岳毅蒙,王 欣,李江濤

(1.商洛學(xué)院數(shù)學(xué)與計算機(jī)應(yīng)用學(xué)院,陜西商洛 726000;2.商洛學(xué)院經(jīng)濟(jì)與管理學(xué)院,陜西商洛 726000; 3.西安武警工程大學(xué)理學(xué)院,陜西西安 710086)

二維風(fēng)險模型帶交易費用的最優(yōu)分紅

岳毅蒙1,王 欣2,李江濤3

(1.商洛學(xué)院數(shù)學(xué)與計算機(jī)應(yīng)用學(xué)院,陜西商洛 726000;2.商洛學(xué)院經(jīng)濟(jì)與管理學(xué)院,陜西商洛 726000; 3.西安武警工程大學(xué)理學(xué)院,陜西西安 710086)

研究帶交易費用的二維風(fēng)險模型的最優(yōu)分紅問題。以最大化分紅與注資的折現(xiàn)之差為目標(biāo),利用隨機(jī)控制理論,通過求解哈密爾頓-雅克比-貝爾曼(HJB)方程,得到了相應(yīng)的最優(yōu)分紅策略。

注資;分紅;HJB方程

二維風(fēng)險模型是由Chan等[1]在2003年首次提出并給出了關(guān)于破產(chǎn)概率的一些重要結(jié)論。文獻(xiàn)[2]中研究了二維風(fēng)險模型的生存概率,文獻(xiàn)[3]中研究了動態(tài)比例再保險和近似二維復(fù)合泊松風(fēng)險模型的破產(chǎn)概率,文獻(xiàn)[4]中研究了二維風(fēng)險模型的最優(yōu)分紅和脈沖控制,文獻(xiàn)[5]中研究了帶注資的二維復(fù)合泊松模型的最優(yōu)分紅。結(jié)合文獻(xiàn)[6-9],在前人研究的基礎(chǔ)上,討論帶最小盈余約束和交易費用情況下,二維風(fēng)險模型的最優(yōu)分紅與注資策略。

1 模型建立

其中:x1x2為初始準(zhǔn)備金;c1,c2(＞0)為保費收入率;{N1(t)}t≥0、{N2(t)}t≥0、{Nc(t)}t≥0分別表示獨立的強(qiáng)度為λ1、λ2、λc的Poisson過程;適應(yīng)最小的右連續(xù)域流{F}t≥0;{Ui}、{Vi}分別表示一列正的、相互獨立同分布的變量序列。在原模型基礎(chǔ)上引入策略π＝{(Dt,Zt)},其中Dt{}和Zt{}分別表示0～t時刻的累積分紅和累積注資。一個策略稱為可行策略需滿足:Dt{}是右連左極的,增的適應(yīng)的過程,且滿足D0－＝0;{Zt}是左連右極的,增的適應(yīng)的過程,且滿足Z0＝0。記所有可允許策略集為

則受控的盈余過程轉(zhuǎn)化為

其中:x＝x1＋x2,c＝c1＋c2。

假設(shè)保險公司有最低盈余要求m＞0,如果赤字太大,注資不合理,破產(chǎn)時刻定義為

2 HJB方程和最優(yōu)策略

引理1值函數(shù)V(x)在[m,∞)單調(diào)遞增且對0≤y≤x有V(x)－V(y)≥β(x－y),且

引理2若V(x)在[m,∞)為凹的,則最優(yōu)注資上限a＝m。

類似于文獻(xiàn)[6]中的討論,對x≥m及任意停時τ,有如下動態(tài)規(guī)劃原理成立:

類似于文獻(xiàn)[5]可得V(x)在[m,∞)上滿足的HJB方程:

可以定義最優(yōu)分紅限b:＝inf{x:V′(x)≤β}和最優(yōu)注資上限a:＝sup{x:V′(x)≥φ}∨m?，F(xiàn)在定義策略如下:

定理1若V(x)在[m,∞)是凹的,則策略(4)是最優(yōu)策略,即Vπ?(x)＝V(x)。

證明設(shè)X?(t)是策略(4)下的受控過程,當(dāng)X?(t)＝b有V′(X?(t))＝1;當(dāng)X?(t)＜b有V′(X?(t))＞1。運(yùn)用It^o公式,有

