﻿ CEV模型下鲁棒最优投资和超额损失再保险问题研究 Research on Robust Optimal Investment and Excess-of-Loss Reinsurance under CEV Model

Statistics and Application
Vol. 07  No. 05 ( 2018 ), Article ID: 27182 , 10 pages
10.12677/SA.2018.75058

Research on Robust Optimal Investment and Excess-of-Loss Reinsurance under CEV Model

Bing Li, Caixia Geng

Hebei University of Technology, Tianjin

Received: Sep. 30th, 2018; accepted: Oct. 12th, 2018; published: Oct. 19th, 2018

ABSTRACT

In this paper, we investigate an optimal investment and excess-of-loss reinsurance strategy for an ambiguity-averse insurer (AAI). The financial market consists of one risk-free asset and one risky asset whose price is modeled by a constant elasticity of variance (CEV) model. The insurer can purchase excess-of-loss reinsurance and invest in the financial market. The surplus process of the insurer is approximated by a Brownian motion with drift. The objective is to maximize the minimal expected exponential utility function of the insurer's terminal wealth. By using the dynamic programming approach, we solve the Hamilton-Jacobi-Bellman (HJB) equation and derive the closed form expression of the optimal strategy and the corresponding value function for exponential utility function. Finally, we present numerical examples to illustrate the effects of model parameters on the optimal investment and reinsurance strategies.

Keywords:Robust Control, Excess-of-Loss Reinsurance, Expected Exponential Utility, HJB Equation

CEV模型下鲁棒最优投资和超额损失再保险问题研究

Copyright © 2018 by authors and Hans Publishers Inc.

1. 引言

2. 模型介绍

$R\left(t\right)={c}_{0}+ct-\underset{i=1}{\overset{N\left(t\right)}{\sum }}{Y}_{i},$

${R}^{\left(a\right)}\left(t\right)={c}_{0}+{c}^{\left(a\right)}t-\underset{i=1}{\overset{N\left(t\right)}{\sum }}{Y}_{i}^{\left(q\right)},$

${c}^{\left(a\right)}=\left(1+\eta \right)\lambda {\mu }_{1}-\left(1+\theta \right)\lambda \left({\mu }_{1}-E{Y}_{i}^{\left(a\right)}\right)=\left(\eta -\theta \right)\lambda {\mu }_{1}+\lambda \left(1+\theta \right)E{Y}_{i}^{\left(a\right)},$

$\theta$ 为再保险公司的安全负荷，并且 $\theta >\eta$ 。为了方便，盈余过程 ${R}^{\left(a\right)}\left(t\right)$ 可近似为扩散模型

$\text{d}{R}^{\left(a\right)}\left(t\right)=\lambda \left[\theta \mu \left(a\right)+\left(\eta -\theta \right){\mu }_{1}\right]\text{d}t+\sqrt{\lambda }\sigma \left(a\right)\text{d}{W}_{0}\left(t\right),$

$\begin{array}{l}\mu \left(a\right)=E{Y}_{i}^{\left(a\right)}={\int }_{0}^{a}\left(1-F\left(u\right)\right)\text{d}u={\int }_{0}^{a}\stackrel{¯}{F}\left(u\right)\text{d}u,\\ {\sigma }^{2}\left(a\right)=E{\left({Y}_{i}^{\left(a\right)}\right)}^{2}={\int }_{0}^{a}2u\left(1-F\left(u\right)\right)\text{d}u={\int }_{0}^{a}2u\stackrel{¯}{F}\left(u\right)\text{d}u,\end{array}$

$\left\{\begin{array}{l}\text{d}B\left(t\right)=rB\left(t\right)\text{d}t,\text{\hspace{0.17em}}t\in \left[0,T\right],\\ B\left(0\right)={b}_{0}>0,\end{array}$

$\left\{\begin{array}{l}\text{d}S\left(t\right)=S\left(t\right)\left[\mu \text{d}t+\sigma {S}^{\beta }\left(t\right)\text{d}{W}_{1}\left(t\right)\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[0,T\right],\\ S\left(0\right)={s}_{0}>0,\end{array}$

