$, where $v_{\pi^{a}}(x)$ is the expected NPV of dividends and capital injections of the double barrier strategy at $a\geq 0$ when the risk process $X$ starts from $x\in\mathbb{R}$ (see Sections 4 and A). 2. (ii)

To prove optimality for the chosen case, we apply a verification lemma to $v_{\pi^{a^{\ast}}}(x)$ as done in [2]. Here, we need to find and use some properties of the derivative of $v_{\pi^{a}}(x)$ with respect to $x$ (see Sections 5, B and C).

A difficulty with this approach is how to obtain some properties of the derivative of the expected NPV of dividends and capital injections of double barrier strategies. In the case of spectrally one-sided Lévy processes, we can represent the expected NPVs of dividends and capital injections of double barrier strategies by using scale functions, as done for [2, (5.4)] and [4, (3.1)]. Since we know many properties of scale functions (see, e.g., [8] or [9, Section 8]), we can obtain some properties of the derivative of each expected NPV. On the other hand, Lévy processes that have two-sided jumps do not have scale functions associated with them. In the case of mixed-exponential jump diffusion processes, I predict we can represent the expected NPVs as the sum of exponential functions in the same way as done in [6] and [18]. However, I expect that the forms of the expected NPVs found in this way will be complicated to analyze. In addition, the expected NPVs cannot be expressed using this approach for more general Levy processes. So, we need to consider a new way to obtain some properties of the derivatives of the expected NPVs. In this paper, we obtain the derivatives of the expected NPVs by observing how the behavior of the sample path changes when either the initial value of the sample path or the value of the barrier is slightly shifted. Specifically, we represent the derivatives of the expected NPVs using the Laplace transforms of hitting times.

This paper is organized as follows. In Section 2, we describe the notation and give some assumptions about Lévy processes. In addition, we give the setting of the optimal dividend problem. In Section 3, we give an overview of the double barrier strategies and confirm that they are admissible. In Section 4, we select the candidate barrier $a^{\ast}$. In Section 5, we prove the optimality of the double barrier strategy for $a^{\ast}$, using the verification lemma. The main result is in this section. In Section 6, we give examples of Lévy processes with unbounded variation paths that satisfy the assumptions given in Section 2. In Section A, we consider the behavior of doubly reflected Lévy processes to compute the derivative of $v_{\pi^{a}}(x)$ with respect to $a$ and select the candidate barrier $a^{\ast}$. In Section B, we give the proof of the verification lemma. In Section C, we consider the behavior of doubly reflected Lévy processes to compute the derivative of $v_{\pi^{a}}(x)$ with respect to $x$.

2 Preliminalies

2.1 Lévy processes

In this section, we describe our notation and give some assumptions about the Lévy processes considered in this paper.

Let $X=\{X_{t}:t\geq 0\}$ be a Lévy process defined on a probability space $(\Omega,\mathbb{F},\mathbb{P})$. For $x\in\mathbb{R}$, we denote by $\mathbb{P}_{x}$ the law of $X$ when it starts at $x$. Let $\Psi$ be the characteristic exponent of $X$ that satisfies

[TABLE]

The characteristic exponent $\Psi$ is known to take the form

[TABLE]

Here, $\gamma\in\mathbb{R}$, $\sigma\geq 0$, and $\Pi$ is a Lévy measure on $\mathbb{R}\backslash\{0\}$ such that

[TABLE]

The process $X$ has bounded variation paths if and only if $\sigma=0$ and $\int_{\left|x\right|<1}\left|x\right|\Pi(dx)<\infty$. When this holds, we can write

[TABLE]

where

[TABLE]

Let $\mathcal{F}=\{\mathcal{F}_{t}:t\geq 0\}$ be the filtration generated by $X$. For $x\in\mathbb{R}$, we write

[TABLE]

We fix the discount factor $q>0$. For $a>0$ and $x\in\mathbb{R}$, we write

[TABLE]

For $a\in\mathbb{R}$, let $Y^{a}$ be a reflected process defined by

[TABLE]

For $x\in\mathbb{R}$, we write

[TABLE]

We impose the following assumptions on $X$.

Assumption 2.1**.**

We assume that $X$ does not have monotone paths, and $X$ satisfies

[TABLE]

By [9, Theorem 3.8], the condition (2.10) holds if and only if

[TABLE]

If the process $X$ has unbounded variation paths, then we assume the maps $\overline{\varphi}_{a,0}$ and $\underline{\varphi}_{0,a}$ have Radon–Nikodym densities $\overline{\varphi}_{a,0}^{\prime}$ and $\underline{\varphi}_{0,a}^{\prime}$ with respect to the Lebesgue measure, which is continuous almost everywhere and locally bounded on $(0,a)$.

Remark 2.2**.**

Because the maps $\overline{\varphi}_{a,0}$ and $\underline{\varphi}_{0,a}$ are monotone functions, these are continuous on $[0,a]$ almost everywhere with respect to the Lebesgue measure. If the process $X$ has unbounded variation paths, it is easy to check that the maps $\overline{\varphi}_{a,0}$ and $\underline{\varphi}_{0,a}$ are continuous on $[0,a]$.

Remark 2.3**.**

In fact, the maps $\overline{\varphi}_{a,0}$ and $\underline{\varphi}_{0,a}$ are continuous on $[0,a]$ if $X$ is not a compound Poisson process. However, we do not give the proof of the fact since the fact is not important in this paper.

For $a>0$, we define

[TABLE]

Note that $E^{(1)}_{a}$ is the null set. In addition, $E^{(1)}_{a}=\emptyset$ when $X$ has unbounded variation paths.

When $X$ has unbounded variation paths, we define

[TABLE]

Note that $E^{(2)}_{a}$ is the null set.

