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An optimal continuous type investment policy for the surplus in a risk model

Seung Kyoung Choi^{a}, and Eui Yong Lee^{1,a}

Received October 17, 2017; Revised November 27, 2017; Accepted November 29, 2017.

- Abstract
In this paper, we show that there exists an optimal investment policy for the surplus in a risk model, in which the surplus is continuously invested to other business at a constant rate

a > 0, whenever the level of the surplus exceeds a given thresholdV > 0. We assign, to the risk model, two costs, the penalty per unit time while the level of the surplus being underV > 0 and the opportunity cost per unit time by keeping a unit amount of the surplus. After calculating the long-run average cost per unit time, we show that there exists an optimal investment ratea ^{*}> 0 which minimizes the long-run average cost per unit time, when the claim amount follows an exponential distribution.**Keywords**: risk model, surplus process, continuous type investment policy, long-run average cost, optimal investment rate

- 1. Introduction
Cho

et al . (2016) introduced a continuous time surplus process in a risk model which involves a continuous type investment. The surplus in the risk model linearly increases at a constant ratec > 0 due to the incoming premium. Meanwhile, claims arrive according to a Poisson process of rateλ > 0 and decrease the level of the surplus jump-wise by random amounts which are independent and identically distributed with distribution functionG of meanμ > 0. Whenever the level of the surplus exceeds a given thresholdV > 0, the investment of the surplus to other business is continuously made at a constant ratea (0 <a <c ), until the surplus process goes belowV > 0. The investment starts again, if the level of the surplus goes overV > 0.It is assumed that

c is larger thanλμ , the expected total amount of claims per unit time, however,c −a is assumed to be less thanλμ to keep the surplus process from being infinitely large. Choet al . (2016) obtained the stationary distribution of the surplus level by forming martingales from the surplus process and applying the optional sampling theorems to the martingales. They also obtained the moment generating function of the stationary distribution by establishing and solving an integro-differential equation for the distribution function of the surplus level.In this paper, we study an optimal investment policy for the surplus in the risk model introduced by Cho

et al . (2016). After assigning, to the risk model, two costs which are the penalty per unit time while the level of the surplus being underV > 0 and the opportunity cost per unit time by keeping a unit amount of the surplus, we show that there exists an optimal investment ratea ^{*}> 0 which minimizes the long-run average cost per unit time.The classical risk model has been studied by many authors, for examples, Gerber (1990), Dufresne and Gerber (1991), and Gerber and Shiu (1997), by assuming that a ruin occurs if the surplus becomes negative. They have studied the ruin probability of the surplus and some interesting characteristics, such as the time of ruin, the surplus before ruin and the deficit at ruin. The core result on the ruin probability is well summarized in Klugman

et al . (2004). Dickson and Willmot (2005) calculated the density of the time of ruin by inverting its Laplace transform.However, in all of the above works, the surplus process is assumed to stop if the ruin occurs. Cho

et al . (2013) introduced a risk model where the surplus process continues to move even though the level of the surplus becomes negative and an investment of the surplus is made, by a fixed amount, to other business jump-wise and instantly, if the level of the surplus reaches a given level. They obtained the characteristic functions of the transient and stationary distributions of the surplus process.Lim

et al . (2016) studied an optimal investment policy in the risk model introduced by Choet al . (2013). After assigning, to the risk model, the reward of the investment, the penalty of the surplus being short and the opportunity cost of keeping the surplus, they showed that there exists a unique amount of the surplus being invested, which minimizes the long-run average cost per unit time.In Section 2, we review some interesting characteristics of the risk model obtained by Cho

et al . (2016) which are needed for the optimization. In Section 3, we assign, to the risk model, two costs which are the penalty per unit time while the level of the surplus being underV > 0 and the opportunity cost per unit time by keeping a unit amount of the surplus, and calculate the long-run average cost per unit time. In Section 4, by assuming that the amount of each claim independently follows an exponential distribution of meanμ > 0, we show that there exists a unique valuea ^{*}> 0 of the investment rate, which minimizes the long-run average cost per unit time.

- 2. Interesting characteristics
In this section, we summarize several interesting characteristics in Cho

et al . (2016), which are necessary to study the optimal investment policy. Choet al . (2016) decomposed {U (t ),t ≥ 0} into two processes {U _{1}(t ),t ≥ 0} and {U _{2}(t ),t ≥ 0}.U _{1}(t ) is formed by separating the periods whereU (t ) ≥V from the original process and connecting them together.U _{2}(t ) is similarly formed by separating the periods whereU (t ) ≤V from the original process and connecting them together.We, first, summarize the interesting characteristics of {

U _{1}(t ),t ≥ 0}. The detailed proofs of the following propositions are given in Choet al . (2016).### Proposition 1

