## Optimal Investment and Proportional Reinsurance Strategies to Minimize the Probability of Drawdown Under Ambiguity Aversion

Zhao Yuying,, Wen Yuzhen,

 基金资助: 国家自然科学基金.  11501319中国博士后科学基金.  2015M582064山东省自然科学基金.  ZR-2020MA035山东省自然科学基金.  ZR2015AL013

 Fund supported: the NSFC.  11501319the China Postdoctoral Science Foundation.  2015M582064the NSF of Shandong Province.  ZR-2020MA035the NSF of Shandong Province.  ZR2015AL013

Abstract

In this paper, we consider the optimal investment and reinsurance control problem for insurers with ambiguity, and we obtain the minimum drawdown probability, optimal robust investment-reinsurance strategies and the associated drift distortion. Moreover, some numerical examples are presented to show the impact of model parameters on the optimal results.

Keywords： Ambiguity aversion ; Probability of drawdown ; Optimal robust investment and reinsurance strategies ; Drift distortion

Zhao Yuying, Wen Yuzhen. Optimal Investment and Proportional Reinsurance Strategies to Minimize the Probability of Drawdown Under Ambiguity Aversion. Acta Mathematica Scientia[J], 2021, 41(4): 1147-1165 doi:

## 1 引言

### 2.1 保险金融市场的模型假设

$$${\rm d}C_t = a{\rm d}t-b{\rm d}W_t,$$$

$\rm(3) $$(2.4) 式关于 U_t^{\pi, q} 有唯一的强解. 则称 (\pi_t, q_t) 是可容许的, 将可容许策略集记为 {\cal D} . ### 2.2 Drawdown概率 定义 t 时刻的最大盈余值 M_t $$M_t = \max\Big\{\sup\limits_{0\leq s\leq t}U_s^{\pi, q}, M_0\Big\},$$ 其中 M_0 = m>0 . 注意我们允许盈余过程具有过去的财务状况, 并且 m 不小于定义中的初始盈余, 即 m\ge u . 这里的drawdown一词是指盈余过程的价值达到其最大值 M_t$$ \alpha\in[0, 1]$倍. 定义相应的击中时

$$$\tau_\alpha = \inf\{t\ge0:U^{\pi, q}(t)\leq\alpha M_t\}.$$$

$$$u_s = \frac{(\eta-\theta)a}{r}.$$$

### 2.3 保险金融市场的模糊性与目标函数

$$${\cal Q}: = \{Q|Q\sim P\}.$$$

$h(Q\parallel P) = E^Q[\ln\frac{{\rm d}Q}{{\rm d}P}]$为相对熵函数, 表示对测度$P$的惩罚.

$$$E^P\left[\exp\left(\frac{1}{2}\int_{0}^{T}[\beta(s)]^2{\rm d}s\right)\right]<\infty,$$$

$$$E^P\left[\exp\left(\frac{1}{2}\int_{0}^{T}[\gamma(s)]^2{\rm d}s\right)\right]<\infty.$$$

$$${\rm d}B_t^S = \beta(t){\rm d}t+{\rm d}B_t^Q,$$$

$$${\rm d}W_t = \gamma(t){\rm d}t+{\rm d}W_t^Q,$$$

### 3.1 验证定理

$\begin{eqnarray} 0& = &\inf\limits_{\pi, q}\sup\limits_{Q}\bigg\{ -\frac{1}{2\varepsilon}\beta^2 -\frac{1}{2\varepsilon}\gamma^2 [ru+(\mu+\sigma\beta-r)\pi{}\\ &&+(q\eta-\eta+\theta)a+q\gamma b]h_u+\frac{1}{2}(q^2b^2+\sigma^2\pi^2)h_{uu}\bigg\}; \end{eqnarray}$

$\rm(3) $$h(\alpha m, m) = 1 , h(u_s, m) = 0$$ m\geq u_s$, $h_m(m, m) = 0 $$m<u_s ; \rm(4)$$ \Pi(u, m)$, ${\bf Q}(u, m)$对任意$u\in(\alpha m, \min\{u_s, m\})$使得(3.1) 式取得下确界, ${\bf B}(\pi, u, m)$对任意$\pi\in{{\Bbb R}}$, $u\in(\alpha m, \min\{u_s, m\})$使得(3.1) 式取得上确界, $\Gamma(q, u, m)$对任意$q\in{{\Bbb R}}$, $u\in(\alpha m, \min\{u_s, m\})$使得(3.1) 式取得上确界;

$\rm(5) $$\Pi(u, m) = {\bf Q}(u, m) = {\bf B}(\pi, u, m) = \Gamma(q, u, m) = 0 , 如果 u\not\in(\alpha m, \min\{u_s, m\}) ; \rm(6)$$ \Pi$, ${\bf Q}$在区间${\cal O}$上有界且Lipschitz连续；${\bf B}$, $\Gamma $${\cal O} 上有界. 则在区间 {\cal O}$$ \psi = h$, 并且$\Pi$, ${\bf Q}$, ${\bf B}$, $\Gamma$为最优马尔可夫控制.