We define sets of functions $C^{(1)}_{\text{line}}$ and $C^{(2)}_{\text{line}}$. Let $C^{(1)}_{\text{line}}$ be the set of function $f\in C(\mathbb{R})$ such that $f(y)-f(x)=\int_{x}^{y}f^{\prime}(u)du$ on $(0,\infty)$ for some locally bounded function $f^{\prime}$ on $(0,\infty)$ and such that $f$ satisfies

[TABLE]

for some $b_{1},b_{2}>0$. Let $C^{(2)}_{\text{line}}$ be the set of function $f$ in $C^{(1)}_{\text{line}}$ such that $f$ is continuously differentiable on $(0,\infty)$, and $f^{\prime}(y)-f^{\prime}(x)=\int_{x}^{y}f^{\prime\prime}(u)du$ on $(0,\infty)$ for some locally bounded function $f^{\prime\prime}$ on $(0,\infty)$. Let $\mathcal{L}$ be the operator applied to $f\in C^{(1)}_{\text{line}}$ (resp., $C^{(2)}_{\text{line}}$) for the case in which $X$ is of bounded (resp., unbounded) variation with

[TABLE]

for a fixed $f^{\prime}$ (resp., $f^{\prime\prime}$).

Remark 2.4**.**

The integral in (2.15) is well defined. We prove this fact here. We have

[TABLE]

By (2.11), (2.14), and the definition of $\Pi$, (2.17) is finite. We have

[TABLE]

Because $f^{\prime}$ is locally bounded, we have $\left|f^{\prime}(x+\cdot)\right|\leq b_{3}$ on $[-1\lor(\frac{-x}{2}),1]$ for some $b_{3}>0$, we have

[TABLE]

and so (2.19) is finite when $X$ has bounded variation paths. When $X$ has unbounded variation paths, because $f^{\prime\prime}$ is locally bounded, we have

[TABLE]

for some $b_{4}>0$. So, we have

[TABLE]

The proof is now complete.

Remark 2.5**.**

By the proof in Remark 2.4, and the dominated convergence theorem, it is easy to verify that the map

[TABLE]

is continuous on $(0,\infty)$ when $X$ has bounded (resp., unbounded) variation paths.

2.2 The optimal dividend problem with capital injection

In this paper, a strategy is a pair of processes $\pi=\{(L^{\pi}_{t},R^{\pi}_{t}):t\geq 0\}$ consisting of the cumulative amount of dividends $L^{\pi}$ and the cumulative amount of capital injection $R^{\pi}$. The corresponding risk process is given by $U^{\pi}_{0-}=X_{0}$, and

[TABLE]

For the dividend strategy, we assume that $L^{\pi}$ is a non-decreasing, right-continuous, and $\mathcal{F}$-adapted process with $L^{\pi}_{0-}=0$.

For the capital injection strategy, we assume that $R^{\pi}$ is a non-decreasing, right-continuous, and $\mathcal{F}$-adapted process with $R^{\pi}_{0-}=0$, and

[TABLE]

The condition (2.28) implies that $U^{\pi}$ never hits $(-\infty,0)$.

For $x\in\mathbb{R}$, we write

[TABLE]

Let $\beta>1$ be the cost per unit of injected capital. The objective is to maximize the expected NPV

[TABLE]

over the set of all admissible strategies $\mathcal{A}$ that satisfy all the constraints described above as well as $v^{R}_{\pi}(x)<\infty$ for $x\in\mathbb{R}$. Hence, the problem is to obtain an optimal strategy $\pi^{\ast}$ satisfying

[TABLE]

3 Double barrier strategies

The objective of this paper is to show the optimality of the double barrier strategy. In this section, we recall details of double barrier strategies, which are constructed in [2, Section 4].

Let $a>0$. The double barrier strategy $\pi^{a}$ at $a$ is the strategy constructed as follows.

• Step 0

Set $T_{0}=\tau^{-}_{0}$, $T_{a}=\tau^{+}_{a}$, and $\eta=X_{T_{0}\land T_{a}}$. For $t<T_{0}\land T_{a}$, set $L^{\pi^{a}}_{t}=R^{\pi^{a}}_{t}=0$ and $U^{\pi^{a}}_{t}=X_{t}$. If $T_{0}<T_{a}$, go to Step 2. If $T_{0}>T_{a}$, go to Step 1.

• Step 1

For $t\geq T_{a}$, we set

[TABLE]

We reset $T_{0}=\inf\{t>0:U^{\prime}_{t}<0\}$ and $\eta=U^{\prime}_{T_{0}}$. For $t\in[T_{a},T_{0})$, we set $U^{\pi^{a}}_{t}=U^{\prime}_{t}$, $L^{\pi^{a}}_{t}=L^{\pi^{a}}_{T_{a}-}+L^{\prime}_{t}$, and $R^{\pi^{a}}_{t}=R^{\pi^{a}}_{T_{a}-}$. Go to Step 2.

• Step 2

For $t\geq T_{0}$, we set

[TABLE]

We reset $T_{a}=\inf\{t>0:U^{\prime}_{t}>a\}$ and $\eta=U^{\prime}_{T_{a}}$. For $t\in[T_{0},T_{a})$, we set $U^{\pi^{a}}_{t}=U^{\prime}_{t}$, $L^{\pi^{a}}_{t}=L^{\pi^{a}}_{T_{0}-}$, and $R^{\pi^{a}}_{t}=R^{\pi^{a}}_{T_{0}-}+R^{\prime}_{t}$. Go to Step 1.

When $X$ has bounded variation paths, we can construct the double barrier strategy $\pi^{0}$ at [math] as follows. For $t\geq 0$,

[TABLE]

Remark 3.1**.**

The doubly reflected processes are standard processes.

We prove the admissibility of the double barrier strategies in the following lemmas:

Lemma 3.2**.**

We have, for $a>0$ and $x\in\mathbb{R}$,

[TABLE]

Proof.

The proof of $v^{L}_{\pi^{a}}(x)<\infty$ is the same as the proof of $v^{R}_{\pi^{a}}(x)<\infty$, so we prove only $v^{R}_{\pi^{a}}(x)<\infty$.

