# Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model

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## Abstract

**:**

## 1. Introduction

**Motivation**. Keeping shareholders satisfied while maintaining the company liquid enough constitutes the core of the control activity of reserves/risk processes. The rough idea is that one should intelligently balance return over invested capital (dividends) and replenishment by capital injections to insure the company’s robustness against claims. The easiest way to conceive such balance can roughly be stated as follows: when below low levels $-a\le 0$ (a acting as a maximally-admitted severity of ruin), reserve processes should be replenished by capital injections at some cost (hereafter, the unitary cost is denoted by $k\ge 1$), and when above high levels $b>0$, they should be taken out of the reserves as dividends–see for example [1,2] and the comprehensive book [3].

**Historical overview**. The first results tackling a related problem, due to de Finetti [4], concerned maximizing the

**expected value of the discounted cumulative dividends**until the

**time of passage below a given level, called ruin**. This can be seen as a particular case of the above-mentioned setting in which the cost of “borrowing” money is $k=\infty $ (or, equivalently, in which the maximal-targeted severity of ruin is $a=0$.

**state-dependent diffusion process**, the dividends problem for the

**reflected process**(RP). This consists in maximizing the functional

- x is the initial surplus;
- q is a discount factor;
- ${L}_{t}$ are the cumulative dividends;
- ${I}_{t}$ are the cumulative
**forced capital injections**each time the process attempts to cross into $(-\infty ,0)$, and $k>1$ is a proportional cost for injecting capital; - ${X}_{t}$ is the process modified by capital injections and dividends, and $\pi =({L}_{t},{I}_{t})$.

**forced bailouts**. To our best knowledge, our paper hereafter seems to be the first to quantify the optimal buffer (severity of ruin) when bankruptcy should replace

**individual bailouts**(to be fair, this possibility has been considered before, but without studying optimality—see for example [12]). For a different approach, quantifying the optimal policy under an expected total bailouts constraint, see [13].

- $k<{k}_{c}\u27f9{V}_{D}\left(x\right)<{V}_{k}\left(x\right)$; the optimal policy is (RP), i.e., pay dividends at an upper barrier ${b}_{k}$ and always inject capital at 0, implying that bankruptcy never occurs;
- $k\ge {k}_{c}\u27f9{V}_{D}\left(x\right)\ge {V}_{k}\left(x\right)$; the optimal policy is (AP), i.e., pay dividends in order to reflect the surplus process at some upper barrier ${b}^{*}\ge 0$, and never inject capital.

**Aim**. Our paper focuses on the optimization of a criterion of type (2) in a Cramér-Lundberg framework without further assumptions on behavior of the value function at $x=0$. We seek a related dichotomy (hereafter referred to as Lokka-Zervos alternative) of optimal policies following the expensiveness of capital injections. Three points are of particular interest

- we
**prove**the optimality of policies such as those described at the very beginning for which, in general, $a\notin \left\{0,\infty \right\}$ (i.e., de Finetti and Shreve, Lehoczky and Gaver policies fail to be optimal); - the resulting value function is
**not of class**${C}^{1}$**at**0 (in the sense that its derivative does not exist at $x=0$ and, more, the right-hand derivative at 0 is not k as assumed, for Brownian claims, in [14], 5.2). This can be found in Remark 14 and it implies, in particular, that the verification Theorem 4 has to be given for absolutely-continuous functions. Working with super-solutions also explains the particular form of the Equation (8) on ${\mathbb{R}}_{-}$. - the optimal parameters ${a}^{*}$ and ${b}^{*}$ are
**completely characterized**by non-trivial equations and so is the alternative-inducing cost ${k}^{*}$.

**Method**. We have organized our paper around the classical guess and verify procedure for solving stochastic control problems:

- Guess a family of policies which yields the optimum for all possible values of the parameters, and compute its expected net present value (EPV) in terms of the $W,Z$ scale functions.

- 2.
- Identify the optimal arguments with respect to the parameters of our policies. This step forced us to restrict to the case of exponential jumps, where the independence of ruin and ruin overshoots leads to simplifying factorizations related to the memoryless property of this distribution.
- 3.
- In the final step, confirm the optimality of the selected candidate optimal policy via a verification theorem (sufficient condition for optimality). If the conjectured value function were sufficiently smooth (this means ${\mathcal{C}}^{\left(1\right)}$ in our problem), this would require only verifying that it satisfies an associated HJB equation (system of variational inequalities).

