Abstract
This paper studies an optimal dividend problem for a company with non-exponential discounting. The surplus process is described by a dual model, and the target is to find a dividend strategy that maximizes the expected discounted value of dividends until ruin. The non-exponential discount function leads to a time-inconsistent problem. We aim at seeking the equilibrium strategy derived by taking our problem as a non-cooperate game, which is a time-consistent strategy. An extended Hamilton–Jacobi–Bellman equation system and a verification theorem are provided to derive the equilibrium strategy and the equilibrium value function. For the case of pseudo-exponential discount function, closed-form expressions for the equilibrium strategy and the equilibrium value function are derived. In addition, some numerical illustrations of our results are showed.
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Notes
A barrier strategy is that, when the controlled surplus is above a barrier, whatever amount exceeds the barrier level is paid out as dividend, but no dividend is paid out when the controlled surplus is below the barrier.
We note that the classical C–L model, \(R_{t}=x+\mu t-S_{t}\), is a Lévy process with sample paths that are skip-free upwards. However, the sample paths of stochastic process, in Eq. (1), are skip-free downwards, and if we turn the sample paths of C–L model upside down (rotate \(180^{\circ }\)) and look at them from right to left, we get the shape of sample paths of model (1) (see [16]). Because of the symmetry of the two processes’ sample paths, the model (1) is called the dual model of the classical C–L model in the actuarial literature. About the more detailed discussion of the connection between the classical C–L model and the dual model, we can refer to Avanzi et al. [13], Afonso et al. [16], Dimitrova et al. [17].
We can also state the time-inconsistent strategy as follows: We determine the optimal strategy \(\tilde{\ell }=\{\tilde{l}(t)\}_{t\ge 0}\) for the objective functional \(J(0, x, \ell )\) at initial point (0, x); and at some intermediate time s, we also determine the optimal strategy \(\bar{\ell }=\{\bar{l}(t)\}_{t\ge s}\) at state \((s, R_s^{\tilde{\ell }})\). If \(\tilde{l}(t)\ne \bar{l}(t)\) at least some \(t\ge s\), then \(\tilde{l}\) is a time-inconsistent strategy.
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Acknowledgments
The authors are grateful to the two anonymous referees for valuable comments for the revision of the paper. This research is partially supported by Grants of the National Natural Science Foundation of China (Nos. 71231008, 71201173), Guangdong Natural Science for Research Team (2014A030312003), Guangdong Natural Science for Distinguished Young Scholar and Natural Science Foundation of Guangdong Province of China (No. S2013010011959).
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Communicated by Moawia Alghalith.
Appendices
Appendix 1: Proof of Theorem 3.1
We first show that V is the value function corresponding to \(\hat{\ell }\), i.e., that \(V (t, x) = J(t, x, \hat{\ell })\). Note that
By Dynkin’s Theorem for \(u^{rs}\), we have
Thus, using (34) and boundary condition \(u^{rs}(t,0)=0\), we have
and
Next, by the extended HJB equation system for V, we have
and (34). Thus we have equation
Using Dynkin’s Theorem for V, we thus have
Hence, we get
By using Dynkin’s Theorem for \(u^s\), we have
Using (35), (36) and boundary conditions for V and \(u^s\), we can derive
We now show that \(\hat{\ell }\) is indeed an equilibrium strategy. For any \(h > 0\) and an arbitrary \(l\in [0, M]\), define the strategy \(\ell _h\) as in Definition 2.2. Without loss of generality, suppose that \(t+h<\tau _{t,x}^{\ell _h}\). Then we have
Using Dynkin’s Theorem for V, we have
Then
Noting the fact that \(V (t, x) = J(t, x, \hat{\ell })\), we obtain
So
The last inequality follows from
\(\square \)
Appendix 2: Proof of Proposition 3.1
Note that
and
Then
Thus,
and
Inserting (37)–(38) into equations of Definition 3.1, we complete the proof. \(\square \)
Appendix 3: Proof of Lemma 4.1
Note that
- (i):
-
If \(\xi _5-\xi _1>0\), then \(A_1<0\).