By the definition of $\pi^{a}$, we have

[TABLE]

Thus it is sufficient to prove that

[TABLE]

We write $R_{t}=-\!\left((\inf_{s\in[0,t]}X_{s})\land 0\right)$ and fix $u>0$. For $n\in\mathbb{N}$, we define

[TABLE]

where $\mu^{0}_{a}=0$. By the strong Markov property, we have

[TABLE]

and by the definition of $\mu^{[1]}_{a}$, we have

[TABLE]

By the Lévy–Itô decomposition, there exists some $\delta^{X}\in\mathbb{R}$, a zero-mean square-integrable martingale $M^{X}$ starting from [math], a Poisson process $N^{X}$, and a sequence of independent and identically distributed (i.i.d.) random variables ${\{J^{X}_{n}\}}_{n\in\mathbb{N}}$ taking values in $(-\infty,-1)\cup(1,\infty)$ such that

[TABLE]

By Doob’s maximal inequality, we have

[TABLE]

By the compensation theorem of the Poisson point processes and (2.11), we have

[TABLE]

where $\mathcal{N}$ is a Poisson random measure on $([0,\infty)\times\mathbb{R},\mathcal{B}[0,\infty)\times\mathcal{B}(\mathbb{R}),ds\times\Pi(dx))$ associated with the jumps of $X$. By (3.15), (3.16), and (3.19), we have

[TABLE]

By (3.14) and (3.20), we obtain (3.8), and the proof is complete. ∎

Lemma 3.3**.**

If $X$ has bounded variation paths, then for $x\in\mathbb{R}$ we have

[TABLE]

Proof.

By the same argument as that in the proof of Lemma 3.2, it is sufficient to prove that $v^{R}_{\pi^{0}}(0)<\infty$. By the definition of $\pi^{0}$, we have

[TABLE]

By the compensation theorem of the Poisson point processes, we have

[TABLE]

Because $X$ has bounded variation paths and (2.11) holds, we have $\eqref{325}<\infty$. The proof is complete. ∎

Remark 3.4**.**

We note that the assumption (2.10) is necessary to prove the optimality of a double barrier strategy. We show this briefly here. Suppose $X$ has bounded (resp., unbounded) variation paths. If $\mathbb{E}_{0}\!\left[X_{1}\land 0\right]=-\infty$, then we can prove that $v^{R}_{\pi^{a}}(x)=\infty$ for $a\in[0,\infty)$ (resp., $a\in(0,\infty)$) and $x\in\mathbb{R}$, and so no double barrier strategies are admissible. On the other hand, if $\mathbb{E}_{0}\!\left[X_{1}\lor 0\right]=\infty$, then we can prove that $v^{L}_{\pi^{a}}(x)=\infty$ for $a\in[0,\infty)$ and $x\in\mathbb{R}$, and so this problem does not make sense.

4 Selection of the candidate barrier $a^{\ast}$

In this section, we focus on the double barrier strategy defined in the previous section and choose a candidate barrier $a^{\ast}$.

We define

[TABLE]

where $\underline{\nu}(a)=\mathbb{E}_{a}\!\left[e^{-q\kappa^{a,-}_{0}}\right]$. Since $X$ has stationary independent increments, the map $a\mapsto\underline{\nu}(a)$ is non-increasing, and we have $\lim_{a\uparrow\infty}\underline{\nu}(a)=0$, so $a^{\ast}<\infty$.

Lemma 4.1**.**

For $a,x\in[0,\infty)$, we have

[TABLE]

We postpone the proof of Lemma 4.1 until after the proof of Lemma 4.3.

For $n\in\mathbb{N}$, we define the hitting times for $n\in\mathbb{N}$ by induction as follows. For $n\in\mathbb{N}$,

[TABLE]

where $\underline{\rho}^{a,(0)}_{0}=0$. For simplicity, we write $\overline{\nu}_{x}(a)=\mathbb{E}_{x}\!\left[e^{-q\overline{\rho}^{a,[1]}_{a}}\right]$. To compute the derivative of $v_{\pi^{a}}(x)$ with respect to $a$, we write $V_{x}(a):=v_{\pi^{a}}(x)$, $V^{L}_{x}(a):=v^{L}_{\pi^{a}}(x)$ and $V^{R}_{x}(a):=v^{R}_{\pi^{a}}(x)$.

Lemma 4.2**.**

For $x\in\mathbb{R}$ and $a\in(0,\infty)$, we have

[TABLE]

Proof.

We estimate $\overline{\lim}_{\epsilon\downarrow 0}\frac{V_{x}^{L}(a+\epsilon)-V_{x}^{L}(a)}{\epsilon}$ and $\underline{\inf}_{\epsilon\downarrow 0}\frac{V_{x}^{L}(a+\epsilon)-V_{x}^{L}(a)}{\epsilon}$. We have

[TABLE]

The behaviors of $L^{\pi^{a}}$ and $L^{\pi^{a+\epsilon}}$ are given in Section A. Because we have (A.4) and (A.7), for $t\in[\underline{\rho}^{a,[k-1]}_{0},\overline{\rho}^{a,[k]}_{a})$, we have, for $k\in\mathbb{N}$,

[TABLE]

Because we have (A.18) for $t\in[\overline{\rho}^{a,[k]}_{a},\underline{\rho}^{a,[k]}_{0})$, we have, for $k\in\mathbb{N}$,

[TABLE]

By (4.7), (4.9), and (4.10), we have

[TABLE]

By (A.10) and (A.19), we have

[TABLE]

So, we have

[TABLE]

By (A.9) and (A.16), $R^{\pi^{a}}-R^{\pi^{a+\epsilon}}$ is non-decreasing. Therefore, $L^{\pi^{a}}-L^{\pi^{a+\epsilon}}$ increases by at least $\epsilon$ on $[\underline{\rho}^{a+\epsilon,[k-1]}_{0},\overline{\rho}^{a+\epsilon,[k]}_{a+\epsilon}]$. By (A.7) and (A.18), $L^{\pi^{a}}-L^{\pi^{a+\epsilon}}$ is non-decreasing, giving us

[TABLE]

By (4.8) and (4.16), we have

[TABLE]

It is easy to check that $\underline{\nu}$ and $\overline{\nu}_{x}$ are right continuous. Thus, by (4.13) and (4.17), we obtain (4.4).