**The model**. We focus here on the classic Cramér-Lundberg model

**Remark**

**1.**

**Related literature**. Among other papers dealing with similar topics like [17,18,19], the most related to ours is [20], which was brought to our attention only after completion of our paper. Ref. [20] deals with the spectrally negative Lévy model, and the expressions for the cost functions provided in [20] are more general than ours. However, we prefer to offer here direct proofs adapted to the exponential case, for the sake of completeness.

**Contents**. The paper is organized as follows. Section 2 gives the necessary framework, mathematically formulates the admissible policies, the targeted cost and the value function. Section 2.2 turns to a structural study of the value function for arbitrarily distributed claims. This is achieved by making precise the regularity of the value function (Lipschitz-continuity, upper and lower-bounds in Proposition 2). Furthermore, the value function is characterized, among a class of linear-growth functions, as the smallest absolutely-continuous super-solution of the associated variational inequality of Hamilton-Jacobi integro-differential type. As it is by now standard, this characterization implies a verification result as to the optimality of policies (cf. Theorem 4).

## 2. Optimizing Dividends and Capital Injections with Proportional Costs

#### 2.1. The Framework

- given a couple $\pi $: $=\left(L,I\right)$ describing dividends and capital injection, the modified surplus process is defined (under the ${\mathbb{P}}_{x}$) by setting$${X}_{t}^{\pi}:={X}_{t}-{L}_{t}+{I}_{t};$$
- the cumulative dividend strategy L is adapted, non-decreasing and càdlàg (right-continuous, left-limits), ${L}_{0-}=0$;
- the cumulative capital injection process I is adapted, non-decreasing and càdlàg, ${I}_{0-}=0$;
- the triplet $\pi :=\left(L,I\right)$ satisfying the previous conditions is referred to as (general) strategy and the family of all such strategies is denoted by ${\Pi}^{+}$.
- (prior to ruin) for every $t\ge 0$, the dividends should satisfy$$\u25b3{L}_{t}:\phantom{\rule{4.pt}{0ex}}={L}_{t}-{L}_{t-}\le {X}_{t-}^{\pi}-\u25b3{\overline{N}}_{t}+\u25b3{I}_{t},\mathrm{where}{\overline{N}}_{t}:=\sum _{i=1}^{{N}_{t}}{C}_{i}.$$

- (after the ruin time ${X}_{t-}^{\pi}-\u25b3{\overline{N}}_{t}+\u25b3{I}_{t}<0$), we set ${I}_{s}={I}_{t-},\phantom{\rule{4pt}{0ex}}{L}_{s}={L}_{t-},\phantom{\rule{4pt}{0ex}}\forall s\ge t.$ (The reader is invited to note that in this case $t={\tau}_{i}$ one of the jumping times for N such that $I,L$ remain adapted, non-decreasing and càdlàg).

- the $\pi \in {\Pi}^{+}$ satisfying the dividends restriction is called an admissible strategy and the family of all such strategies is denoted by $\tilde{\Pi}\left(x\right)$.
- for an admissible strategy $\pi $,
- (a)
- we consider the ruin time ${\sigma}_{0-}^{x,\pi}:=inf\left\{t>0:\phantom{\rule{4pt}{0ex}}{X}_{t-}^{\pi}-\u25b3{\overline{N}}_{t}+\u25b3{I}_{t}<0\right\}$ (if a “very large” claim occurs, bankruptcy is declared; as we will see in the main result, it is never optimal to take ${\sigma}_{0-}^{x,\pi}=\infty $ and modify the equity I accordingly).
- (b)
- the associated cost is$$\tilde{J}\left(x,\pi \right):={\mathbb{E}}_{x}[{\int}_{\left[0,{\sigma}_{0-}^{x,\pi}\right]}{e}^{-qt}\left(d{L}_{s}-kd{I}_{s}\right)],$$
- (c)
- Every strategy $\pi $ can be replaced with $\tilde{\pi}$ by modifying $\u25b3{\tilde{I}}_{t}:\phantom{\rule{4.pt}{0ex}}=0$ if ${X}_{t-}^{\pi}-\u25b3{\overline{N}}_{t}+\u25b3{I}_{t}<0$ such that ${\sigma}_{0-}^{x,\pi}={\sigma}_{0-}^{x,\tilde{\pi}}$ and improving the associated cost $\tilde{J}\left(x,\pi \right)$. From now on, whenever a policy $\pi $ is considered, we identify it with $\tilde{\pi}$ and reason accordingly.
- (d)
- In connection to this type of policies and the related cost, we set$$\tilde{V}\left(x\right):\phantom{\rule{4.pt}{0ex}}=\underset{\pi \in \tilde{\Pi}\left(x\right)}{sup}\tilde{J}\left(x,\pi \right),\phantom{\rule{4pt}{0ex}}x\in \mathbb{R}.$$