Let T _{1}denote the length of a cycle between two successive regeneration points where U _{1}(t ) =V, then $$E({T}_{1})=\frac{E\left({Y}^{2}\right)}{2\mu [\lambda \mu -(c-a)]}.$$ where Y denotes the claim amount following distribution function G. ### Proposition 2

Let ${P}_{x}^{u}$ be the probability that U _{1}(t ), starting from u (V ≤u ≤x ), reaches x ≥V before it goes below V, and let ${P}_{V}^{u}$ be the probability that U _{1}(t )goes below V without reaching x ≥V, then $${P}_{x}^{u}=\frac{{e}^{\theta u}-{e}^{\theta V}{M}_{{Y}_{e}}(-\theta )}{{e}^{\theta x}-{e}^{\theta V}{M}_{{Y}_{e}}(-\theta )}=1-{P}_{V}^{u},$$ where M (−_{Ye}θ ) = {1/(μθ )}[1 −M (−_{Y}θ )], ${M}_{Y}(r)={\int}_{0}^{\infty}{e}^{ry}dG(y)$ , and θ > 0is the solution of $$d(r)=r(c-a)+\lambda \{{M}_{Y}(-r)-1\}=0.$$ ### Proposition 3

Let F _{1}(x )be the stationary distribution function of U _{1}(t ), then $${F}_{1}(x)=1-\frac{{P}_{x}^{V}}{{P}_{V}^{(x-{Y}_{e})\vee V}},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for\hspace{0.17em}}V\le x<\infty ,$$ where ${P}_{V}^{(x-{Y}_{e})\vee V}=1-{G}_{e}(x-V)+{\int}_{0}^{x-V}{P}_{V}^{x-y}d{G}_{e}(y)$ with G _{e}being the equilibrium distribution function of G, Y _{e}is the random variable having G _{e}as its distribution function, and a ∨b denotes the larger one of a and b. **Remark 1**In the earlier analysis of Cho

et al . (2016),F _{1}(x ) was calculated as$${F}_{1}(x)=1-\frac{{P}_{x}^{V}}{{P}_{V}^{x-{Y}_{e}}},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for\hspace{0.17em}}V\le x<\infty ,$$ which is wrong, however, since

F _{1}(V ) ≠ 0 in this formula. Observe thatU _{1}(t ) ≥V , almost everywhere, except the points whereU _{1}(t ) goes belowV due to the claims.We, now, summarize the interesting characteristics of {

U _{2}(t ),t ≥ 0}. The detailed proofs of the following propositions are given in Choet al . (2016).### Proposition 4

Let T _{2}denote the length of a cycle between two successive regeneration points where U _{2}(t ) =V, then $$E({T}_{2})=\frac{E\left({Y}^{2}\right)}{2\mu (c-\lambda \mu )}.$$ ### Proposition 5

Let M _{2}(r )be the moment generating function of the stationary distribution function F _{2}(x )of U _{2}(t ), then $${M}_{2}(r)=\frac{2{e}^{rV}(c-\lambda \mu )[\mu r-1+{M}_{Y}(-r)]}{rE({Y}^{2})[cr-\lambda +\lambda {M}_{Y}(-r)]}.$$

- 3. Long-run average cost
In this section, we assign the following two costs to the risk model:

b is the penalty per unit time while the surplus is under levelV .V may be considered as the required level of the surplus for payments by the government.h is the opportunity cost incurred by keeping a unit amount of the surplus per unit time without using (investing) it.

These two are typical costs when we manage the surplus in a risk model. The first one increases as the investment rate

a increases, meanwhile, the second one decreases asa increases. With these two costs being assigned to the risk model, we will study whether there exists an optimal investment ratea .By making use of the propositions in Section 2, we can calculate the long-run average cost per unit time. To do that, we, first, obtain the long-run average level of the original surplus process {

U (t ),t ≥ 0}.Let

E (U ) = lim_{t}_{→∞}E [U (t )] the stationary expectation of {U (t ),t ≥ 0}. Observe that the original process,U (t ), is also a regenerative process with cycles in which the cycles ofU _{1}(t ) andU _{2}(t ) alternate with weightsE (T _{1}) andE (T _{2}), which are the expected lengths of cycles ofU _{1}(t ) andU _{2}(t ), respectively. Hence, from the renewal reward theorem of Ross (1996, pp. 133–135), the stationary expectation ofU (t ) is given by$$E[U]=\frac{E({T}_{1})E({U}_{1})+E({T}_{2})E({U}_{2})}{E({T}_{1})+E({T}_{2})},$$ where

E (U _{1}) andE (U _{2}) are the stationary expectations of {U _{1}(t ),t ≥ 0} and {U _{2}(t ),t ≥ 0}, respectively.E (U _{1}) can be obtained from Proposition 3 in Section 2, which is$$E({U}_{1})={\int}_{V}^{\infty}xd{F}_{1}(x).$$ In the next section, we will calculate