By the same argument as above, we obtain (4.5). The proof is complete. ∎

Lemma 4.3**.**

Suppose $X$ has bounded variation paths. Then for $x\in\mathbb{R}$ we have

[TABLE]

Proof.

We assume that the drift parameter $\delta$ is less than [math]. We prove $\lim_{a\downarrow 0}V^{L}_{x}(a)=V^{L}_{x}(0)$ and $\lim_{a\downarrow 0}V^{R}_{x}(a)=V^{R}_{x}(0)$. By the construction of $\pi^{a}$, $L^{\pi^{a}}$ increases only when $X$ takes positive jumps, and we have

[TABLE]

Because

[TABLE]

we have

[TABLE]

By the construction of $\pi^{a}$, we can write

[TABLE]

By (4.21), we have

[TABLE]

so we have

[TABLE]

When $\delta$ is assumed to be positive, the proof is almost the same as above. The proof is now complete. ∎

Proof of Lemma 4.1.

Since $\frac{-\overline{\nu}_{x}(a)}{1-\underline{\nu}(a)\overline{\nu}_{0}(a)}$ and $\frac{-\overline{\nu}_{x}(a)\underline{\nu}(a)}{1-\underline{\nu}(a)\overline{\nu}_{0}(a)}$ are non-decreasing, and by Lemma 4.2, both $V^{L}_{x}$ and $V^{R}_{x}$ are convex functions having Radon–Nikodym densities

[TABLE]

with respect to the Lebesgue measure. This implies that $V_{x}$ has Radon–Nikodym density

[TABLE]

By the definition of $a^{\ast}$, $V_{x}$ is non-decreasing on $(0,a^{\ast})$ and non-increasing on $(a^{\ast},\infty)$. By Lemma 4.3, the proof is complete. ∎

5 Verification

In this section, we show the optimality of the strategy $\pi^{a^{\ast}}$ for the value $a^{\ast}$ selected in the previous section. The main theorem is the following.

Theorem 5.1**.**

The strategy $\pi^{a^{\ast}}$ is optimal, and the value function of the problem (2.31) is given by $v=v_{\pi^{a^{\ast}}}$.

We postpone the proof of Theorem 5.1 until after the proof of Lemma 5.8. To prove Theorem 5.1, we use the following verification lemma.

Proposition 5.2**.**

Suppose that $X$ has bounded (resp., unbounded) variation paths. Let $w$ be a function on $\mathbb{R}$ belonging to $C^{(1)}_{\text{line}}$ (resp., $C^{(2)}_{\text{line}}$) and satisfying

[TABLE]

Then we have $w(x)\geq v(x)$ for all $x\geq 0$.

We give the proof of Proposition 5.2 in Section B. The purpose of this section is to prove that the strategy $\pi^{a^{\ast}}$ satisfies the conditions in Proposition 5.2.

To apply Proposition 5.2 to $v_{\pi^{\ast}}$, we give Lemmas 5.3, 5.4, and 5.5, which are lemmas about the smoothness of $v_{\pi^{a^{\ast}}}$ and some properties of the derivative of $v_{\pi^{a^{\ast}}}$.

Lemma 5.3**.**

We fix $a>0$. For $x\in(0,a)\backslash E^{(1)}_{a}$, we can take a Radon–Nikodym density of $v_{\pi^{a}}$ as

[TABLE]

Proof.

We compute the derivative of $v^{L}_{\pi^{a}}$. For $y\in\mathbb{R}$ and $t\geq 0$, we write $X^{(y)}_{t}=X_{t}+y$. For $b\in\mathbb{R}$, we write $\tau^{(y),+}_{b}=\inf\{t>0:X^{(y)}_{t}\geq b\}$ and $\tau^{(y),-}_{b}=\inf\{t>0:X^{(y)}_{t}\leq b\}$. We write $L^{(y)}$ for the process that represents the cumulative amount of dividends of $X^{(y)}$ on which the double barrier strategy at $a$ has been imposed. We write $U^{(y)}$ for its surplus process.

For $x\in[0,a-\epsilon]$, we can rewrite $v^{L}_{\pi^{a}}(x+\epsilon)-v^{L}_{\pi^{a}}(x)$ as

[TABLE]

The behaviors of $U^{(x)}$ and $U^{(x+\epsilon)}$ are summarized in Section C. From Section C, we have

[TABLE]

Because $L^{(x+\epsilon)}_{\tau^{(x+\epsilon),+}_{a}-}=0$ and (5.5) holds, we have

[TABLE]

On $\{\tau^{(x+\epsilon),+}_{a}>\tau^{(x+\epsilon),-}_{0}\}$, we have, by (5.8) and (5.7),

[TABLE]

and so by (5.6) and (5.7), we have

[TABLE]

By (5.4), (5.5), (5.8), and (5.10), we have

[TABLE]

Because $R^{(x)}_{\tau^{(x),-}_{0}-}=0$ and (5.6) holds, we have

[TABLE]

On $\{\tau^{(x),+}_{a}<\tau^{(x),-}_{0}\}$, by (5.12) and (5.7), we have

[TABLE]

which implies that

[TABLE]

By (5.5) and (5.14), we have

[TABLE]

Following the logic of the proofs of (5.11) and (5.15), we have, for $x\in[\epsilon,a]$,

[TABLE]

From (5.11), (5.15), and (5.16), for $x\in(0,a)\backslash E^{(1)}_{a}$, we obtain

[TABLE]

By a computation similar to that for the derivative of $v^{L}_{\pi^{a}}$, for $x\in(0,a)\backslash E^{(1)}_{a}$, we obtain

[TABLE]

In addition, $v^{L}_{\pi^{a}}$ and $v^{R}_{\pi^{a}}$ are continuous concave functions on $[0,a]$ from (5.11), (5.15), (5.16), and the other computations, and so $v_{\pi^{a}}$ has a Radon–Nikodym density (5.3). The proof is complete. ∎

Lemma 5.4**.**

For $a\geq 0$, the function $v_{\pi^{a}}$ is a continuous function.

Proof.