#### 2.2. The Value Function

- First, we focus on the regularity properties of the value function $\tilde{V}$ (lower and upper-bound and Lipschitz-continuity) in Proposition 2.
- Second, we prove the connection between this value function and the associated partial-integral differential system (of HJB-type) in Proposition 3.
- Third, we show in Theorem 4 that the value function $\tilde{V}$ is the lowest absolutely-continuous super-solution of the associated equation and, as a verification result, give the optimality condition for candidate policies.

#### 2.3. Some Elementary Properties of the Value Function

**Proposition**

**2.**

- For every $x\in {\mathbb{R}}_{+}$, the set of admissible strategies $\tilde{\Pi}\left(x\right)$ is non-empty. If $x,{x}^{\prime}\in {\mathbb{R}}_{+}$, then $\tilde{\Pi}\left(x\right)\subset \tilde{\Pi}\left({x}^{\prime}\right)$ and, for every $\pi \in \tilde{\Pi}\left(x\right)$, ${\sigma}_{0-}^{x,\pi}\le {\sigma}_{0-}^{{x}^{\prime},\pi},\phantom{\rule{4pt}{0ex}}\mathbb{P}-a.s.$.
- For every $x\ge 0$ and every $\pi \in \tilde{\Pi}\left(x\right)$, one has $\tilde{V}\left(x\right)\ge x+\frac{p}{\lambda +q}$.
- If $q>0$, then, for every $x\ge 0$ and every policy $\pi \in \tilde{\Pi}\left(x\right)$, one has $\tilde{J}(x,\pi )\le x+\frac{p}{q}$.
- For every $x\ge 0$ and every $\epsilon >0$, $\tilde{V}\left(x\right)+\epsilon \le \tilde{V}\left(x+\epsilon \right)\le \tilde{V}\left(x\right)+k\epsilon $.

#### 2.4. The HJB System

#### 2.5. The Value Function as Smallest $\mathcal{AC}$ Super-Solution

**Theorem**

**4.**

- The value function $\tilde{V}$ is non-negative, $x+\frac{p}{\lambda +q}\le \tilde{V}\left(x\right)\le x+\frac{p}{q},$ for $x\ge 0$ and $\tilde{V}\left(x\right)=max\left\{\tilde{V}\left(0\right)+kx,0\right\},\phantom{\rule{4pt}{0ex}}\forall x\le 0$.
- Every non-negative $\mathcal{AC}$-regular of growth $\varphi \left(x\right)\le max\left\{x+\alpha ,0\right\}$ viscosity super-solution of (8), where α is introduced in Proposition 15, is greater than or equal to $\tilde{V}$ on ${\mathbb{R}}_{+}$.
- If ${\pi}^{*}:=\left({\pi}^{x}={\left(L,I\right)}^{x}\in \tilde{\Pi}\left(x\right),\phantom{\rule{4pt}{0ex}}x\ge 0\right)$ is a family of admissible strategies such that the associated costs $\tilde{J}(\xb7,{\pi}^{*})$ is an $\mathcal{AC}$ super-solution for (8), then ${\pi}^{*}$ is optimal and $\tilde{V}\left(x\right)=\tilde{J}(x,{\pi}^{*}),\phantom{\rule{4pt}{0ex}}\forall x\in {\mathbb{R}}_{+}$.

**Remark**

**5.**

## 3. The Value Function for the Cramér-Lundberg Model

#### 3.1. The Guess Step: Severity-Constrained Double Barrier Policies, for the Cramér-Lundberg Process with Exponential Jumps

- the inverse Laplace transform of $\frac{1}{\kappa \left(s\right)-q}$, where $\kappa \left(s\right)$ is the Laplace exponent, and
- ${Z}_{q}\left(x\right)=1+q{\int}_{0}^{x}{W}_{q}\left(y\right)dy$.