${X}^{\text{π}}\left(t\right)$ 表示t时刻保险公司的财富， $a\left(t\right)\in \left[0,D\right]$ 表示t时刻的超额损失自留水平， ${\text{π}}_{\text{1}}\left(t\right)$ 表示投资到风险资产中的数额。 $\text{π}={\left(a\left(t\right),{\text{π}}_{1}\left(t\right)\right)}_{t\in \left[0,T\right]}$ 表示一个交易策略。在策略 $\text{π}$ 下，财富过程 ${\left\{{X}^{\pi }\left(t\right)\right\}}_{t\in \left[0,T\right]}$ 表示为

$\begin{array}{c}\text{d}{X}^{\text{π}}\left(t\right)=\left[r{X}^{\pi }\left(t\right)+\left(\mu -r\right){\text{π}}_{1}\left(t\right)+\lambda \theta \mu \left(a\right)+\lambda \left(\eta -\theta \right){\mu }_{1}\right]\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\sqrt{\lambda }\sigma \left(a\right)\text{d}{W}_{0}\left(t\right)+{\text{π}}_{1}\left(t\right)\sigma {S}^{\beta }\left(t\right)\text{d}{W}_{1}\left(t\right)\end{array}$ (1)

3. 效用函数下的鲁棒问题

$U\left(x\right)=-\frac{1}{\gamma }\mathrm{exp}\left\{-\gamma x\right\},$ (2)

$\underset{\text{π}\in \prod }{\mathrm{sup}}E\left[U\left({X}^{\text{π}}\left(T\right)\right)\right]=\underset{\text{π}\in \prod }{\mathrm{sup}}E\left[-\frac{1}{\gamma }\mathrm{exp}\left\{-\gamma {X}^{\text{π}}\left(T\right)\right\}\right],$ (30

${\Lambda }^{\omega }\left(t\right):=\mathrm{exp}\left\{-{\int }_{0}^{t}{\omega }_{1}\left(s\right)\text{d}{W}_{0}\left(s\right)-\frac{\text{1}}{\text{2}}{\int }_{0}^{t}{\omega }_{1}^{2}\left(s\right)\text{d}s-{\int }_{0}^{t}{\omega }_{2}\left(s\right)\text{d}{W}_{1}\left(s\right)-\frac{\text{1}}{\text{2}}{\int }_{\text{0}}^{t}{\omega }_{2}^{2}\left(s\right)\text{d}s\right\}.$ (4)

$\text{d}{W}_{0}^{Q}\left(t\right)=\text{d}{W}_{0}\left(t\right)+{\omega }_{1}\left(t\right)\text{d}t,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{d}{W}_{1}^{Q}\left(t\right)=\text{d}{W}_{1}\left(t\right)+{\omega }_{2}\left(t\right)\text{d}t.$ (5)

CEV模型描述为

(6)

$\begin{array}{c}\text{d}{X}^{\text{π}}\left(t\right)=\left[r{X}^{\text{π}}\left(t\right)+\left(\mu -r\right){\text{π}}_{1}\left(t\right)+\lambda \theta \mu \left(a\right)+\lambda \left(\eta -\theta \right){\mu }_{1}-{\omega }_{1}\left(t\right)\sqrt{\lambda }\sigma \left(a\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\sigma {\omega }_{2}\left(t\right){\text{π}}_{1}\left(t\right){S}^{\beta }\left(t\right)\right]\text{d}t+\sqrt{\lambda }\sigma \left(a\right)\text{d}{W}_{0}^{Q}\left(t\right)+{\text{π}}_{1}\left(t\right)\sigma {S}^{\beta }\left(t\right)\text{d}{W}_{1}^{Q}\left(t\right).\end{array}$ (7)