By the definition of $\pi^{a}$, we have

[TABLE]

So $v_{\pi^{a}}$ is continuous on $(-\infty,0]\cup[a,\infty)$. In addition, $v_{\pi^{a}}$ is continuous on $[0,a]$ by the proof of Lemma 5.3. The proof is complete. ∎

Since we have (5.19), we define $v_{\pi^{a^{\ast}}}^{\prime}$ as

[TABLE]

Lemma 5.5**.**

We have

[TABLE]

and $v_{\pi^{a^{\ast}}}$ is a concave function on $(0,\infty)$. In addition, we have

[TABLE]

when $X$ has unbounded variation paths.

Proof.

i) It is easy to check that $\underline{\nu}$ is right-continuous, and so we have $\beta\underline{\nu}(a^{\ast})\leq 1$. In this step, we define a constant $p^{\ast}$, and stopping times $K^{p^{\ast}}_{0}$ and $T^{p^{\ast}}_{0}$ for two cases. One is the case in which $\beta\underline{\nu}(a^{\ast})=1$. The other is the case in which $\beta\underline{\nu}(a^{\ast})<1$.

Suppose $\beta\underline{\nu}(a^{\ast})=1$. Then we define $p^{\ast}=1$, $K^{p^{\ast}}_{0}=K^{1}_{0}=\kappa^{a^{\ast},-}_{0}$, and $T^{p^{\ast}}_{0}=T^{1}_{0}=\tau^{-}_{0}$. Here, we have

[TABLE]

Suppose $\beta\underline{\nu}(a^{\ast})<1$. Then $X$ has bounded variation paths. For $n\in\mathbb{N}$, we write

[TABLE]

where $T(0)=S(0)=0$. For $p\in[0,1]$, we define i.i.d. random variables ${\{A^{[n]}_{p}\}}_{n\in\mathbb{N}}$ as

[TABLE]

We write

[TABLE]

Then, we have

[TABLE]

By the strong Markov property, we have

[TABLE]

and so $p\mapsto\mathbb{E}_{a^{\ast}}\!\left[e^{-qK^{p}_{0}}\right]$ is continuous on $p\in[0,1]$. Therefore, we can take some $p^{\ast}\in[0,1)$ that satisfies (5.24).

ii) In this step, we rewrite $v_{\pi^{a^{\ast}}}$ in a convenient form. For $x,a\in(0,\infty)$ with $x\leq a$ and $\epsilon>0$, we have

[TABLE]

By the continuity of $\underline{\varphi}_{0,a}$, for $a\in(0,\infty)$ and $x\in(0,a]\backslash E^{(1)}_{a}$, we have

[TABLE]

By the same argument as above, for $a\in(0,\infty)$ and $x\in(0,a]\backslash E^{(1)}_{a}$, we have

[TABLE]

So, by Lemma 5.3, for $a\in(0,\infty)$ and $x\in(0,a)\backslash E^{(1)}_{a}$, we have

[TABLE]

Because we have

[TABLE]

for $x\in(0,a^{\ast}]$, we also have

[TABLE]

and

[TABLE]

By (5.37), (5.39), and (5.41), for $x\in(0,a^{\ast})\backslash E^{(1)}_{a^{\ast}}$, we have

[TABLE]

By (5.24), we have

[TABLE]

From (5.24) and (5.43), we obtain (5.22) and that $v_{\pi^{a^{\ast}}}$ is a concave function.

Suppose that $X$ has unbounded variation paths. Then it is easy to check that the map $x\mapsto\beta\mathbb{E}_{x}\!\left[e^{-qK^{p^{\ast}}_{0}}\right]$ is continuous. Thus we have

[TABLE]

and we obtain (5.23). The proof is now complete. ∎

We define the Radon–Nikodym density $v_{\pi^{a^{\ast}}}^{\prime}$ by

[TABLE]

In addition, when $X$ has unbounded variation paths, we define the Radon–Nikodym density $v_{\pi^{a^{\ast}}}^{\prime\prime}$ by

[TABLE]

Lemma 5.6**.**

The function $v_{\pi^{a^{\ast}}}$ belongs to $C^{(1)}_{\text{line}}$. Furthermore, if $X$ has unbounded variation paths, then $v_{\pi^{a^{\ast}}}\in C^{(2)}_{\text{line}}$.

Proof.

By Assumption 2.1, Lemma 5.3, Lemma 5.4, and Lemma 5.5, it is obvious that $v_{\pi^{a^{\ast}}}$ belongs to $C^{(1)}_{\text{line}}$. In addition, we know that $v_{\pi^{a^{\ast}}}$ is continuously differentiable and $v_{\pi^{a^{\ast}}}^{\prime}$ has a Radon–Nikodym density on $(0,\infty)$ when $X$ has unbounded variation paths. So, it is enough to check that $v_{\pi^{a^{\ast}}}^{\prime}$ has a locally bounded density on $(0,\infty)$ when $X$ has unbounded variation paths.

Suppose $X$ has unbounded variation paths. We use the same notation as in the proof of Lemma 5.3. For $x\in\mathbb{R}$, let $Y^{(x)}$ be a reflected process defined by

[TABLE]

and write

[TABLE]

We fix $a^{\dagger}\in(a^{\ast},\infty)$. From (5.45) and because

[TABLE]

for $x\in(0,a^{\ast})\backslash(E^{(2)}_{a^{\ast}}\cup E^{(2)}_{a^{\dagger}})$, we have

[TABLE]

Because $Y^{(x+\epsilon)}_{t}=Y^{(x)}_{t}$ for $t\in[\tau^{(x),+}_{a^{\ast}},\infty)$, we have

[TABLE]

and so

[TABLE]

On $\{\tau^{(x),-}_{0}<\tau^{(x),+}_{a^{\ast}}\}$, we have

[TABLE]

and so we have

[TABLE]

Because

[TABLE]

we have

[TABLE]

This implies that $v_{\pi^{a^{\ast}}}^{\prime\prime}$ is bounded on $(a^{\ddagger},a^{\ast}){\backslash(E^{(2)}_{a^{\ast}}\cup E^{(2)}_{a^{\dagger}})}$ for $a^{\ddagger}\in(0,a^{\ast})$. Because $v_{\pi^{a^{\ast}}}^{\prime\prime}=0$ on ${[}a^{\ast},\infty)$, the proof is complete. ∎

Lemma 5.7**.**

*Suppose $a^{\ast}>0$. When $X$ has bounded variation paths, for $x\in(0,a^{\ast})$, *

[TABLE]

In addition, when $X$ has unbounded variation paths, we redefine $v_{\pi^{a^{\ast}}}^{\prime\prime}$ on $(0,a^{\ast})$ to satisfy (5.62).

