Optimal ratcheting of dividends in a Brownian risk model

The identification of the optimal way to pay out dividends from a surplus process to shareholders is a classical topic in actuarial science and mathematical finance. There is a natural trade-off between paying out gains as dividends to shareholders early and at the same time leaving sufficient surplus in order to safeguard future adverse developments and avoid ruin. Depending on risk preferences, the concrete situation and the simultaneous exposure to other risk factors such a problem can be formally stated in various different ways in terms of objective functions and constraints. In this paper we would like to follow the actuarial tradition of considering the surplus process as the free capital in an insurance portfolio at any point in time, and the goal is to maximize the expected sum of discounted dividend payments that can be paid until the surplus process goes below 0 (which is called the time of ruin). In such a formulation, the problem goes back to de Finetti [15] and Gerber [18], and has since then been studied in many variants concerning the nature of the underlying surplus process and constraints on the type of admissible dividend payment strategies, see e.g. Albrecher & Thonhauser [4] and Avanzi [8] for an overview. From a mathematical perspective, the problem turns out to be quite challenging, and was cast into the framework of modern stochastic control theory and the concept of viscosity solutions for corresponding Hamilton-Jacobi-Bellman equations over the last years, cf. Schmidli [25] and Azcue & Muler [10].

Among the variants of the general problem is to look for the optimal dividend payment strategy if the rate at which dividends are paid can never be reduced. This ratcheting constraint has often been brought up by practitioners and is in part motivated by the psychological effect that shareholders are likely to be unhappy about a reduction of dividend payments over time (see e.g. Avanzi et al. [9] for a discussion). One crucial question in this context is how much of the expected discounted dividends until ruin is lost if one respects such a ratcheting constraint, if that ratcheting is done in an optimal way. A first step in that direction was done in Albrecher et al. [3], where the consequences of ratcheting were studied under the simplifying assumption that a dividend rate can be fixed in the beginning and can be augmented only once during the lifetime of the process (concretely, when the surplus process hits some optimally chosen barrier for the first time). The analysis in that paper was both for a Brownian surplus process as well as for a surplus process of compound Poisson type. In our recent paper [2], we then provided the analysis and solution for the general ratcheting problem for the latter compound Poisson process, and it turned out that the optimal ratcheting dividend strategy does not lose much efficiency compared to the unconstrained optimal dividend payout performance, and also that the one-step ratcheting strategy studied earlier compares remarkably well to the overall optimal ratcheting solution. In this paper we would like to address the general ratcheting problem for the Brownian risk model. Such a model can be seen as a diffusion approximation of the compound Poisson risk model, but is also interesting in its own right. In particular, the fact that ruin with zero initial capital is immediate often leads to a more amenable analysis of stochastic control problems. In addition, the convergence of optimal strategies from a compound Poisson setting to the one for the diffusion approximation can be quite delicate, see e.g. Bäuerle [12], see also Cohen and Young [13] for a recent convergence rate analysis of simple uncontrolled ruin probabilities towards its counterparts for the diffusion limit.

On the mathematical level, the general ratcheting formulation leads to a fully two-dimensional stochastic control problem with all its related challenges, and it is only recently that in the context of insurance risk theory some first two-dimensional problems became amenable for analysis, see e.g. Albrecher et al. [1], Gu et al. [22], Grandits [21] and Azcue et al. [11]. In the present contribution we would like to exploit the more amenable nature of the ratcheting problem in the diffusion setting that push the analysis considerably further than was possible in [2]. In particular, we will use calculus of variation techniques to identify quite explicit formulas for the candidates of optimal strategies and provide various optimality results.

We would like to mention that optimal ratcheting strategies have been investigated in the framework of lifetime consumption in the mathematical finance literature, see e.g. Dybvig [16], Elie and Touzi [17], Jeon et al. [23] and more recently Angoshtari et al. [5]. However, the concrete model setup and correspondingly also the involved techniques are quite different to the one of the present paper.

