Optimal Dynamic Premium Control in Non-life Insurance. Maximizing Dividend Pay-outs

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Optimal Dynamic Premium Controlin Non-life Insurance. MaximizingDividend Pay-outsBjarne Højgaard

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Scand. Actuarial J. 2002; 4: 225–245 ORIGINAL ARTICLE

Optimal Dynamic Premium Control in Non-life Insurance.Maximizing Dividend Pay-outs

BJARNE HØJGAARD

Højgaard B. Optimal dynamic premium control in non-life insurance.Maximizing dividend pay-outs. Scand. Actuarial J. 2002; 4: 225–245.

In this paper we consider the problem of � nding optimal dynamicpremium policies in non-life insurance. The reserve of a company ismodeled using the classical Cramer-Lundberg model with premium ratescalculated via the expected value principle. The company controls dynam-ically the relative safety loading with the possibility of gaining or loosingcustomers. It distributes dividends according to a ‘barrier strategy’ andthe objective of the company is to � nd an optimal premium policy anddividend barrier maximizing the expected total, discounted pay-out ofdividends. In the case of exponential claim size distributions optimalcontrols are found on closed form, while for general claim size distribu-tions a numerical scheme for approximations of the optimal control isderived. Based on the idea of De Vylder going back to the 1970s, thepaper also investigates the possibilities of approximating the optimalcontrol in the general case by using the closed form solution of anapproximating problem with exponential claim size distributions. Keywords: Dynamic control, dividend distributions, Bellman equation, barrierstrategies, De Vylder approximations.

1. INTRODUCTION

Consider a non-life insurance company. For a given portfolio of insured let thereserve of the company {Rt } be given by

Rt ¾x»pt¼N t

b

i ¾ 1

Ui, (1.1)

where xÏ is the initial reserve and the arrival process {N tb} is Poisson process with

constant intensity b. The random variables Ui, i¾1, 2, . . . are iid. claim sizesindependent of Nt. We let G denote the claim size distribution with � nite mean m.The constant p is the premium rate, which is calculated via the expected valueprinciple, that is

p¾(1 »h)bm, (1.2)

where h\0 is the relative safety loading. The intensity b obviously depends on thesize of the portfolio in such a way that a large portfolio will give a large intensity.

© 2002 Taylor & Francis. ISSN 0346-1238 DOI: 0346123011010629 1

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Scand. Actuarial J. 4B. Højgaard226

Assume that the company wants to control its premium in an optimal way butis restricted to keeping the premium principle � xed. Then the only controllableparameter in the premium rate is the safety loading h. Intuitively the result ofchanging the premium will be a change in the size of the portfolio and therefore achange in the intensity. This means the intensity is a function of the safety loading,say b¾d(h). The control region for the parameter h is the non-negative half line[0, Ä) and it seems reasonable to assume that d(h) will be strictly decreasing.Furthermore we assume, that there is only a � nite number of potential customers,which implies that d (0)¾bmax BÄ. Finally d (h) must tend to 0 when h“Ä andthe convergence must be so fast that hd(h)“0. If this is not ful� lled, the companycan choose h¾Ä then no claims arrive but the premium rate is positive. This is anexample of ‘free lunch’ and will not be allowed in the model.

Since d(h) is assumed to be monotone is has an inverse h such that h¾h (b ),which implies that we can write p¾ f(b ) (1 »h(b ))bm. Therefore instead ofcontrolling h we can consider bÏ [0, bmax] as our control parameter. Throughout thepaper we only consider functions h for which f(b) is strictly concave.

The company distributes dividends to shareholders according to a barrier strat-egy, which means there exists a barrier bE0, such that the company pays out allof the reserve in excess of b as dividend and when the reserve level is below b nodividends are distributed. The objective for the company is to � nd an optimalintensity process {b*(t)} and barrier b* which maximizes the expected discountedfuture pay-out of dividend until the time of ruin. In Gerber (1969) it is shown thatfor initial reserve x0b* the optimal barrier strategy is optimal among all dividendstrategies, but if x\b* it is only optimal in case of exponential claim sizedistributions.