Proof.

For $x\in(0,a)$, we have

[TABLE]

For $x\in(0,a^{\ast})$, the process $\!\left\{M^{[1]}_{t}:t\geq 0\right\}$ where

[TABLE]

is a martingale under $\mathbb{P}_{x}$ because

[TABLE]

By the same argument, for $x\in(0,a^{\ast})$, $\!\left\{M^{(2)}_{t}:t\geq 0\right\}$ and $\!\left\{M^{(3)}_{t}:t\geq 0\right\}$ where

[TABLE]

are martingales under $\mathbb{P}_{x}$. By (5.63), the process ${\!\left\{M^{(4)}_{t}:t\geq 0\right\}}$ where

[TABLE]

is a martingale under $\mathbb{P}_{x}$. By the argument used for the proof of (5.70), the processes ${\!\left\{M^{(5)}_{t}:t\geq 0\right\}}$ and ${\!\left\{M^{(6)}_{t}:t\geq 0\right\}}$, where

[TABLE]

are martingales under $\mathbb{P}_{x}$. By the same reasoning as that of the proof of [5, (12)], for $x\in(0,a^{\ast})\backslash E^{(1)}_{a^{\ast}}$ (resp., $(0,a^{\ast})\backslash E^{(2)}_{a^{\ast}}$), we have (5.62). Here, we used the continuity of the map $x\mapsto\mathcal{L}v_{\pi^{a^{\ast}}}(x)$ on $(0,a^{\ast})\backslash E^{(1)}_{a^{\ast}}$ (resp., $(0,a^{\ast})\backslash E^{(2)}_{a^{\ast}}$).

When $X$ has bounded variation paths, we obtain (5.62) for $x\in(0,a^{\ast})$ since Remark 2.5 holds and $v_{\pi^{a^{\ast}}}^{\prime}$ is right continuous.

When $X$ has unbounded variation paths, by Remark 2.5, we can redefine the locally bounded Radon–Nikodym density $v_{\pi^{a^{\ast}}}^{\prime\prime}$ on $(0,a^{\ast})$, which is continuous almost everywhere and satisfies (5.62) for $x\in(0,a^{\ast})$.

The proof is complete. ∎

Lemma 5.8**.**

For $x\leq a^{\ast}$, we have

[TABLE]

Proof.

This proof is almost the same as that of [2, Lemma 5]. We write $g(x)=\mathcal{L}v_{\pi^{a^{\ast}}}(x)-qv_{\pi^{a^{\ast}}}(x)$ for $x\geq 0$.

From the form of the operator $\mathcal{L}$, (5.20), and (5.21), for $x{\geq}a^{\ast}$, we have

[TABLE]

where $b=v_{\pi^{a^{\ast}}}(a^{\ast})-a^{\ast}$. By the concavity of $v_{\pi^{a^{\ast}}}$ (see Lemma 5.5) and the form of (5.74), $g(x)$ is a continuous concave function on $(a^{\ast},\infty)$.

We prove that, for $a>a^{\ast}$,

[TABLE]

We write $v_{\pi^{a^{\ast}}}^{(\epsilon)}(x)=v_{\pi^{a^{\ast}}}(x+\epsilon)$ for $x\in\mathbb{R}$. Then, we can define $\mathcal{L}v_{\pi^{a^{\ast}}}^{(\epsilon)}(x):=\mathcal{L}v_{\pi^{a^{\ast}}}(x+\epsilon)$ for $x>-\epsilon$. Let $L^{\pi,c}$ be the continuous part of $L^{\pi}$ and let $R^{\pi,c}$ be the continuous part of $R^{\pi}$ for $\pi\in\mathcal{A}$. By an application of the Meyer–Itô formula (see [16, Theorem IV.70 or IV.71]), we have

[TABLE]

Because

[TABLE]

we have

[TABLE]

Rewriting the above equation leads to

[TABLE]

Here, $\{M^{\pi^{a}}_{t}:t\geq 0\}$ is a local martingale such that

[TABLE]

where $B$ is a standard Brownian motion. Because $M^{\pi^{a}}$ is a local martingale, we can take a sequence of stopping times ${\{T^{\pi^{a}}_{n}\}}_{n\in\mathbb{N}}$, which is a localizing sequence for $M$ with $T_{n}\uparrow\infty$ almost surely. We take the expectation of (5.83) at time $t\land T^{\pi^{a}}_{n}$ and take the limit as $t\uparrow\infty$ and $n\uparrow\infty$. By Lemma 5.8, we have

[TABLE]

By (5.45), $v_{\pi^{a^{\ast}}}^{\prime}(0+)=\beta$ when [math] is regular for $(-\infty,0)$. By contrast, $R^{\pi^{a},c}\equiv 0$ when [math] is irregular for $(-\infty,0)$. So, we have

[TABLE]

By taking the limit as $\epsilon\downarrow 0$ and applying Lemma 5.5, (5.90), and the continuity of $\mathcal{L}v_{\pi^{a^{\ast}}}$, we have (5.75).

By Lemma 4.1 and (5.75), we have

[TABLE]

By the continuity and the concavity of $g$ along with (5.91), we have (5.73) for $x\in[a^{\ast},\infty)$. The proof is complete. ∎

Proof of Theorem 5.1.

By Lemmas 5.5, 5.6, 5.7, 5.8, and (5.45) $v_{\pi^{a^{\ast}}}\in C^{(1)}_{\text{line}}$ (resp., $C^{(2)}_{\text{line}}$) satisfies (5.1) and (5.2) when $X$ has bounded (resp., unbounded) variation paths. From Proposition 5.2, the proof is complete. ∎

6 Examples

We assumed the continuity of $\overline{\varphi}_{a,0}$ and $\underline{\varphi}_{0,a}$ in Assumption 2.1 when $X$ has unbounded variation paths. In this section, we present examples of Lévy processes having unbounded variation paths that satisfy Assumption 2.1.