The remainder of the paper is organized as follows. Section 2 introduces the model and the detailed formulation of the problem. It also provides some first basic results on properties of the value function under consideration. Section 3 derives the Hamilton-Jacobi-Bellman equations and characterization theorems for the value function for both a closed interval as well as a finite discrete set of admissible dividend payment rates. In Section 4 we prove that the optimal value function of the problem for discrete sets convergences to the one for a continuum of admissible dividend rates as the mesh size of the finite set tends to zero. In Section 5 we show that for finitely many admissible dividend rates, there exists an optimal strategy for which the change and non-change regions have only one connected component (this corresponds to the extension of one-dimensional threshold strategies to the two-dimensional case). We also provide an implicit equation defining the optimal threshold function for this case. Subsequently, we turn to the case of a continuum of admissible dividend rates and use calculus of variation techniques to identify the optimal curve splitting the state space into a change and a non-change region as the unique solution of an ordinary differential equation. We show that the corresponding dividend strategy is $\varepsilon$-optimal, in the sense that there exists a known sequence of curves such that the corresponding value functions converge uniformly to the optimal value function of the problem. Section 6 contains a numerical illustration of the optimal strategy and its performance relative to the one of for the unconstrained dividend problem and the one where the dividend rate can only be increased once. Section 7 concludes.

Some technical proofs are delegated to an appendix.

Assume that the surplus process of a company is given by a Brownian motion with drift

$$X_{t}=x+\mu t+\sigma W_{t}$$

where $W_{t}$ is a standard Brownian motion, and $\mu,\sigma>0$ are given constants. Let $(\Omega,\mathcal{F},\left(\mathcal{F}_{t}\right)_{t\geq 0},\mathcal{P})$ be the complete probability space generated by the process $X_{t}$.

The company uses part of the surplus to pay dividends to the shareholders with rates in a set $S\subset[0,\overline{c}]$, where $0\leq\overline{c}\in S$ is the maximum dividend rate possible. Let us denote by $C_{t}$ the rate at which the company pays dividends at time $t$. Given an initial surplus $X_{0}=x$ and a minimum dividend rate $c\in S$ at $t=0$, a dividend ratcheting strategy is given by $C=\left(C_{t}\right)_{t\geq 0}$ and it is admissible if it is non-decreasing, right-continuous, adapted with respect to the filtration $\left(\mathcal{F}_{t}\right)_{t\geq 0}$ and it satisfies $C_{t}\in S$ for all $t$. The controlled surplus process can be written as

$$X_{t}^{C}=X_{t}-\int_{0}^{t}C_{s}ds.$$

(1)

Define $\Pi_{x,c}^{S}$ as the set of all admissible dividend ratcheting strategies with initial surplus $x\geq 0$ and minimum initial dividend rate $c\in S$.  Given $C\in\Pi_{x,c}^{S}$, the value function of this strategy is given by

$$J(x;C)=\mathbb{E}\left[\int_{0}^{\tau}e^{-qs}C_{s}ds\right],$$

where $q>0$ and $\tau=\inf\left\{t\geq 0:X_{t}^{C}<0\right\}$ is the ruin time. Hence, for any initial surplus $x\geq 0$ and initial dividend rate $c$, our aim is to maximize

$$V^{S}(x,c)=\sup_{C\in\Pi_{x,c}^{S}}J(x;C).$$

(2)

It is immediate to see that $V^{S}(0,c)=0$ for all $c\in S.$

The following Lemma states the dynamic programming principle, its proof is similar to the one of Lemma 1.2 in Azcue and Muler [10].

We now state a straightforward result regarding the boundedness and monotonicity of the optimal value function.

Proof. Since the discounted value of paying the maximum rate $\overline{c}$ up to infinity is $\overline{c}/q,$ we conclude the boundedness result.

On the one hand $V^{S}(x,c)$ is non-increasing in $c$ because given $c_{1}<c_{2}$ we have $\Pi_{x,c_{2}}^{S}\subset\Pi_{x,c_{1}}^{S}$ for any $x\geq 0$. On the other hand, given $x_{1}<x_{2}$ and an admissible ratcheting strategy $C^{1}\in\Pi_{x_{1},c}^{S}$ for any $c\in S$, let us define $C^{2}\in\Pi_{x_{2},c}^{S}$ as $C_{t}^{2}=C_{t}^{1}$ until the ruin time of the controlled process $X_{t}^{C^{1}}$ with $X_{0}^{C^{1}}=x_{1}$, and pay the maximum rate $\overline{c}$ afterwards. Thus, $J(x;C_{1})\leq J(x;C_{2})$ and we have the result. $\blacksquare$

The Lipschitz property of the function $V_{NR}$ introduced in Remark 2.1 can now be used to prove a first result on the regularity of the function $V^{S}.$

The proof is given in the appendix.