- (ii):
-
If \(\xi _5-\xi _1<0\), then \((\xi _5-\xi _1)\hbox {e}^{\xi _1b}>\xi _5-\xi _1\) and \((\xi _2-\xi _5)\hbox {e}^{\xi _2b}>\xi _2-\xi _5\).
So
and then \(A_1<0\).
Similarly, we can prove \(A_2<0,\;\; B_1<0\), and \(B_2<0.\) \(\square \)
Appendix 4: Proof of Lemma 4.2
\((\mathbf {i})\) We only prove the first inequality \((P+1)\xi _2 > \xi _5\). The second one is similar. Inequality \((P+1)\xi _2 > \xi _5\) is equivalent to \(\frac{M}{\rho _1} < \frac{1}{\xi _2} - \frac{1}{\xi _5}.\)
Firstly, note that
In addition, according to Cauchy inequality, we have
Next, we split the problem into the following three situations.
(1) If \((\mu +M)\delta -\lambda -\rho _1>0\) and \(\mu \delta -\lambda -\rho _1<0,\) then we have
i.e.,
(2) If \((\mu +M)\delta -\lambda -\rho _1\le 0\), we have \(\mu \delta -\lambda -\rho _1<0.\) Hence, we have
i.e.,
(3) If \(\mu \delta -\lambda -\rho _1\ge 0,\) we have \((\mu +M)\delta -\lambda -\rho _1>0.\) Thus
Combining (1), (2) and (3), we complete the proof of \((\mathbf {i})\).
\((\mathbf {ii})\) If \(\omega P+(1-\omega )Q+1<0\), we have \(F(0)>0.\) Note
According to \((\mathbf {i})\), we have
Furthermore,
Therefore, \(F(b)=0\) has a unique positive root. The proof of \((\mathbf {ii})\) is completed. \(\square \)
Appendix 5: Proof of Theorem 4.1
\((\mathbf {i})\) We can verify that V(x) defined by (32) is a continuously differentiable concave function and satisfies \(V(0)=0\). Since
\(V^{\prime }(x)\le 1\) for all \(x\ge 0\) and
In addition, we know that the function V given by (32) satisfies
Adding the inequality (39) to the equality (40), we can obtain (12).
\((\mathbf {ii})\) \(V(0)=0\) is obvious. The first-order derivative of V(x) given by (33) is:
From Lemma 4.1, we have that \(V^{\prime }(x)>0\) for all \(x>0\), which implies that V(x) is a strictly increasing function.
In addition, we can derive the second-order derivative of V(x) as follows:
From Lemma 4.1, we have that \(V^{\prime \prime }(x)<0\) for all \(x\ge b\). Now, we consider the case when \(0<x<b\). Note that
Thus \(V^{\prime \prime \prime }(x)>0\) and \(V^{\prime \prime }(x)\) is increasing on interval (0, b). Then we have that \(V^{\prime \prime }(x)< V^{\prime \prime }(b-)\) for \(0<x<b\).
Besides, from (16), (18) and (19), we have
and
Noting that \(V^{\prime }(b)=1\), we have \(\mu V^{\prime \prime }(b-)=(\mu +M)V^{\prime \prime }(b+)\le 0. \) Therefore, \(V^{\prime \prime }(x)< 0\) for \(0<x<b\).
In conclusion, V(x) defined by (33) is an increasing and continuously differentiable concave function on \(]0, +\infty [\). This implies the uniqueness of b.
In addition, we can adopt the same way as (i) to verify that V(x) given by (33) satisfies Eq. (12), and here we omit it. \(\square \)
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Li, Y., Li, Z. & Zeng, Y. Equilibrium Dividend Strategy with Non-exponential Discounting in a Dual Model. J Optim Theory Appl 168, 699–722 (2016). https://doi.org/10.1007/s10957-015-0742-8
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DOI: https://doi.org/10.1007/s10957-015-0742-8
Keywords
- Non-exponential discount function
- Equilibrium strategy
- Dividend payment
- Dual model
- Hamilton–Jacobi–Bellman equation