Let $X$ be a Lévy process with characteristic exponent (2.2) having unbounded variation paths. We additionally assume that $\Pi(-\infty,0)<\infty$ or $\Pi(0,\infty)<\infty$. Then, for $a>0$, both $\overline{\varphi}_{a,0}$ and $\underline{\varphi}_{0,a}$ are continuously differentiable on $(0,a)$. We check this fact.

We assume without loss of generality that $\Pi(0,\infty)<\infty$. Then there exist a spectrally negative Lévy process $Z$ with unbounded variation paths, a Poisson process $N^{(r)}$ with rate $r>0$, and i.i.d. positive random variables ${\{J_{n}\}}_{n\in\mathbb{N}}$ such that

[TABLE]

Here, $Z$ has the Laplace exponent $\psi_{Z}$, which satisfies

[TABLE]

where $\mathbb{P}^{Z}_{x}$ is the law of $Z$ when it starts at $x\in\mathbb{R}$. Then, $\psi_{Z}$ takes the form

[TABLE]

where $\Pi_{Z}(\cdot)=\Pi(\cdot\cap(-\infty,0))$.

We recall the definition of scale functions and some properties of these functions. For $p\geq 0$, let $W_{Z}^{(p)}$ be the scale function of $Z$, which is the function from $\mathbb{R}$ to $[0,\infty)$ such that $W_{Z}^{(p)}=0$ on $(-\infty,0)$ and $W_{Z}^{(p)}$ on $[0,\infty)$ is continuous, satisfying

[TABLE]

where $\Phi_{Z}(p)=\inf\{s\geq 0:\psi_{Z}(s)>p\}$. For proofs of uniqueness, existence, and the basic facts listed below, see, for example, [9, Section 8]. For $a>0$, $x\in[0,a]$, and a non-negative measurable function $f$, we have

[TABLE]

The function $W_{Z}^{(p)}$ is continuously differentiable on $(0,\infty)$ because $Z$ has unbounded variation paths.

We prove that $\overline{\varphi}_{a,0}$ is continuously differentiable. By the strong Markov property, we have, for $x\in(0,a)$,

[TABLE]

where $T^{(r)}[1]$ is the first jump time of $N^{(r)}$. By the definitions of $Z$ and $T^{(r)}[1]$, we have

[TABLE]

By (6.5) and (6.6), we have

[TABLE]

We know that $\frac{W_{Z}^{(q+r)}(x)}{W_{Z}^{(q+r)}(a)}$ is continuously differentiable on $(0,a)$, so we consider the differentiability of

[TABLE]

Here, we cannot obtain the derivative of (6.10) right away using the dominated convergence theorem because the derivative of $W_{Z}^{(q+r)\prime}$ may not be bounded. By (6.9) and Fubini’s theorem, we have

[TABLE]

where

[TABLE]

Here, $h\geq 0$ almost everywhere with respect to the Lebesgue measure because $\overline{\varphi}_{a,0}$ is non-decreasing. So, by Fubini’s theorem, we have

[TABLE]

Because we have

[TABLE]

applying the dominated convergence theorem gives the derivative of (6.17) as

[TABLE]

which is continuous on $(0,a)$. Therefore, $\overline{\varphi}_{a,0}$ is continuously differentiable on $(0,a)$.

We can prove that $\underline{\varphi}_{0,a}$ is continuously differentiable on $(0,a)$ by the same argument as above.

Appendix A The behavior of $U^{\pi^{a}}$ and $U^{\pi^{a+\epsilon}}$

For $t\in[\underline{\rho}^{a,[k-1]}_{0},\overline{\rho}^{a,[k]}_{a})$, the processes $U^{\pi^{a}}$ and $U^{\pi^{a+\epsilon}}$ behave as follows. We have

[TABLE]

By the definitions of $\pi^{a}$ and $\overline{\rho}^{a,[k]}_{a}$, for $t\in[\underline{\rho}^{a,[k-1]}_{0},\overline{\rho}^{a,[k]}_{a})$, processes $U^{\pi^{a}}$, $L^{\pi^{a}}$, and $R^{\pi^{a}}$ satisfy

[TABLE]

Additionally, by the definition of $\pi^{a+\epsilon}$, processes $U^{\pi^{a+\epsilon}}$, $L^{\pi^{a+\epsilon}}$, and $R^{\pi^{a+\epsilon}}$ satisfy

[TABLE]

before the right-hand side of (A.6) hits $(a+\epsilon,\infty)$. From (A.1), (A.2), and (A.5), for $t\in[\underline{\rho}^{a,[k-1]}_{0},\overline{\rho}^{a,[k]}_{a})$,

[TABLE]

So the right-hand side of (A.6) is no more than $a+\epsilon$ on $[\underline{\rho}^{a,[k-1]}_{0},\overline{\rho}^{a,[k]}_{a})$, which implies that each of (A.5), (A.6), and (A.7) holds for $t\in[\underline{\rho}^{a,[k-1]}_{0},\overline{\rho}^{a,[k]}_{a})$. From (A.1), (A.2), and (A.5), for $t\in[\underline{\rho}^{a,[k-1]}_{0},\overline{\rho}^{a,[k]}_{a})$,

[TABLE]

From (A.1), (A.3), (A.6), and (A.9), for $t\in[\underline{\rho}^{a,[k-1]}_{0},\overline{\rho}^{a,[k]}_{a})$,

[TABLE]

For $t\in[\overline{\rho}^{a,[k]}_{a},\underline{\rho}^{a,[k]}_{0})$, the processes $U^{\pi^{a}}$ and $U^{\pi^{a+\epsilon}}$ behave as follows. By the definitions of $\pi^{a}$ and $\underline{\rho}^{a,[k]}_{0}$, for $t\in[\overline{\rho}^{a,[k]}_{a},\underline{\rho}^{a,[k]}_{0})$, we have