In this section we introduce the Hamilton-Jacobi-Bellman (HJB) equation of the ratcheting problem for $S\subset[0,\infty)$, when $S$ is either a closed interval or a finite set. We show that the optimal value function $V$ defined in (2) is the unique viscosity solution of the corresponding HJB equation with boundary condition $\overline{c}/q$ when $x$ goes to infinity, where $\overline{c}=\max S.$

First, consider the case $S=\left\{c\right\}$. In this case, the unique admissible strategy consists of paying a constant dividend rate $c$ up to the ruin time. Correspondingly, the value function $V^{\left\{c\right\}}(x,c)$ is the unique solution of the second order differential equation

$$\mathcal{L}^{c}(W):=\frac{\sigma^{2}}{2}\partial_{xx}W+(\mu-c)\partial_{x}W-qW+c=0$$

(3)

with boundary conditions $V^{\left\{c\right\}}(0,c)=0$ and $\lim_{x\rightarrow\infty}$ $V^{\left\{c\right\}}(x,c)=c/q.$ The solutions of (3) are of the form

$$\frac{c}{q}+a_{1}e^{\theta_{1}(c)x}+a_{2}e^{\theta_{2}(c)x}\text{ with }a_{1},a_{2}\in{\mathbb{R}},$$

(4)

where $\theta_{1}(c)>0$ and $\theta_{2}(c)<0$ are the roots of the characteristic equation

$$\frac{\sigma^{2}}{2}z^{2}+(\mu-c)z-q=0$$

associated to the operator $\mathcal{L}^{c}$, that is

$$\theta_{1}(c):=\frac{c-\mu+\sqrt{(c-\mu)^{2}+2q\sigma^{2}}}{\sigma^{2}},\quad\theta_{2}(c):=\text{$\frac{c-\mu-\sqrt{(c-\mu)^{2}+2q\sigma^{2}}}{\sigma^{2}}$.}$$

(5)

In the following remark, we state some basic properties of $\theta_{1}$ and $\theta_{2}.$

$$\frac{c}{q}\left(1-e^{\theta_{2}(c)x}\right)+a(e^{\theta_{1}(c)x}-e^{\theta_{2}(c)x})\ \text{with }a\in{\mathbb{R}}.$$

(6)

And finally, the unique solution of $\mathcal{L}^{c}(W)=0$ with boundary conditions $W(0)=0$ and $\lim_{x\rightarrow\infty}$ $W(x)=c/q\$corresponds to $a=0,$ so that

$$V^{\left\{c\right\}}(x,c)=\frac{c}{q}\left(1-e^{\theta_{2}(c)x}\right).$$

(7)

We have that $V^{\left\{c\right\}}(\cdot,c)$ is increasing and concave.

In this section we prove that the optimal value functions of finite ratcheting strategies approximate the optimal value function of the continuous case as the mesh size of the finite sets goes to zero.

Consider for $n\geq 0$, a sequence of sets $\mathcal{S}^{n}$ (with $k_{n}$ elements) of the form 

$$\mathcal{S}^{n}=\left\{c_{1}^{n}=\underline{c}<c_{2}^{n}<\cdots<c_{k_{n}}^{n}=\overline{c}\right\}.$$

satisfying $\mathcal{S}^{0}=\left\{\underline{c},\overline{c}\right\}$, $\mathcal{S}^{n}\subset\mathcal{S}^{n+1}$ and mesh-size $\delta(\mathcal{S}^{n}):=\max_{k=2,k_{n}}\left(c_{k}^{n}-c_{k-1}^{n}\right)\searrow 0$ as $n$ goes to infinity.

Let us extend the definition of $V^{\mathcal{S}^{n}}$ to the function $V^{n}:[\underline{c},\infty)\times[0,\overline{c}]\rightarrow{\mathbb{R}},$ as

$$V^{n}(x,c)=V^{\mathcal{S}^{n}}(x,\widetilde{c}^{n}),$$

(12)

where

$$\widetilde{c}^{n}=\min\{c_{i}^{n}\in\mathcal{S}^{n}:c_{i}^{n}\geq c\}.$$

(13)

We will prove that $\lim_{n\rightarrow\infty}V^{n}(x,c)=V^{[\underline{c},\overline{c}]}(x,c)$ for any $(x,c)\in[0,\infty)\times[\underline{c},\overline{c}]$ and we will study the uniform convergence of this limit.