见附录B.

### 3.2 模糊厌恶下的最小Drawdown概率

$$$\inf\limits_{\pi, q\in{\cal D}}\bigg\{\frac{1}{2}\sigma^2(\varepsilon\psi_u^2+\psi_{uu})\pi^2 +(\mu-r)\psi_u\pi+ru\psi_u +\frac{1}{2}b^2(\varepsilon\psi_u^2+\psi_{uu})q^2 +(q\eta-\eta+\theta)a\psi_u\bigg\} = 0.$$$

$$$\pi^\ast = -\frac{\mu-r}{\sigma^2}\cdot\frac{\psi_u}{\varepsilon\psi_u^2+\psi_{uu}},$$$

$$$q^\ast = -\frac{a\eta}{b^2}\cdot\frac{\psi_u}{\varepsilon\psi_u^2+\psi_{uu}}.$$$

$$$\beta^\ast = -\frac{\mu-r}{\sigma}\cdot\frac{\varepsilon\psi_u^2}{\varepsilon\psi_u^2+\psi_{uu}},$$$

$$$\gamma^\ast = -\frac{a\eta}{b}\cdot\frac{\varepsilon\psi_u^2}{\varepsilon\psi_u^2+\psi_{uu}}.$$$

$$$(R+G)\cdot\frac{\psi_u^2}{\varepsilon\psi_u^2+\psi_{uu}}-(ru-a\eta+a\theta)\psi_u = 0,$$$

$\rm(1)$如果$u_s\leq m$, 则盈余过程(2.14) 的模糊厌恶下最小drawdown概率为

$$$\psi(u, m) = \left\{\begin{array}{ll} { } \frac{1}{\varepsilon}\ln\left[(1-e^\varepsilon)\frac{g_{1}(u, m)}{g_{2}(u_s, m)}+e^\varepsilon\right], \alpha m\leq u<\max\{\alpha m, u_1\}, \\ { } \frac{1}{\varepsilon}\ln\left[(1-e^\varepsilon)\frac{g_{2}(u, m)}{g_{2}(u_s, m)}+e^\varepsilon\right], \max\{\alpha m, u_1\}\leq u\leq u_s\leq m. \end{array}\right.$$$