[TABLE]

Additionally, by the definition of $\pi^{a+\epsilon}$, the processes $U^{\pi^{a+\epsilon}}$, $L^{\pi^{a+\epsilon}}$, and $R^{\pi^{a+\epsilon}}$ satisfy

[TABLE]

before the right-hand side of (A.15) hits $(-\infty,0)$. From (A.10), (A.11), and (A.14), for $t\in[\overline{\rho}^{a,[k]}_{a},\underline{\rho}^{a,[k]}_{0})$,

[TABLE]

So the right-hand side of (A.15) is non-negative on $[\overline{\rho}^{a,[k]}_{a},\underline{\rho}^{a,[k]}_{0})$, which implies that each of (A.14), (A.15), and (A.16) holds for $t\in[\overline{\rho}^{a,[k]}_{a},\underline{\rho}^{a,[k]}_{0})$. From (A.10), (A.11), and (A.14), for $t\in[\overline{\rho}^{a,[k]}_{a},\underline{\rho}^{a,[k]}_{0})$,

[TABLE]

From (A.10), (A.12), (A.15), and (A.18), for $t\in[\overline{\rho}^{a,[k]}_{a},\underline{\rho}^{a,[k]}_{0})$,

[TABLE]

Appendix B Proof of Proposition 5.2

The proof of Proposition 5.2 is almost the same as that of [2, Proposition 4 (ii)].

Let $\pi\in\Pi$ be any admissible strategy. Then, $U^{\pi}$ is a $[0,\infty)$-valued process. We fix $\epsilon>0$ and define $w_{\epsilon}(x)=w(x+\epsilon)$ for $x\in\mathbb{R}$. Then, we can define $\mathcal{L}w_{\epsilon}(x)=\mathcal{L}w(x+\epsilon)$ for $x>-\epsilon$. By (5.1) and (5.2), we have

[TABLE]

By an application of the Meyer–Itô formula (see [16, Theorem II.31 and IV.71]) and by the same calculation as used for (5.83), we have

[TABLE]

Here, $\{M_{t}:t\geq 0\}$ is a local martingale satisfying

[TABLE]

By (B.2), we have

[TABLE]

and so

[TABLE]

Because $M$ is a local martingale, we can take a sequence of stopping times ${\{T_{n}\}}_{n\in\mathbb{N}}$ that is a localizing sequence for $M$ with $T_{n}\uparrow\infty$ almost surely. Then, taking an expectation, we have

[TABLE]

where in (B.14) we used (B.1). By taking the the limit as $t\uparrow\infty$, $n\uparrow\infty$, and $\epsilon\downarrow 0$, the proof is complete.

Appendix C The behavior of $U^{(x)}$ and $U^{(x+\epsilon)}$ under $\mathbb{P}_{0}$

In this section, we describe the behavior of $U^{(x)}$ and $U^{(x+\epsilon)}$ under $\mathbb{P}_{0}$, which is necessary for the proof of Lemma 5.3. We define the hitting times inductively as follows. For $n\in\mathbb{N}$,

[TABLE]

Then, we have the following by induction.

For $t\in[\underline{\kappa}^{[n-1]}_{0},\overline{\kappa}^{[n]}_{a})$, the processes $U^{(x)}$ and $U^{(x+\epsilon)}$ behave as follows. We have

[TABLE]

By the definitions of $\pi^{a}$ and $\overline{\kappa}^{[n]}_{a}$, for $t\in[\underline{\kappa}^{[n-1]}_{0},\overline{\kappa}^{[n]}_{a})$, we have

[TABLE]

Additionally, by the definition of $\pi^{a}$, the processes $U^{(x)}$, $L^{(x)}$, and $R^{(x)}$ satisfy

[TABLE]

before the right-hand side of (C.12) hits to $[a,\infty)$. By (C.6), (C.8), and (C.11), for $t\in[\underline{\kappa}^{[n-1]}_{0},\overline{\kappa}^{[n]}_{a})$,

[TABLE]

and so (C.10), (C.11), and (C.12) hold for $t\in[\underline{\kappa}^{[n-1]}_{0},\overline{\kappa}^{[n]}_{a})$. From (C.5), (C.6), (C.8), (C.9), (C.11), and (C.12), for $t\in[\underline{\kappa}^{[n-1]}_{0},\overline{\kappa}^{[n]}_{a})$, we have

[TABLE]

From (C.7), (C.10), (C.14), and (C.15), we have

[TABLE]

For $t\in[\overline{\kappa}^{[n]}_{a},\underline{\kappa}^{[n]}_{0})$, processes $U^{(x)}$ and $U^{(x+\epsilon)}$ behave as follows. By the definitions of $\pi^{a}$ and $\underline{\kappa}^{[n]}_{0}$, for $t\in[\overline{\kappa}^{[n]}_{a},\underline{\kappa}^{[n]}_{0})$, we have

[TABLE]

Additionally, by the definition of $\pi^{a}$, the processes $U^{(x+\epsilon)}$, $L^{(x+\epsilon)}$, and $R^{(x+\epsilon)}$ satisfy

[TABLE]

before the right-hand side of (C.24) hits $(-\infty,0]$. From (C.18), (C.19), and (C.22), for $t\in[\overline{\kappa}^{[n]}_{a},\underline{\kappa}^{[n]}_{0})$,

[TABLE]

and so (C.22), (C.23), and (C.24) hold for $t\in[\overline{\kappa}^{[n]}_{a},\underline{\kappa}^{[n]}_{0})$. From (C.16), (C.18), (C.19), (C.21), (C.22), and (C.24), for $t\in[\overline{\kappa}^{[n]}_{a},\underline{\kappa}^{[n]}_{0})$, we have

[TABLE]

From (C.20), (C.23), (C.26), and (C.27), we have

[TABLE]

Acknowledgments

I express my deepest gratitude to Prof. Kazutoshi Yamazaki and Prof. Kouji Yano for their comments and improvements. The author was supported by a KAKENHI grant, No. JP18J12680, from the Japan Society for the Promotion of Science.