理賠到達(dá)時,有

為均值為0的鞅。等價地,過程

為鞅。又V(x)為凹的且左右可導(dǎo)。進(jìn)一步我們假設(shè)FU,V(u,v)連續(xù),因此V(x)滿足式(3),且連續(xù)可微。

3 解的特征

定理2V(x)是HJB方程(3)的最小非負(fù)解。

證明設(shè)f(x)是HJB方程的非負(fù)解,那么f(x)遞增。令X?為最優(yōu)策略下的受控過程,由定理1得

為均值為0的鞅。由HJB方程得

[1] Chan W C,Yang H L,Zhang L Z.Some Results on Ruin Probabilities in a Two-dimensional Risk Model[J].Insurance:Mathematics and E-conomics,2003,32(3):345-358.

[2] Dang L F,Zhu N,Zhang H M.Survival Probability for a Two-dimensional Risk Model[J].Insurance:Mathematics and Economics,2009,44 (3):491-496.

[3] Corporation H P.Dynamic Proportional Reinsurance and Approximations for Ruin Probabilities in the Two-dimensional Compound Poisson Risk Model[J].Discrete Dynamics in Nature&Society,2012,12(12):1 951-1 965.

[4] Czarna I,Palmowski Z.De Finetti＇s Dividend Problem and Impulse Control for a Two-dimensional Insurance Risk Process[J].Stochastic Models,2011,27(2):220-250.

[5] Zhang S Q.Optimal Dividend Payments of the Two-dimensional Compound Poisson Risk Model with Capital Injection[J].Operations Research Transactions,2012,16(3):119-131.

[6] Azcue P,Muler N.Optimal Reinsurance and Dividend Distribution Policies in the Cramer-Lundberg Model[J].Math.Finance,2005,15(2): 261-308.

[7] Bayraktar E,Kyprianou A E,Yamazaki K.Optimal Dividends in the Dual Model under Transaction Costs[J].Insurance:Mathematics and Economics,2014,54:133-143.

[8] Eisenberg J,Schmidli H.Minimising Expected Discounted Capital Injections by Reinsurance in a Classical Risk Model[J].Scandinavian Actuarial Journal,2011,3:155-176.

[9] 岳毅蒙.最小盈余約束下風(fēng)險模型的最優(yōu)分紅策略[J].甘肅科學(xué)學(xué)報,2015,27(2):19-24.

Optimized Participation in Profit of Transaction Expenses in Two-dimension Risk Model Region

Yue Yimeng1,Wang Xin2,Li Jiangtao3
(1.School of Mathematics and Computer Application,Shangluo University,Shangluo726000,China; 2.School of Economics and Management,Shangluo University,Shangluo726000,China; 3.College of Sciences,Engineering University of CAPF,Xi’an710086,China)

Study the problem of the optimized participation in profit of transaction expenses in two-dimension risk model region.Aiming for the difference between the optimized participation in profit and discounting of capital injection,the corresponding strategy of the optimized participation in profit can be got by stochastic control theory and solve HJB equation.

Capital injection;Participation in profit;Hamilton Jocabi Bellman equation

O211.6

:A

:1004-0366(2016)05-0026-04

2015-08-07;

:2015-10-08.

陜西省教育科學(xué)“十二五”規(guī)劃課題(SGH13406);商洛學(xué)院科研項目(15SKY011).

岳毅蒙(1984-),男,陜西富平人,碩士,講師,研究方向為金融數(shù)學(xué).E-mail:18740586401＠126.com.

Yue Yimeng,Wang Xin,Li Jiangtao.Optimized Participation in Profit of Transaction Expenses in Two-dimension Risk Model Region[J].Journal of Gansu Sciences,2016,28(5):26-29.[岳毅蒙,王欣,李江濤.二維風(fēng)險模型帶交易費用的最優(yōu)分紅[J].甘肅科學(xué)學(xué)報,2016,28(5):26-29.]

10.16468/j.cnkii.ssn1004-0366.2016.05.007.