**mean function of our claims cut at level**$s\ge 0$.

**Proposition**

**6.**

- ${\Phi}_{q}+{\rho}_{-}=-\frac{p\mu -\lambda -q}{p};\phantom{\rule{4pt}{0ex}}{\Phi}_{q}{\rho}_{-}=-\frac{\mu q}{p}$;
- $p\rho \left(\rho +\mu \right)=\left(\lambda +q\right)\rho +\mu q,\phantom{\rule{4pt}{0ex}}\rho \in \left\{{\Phi}_{q},{\rho}_{-}\right\}$;
- ${\rho}_{-}+\mu \ge 0,\mathrm{and}p\left({\Phi}_{q}+\mu \right)\left({\rho}_{-}+\mu \right)=\lambda \mu $;
- $p{W}_{q}\left(x\right)-{Z}_{q}\left(x\right)=\frac{\lambda}{p\left({\Phi}_{q}-{\rho}_{-}\right)}\left({e}^{{\Phi}_{q}x}-{e}^{{\rho}_{-}x}\right)\ge 0$, for all $x\ge 0$;
- For all $b\ge 0$, ${W}_{q}$ and ${Z}_{q}$ (extended by ${W}_{q}\left(x\right)=0,\phantom{\rule{4pt}{0ex}}{Z}_{q}\left(x\right)=1,\phantom{\rule{4pt}{0ex}}\forall x<0$) satisfy the equation $\mathcal{L}\varphi \left(b\right)=0$, where $\mathcal{L}$ is given by (41). More precisely,$$\begin{array}{cc}\hfill & p{W}_{q}^{\prime}(b+)+\lambda {\int}_{0}^{b}{W}_{q}\left(y\right)\mu {e}^{-\mu (b-y)}dy-\left(\lambda +q\right){W}_{q}\left(b\right)=0;\\ & p{Z}_{q}^{\prime}(b+)+\lambda {\int}_{-\infty}^{b}{Z}_{q}\left(y\right)\mu {e}^{-\mu (b-y)}dy-\left(\lambda +q\right){Z}_{q}\left(b\right)=0.\end{array}$$

**Proposition**

**7.**

- The cost function satisfies, for $0\le x\le b$,$${J}_{x}=\frac{{W}_{q}\left(x\right)}{{W}_{q}\left(b\right)}{J}_{b}+\left({Z}_{q}\left(x\right)-\frac{{W}_{q}\left(x\right)}{{W}_{q}\left(b\right)}{Z}_{q}\left(b\right)\right)z\left(a,{J}_{0}\right).$$In particular, if we set$${j}_{x}:={J}_{x}-z(a,{J}_{0}){Z}_{q}\left(x\right),$$then$${j}_{x}=\frac{{W}_{q}\left(x\right)}{{W}_{q}\left(b\right)}{j}_{b},\phantom{\rule{0.277778em}{0ex}}0\le x\le b.$$
- Moreover, the cost can be explicitly written as$${J}_{x}=\left\{\begin{array}{cc}0\hfill & ,\phantom{\rule{4pt}{0ex}}x<-a\hfill \\ kx+{J}_{0},\hfill & ,\phantom{\rule{4pt}{0ex}}x\in [-a,0]\hfill \\ kG(a,x)+{J}_{0}S(a,x)=kG(a,x)+\frac{1-k\phantom{\rule{0.277778em}{0ex}}{\partial}_{b}G(a,b+)}{{\partial}_{b}S(a,b+)}S(a,x)\hfill & ,\phantom{\rule{4pt}{0ex}}x\in [0,b]\hfill \end{array}\right.,$$and$${J}_{0}=\frac{1-k\phantom{\rule{0.277778em}{0ex}}{\partial}_{b}G(a,b+)}{{\partial}_{b}S(a,b+)}=\frac{1-k\phantom{\rule{0.277778em}{0ex}}[m\left(a\right){C}_{q}^{\prime}(b+)]}{{e}^{-\mu a}{C}_{q}^{\prime}(b+)+q{W}_{q}\left(b\right)}.$$