1) $\left(a\left(t\right),{\text{π}}_{1}\left(t\right)\right)$ 是可测的；

2) $a\left(t\right)\in \left[0,D\right]$${\text{π}}_{\text{1}}\left(t\right)\in \left[0,+\infty \right)$${E}^{{Q}^{*}}\left[{\int }_{0}^{T}{\text{π}}_{1}{\left(t\right)}^{2}{S}^{2\beta }\text{d}t\right]<+\infty$

3) $\forall \left(t,x,s\right)\in \left[0,T\right]×R×{R}_{+}$ ，SDE (7)有唯一(强)解 $\left\{{X}^{\text{π}}\left(t\right):t\in \left[0,T\right]\right\}$${E}_{t,x,s}^{{Q}^{*}}\left[U\left({X}^{\text{π}}\left(T\right)\right)\right]<\infty$${Q}^{*}$ 是描述最坏情况下的备选模型。

(8)

$\frac{\text{1}}{\text{2}}{\omega }_{\text{1}}^{\text{2}}\left(t\right)\text{d}t+\frac{1}{2}{\omega }_{2}^{2}\left(t\right)\text{d}t.$

${\phi }_{i}\left(t,x,s\right)=-\frac{{\beta }_{i}}{\gamma V\left(t,x,s\right)},i=1,2,$ (9)

$\underset{\text{π}\in \prod }{\mathrm{sup}}\underset{Q\in {Q}_{1}}{\mathrm{inf}}\left\{{Α}^{\text{π},\omega }V+\frac{{\omega }_{1}^{2}}{2{\phi }_{1}}+\frac{{\omega }_{2}^{2}}{2{\phi }_{2}}\right\}=0$ (10)

$\begin{array}{c}{Α}^{\text{π},\omega }V={V}_{t}+\left[rx+\left(\mu -r\right){\text{π}}_{1}+\lambda \theta \mu \left(a\right)+\lambda \left(\eta -\theta \right){\mu }_{1}-{\omega }_{1}\sqrt{\lambda }\sigma \left(a\right)-\sigma {\omega }_{2}{\text{π}}_{1}{s}^{\beta }\right]{V}_{x}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\text{1}}{\text{2}}\left({\sigma }^{\text{2}}{\text{π}}_{\text{1}}^{\text{2}}{s}^{2\beta }+\lambda \sigma {\left(a\right)}^{2}\right){V}_{xx}+s\left(\mu -\sigma {\omega }_{2}{s}^{\beta }\right){V}_{s}+\frac{\text{1}}{\text{2}}{\sigma }^{2}{s}^{2\beta +2}{V}_{ss}+{\sigma }^{2}{s}^{2\beta +1}{\text{π}}_{1}{V}_{xs}.\end{array}$ (11)

$V\left(t,x,s\right)=-\frac{\text{1}}{\gamma }\mathrm{exp}\left\{-\gamma \left[x{\text{e}}^{r\left(T-t\right)}+h\left(t,s\right)+g\left(t\right)\right]\right\}\text{.}$ (12)

$\begin{array}{l}{V}_{t}=-\gamma \left[{g}_{t}+{h}_{t}-r{\text{e}}^{r\left(T-t\right)}x\right]V,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{V}_{x}=-\gamma {\text{e}}^{r\left(T-t\right)}V,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{V}_{s}=-\gamma {h}_{s}V,\\ {V}_{xx}={\gamma }^{2}{\text{e}}^{2r\left(T-t\right)}V,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{V}_{ss}=-\gamma \left[{h}_{ss}-\gamma {h}_{s}^{2}\right]V,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{V}_{xs}={\gamma }^{2}{\text{e}}^{r\left(T-t\right)}{h}_{s}V.\end{array}$ (13)

${\omega }_{\text{1}}^{\text{*}}={\beta }_{\text{1}}\sqrt{\lambda }\sigma \left(a\right){\text{e}}^{r\left(T-t\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\omega }_{\text{2}}^{\text{*}}={\beta }_{\text{2}}\sigma {s}^{\beta }\left({\text{π}}_{1}{\text{e}}^{r\left(T-t\right)}+s{h}_{s}\right).$ (14)