$\rm(2) $$m<u_s 时, 如果 \max\{\alpha m, u_1\}\leq m<u_s , 则盈余过程(2.14) 的模糊厌恶下最小drawdown概率为 $$\psi(u, m) = \left\{\begin{array}{ll} { }\frac{1}{\varepsilon}\ln\left[(1-e^\varepsilon)k_{1}(m)\frac{g_{1}(u, m)}{g_{2}(u_s, u_s)}+e^\varepsilon\right], \alpha m\leq u<\max\{\alpha m, u_1\}, \\ { }\frac{1}{\varepsilon}\ln\left[(1-e^\varepsilon)k_{1}(m)\frac{g_{2}(u, m)}{g_{2}(u_s, u_s)}+e^\varepsilon\right], \max\{\alpha m, u_1\}\leq u\leq m\leq u_s, \end{array}\right.$$ 其中 如果 \alpha m<m<\max\{\alpha m, u_1\} , 则盈余过程(2.14) 的模糊厌恶下最小drawdown概率为 $$\psi(u, m) = \frac{1}{\varepsilon}\ln\left[(1-e^\varepsilon)k_{2}(m) \frac{g_{1}(u, m)}{g_{2}(u_s, u_s)}\right],$$ \forall u\in[\alpha m, m] . 其中 模糊厌恶下的最优鲁棒投资策略和最优鲁棒再保险策略为 $$(\pi^\ast(u), q^\ast(u)) = \left\{\begin{array}{ll} { } (\tilde{\pi}(u), 1), &\alpha m\leq u<\max\{\alpha m, u_1\}, \\ (\hat{\pi}(u), \hat{q}(u)), &\max\{\alpha m, u_1\}\leq u\leq\min\{m, u_s\}. \end{array}\right.$$ 最优扭曲漂移分别为 $$\beta^\ast = \left\{\begin{array}{ll} { }-\frac{(ru+a\theta)-\sqrt{(ru+a\theta)^2+2b^2R}}{\mu-r}\cdot\varepsilon\psi_u\sigma, &\alpha m\leq u<\overline{M}, \\ { }-\frac{(\mu-r)(e^\varepsilon-1)(R+G+r)(ru-a\eta+a\theta)^{\frac{R+G+r}{r}}} {\sigma(R+G)\left[(r\alpha m-a\eta+a\theta)^{\frac{R+G+r}{r}} +(e^\varepsilon-1)(ru-a\eta+a\theta)^{\frac{R+G+r}{r}}\right]}, &\overline{M}\leq u\leq\underline{M}. \end{array}\right.$$ $$\gamma^\ast = \left\{\begin{array}{ll} { }\varepsilon\psi_ub, &\alpha m\leq u<\overline{M}, \\ [2mm] { }-\frac{a\eta(e^\varepsilon-1)(R+G+r)(ru-a\eta+a\theta)^{\frac{R+G+r}{r}}} {b(R+G)\left[(r\alpha m-a\eta+a\theta)^{\frac{R+G+r}{r}}+(e^\varepsilon-1) (ru-a\eta+a\theta)^{\frac{R+G+r}{r}}\right]}, &\overline{M}\leq u\leq\underline{M}. \end{array}\right.$$ 其中, 我们记 \overline{M} = \max\{\alpha m, u_1\}, \; \underline{M} = \min\{m, u_s\} . (1)首先对 u_s\leq m 时的最优解进行证明. 为方便计算, 前文我们对值函数 \psi 满足的HJB方程进行线性变换得到(3.11)式和(3.16) 式. 其中线性变换后的边界值条件为 首先猜解 \varphi 有以下形式 其中 c_{1i}\;(i = 1, 2, 3, 4) 根据边界值条件以及下述光滑性条件得出 经过计算得到 最后, 根据 \psi = \frac{\ln\varphi}{\varepsilon} 即可得出模糊厌恶下的最小drawdown概率. (2) 接下来对 m>u_s 时的情形证明. 此时线性变换后的边界值条件为 我们给出 m\in[\max\{\alpha m, u_1\}, u_s] 时的证明, 而 m\in[\alpha m, \max\{\alpha m, u_1\}] 时的证明可类似得出. 为简化过程, 我们假设 \alpha m<u_1 ( \alpha m\geq u_1 时的结果可类似得出). 首先猜解 \varphi 有以下形式 其中 c_{2i}\;(i = 1, 2, 3, 4) 根据边界值条件以及下述光滑性条件得出 经过计算得到 最后, 根据 \psi = \frac{\ln\varphi}{\varepsilon} 即可得出模糊厌恶下的最小drawdown概率. 证毕. 注3.2 我们给出特殊情形下的最优鲁棒值函数. u_1<\alpha m\leq u\leq u_s 时, 模糊厌恶下的最小drawdown概率为 $$\psi(u, m) = \frac{\ln\varphi}{\varepsilon} = \frac{1}{\varepsilon}\ln\left[1+(e^\varepsilon-1)\left(\frac{ru-a\eta+a\theta}{r\alpha m-a\eta+a\theta}\right)^{\frac{R+G+r}{r}}\right].$$ u_1<\alpha m\leq u\leq m\leq u_s 时, 模糊厌恶下的最小drawdown概率为 $$\psi(u, m) = \frac{1}{\varepsilon}\ln\left\{e^\varepsilon+(e^\varepsilon-1) \frac{r\alpha m-a\eta+a\theta}{r\alpha u_s-a\eta+a\theta} \left[\left(\frac{ru-a\eta+a\theta}{r\alpha m-a\eta+a\theta}\right)^{\frac{R+G+r}{r}}-1\right]\right\}.$$ 注3.3 验证Novikov条件. 结合(2.10), (2.11) 式和(3.5), (3.6) 式有 不等式成立是由于 T<\infty . ## 4 规律和性质 本小节我们将要说明反馈形式的一些性质, 观察(3.12), (3.14)和(3.20) 式不难发现, \pi^\ast , q^\ast 在区间 (\alpha m, \min\{m, u_s\}) 上有界且Lipschitz连续, 接下来讨论其他性质. 定理4.1 由(3.20) 式的 q^\ast 使得(3.2)-(3.3) 式的HJB方程取到下确界. 根据(3.2)式的HJB方程, 设 所以 并且 所以有 g(q) 为凸函数, 根据引理3.1定理得证. 证毕. 注4.1 因为 \pi^\ast , q^\ast 有解析解, 不难发现其函数有界性和Lipschitz连续性. 但是对于最优再保险保留水平 q^\ast , 我们应该确保它在 [0, 1] 区间内, 同时由于验证定理要求 q^\ast 的最优性, 所以有上述定理. 命题4.1 \psi(u, m) 关于 u 在区间 [\alpha m, \min\{u_s, m\}] 上至多改变一次凹凸性. (1) 当 \alpha m\leq u\leq u_s\leq m 时, 如果 0\leq\varepsilon\leq-\frac{(R+G)g_{2}(u_s, m)}{r\alpha m-a\eta+a\theta} , 则 \psi(\cdot, m) 在区间 [\alpha m, \min\{u_s, m\}] 上为严格凸函数; \max\{\alpha m, u_1\}\leq m\leq u_s 时, 如果 0\leq\varepsilon\leq-\frac{(R+G)\cdot g_{2}(u_s, u_s)}{(r\alpha m-a\eta+a\theta)\cdot k_{2}(m)} , 则 \psi(\cdot, m) 在区间 [\alpha m, \min\{u_s, m\}] 上为严格凸函数; \alpha m\leq m<\max\{\alpha m, u_1\} 时, 如果 0\leq\varepsilon\leq-\frac{(R+G)\cdot g_{2}(u_s, u_s)}{(r\alpha m-a\eta+a\theta)\cdot k_{1}(m)} , 则 \psi(\cdot, m) 在区间 [\alpha m, \min\{u_s, m\}] 上为严格凸函数. (2) 如果 \varepsilon>\frac{R+G}{r} , \psi(\cdot, m) 凹凸性发生变化. \psi 在区间 [\alpha m, u_0] 上严格凸, 在 (u_0,$$ \min\{u_s, m\})$上严格凹, 其中$u_0$是区间$[\alpha m, \min\{u_s, m\}]$上满足$(ru_0-a\eta+a\theta)\psi_u(u_0) = \frac{R+G}{\varepsilon}$的唯一点.