**Remark**

**8.**

- The first step of the previous proposition applies also to the perturbed Cramér-Lundberg process obtained by adding a Brownian term $\sigma {B}_{t}$ to (3). The second and third however use specific features available only if $\sigma =0$ (in particular that ${W}_{q}\left(0\right)=\frac{1}{p}>0$).
- Note that$$(S\left(x\right),G\left(x\right))=\left\{\begin{array}{cc}(p{W}_{q}\left(x\right),0)\hfill & ,\phantom{\rule{4pt}{0ex}}a=0\hfill \\ ({Z}_{q}\left(x\right),\frac{{C}_{q}\left(x\right)}{\mu})=({Z}_{q}\left(x\right),{Z}_{q,1}\left(x\right))\hfill & ,\phantom{\rule{4pt}{0ex}}a=\infty \hfill \end{array}\right.,$$where the last identity $\frac{{C}_{q}\left(x\right)}{\mu}={Z}_{q,1}\left(x\right)$ as well as the notation ${Z}_{q,1}$ appear in [15]. Our formula interpolates between the de Finetti and Shreve, Lehoczky and Gaver cases:$${J}_{x}^{a,b}=\left\{\begin{array}{cc}{V}^{b]}\left(x\right):=\frac{{W}_{q}\left(x\right)}{{W}_{q}^{\prime}(b+)}\hfill & ,\phantom{\rule{4pt}{0ex}}a=0\hfill \\ {V}_{k}^{b]}\left(x\right):=k{Z}_{q,1}\left(x\right)+{Z}_{q}\left(x\right)\frac{1-k{Z}_{q,1}^{\prime}\left(b\right)}{q{W}_{q}\left(b\right)}\hfill & ,\phantom{\rule{4pt}{0ex}}a=\infty \hfill \end{array}\right..$$(Again, for details, the reader is guided to [15] and references therein).
- The Equation (20) may also be expressed as$${J}_{x}^{a,b}=\frac{S(a,x)}{{\partial}_{b}S(a,b+)}-k\phantom{\rule{0.277778em}{0ex}}\left[{\partial}_{b}G(a,b+)\frac{S(a,x)}{{\partial}_{b}S(a,b+)}-G(a,x)\right],$$

**Remark**

**9.**

- a term depending on b, which is multiplied by the scale function ${Z}_{q}\left(x\right)$, and
- a term independent of b, ${Z}_{q,1}\left(x\right)$, which has been called sometimes “smooth Gerber-Shiu function”—see for example [35], and also “smooth harmonic extension of $w\left(x\right)=x,x\le 0$, see [2]. It is striking that the scale function and the Gerber-Shiu function are the same for these three problems. This begs for a formula for the Gerber-Shiu function which does not depend on the problem, and such a candidate is offered by the “LRZ harmonic extension” obtained in [36,37]. However, this has only been rigorously proved in particular examples – see for example [12].

## 4. The Optimal ${\mathit{a}}^{*},{\mathit{b}}^{*}$ for the Cramér-Lundberg Model with Exponential Claims

#### 4.1. Preliminary Remarks

**${b}^{*}=\infty $ can never be optimal**. For this purpose, one notes that ${lim}_{x\to \infty}\gamma \left(x\right)=0$. It follows that either ${b}^{*}=0$ or it is a critical point of $\left(0,\infty \right)\ni b\mapsto {J}_{0}^{a,b}$.

**Lemma**

**10.**

- As a consequence, picking $a\in \left\{0,\infty \right\}$ can never be optimal.
- For fixed$k\ge 1$ and $b\ge 0$, there exists a unique critical point ${a}_{k,b}\ne 0$ satisfying $\frac{\partial}{\partial a}{J}_{0}^{{a}_{k,b},b}=0.$

**our solution of the Lokka-Zervos problem provided in [15], where we restricted to these two types of policies, is irrelevant to the global optimization problem!**

#### 4.2. Determining the Candidate Maximal Arguments

**Proposition**

**11.**

- either ${b}^{*}=0$, in which case (27) implies that ${a}^{*}={a}_{k}:={a}_{k,0}$ satisfies$$\begin{array}{c}\hfill -kq{a}_{k}+p+\lambda k\frac{-1+{e}^{-\mu {a}_{k}}}{\mu}=0;\end{array}$$
- or ${b}^{*}\in \left(0,\infty \right)$ satisfies$$-\frac{p{\gamma}^{\prime}\left(b\right)}{\mu q{\theta}^{\prime}\left(b\right)}\left(q+\mu q\theta \left(b\right)\right)+p\gamma \left(b\right)+\lambda k\frac{-1+{e}^{-\frac{p{\gamma}^{\prime}\left(b\right)}{kq{\theta}^{\prime}\left(b\right)}}}{\mu}=0,\phantom{\rule{4pt}{0ex}}{\gamma}^{\u2033}\left(b\right)-\frac{{\gamma}^{\prime}\left(b\right)}{{\theta}^{\prime}\left(b\right)}{\theta}^{\u2033}\left(b\right)\le 0,$$$\frac{p{\gamma}^{\prime}\left(b\right)}{\mu q{\theta}^{\prime}\left(b\right)}>0,$ and$${a}^{*}=\frac{p{\gamma}^{\prime}\left({b}^{*}\right)}{k\mu q{\theta}^{\prime}\left({b}^{*}\right)}.$$