${\text{π}}_{\text{1}}^{\ast }=\frac{\mu -r-{\sigma }^{2}{s}^{2\beta +1}{h}_{s}\left(\gamma +{\beta }_{2}\right)}{{\sigma }^{2}{s}^{2\beta }{\text{e}}^{r\left(T-t\right)}\left(\gamma +{\beta }_{2}\right)}.$ (15)

$\left[\theta \gamma {\text{e}}^{r\left(T-t\right)}-\left({\gamma }^{2}{\text{e}}^{2r\left(T-t\right)}+\gamma {\beta }_{1}{\text{e}}^{2r\left(T-t\right)}\right)a\left(t\right)\right]\lambda \stackrel{¯}{F}\left(a\left(t\right)\right)=0.$ (16)

${a}^{\ast }\left(t\right)=\frac{\theta }{\gamma +{\beta }_{1}}{\text{e}}^{-r\left(T-t\right)}.$ (17)

$\begin{array}{l}{g}_{t}+\lambda \left(\eta -\theta \right){\mu }_{1}{\text{e}}^{r\left(T-t\right)}+\lambda \theta {\text{e}}^{r\left(T-t\right)}{\int }_{\text{0}}^{\frac{\theta {\text{e}}^{-r\left(T-t\right)}}{\gamma +{\beta }_{1}}}\stackrel{¯}{F}\left(u\right)\text{d}u\\ -\frac{\text{1}}{\text{2}}\left(\lambda \gamma {\text{e}}^{2r\left(T-t\right)}+{\beta }_{1}\lambda {\text{e}}^{2r\left(T-t\right)}\right){\int }_{0}^{\frac{\theta {\text{e}}^{-r\left(T-t\right)}}{\gamma +{\beta }_{1}}}2u\stackrel{¯}{F}\left(u\right)\text{d}u=0,\\ {h}_{t}+sr{h}_{s}+\frac{\gamma }{2}{\sigma }^{2}{s}^{2\beta +2}{h}_{ss}+\frac{{\left(\mu -r\right)}^{2}}{2\left(\gamma +{\beta }_{2}\right){\sigma }^{2}{s}^{2\beta }}=0.\end{array}$ (18)

$g\left(t\right)=\frac{\lambda \left(\eta -\theta \right){\mu }_{1}}{r}\left({\text{e}}^{r\left(T-t\right)}-1\right)+{\int }_{t}^{T}f\left(s\right)\text{d}s,$ (19)

$f\left(t\right)=\lambda \theta {\text{e}}^{r\left(T-t\right)}{\int }_{0}^{\frac{\theta {\text{e}}^{-r\left(T-t\right)}}{\gamma +{\beta }_{1}}}\stackrel{¯}{F}\left(u\right)\text{d}u-\frac{1}{2}\left(\lambda \gamma {\text{e}}^{2r\left(T-t\right)}+{\beta }_{1}\lambda {\text{e}}^{2r\left(T-t\right)}\right){\int }_{0}^{\frac{\theta {\text{e}}^{-r\left(T-t\right)}}{\gamma +{\beta }_{1}}}2u\stackrel{¯}{F}\left(u\right)\text{d}u.$

$h\left(t,s\right)=A\left(t\right)+B\left(t\right){s}^{-2\beta },$ (20)

$\begin{array}{l}{A}_{t}+\gamma {\sigma }^{2}\beta \left(2\beta +1\right)B\left(t\right)=0,\\ {B}_{t}-2r\beta B\left(t\right)+\frac{{\left(\mu -r\right)}^{2}}{2\left(\gamma +{\beta }_{2}\right){\sigma }^{2}}=0.\end{array}$ (21)

 (22)

(23)