Since $V^{n}\leq V^{n+1}\leq V^{[\underline{c},\overline{c}]}$, there exists the limit function

$$\overline{V}(x,c):=\lim_{n\rightarrow\infty}V^{n}(x,c).$$

(14)

Later on, we will show that $\overline{V}=V^{[\underline{c},\overline{c}]}$. Note that $\overline{V}(x,c)$ is non-increasing in $c$ with $\overline{V}(x,\overline{c})=V(x,\overline{c})$, and non-decreasing in $x$ with $\lim_{x\rightarrow\infty}$ $\overline{V}(x,c)=\overline{c}/q$. With the same proof the one for Proposition 6.1 of [2], we have the following proposition:

With this, we can obtain the main result of this section.

Proof. Note that $\overline{V}(x,c)$ is a limit of value functions of admissible strategies, so in order to satisfy the assumptions of Theorem 3.4, it remains to see that $\overline{V}$ is a viscosity supersolution of (8) at any point $(x_{0},c_{0})$ with $x_{0}>0.$ $\partial_{c}\overline{V}(x_{0},c_{0})\leq 0$ in the viscosity sense because $\overline{V}$ is non-increasing in $c$; so it is sufficient to show that $\mathcal{L}^{c_{0}}(\overline{V})(x_{0},c_{0})\leq 0$ in the viscosity sense. Let $\varphi$ be a test function for viscosity supersolution of (8) at $(x_{0},c_{0})$, i.e. a (2,1)-differentiable function $\varphi$ with

$$\overline{V}(x,c)\geq\varphi(x,c)\text{ and }\overline{V}(x_{0},c_{0})=\varphi(x_{0},c_{0})\text{.}$$

(15)

In order to prove that $\mathcal{L}^{c}(\varphi)(x_{0},c_{0})\leq 0$, consider now, for $\gamma>0$ small enough,

$$\varphi_{\gamma}(x,c)=\varphi(x,c)-\gamma(x-x_{0})^{4}.$$

Given $n\geq 1,$ let us consider $\widetilde{c}_{0}^{n}$ as defined in (13),

$$a_{n}^{\gamma}:=\min\{V^{n}(x,\widetilde{c}_{0}^{n})-\varphi_{\gamma}(x,\widetilde{c}_{0}^{n}):x\in[0,x_{0}+1]\},$$

$$x_{n}^{\gamma}:=\arg\min\{V^{n}(x,\widetilde{c}_{0}^{n})-\varphi_{\gamma}(x,\widetilde{c}_{0}^{n}):x\in[0,x_{0}+1]\},$$

and

$$b_{n}^{\gamma}:=\max\{\overline{V}(x,\widetilde{c}_{0}^{n})-V^{n}(x,\widetilde{c}_{0}^{n}):x\in[0,x_{0}+1]\}.$$

Since $\widetilde{c}_{0}^{n}\searrow c_{0}$ and, from Proposition 4.1, $\lim_{n\rightarrow\infty}a_{n}^{\gamma}=0$ and $\lim_{n\rightarrow\infty}b_{n}^{\gamma}=0$, we also have that $\lim_{n\rightarrow\infty}~{}x_{n}^{\gamma}=x_{0}$ because

$$\begin{array}[c]{lll}0&=&V^{n}(x_{n}^{\gamma},\widetilde{c}_{0}^{n})-\left(\varphi_{\gamma}(x_{n}^{\gamma},\widetilde{c}_{0}^{n})+a_{n}^{\gamma}\right)\\ &=&\left(V^{n}(x_{n}^{\gamma},\widetilde{c}_{0}^{n})-\overline{V}(x_{n}^{\gamma},\widetilde{c}_{0}^{n})\right)+\left(\overline{V}(x_{n}^{\gamma},\widetilde{c}_{0}^{n})-\varphi_{\gamma}(x_{n}^{\gamma},\widetilde{c}_{0}^{n})\right)-a_{n}^{\gamma}\\ &\geq&-b_{n}^{\gamma}+0-a_{n}^{\gamma}+\gamma(x_{n}^{\gamma}-x_{0})^{4}\end{array}$$

and then

$$(x_{n}^{\gamma}-x_{0})^{4}\leq\frac{a_{n}^{\gamma}+b_{n}^{\gamma}}{\gamma}\rightarrow 0~{}\text{as }n\rightarrow\infty.$$