E (U _{1}), whenG is an exponential distribution of meanμ .E (U _{2}) can be obtained from Proposition 5 in Section 2, which is$$E({U}_{2})=\frac{d}{dr}{M}_{2}(r){\mid}_{r=0}.$$ In the next section, we will calculate

E (U _{2}), whenG is an exponential distribution of meanμ .Finally, noting that

T _{1}+T _{2}forms a regeneration cycle of {U (t ),t ≥ 0}, we can show, from the renewal reward theorem of Ross (1996, pp. 133–135), that the long-run average cost per unit time, as a function of the investment ratea , is given by$$C(a)=\frac{bE({T}_{2})}{E({T}_{1})+E({T}_{2})}+hE(U),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for\hspace{0.17em}}c-\lambda \mu <a<c.$$ Recall that the investment rate

a is assumed to be less than the premium ratec and thatc −a is assumed to be less thanλμ , the expected total amount of claims per unit time, to keep the surplus process from being infinitely large.If the distribution function

G of the claim amount is given, we may obtainC (a ) in a more closed form, and hence, we can find, numerically, the optimal investment ratea , even though the formula ofC (a ) may be complicate. IfG is an exponential distribution, however, we can find the explicit formula of the optimal investment ratea which minimizesC (a ).

- 4. An optimal investment policy
In this section, we assume that the amount

Y of each claim, independently, follows an exponential distribution of meanμ > 0, and show that there exists an optimal investment ratea ^{*}> 0 which minimizes the long-run average cost per unit time.We, first, calculate

F _{1}(x ) andE (U _{1}). SinceM (_{Y}r ) = 1/(1 −μr ), forr < 1/μ , the unique solution ofd (r ) = 0 in Proposition 2 is given by$$\theta =\frac{\lambda \mu -(c-a)}{(c-a)\mu}>0.$$ Moreover,

M (−_{Ye}θ ) = (c −a )/λμ , and hence,${P}_{x}^{u}$ and${P}_{V}^{u}$ , in Proposition 2, are$${P}_{x}^{u}=\frac{{e}^{\theta u}+\mu \theta {e}^{\theta u}-{e}^{\theta V}}{{e}^{\theta x}+\mu \theta {e}^{\theta x}-{e}^{\theta V}}=1-{P}_{V}^{u},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for\hspace{0.17em}}V\le u\le x.$$ To obtain

F _{1}(x ) in Proposition 3, we need to find${P}_{x}^{V}$ and${P}_{V}^{(x-{Y}_{e})\vee V}$ . From the above equation,$${P}_{x}^{V}=\frac{\mu \theta {e}^{\theta V}}{{e}^{\theta x}+\mu \theta {e}^{\theta x}-{e}^{\theta V}}$$ and from Proposition 3,

$${P}_{V}^{(x-{Y}_{e})\vee V}=1-{G}_{e}(x-V)+{\int}_{0}^{x-V}{P}_{V}^{x-y}d{G}_{e}(y),$$ where

G (_{e}y ) = 1 −e ^{−}^{y}^{/} , since^{μ}G is the exponential distribution with meanμ , and$${P}_{V}^{x-y}=\frac{(1+\mu \theta )\left({e}^{\theta x}-{e}^{\theta (x-y)}\right)}{{e}^{\theta x}+\mu \theta {e}^{\theta x}-{e}^{\theta V}}.$$ After tedious calculation, we can show that

$${P}_{V}^{(x-{Y}_{e})\vee V}=\frac{\mu \theta {e}^{\theta x}}{{e}^{\theta x}+\mu \theta {e}^{\theta x}-{e}^{\theta V}}.$$ Hence, from Proposition 3,

F _{1}(x ) is given by$${F}_{1}(x)=1-{e}^{-\theta (x-V)},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for\hspace{0.17em}}x\ge V,$$ which turns out to be a shifted exponential distribution. Therefore,

E (U _{1}) is$$E({U}_{1})=V+\frac{1}{\theta}=V+\frac{(c-a)\mu}{\lambda \mu -(c-a)}.$$ Now, to obtain

E (U _{2}), we differentiateM _{2}(r ), in Proposition 5, once with respect tor and letr → 0. By applying the L’Hôspital’s rule twice, we have$$E({U}_{2})=\frac{(c-\lambda \mu )\left[2VE(Y)-E\left({Y}^{2}\right)\right]-\lambda E(Y)E\left({Y}^{2}\right)}{2(c-\lambda \mu )E(Y)}.$$ Since