$\begin{array}{c}{h}_{1}\left(t,s\right)=\frac{\left(2\beta +1\right){\left(\mu -r\right)}^{2}}{8{r}^{2}\beta \left(\gamma +{\beta }_{2}\right)}{\text{e}}^{2r\beta t}{C}_{2}-\frac{\left(2\beta +1\right){\left(\mu -r\right)}^{2}}{4r\left(\gamma +{\beta }_{2}\right)}t+{C}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[{C}_{4}-{C}_{5}\frac{{\left(\mu -r\right)}^{2}}{4r\beta \left(\gamma +{\beta }_{2}\right){\sigma }^{2}}{\text{e}}^{-2r\beta t}\right]{s}^{-2\beta },\end{array}$ (24)

$V\left(t,x,s\right)=-\frac{\text{1}}{\gamma }\mathrm{exp}\left\{-\gamma \left[x{\text{e}}^{r\left(T-t\right)}+{h}_{\text{2}}\left(t,s\right)+{g}_{\text{2}}\left(t\right)\right]\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{t}_{0}\le t\le T,$

$\begin{array}{l}{h}_{2}\left(t,s\right)=h\left(t,s\right),\\ {g}_{2}\left(t\right)=\frac{\lambda \eta {\mu }_{1}}{r}\left({\text{e}}^{r\left(T-t\right)}-1\right)-\frac{\lambda {\mu }_{2}^{2}}{4r}\left(\gamma +{\beta }_{1}\right)\left({\text{e}}^{2r\left(T-t\right)}-1\right).\end{array}$ (25)

${h}_{1}\left({t}_{0},s\right)+{g}_{1}\left({t}_{0}\right)={h}_{2}\left({t}_{0},s\right)+{g}_{2}\left({t}_{0}\right),$

${C}_{1}={g}_{2}\left({t}_{0}\right)-\frac{\lambda \left(\eta -\theta \right){\mu }_{1}}{r}{\text{e}}^{-r{t}_{0}}+{\int }_{0}^{{t}_{0}}f\left(s\right)\text{d}s,$

${C}_{2}=\frac{\left(2\beta +1\right){\left(\mu -r\right)}^{2}}{8{r}^{2}\beta \left(\gamma +{\beta }_{2}\right)}{\text{e}}^{-2r\beta T},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{3}=-\frac{\left(2\beta +1\right){\left(\mu -r\right)}^{2}}{8{r}^{2}\beta \left(\gamma +{\beta }_{2}\right)}+\frac{\left(2\beta +1\right){\left(\mu -r\right)}^{2}}{4r\left(\gamma +{\beta }_{2}\right)}T,$

${C}_{4}=\frac{{\left(\mu -r\right)}^{2}}{4r\beta \left(\gamma +{\beta }_{2}\right){\sigma }^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{5}=\frac{{\left(\mu -r\right)}^{2}}{4r\beta \left(\gamma +{\beta }_{2}\right){\sigma }^{2}}{\text{e}}^{-2r\beta T}.$

1) 如果 $D\left(\gamma +{\beta }_{1}\right)\ge \theta$

${\text{π}}_{\text{1}}^{\ast }\left(t\right)=\frac{\mu -r+2\beta {\sigma }^{2}\left(\gamma +{\beta }_{2}\right)B\left(t\right)}{{\sigma }^{2}{s}^{2\beta }{\text{e}}^{r\left(T-t\right)}\left(\gamma +{\beta }_{2}\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}^{\ast }\left(t\right)=\frac{\theta }{\gamma +{\beta }_{1}}{\text{e}}^{-r\left(T-t\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le t\le T,$

$J\left(t,x,s\right)=-\frac{\text{1}}{\gamma }\mathrm{exp}\left\{-\gamma \left[x{\text{e}}^{r\left(T-t\right)}+h\left(t,s\right)+g\left(t\right)\right]\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{0}\le t\le T,$ (26)