## 5. From Guess to Optimality

**Theorem**

**12.**

- A.
- If ${\left(\lambda +q\right)}^{2}<\lambda p\mu $, then, we have the following dichotomy.
- A1.
- The “cheap” equity regime, with ${b}^{*}=0$, holds for $1\le k\le {k}^{*}$, where ${k}^{*}$ is the unique solution of$$\begin{array}{cc}& \delta \left(k\right)=0,\mathrm{with}\left[1,\infty \right)\ni k\mapsto \delta \left(k\right):=\frac{\lambda +q}{\mu}+\lambda k\frac{-1+{e}^{-\frac{p\mu -\lambda -q}{qk}}}{\mu}.\hfill \end{array}$$$$\tilde{V}\left(x\right)=k{\left({a}_{k}+x\right)}^{+}{\mathbf{1}}_{x\le 0}+\left(x+k{a}_{k}\right){\mathbf{1}}_{x>0}.$$
- A2.
- For “expensive” equity i.e., $k>{k}^{*}$,
- The structure equation$$-\frac{p{\gamma}^{\prime}\left(b\right)}{\mu q{\theta}^{\prime}\left(b\right)}\left(q+\mu q\theta \left(b\right)\right)+p\gamma \left(b\right)+\lambda k\frac{-1+{e}^{-\frac{p{\gamma}^{\prime}\left(b\right)}{kq{\theta}^{\prime}\left(b\right)}}}{\mu}=0$$$$\begin{array}{c}\hfill {a}^{*}=\frac{p{\gamma}^{\prime}\left({b}^{*}\right)}{k\mu q{\theta}^{\prime}\left({b}^{*}\right)}.\end{array}$$
- The strategy consisting in injecting capital to take reserve to 0 (for levels above $-{a}^{*}$ satisfying (33)) and paying dividends with the barrier ${b}^{*}$ is optimal. The optimal function is$$\begin{array}{c}\hfill \tilde{V}\left(x\right)=\left\{\begin{array}{cc}\hfill k{\left({a}^{*}+x\right)}^{+},& \mathrm{if}x0,\hfill \\ \hfill kG({a}^{*},x)+k{a}^{*}S({a}^{*},x),& \mathrm{if}x\in \left[0,{b}^{*}\right],\hfill \\ \hfill \tilde{V}\left({b}^{*}\right)+x-{b}^{*},& \mathrm{if}x{b}^{*},\hfill \end{array}\right.\end{array}$$

- B.
- In the remaining case ${\left(\lambda +q\right)}^{2}\ge \lambda p\mu $, independently of $k\ge 1$, ${b}^{*}=0$ and the value function is got as in A1:$$\tilde{V}\left(x\right)=k{\left({a}_{k}+x\right)}^{+}{\mathbf{1}}_{x\le 0}+\left(x+k{a}_{k}\right){\mathbf{1}}_{x>0},$$

**Remark**

**13.**

**Remark**

**14.**

- A careful look at the upper-bound in 2.2.2 (in the proof) shows that ${\tilde{V}}^{\prime}(0+)<k={\tilde{V}}^{\prime}(0-)$. As a consequence:
- the heuristic intuition in ([14], 5.2) no longer holds true for our case ($\tilde{V}\left(0\right)>0$ and ${\tilde{V}}^{\prime}\left(0\right)=k$ fails to hold true);
- $\tilde{V}$ is only absolutely continuous (but not ${C}^{1}$) such that the comparison in Theorem 4 must be given among absolutely continuous super-solutions.

- The equality in (50) gives a way to characterize ${b}^{*}$ when $\tilde{V}$ is known.

- As $\mu \to 0$, the claims approach ∞ average. In a highly impacted market, the notion of equity expensiveness vanishes and it is not optimal to wait for any amount of time $b>0$ (Figure 2a).
- On the other hand (Figure 3a), as the discount parameter $q\to 0+$, for all p that may give a dichotomy (cf. A), $p\mu -\lambda >0$ such that the Equation (31) leads to ${k}^{*}=1$. In other words, absence of inflation makes every equity $k>1$ expensive. Conversely (Figure 3c), high inflation (large q) makes the notion of equity expensiveness vanish.