Y is an exponential random variable with meanμ ,E (Y ) =μ , andE (Y ^{2}) = 2μ ^{2}, and hence,E (U _{2}) becomes$$E({U}_{2})=V-\frac{c\mu}{c-\lambda \mu}.$$ E (T _{1}) in Proposition 1 andE (T _{2}) in Proposition 4 are, now, respectively,$$E({T}_{1})=\frac{\mu}{\lambda \mu -(c-a)}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{and\hspace{0.28em}}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}E({T}_{2})=\frac{\mu}{c-\lambda \mu}.$$ Therefore,

E (U ), in Section 3, is given by$$E[U]=V+\frac{\mu \left[c(c-a)-{\lambda}^{2}{\mu}^{2}\right]}{(c-\lambda \mu )[\lambda \mu -(c-a)]}.$$ Finally, the long-run average cost per unit time

C (a ), in Section 3, becomes$$C(a)=\frac{b[\lambda \mu -(c-a)]}{a}+hE(U),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for\hspace{0.17em}}c-\lambda \mu <a<c.$$ We, now, show that there exists a unique value

a ^{*}> 0 of the investment rate, which minimizesC (a ).### Theorem 1

If b (c −λμ ) >hc ^{2}/λ, C (a )is minimized at $${a}^{*}=\frac{b{(c-\lambda \mu )}^{2}+\sqrt{bh\lambda {\mu}^{2}{(c-\lambda \mu )}^{3}}}{b(c-\lambda \mu )-h\lambda {\mu}^{2}},$$ otherwise, C (a )is minimized at a ^{*}=c. **Proof**Differentiating

C (a ) with respect toa gives$${C}^{\prime}(a)=\frac{b(c-\lambda \mu )}{{a}^{2}}-\frac{h\lambda {\mu}^{2}}{{[\lambda \mu -(c-a)]}^{2}}.$$ Reducing two fractions to a common denominator, we have

$${C}^{\prime}(a)=\frac{\left[b(c-\lambda \mu )-h\lambda {\mu}^{2}\right]{a}^{2}-2b{(c-\lambda \mu )}^{2}a+b{(c-\lambda \mu )}^{3}}{{a}^{2}{[\lambda \mu -(c-a)]}^{2}}.$$ Let

N (a ) denote the numerator ofC ′(a ). Observe thatN (a ) is a quadratic function andN (a ) = 0 has two real valued solutions which are$${a}_{1}=\frac{b{(c-\lambda \mu )}^{2}+\sqrt{bh\lambda {\mu}^{2}{(c-\lambda \mu )}^{3}}}{b(c-\lambda \mu )-h\lambda {\mu}^{2}}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{and\hspace{0.28em}}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{a}_{2}=\frac{b{(c-\lambda \mu )}^{2}-\sqrt{bh\lambda {\mu}^{2}{(c-\lambda \mu )}^{3}}}{b(c-\lambda \mu )-h\lambda {\mu}^{2}}.$$ Note that

a _{1}is positive if and only if the coefficient of the quadratic term,b (c −λμ ) −hλμ ^{2}, sayQ , is positive and thata _{2}is always positive regardless of the sign ofQ . Moreover, observe that$$\begin{array}{ll}\hfill N(0)& =b{(c-\lambda \mu )}^{3}>0,\\ \hfill N(c-\lambda \mu )& =-h\lambda {\mu}^{2}{(c-\lambda \mu )}^{2}<0,\\ \hfill N(c)& =\left[b\lambda (c-\lambda \mu )-h{c}^{2}\right]\hspace{0.17em}\lambda {\mu}^{2}.\end{array}$$ Therefore, we may consider the following four exclusive cases:

When

b (c −λμ ) >hc ^{2}/λ ,N (c ) > 0 and the coefficient of the quadratic termQ > 0 (sincec >λμ ), hence, 0 <a _{2}<c −λμ <a _{1}<c andN (a ) is strictly increasing inc −λμ <a <c , that is,C (a ) is minimized ata =a _{1}.When

hλμ ^{2}<b (c −λμ ) ≤hc ^{2}/λ ,N (c ) ≤ 0 andQ > 0, hence, 0 <a _{2}<c −λμ <c ≤a _{1}andN (a ) ≤ 0 inc −λμ <a <c , that is,C (a ) is minimized ata =c .When

b (c −λμ ) =hλμ ^{2},N (c ) < 0 andQ = 0, hence,N (a ) is linearly decreasing andN (a ) < 0 inc −λμ <a <c , that is,C (a ) is minimized ata =c .When

b (c −λμ ) <hλμ ^{2},N (c ) < 0 andQ < 0, hence,a _{1}< 0 <a _{2}<c −λμ andN (a ) < 0 inc −λμ <a <c , that is,C (a ) is minimized ata =c .

In summary, when

b (c −λμ ) >hc ^{2}/λ, C (a ) is minimized ata =a _{1}, otherwise,C (a ) is minimized ata =c .

- Acknowledgements
This research was supported by the Sookmyung Women’s University Research Grants 2015.

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