2) 如果 $D\left(\gamma +{\beta }_{1}\right)<\theta$

$J\left(t,x,s\right)=\left\{\begin{array}{l}-\frac{1}{\gamma }\mathrm{exp}\left\{-\gamma \left[x{\text{e}}^{r\left(T-t\right)}+h\left(t,s\right)+{g}_{1}\left(t\right)\right]\right\},t<{t}_{0}\le T,\\ -\frac{1}{\gamma }\mathrm{exp}\left\{-\gamma \left[x{\text{e}}^{r\left(T-t\right)}+h\left(t,s\right)+{g}_{2}\left(t\right)\right]\right\},{t}_{0}\le t (27)

(1) ${\text{π}}^{\ast }$ 是可行策略；

(2) ${E}^{{Q}^{*}}\left[{\mathrm{sup}}_{t\in \left[0,T\right]}{|J\left(t,{X}^{{\text{π}}^{*}}\left(t\right),S\left(t\right)\right)|}^{4}\right]<\infty$

(3) ${E}^{{Q}^{*}}\left[{\mathrm{sup}}_{t\in \left[0,T\right]}{|\frac{{\omega }_{1}^{*}{\left(t\right)}^{2}}{2{\phi }_{1}\left(t,{X}^{{\text{π}}^{*}}\left(t\right),S\left(t\right)\right)}+\frac{{\omega }_{2}^{*}{\left(t\right)}^{2}}{2{\phi }_{2}\left(t,{X}^{{\text{π}}^{*}}\left(t\right),S\left(t\right)\right)}|}^{2}\right]<\infty$

$\begin{array}{c}{X}^{{\text{π}}^{*}}\left(t\right)={\text{e}}^{rt}{X}^{{\text{π}}^{*}}\left(0\right)+{\int }_{0}^{t}{\text{e}}^{-r\left(u-t\right)}\left[\left(\mu -r\right){\text{π}}_{1}^{*}\left(u\right)+\lambda \theta \mu \left({a}^{*}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\lambda \left(\eta -\theta \right){\mu }_{1}-{\omega }_{1}^{*}\left(u\right)\sqrt{\lambda }\sigma \left({a}^{*}\right)-\sigma {\omega }_{2}^{\ast }\left(u\right){\text{π}}_{1}^{*}\left(u\right){s}^{\beta }\left(u\right)\right]\text{d}u\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{t}{\text{e}}^{-r\left(u-t\right)}\sqrt{\lambda }\sigma \left({a}^{*}\right)\text{d}{W}_{0}^{{Q}^{*}}\left(u\right)+{\int }_{0}^{t}{\text{e}}^{-r\left(u-t\right)}{\text{π}}_{1}^{*}\left(u\right)\sigma {s}^{\beta }\left(u\right)\text{d}{W}_{1}^{{Q}^{*}}\left(u\right).\end{array}$ (28)

$\begin{array}{l}{|J\left(t,{X}^{{\text{π}}^{*}}\left(t\right),S\left(t\right)\right)|}^{4}=\frac{\text{1}}{{\gamma }^{\text{4}}}\mathrm{exp}\left\{-4\gamma \left[{\text{e}}^{r\left(T-t\right)}{X}^{{\text{π}}^{*}}\left(t\right)+A\left(t\right)+B\left(t\right){s}^{-2\beta }+g\left(t\right)\right]\right\}\\ \le {N}_{1}\mathrm{exp}\left\{-4\gamma {\text{e}}^{r\left(T-t\right)}{X}^{{\text{π}}^{*}}\left(t\right)\right\}\\ \le {N}_{2}\mathrm{exp}\left\{-4\gamma \left(-{\int }_{0}^{t}{\text{e}}^{-r\left(u-T\right)}\sigma {\omega }_{2}^{*}\left(u\right){\text{π}}_{1}^{*}\left(u\right){s}^{\beta }\left(u\right)\text{d}u\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{0}^{t}{\text{e}}^{-r\left(u-T\right)}\sqrt{\lambda }\sigma \left({a}^{*}\right)\text{d}{W}_{0}^{{Q}^{*}}\left(u\right)+{\int }_{0}^{t}{\text{e}}^{-r\left(u-T\right)}{\text{π}}_{1}^{*}\left(u\right)\sigma {s}^{\beta }\left(u\right)\text{d}{W}_{1}^{{Q}^{*}}\left(u\right)\right)\right\}\\ ={N}_{2}\left({F}_{1}\left(t\right)+{F}_{2}\left(t\right)\right),\end{array}$ (29)