## 6. Proofs of the Results

#### 6.1. Proofs for the Value Function: Estimates, Dynamic Programming, Solution

**Proof**

**of Proposition 2.**

- The classical (no dividend, no injection) $\left(0,0\right)$ policy is an admissible strategy. The second result is a mere consequence of the definition of ${X}^{\pi}$.
- For $x\ge 0$, we consider the strategy consisting in no capital injection ($I=0$), then paying ${L}_{0}=x$ and (continuously) the premium and declare bankruptcy at the first (positive) claim. For this admissible strategy ${\pi}^{0}$, the ruin time ${\sigma}_{0-}^{x,{\pi}^{0}}={\tau}_{1},\phantom{\rule{4pt}{0ex}}\mathbb{P}-a.s.$ (the first jumping time except on the 0-probability event ${C}_{1}=0$). We have$$\tilde{V}\left(x\right)\ge \tilde{J}\left(x,{\pi}^{0}\right)=x+p\mathbb{E}[{\int}_{0}^{{\tau}_{1}}{e}^{-qt}dt]=x+\frac{p}{\lambda +q}.$$
- One merely notes that, for every $\pi \in \tilde{\Pi}\left(x\right)$,$$\left\{\begin{array}{cc}\hfill max\left\{{X}_{t-}^{\pi},0\right\}& \le x+pt+{\int}_{\left[0,t\right)}\left(d{I}_{s}-d{L}_{s}\right)\hfill \\ \hfill max\left\{{X}_{t}^{\pi},0\right\}& \le x+pt+{\int}_{[0,t]}\left(d{I}_{s}-d{L}_{s}\right)\hfill \end{array}\right.,\phantom{\rule{4pt}{0ex}}\forall t\in \left[0,{\sigma}_{0-}^{x,\pi}\right],\phantom{\rule{4pt}{0ex}}\mathbb{P}-a.s.,$$$${\int}_{[0,{\sigma}_{0-}^{x,\pi}}]{e}^{-qs}\left(d{L}_{s}-kd{I}_{s}\right)\le {\int}_{\left[0,{\sigma}_{0-}^{x,\pi}\right]}{e}^{-qs}\left(d{L}_{s}-d{I}_{s}\right)\le x+\frac{p}{q},\phantom{\rule{4pt}{0ex}}{\mathbb{P}}_{x}-a.s.$$

- 4.
- The first inequality follows by modifying an arbitrary strategy $\pi \in \tilde{\Pi}\left(x\right)$ into ${\pi}^{\epsilon}:=\left(L+\epsilon ,I\right)$ on ${\mathbb{R}}_{+}$ to get ${\pi}^{\epsilon}\in \tilde{\Pi}\left(x+\epsilon \right)$. For the last inequality, one modifies (on ${\mathbb{R}}_{+}$) an admissible strategy $\pi \in \tilde{\Pi}\left(x+\epsilon \right)$ into ${\pi}_{\epsilon}:=\left(L,I+\epsilon \right)\in \tilde{\Pi}\left(x\right)$.

**Proof of**

**Proposition 3.**

**Proposition**

**15.**

- $\varphi \left(x\right)\le {\varphi}_{n}\left(x\right)\le \varphi \left(x\right)+\frac{k}{n}$ on $\mathbb{R}$;
- ${\varphi}_{n}^{\prime}\in \left[1,k\right]$ on $\left[{a}_{\varphi},\infty \right)$;
- $\underset{n\to \infty}{lim}}{\varphi}_{n}^{\prime}={\varphi}^{\prime$ (pointwise a.e. on $\mathbb{R}$).