Dividend optimization is a classical problem in actuarial mathematics, goingback to de Finetti (1957) and being treated in eg. Buzzi (1974), Gerber (1979),Buhlmann (1970), Paulsen & Gjessing (1997). The novelty of the present paper isthe incorporation of a dynamic control parameter in the dividend optimizationproblem. The combination of dynamic control and dividend optimization ininsurance has recently gained a lot of interest, eg in Højgaard & Taksar (1998a,b,1999), Asmussen et al. (2000). The model in those papers is the diffusion model,which means that the reserve process is modeled as a Brownian motion withpositive drift. The diffusion model can be regarded as an limiting approximation tothe classical model treated in this paper.

To give a mathematical formulation of the problem we start with a probabilityspace (V, F ) equipped with a � ltration {F t } and a family of stochastic processes{R t

b} indexed by bE0, such that for each � xed b, R tb given by (1.1) is F t

measurable. An admissible control is a stochastic process {b(t)} satisfying that foreach tE0, 00b(t)0bmax and b (t) is F t -measurable for t\0. We let B denotethe set of all admissible controls. For any admissible b¾b(t) and barrier bE0 welet Lb,b(t) denote the cumulative dividend pay-outs up to time t. The controlledreserve {R t

b,b) is given by

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Optimal dynamic premium control in non-life insuranceScand. Actuarial J. 4 227

R tb,b ¾x»

t

0

f(b(s )) ds¼N t

b

i ¾ 1

Ui ¼t

0

dL b,b(s ), (1.3)

where

Lb,b(t)¾ (x¼b)» »t

0

f(b (s ))I (R sb,b ¾b) ds.

Note that the controlled arrival process {N tb} is a time-inhomogeneous Poisson

process. We introduce a value function

V bb(x)¾

tb,b

0

e¼ ct dLb,b(t), (1.4)

where c\0 is an exogenously given discount rate and tb,b ¾ inf{t\0: R tb,b B0} is

the time of ruin. Let

Vb (x )¾ supb( · )ÏB

V bb(x) (1.5)

and let

V(x)¾ supb E0

Vb (x). (1.6)

The objective is then to � nd V(x) and b*(t), b* satisfying V(x )¾V b*b*(x ).

The rest of the paper is organized as follows: In Section 2 we show how to � ndVb (x ) via solving a non-linear integro-differentia l equation known as the Bellmanequation and how to use the solution of the Bellman equation to construct theoptimal premium policy and the optimal barrier b*. In Section 3 we considerexponential claim size distributions and show that close form solution to theBellman equation can be found, which also gives closed form solutions of theoptimal premium control and dividend barrier. In Section 4 we show that forgeneral claim size distributions the solution and optimal controls can be approxi-mated numerically via a Dynamic Programming approach. In Section 5 we investi-gate the computationally much faster approximations based on the idea of DeVylder, in which the original problem is approximated by a problem with exponen-tial claim size distributions. The closed form solutions of the latter problem areused as approximations of the solutions to the original problem. Finally in Section6 some closing remarks and conclusions are given.

2. HJB-EQUATION

To � nd the optimal return function V given by (1.6) we � rst (in principle) � ndVb (x ) for all bE0 and then maximize over b.

DEFINITION 2.1. Let DÉ01 denote the class of functions f : [0, Ä], which is right

continuous with limits from the left and differentiable on (0, Ä) with right continuousderivative.

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It is well-known from stochastic control theory that if Vb (x)ÏDÉ01, it satis� es the

following Bellman-equation.

maxbÏ [0,b max]

f (b)V Æb (x )»b

x

0

Vb (x¼y)G(dy)¼Vb (x ) ¼cVb (x) ¾0 (2.1)

for all 0BxBb with

Vb (x )¾0 (2.2)

for xB0,

V Æb (b )¾1 (2.3)

and

Vb (x )¾Vb (b)»x¼b (2.4)

for x\b.We will not prove the above result. Proofs of similar results can be found in e.g.

Højgaard & Taksar (1998a, 1999). Instead we verify that if we have a solution Wof (2.1)–(2.3), then W and Vb coincides.