Note that $\overline{\varphi}^{n}(\cdot)=\varphi_{\gamma}(\cdot,\widetilde{c}_{0}^{n})+a_{n}^{\gamma}$ is a test function for viscosity supersolution of $V^{n}(\cdot,\widetilde{c}_{0}^{n})$ in Equation (11) at the point $x_{n}^{\gamma}$ because

$$\varphi_{\gamma}(x_{n}^{\gamma},\widetilde{c}_{0}^{n})+a_{n}^{\gamma}=V^{n}(x_{n}^{\gamma},\widetilde{c}_{0}^{n})\text{ and }\varphi_{\gamma}(x,\widetilde{c}_{0}^{n})+a_{n}^{\gamma}\leq V^{n}(x,\widetilde{c}_{0}^{n})\text{ for }x\in[0,x_{0}+1].$$

And so

$$\mathcal{L}^{\widetilde{c}_{0}^{n}}(\varphi_{\gamma})(x_{n}^{\gamma},\widetilde{c}_{0}^{n})=\mathcal{L}^{\widetilde{c}_{0}^{n}}(\overline{\varphi}^{n})(x_{n}^{\gamma})+qa_{n}^{\gamma}\leq qa_{n}^{\gamma}.$$

Since $(x_{n}^{\gamma},c_{n})\rightarrow(x_{0},c_{0})$, $\overline{\varphi}^{n}(\cdot)=\varphi_{\gamma}(\cdot,\widetilde{c}_{0}^{n})+a_{n}^{\gamma}\rightarrow\varphi_{\gamma}(\cdot,c_{0})$ as $n\rightarrow\infty$ and $\varphi_{\gamma}$ is (2,1)-differentiable, one gets

$$\mathcal{L}^{c_{0}}(\varphi_{\gamma})(x_{0},c_{0})=\lim_{n\rightarrow\infty}\mathcal{L}^{\widetilde{c}_{0}^{n}}(\overline{\varphi}^{n})(x_{n}^{\gamma})\leq 0.$$

Finally, as

$$\partial_{x}\varphi_{\gamma}(x_{0},c_{0})=\partial_{x}\varphi(x_{0},c_{0})~{}\text{and }\partial_{xx}\varphi_{\gamma}(x_{0},c_{0})=\partial_{xx}\varphi(x_{0},c_{0})$$

and $\varphi_{\gamma}\nearrow\varphi$ as $\gamma\searrow 0$, we obtain that $\mathcal{L}^{c_{0}}(\varphi)(x_{0},c_{0})\leq 0$ and the result follows. $\blacksquare$

We show first that, regardless whether $S$ is finite or an interval with $\max S=\overline{c}$, the optimal strategy for sufficiently small $\overline{c}$ is to immediately start paying dividends at the maximum rate $\overline{c}$.

Proof. If we call $W(x,c):=V^{\{\overline{c}\}}(x,\overline{c}),$ then we know that $\mathcal{L}^{\overline{c}}(W)(x,c)=0$. Since $W(0,\overline{c})=0,$ $\lim_{x\rightarrow\infty}W(x,c)=\overline{c}/q$ and $\partial_{c}W(x,c)=0$, then by Theorem 3.3 and Theorem 3.5 it is enough to prove that $\mathcal{L}^{c}(W)(x,c)\leq 0$ for $c\in S.$ But, by (7) and (5)

$$\begin{array}[c]{lll}\mathcal{L}^{c}(W)(x,c)&=&\mathcal{L}^{\overline{c}}(W)(x,c)+(\overline{c}-c)(\partial_{x}W(x,c)-1)\\ &=&(\overline{c}-c)(-\frac{\overline{c}}{q}\theta_{2}(\overline{c})e^{\theta_{2}(\overline{c})x}-1)\\ &\leq&(\overline{c}-c)(-\frac{\overline{c}}{q}\theta_{2}(\overline{c})-1)\\ &\leq&0\end{array}$$

for $c\leq\overline{c}\leq\frac{q\sigma^{2}}{2\mu}$. $~{}\blacksquare$

Let us first address the case of $S=\left\{c_{1},c_{2},....,c_{n}\right\}$ with $0\leq c_{1}<c_{2}<....<c_{n}=\overline{c}$. We introduce the concept of strategies with a threshold structure for each level $c_{i}\in S$ and prove that there exists an optimal dividend payment strategy and has this form. Later we extend the concept of strategies with this type of structure to the case $S=[\underline{c},\overline{c}]$ by means of a curve in the state space $[0,\infty)\times[\underline{c},\overline{c}]$ and look for the curve which maximizes the expected discounted cumulative dividends.