**Proof.**

**Proof of**

**Theorem 4.**

#### 6.2. Proofs for the Guess Step

**Proof of**

**Proposition 6.**

**Proof**

**of Lemma 7.**

#### 6.3. Proofs for the Optimal Couple

**Proof**

**of Lemma 10.**

**Proof**

**of Proposition 11.**

#### 6.4. Proof of the Main Theorem

**Proof**

**of Theorem 12**.

- $\frac{{\gamma}^{\prime}\left(b\right)}{{\theta}^{\prime}\left(b\right)}=-\frac{{\Phi}_{q}^{2}{e}^{{\Phi}_{q}b}-{\rho}_{-}^{2}{e}^{{\rho}_{-}b}}{\left({\Phi}_{q}-{\rho}_{-}\right){e}^{\left({\Phi}_{q}+{\rho}_{-}\right)b}}>0$ if and only if $b<\overline{b}:=\frac{1}{{\Phi}_{q}-{\rho}_{-}}log\left(\frac{{\rho}_{-}^{2}}{{\Phi}_{q}^{2}}\right)$;
- ${\partial}_{b}\frac{{\gamma}^{\prime}}{{\theta}^{\prime}}\left(b\right)=\frac{-1}{{\Phi}_{q}-{\rho}_{-}}\left(-{\rho}_{-}{\Phi}_{q}^{2}{e}^{-{\rho}_{-}b}+{\rho}_{-}^{2}{\Phi}_{q}{e}^{-{\Phi}_{q}b}\right)<0$, for all $b\in \left[0,\overline{b}\right)$.
- The continuous function $\left[0,\overline{b}\right]\ni b\mapsto \eta \left(b\right):=-\frac{p{\gamma}^{\prime}\left(b\right)}{\mu q{\theta}^{\prime}\left(b\right)}\left(q+\mu q\theta \left(b\right)\right)+p\gamma \left(b\right)+$$\lambda k\frac{-1+{e}^{-\frac{p{\gamma}^{\prime}\left(b\right)}{kq{\theta}^{\prime}\left(b\right)}}}{\mu}$ satisfies $\eta \left(0\right)=\delta \left(k\right)<0,\forall k>{k}^{*}$ and $\eta \left(\overline{b}\right)=p\gamma \left(\overline{b}\right)>0$. We let ${b}^{*}$ denote the first solution of $\eta \left(b\right)=0$.
- We consider $\left[0,\overline{b}\right]\ni b\mapsto \eta (b,a):=-qka-\frac{p{\gamma}^{\prime}\left(b\right)}{\mu q{\theta}^{\prime}\left(b\right)}\mu q\theta \left(b\right)+p\gamma \left(b\right)+\lambda k\frac{-1+{e}^{-\mu a}}{\mu}$. One notes that $\eta \left(b,{a}_{k}\right)=-p+p\gamma \left(b\right)-\frac{p{\gamma}^{\prime}\left(b\right)}{{\theta}^{\prime}\left(b\right)}\theta \left(b\right)$. Moreover, ${\partial}_{b}\eta \left(b,{a}_{k}\right)=-{\partial}_{b}\frac{p{\gamma}^{\prime}\left(b\right)}{{\theta}^{\prime}\left(b\right)}\theta \left(b\right)>0,\forall 0\le x<\overline{b}$ such that $\eta \left(b,{a}_{k}\right)>\eta \left(0,{a}_{k}\right)=0$, for all $\overline{b}>b>0$. In particular, $\eta \left({b}^{*},{a}_{k}\right)>0$. Owing to the monotonicity of $a\mapsto \eta \left({b}^{*},a\right)$, one deduces that ${a}^{*}>{a}_{k}$.

## Author Contributions

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**Figure 1.**Value function ${J}_{0}^{a,b*}$ as a function of a, with $b*=6.91$ fixed to the optimal de Finetti barrier, and $k=1.9488$ chosen so that $b*$ is also the optimal Shreve, Lehoczky and Gaver barrier. The parameters are $p=4,\lambda =1,q=1/10,\mu =2/5$. The optimal a for $b*$ fixed is $3.7353375$ and the value is ${J}_{0}^{a,b*}=7.2794258$. Even so, the guess is below the optimum obtained for $\left({a}^{*},{b}^{*}\right)$: $=\left(3.8473818,4.7859775\right)$ and leading to ${J}_{0}^{{a}^{*},{b}^{*}}=7.4977776$.

**Figure 4.**The optimal surface $({a}^{*},{b}^{*})$ as function of equity k for λ = q = μ = 1 and p = 5.

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Avram, F.; Goreac, D.; Li, J.; Wu, X.
Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model. *Mathematics* **2021**, *9*, 931.
https://doi.org/10.3390/math9090931

**AMA Style**

Avram F, Goreac D, Li J, Wu X.
Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model. *Mathematics*. 2021; 9(9):931.
https://doi.org/10.3390/math9090931

**Chicago/Turabian Style**

Avram, Florin, Dan Goreac, Juan Li, and Xiaochi Wu.
2021. "Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model" *Mathematics* 9, no. 9: 931.
https://doi.org/10.3390/math9090931