LEMMA 2.1 (Veri� cation Lemma). Fix bE0 and assume W (x)ÏDÉ01 satis� es

(2.1)–(2.4). Let b(x) be the maximizer in the left hand side of (2.1) and de� ne anintensity control by b*(t)¾b(R t

b*,b). Then W(x)¾Vb (x)¾Vbb*(x) for all x0b.

Proof. It is obvious that Vb (x ) must satisfy (2.4) for x\b. Therefore let x0b.For any b de� ne the operator A b by

A bz(x)¾ f(b)z Æ(x)»b

x

0

z(x¼y )G (dy)¼z(x ) .

Choose an arbitrary admissible control b (s ). Choose e\0 arbitrary and lette

b,b ¾ inf{t\0: R tb,b Be}. By Ito’s formula and W (x)ÏCÉ 1

[e¼ c(t ‚Teb,b )W (R t ‚t

eb,b

b,b )]¾W(x )»t ‚t

eb,b

0

e¼ cs[A b(s)W (R sb,b)¼cW(R s

b,b)] ds

¼t ‚t

eb,b

0

e¼ csW Æ(R sb,b) dL b,b(s ).

Since R tb,b \0 for all tBt e

b,b it follows from W satisfying (2.1) that

[e¼ c(t ‚teb,b)W (R t ‚t

eb,b

b,b )]0W (x )¼t ‚t

eb,b

0

e¼ csW Æ(R sb,b)I(R s

b,b ¾b ) dL b,b(s )

¾W (x )¼t ‚t

eb,b

0

e¼ cs dL b,b(s ), (2.5)

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where the last equality follows from W satisfying (2.3). Letting e“0 then teb,b

tb,b.The process {R t

b,b} is re� ected at b and it is well-known that tb,b BÄ a.s. Henceletting t“Ä the last term on the right hand side of the above inequality tends toV b

b(x) and since W satis� es (2.2) the left hand side of the above inequality tendszero. Therefore we obtain

V bb(x)5W(x).

Replacing b (s) by b*(s ) we obtain equality in (2.5) and thereby Vb (x)¾W (x)¾V b

b*(x).

LEMMA 2.2. Let b\0. Assume there exists a solution F to (2.1) then F is uniquelydetermined up to a multiplicative constant.

Proof. Since f is strictly concave, the maximizer b(x) for F in (2.1) is uniquelydetermined. Obviously b(x)¾0 can not be a maximizer. Therefore we can split[0, b ) into regions in which either b(x)¾bmax or b (x )Ï (0, bmax). Without loss ofgenerality we can assume b(x) to be non-decreasing, which implies there existx0 Ï [0, b ] such that b(x)Ï (0, bmax) for x0x0 and b (x )¾bmax for x\x0. Assumeb(x)Ï (0, bmax). Then b (x) must satisfy

f Æ(b (x ))F Æ(x)»x

0

F(x¼y)G(dy)¼F(x) ¾0 (2.6)

Insert this in (2.1) we get

f(b(x))F Æ(x )¼b (x)f Æ(b (x ))F Æ(x)¼cF(x )¾0. (2.7)

Let K(b)¾ f(b)¼bf Æ(b ) then for x0x0, F must satisfy

F Æ(x )F (x)

¾c

K(b (x ))

or that

F(x)¾C ex0 (c:K (b(y ))) dy. (2.8)

For x\x0 the equation (2.1) with b¾bmax is well-known to have a solution, whichis unique up to a multiplicative constant, see e.g. Buhlmann (1970). Implyingcontinuity at x0 we then get the solution F to be unique up to a multiplicativeconstant.

Assume that b (0 »)¾b0 with b0 Ï (0, bmax). Letting x 0 in (2.1) and (2.6) we have

f(b0)V Æb (0 »)¼ (b0 »c)Vb (0 »)¾0

f Æ(b0)V Æb (0 »)¼Vb (0 »)¾0

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so since V Æ(0 »)\0, b0 must satisfy

f(b0)¼ (b0 »c)f Æ(b0)¾0. (2.9)

From this we can conclude that

b(0 »)¾min(b0, bmax). (2.10)

Here we should note that since f(b)¾ f Ñ(b )m for some function f Ñ independent of theclaim size distribution, the constant b0 does not depend on the claim sizedistribution.