$\begin{array}{c}{F}_{1}\left(t\right)=\mathrm{exp}\left\{{\int }_{0}^{t}8{\gamma }^{2}{\text{e}}^{-2r\left(u-T\right)}\lambda \sigma {\left({a}^{*}\right)}^{2}\text{d}u\right\}\cdot \mathrm{exp}\left\{{\int }_{0}^{t}-8{\gamma }^{2}{\text{e}}^{-2r\left(u-T\right)}\lambda \sigma {\left({a}^{*}\right)}^{2}\text{d}u\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{t}-4\gamma {\text{e}}^{-r\left(u-T\right)}\sqrt{\lambda }\sigma \left({a}^{*}\right)\text{d}{W}_{0}^{{Q}^{*}}\left(u\right)\right\}.\end{array}$ (30)

(30)中的第一部分是一个常数，第二部分是鞅。所以 ${E}^{{Q}^{*}}\left[{F}_{1}\left(t\right)\right]<\infty$ 。根据参考文献Zheng等 [10] 中的Lemma 3.6，我们知道 ${E}^{{Q}^{*}}\left[{F}_{2}\left(t\right)\right]<\infty$ 。因此，

${E}^{{Q}^{*}}\left[{\mathrm{sup}}_{t\in \left[0,T\right]}{|J\left(t,{X}^{{\pi }^{*}}\left(t\right),S\left(t\right)\right)|}^{4}\right]<\infty .$ (31)

$\begin{array}{l}{E}^{{Q}^{*}}\left[\underset{t\in \left[0,T\right]}{\mathrm{sup}}{|\frac{{\omega }_{1}^{*}{\left(t\right)}^{2}}{2{\phi }_{1}\left(t,{X}^{{\text{π}}^{*}}\left(t\right),S\left(t\right)\right)}+\frac{{\omega }_{2}^{*}{\left(t\right)}^{2}}{2{\phi }_{2}\left(t,{X}^{{\text{π}}^{*}}\left(t\right),S\left(t\right)\right)}|}^{2}\right]\\ ={E}^{{Q}^{*}}\left[\underset{t\in \left[0,T\right]}{\mathrm{sup}}{|\frac{{\omega }_{1}^{*}{\left(t\right)}^{2}\gamma }{2{\beta }_{1}}+\frac{{\omega }_{2}^{*}{\left(t\right)}^{2}\gamma }{2{\beta }_{2}}|}^{2}{|J\left(t,{X}^{{\text{π}}^{*}}\left(t\right),S\left(t\right)\right)|}^{2}\right]\\ \le {\left({E}^{{Q}^{*}}\left[\underset{t\in \left[0,T\right]}{\mathrm{sup}}{|\frac{{\omega }_{1}^{*}{\left(t\right)}^{2}\gamma }{2{\beta }_{1}}+\frac{{\omega }_{2}^{*}{\left(t\right)}^{2}\gamma }{2{\beta }_{2}}|}^{4}\right]\right)}^{1/2}{\left({E}^{{Q}^{*}}\left[\underset{t\in \left[0,T\right]}{\mathrm{sup}}{|J\left(t,{X}^{{\text{π}}^{*}}\left(t\right),S\left(t\right)\right)|}^{4}\right]\right)}^{1/2}\\ <\infty .\end{array}$ (32)

4. 总结

Research on Robust Optimal Investment and Excess-of-Loss Reinsurance under CEV Model[J]. 统计学与应用, 2018, 07(05): 495-504. https://doi.org/10.12677/SA.2018.75058

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