当$\varepsilon = 0$时, $\psi(\cdot, m)$为严格凸函数, 故现假设$\varepsilon>0$.

$y(u, m) = (ru-a\eta+a\theta)\psi_u$. 由引理1可知$\psi_u<0$, 所以在区间$[\alpha m, \min\{u_s, m\}]$上有$y\geq0$. 又因为$\psi$满足

$\psi_{uu}$移到等号左侧, 其他项移到等号右侧, 得到

$$$\left(\frac{R+G}{y}-\varepsilon\right)\psi_u^2 = \psi_{uu}.$$$

## A 辅助函数

$h(u, m)$应用It$\hat{\rm o}$引理有

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Browne S .

Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin

Math Oper Res, 1995, 20, 937- 958

Bai L , Guo J .

Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint

Insur Math Econ, 2008, 42 (3): 968- 975

Liang Z , Bai L , Guo J .

Optimal investment and proportional reinsurance with constrained control variables

Optim Contr Appl Met, 2011, 32 (5): 587- 608

DOI:10.1002/oca.965

Chen M , Guo J .

Optiaml investment and proportional reinsurance under exponential premium calculation

Acta Math Sci, 2014, 34A (5): 1161- 1172

Maenhout P J .

Robust portfolio rules and asset pricing

Rev Financ Stud, 2004, 17 (4): 951- 983

Bordigoni G, Matoussi A, Schweizer M. A Stochastic Control Approach to a Robust Utility Maximization Problem//Benth F E, Di Nunno G, Lindstrom T, et al. Stoch Anal Appl. Berlin: Springer, 2007: 125-151

CEV模型下鲁棒最优投资和超额损失再保险问题研究

Li B , Geng C .

Research on robust optimal investment and excess-of-loss reinsurance under CEV model

Statistics and Applications, 2018, 7 (5): 495- 504

Liu B , Zhou M , Li P .

Optimal investment and premium control for insurers with ambiguity

Commun Stat Theor M, 2020, 49 (9): 2110- 2130

Bayraktar E , Zhang Y .

Minimizing the probability of lifetime ruin under ambiguity aversion

SIAM J. Control Optim, 2015, 53 (1): 58- 90

Schmidli H .

Optimal proportional reinsurance policies in a dynamic setting

Scand Actuar J, 2001, 2001 (1): 55- 68

Young V .

Optimal investmet strategy to minimize the probability of lifetime ruin

North Amer Actuar J, 2004, 8 (4): 105- 126

David P S , Young V R .

Minimizing the probability of ruin when claims follow Brownian motion with drift

North Amer Actuar J, 2005, 9 (3): 110- 128

Angoshtari B , Bayraktar E , Young V .

Optimal investment to minimize the probability of drawdown

Stochastics, 2016, 88 (6): 946- 958

Han X , Liang Z , Yuen K C .

Optimal proportional reinsurance to minimize the probability of drawdown under thinning-dependence structure

Scand Actuar J, 2018, 10, 863- 889

Angoshtari B , Bayraktar E , Young V .

Minimizing the probability of lifetime drawdown under constant consumption

Insur Math Econ, 2016, 69, 210- 223

Han X , Liang Z .

Optimal reinsurance and investment in danger-region and safe-region

Optim Contr Appl Met, 2020, 41, 765- 792

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 〈 〉