Note that the function K (b ) is a non-negative increasing function since K (0)¾f(0)E0 and K Æ(b )¾ ¼bf (b )\0.

Let F (x ) be a solution to (2.1) then Vb (x )¾CF(x ). Since V Æb (b)¾1 we getVb (x )¾F (x):F Æ(b). To maximize Vb (x ) we therefore must minimize F Æ(b ). Let b*be the minimizer then if b*\0

V(x)¾F(x):F Æ(b*) xBb*

F(b*):F Æ(b*)»x¼b* x\b*.(2.11)

If b*¾0, the function V(x) does not satisfy (2.1). In this case

V(x)¾x»V(0)¾x»maxb

T1

0

e¼ ctf (b ) dt¾x»maxb

f(b)c»b

¾x»f(b0)c»b0

,

with b0 given by (2.9). That is, b0 is the value of b which maximizes V 0b(0).

This implies, that in the rest of the paper we can restrict our search for solutionsF to (2.1), which satis� es F (0)¾1. This corresponds to letting C in (2.8) be equalto 1.

3. EXPONENTIAL CLAIM SIZE DISTRIBUTION

We start by considering the case of exponential claim size distribution, which meansGÉ (x)¾e¼ ax for some a\0. Assume that the maximizer b (x)Ï (0, bmax). In thatcase we know from (2.8) that a solution F (x) to (2.1) is given by

F(x)¾e 0x (c:K (b(y))) dy.

Still we need to determine the unknown maximizer b (x). Assume F is twicecontinuously differentiable. Then differentiating the above expression leads to

F Æ(x )¾c

K(b (x ))F(x )E0 (3.1)

and

F (x)¾c

K 2(b (x ))F(x)(c¼K Æ(b (x ))b Æ(x)). (3.2)

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This implies that

F (x)F Æ(x)

¾c¼K Æ(b (x ))b Æ(x)

K (x )(3.3)

Since GÉ (x)¾e¼ ax,

d

dx

x

0

F (x¼y )G (dy )¼F (x ) ¾ae¼ ax »x

0

F Æ(x¼y)G(dy)¼F Æ(x )

¾aF (x )¼a

x

0

F (x¼y )G (dy)¼F Æ(x)

¾ ¼a

x

0

F (x¼y)G(dy )¼F (x) ¼F Æ(x)

¾af Æ(b (x ))F Æ(x)¼F Æ(x ) (3.4)

by (2.6). Let

H(b )¾b»c¼baf Æ(b ). (3.5)

Replace Vb (x ) and b in (2.1) with F(x) and the maximizer b(x). Then differentiate(2.1) with respect to x and insert (3.4) to get

F (x)F Æ(x)

¾H (b (x))f (b (x))

. (3.6)

Equalizing the right hand size of (3.3) and (3.6) implies that b(x) must satisfy

b Æ¾f(b )c¼H (b)K(b)

f(b)K Æ(b )Åg(b). (3.7)

We know that b(0)¾b0. Therefore let

G(u)¾u

b 0

1g (y)

dy (3.8)

and b1 ¾b(b*). Assume b1 Eb0 and that g(y)\0 for b0 0y0b1. Then

b(x)¾G¼ 1(x). (3.9)

Note that since K is positive and increasing a suf� cient condition for g(y)\0 isH(y)B0. The optimal barrier b*¾G (b1), but b1 is unknown. To � nd b* minimiz-ing F Æ(x ) we have from (3.2) and (3.7)

F (x)¾c

K 2(b (x ))F(x)

H(b(x ))K (b (x))f(b(x ))

¾cH (b(x))

f (b (x))K(b(x))F (x). (3.10)

Hence sign (F (x))¾ sign(H (b(x))). This leads to three possible cases

1. b*¾0 implying b1 ¾b0.2. There exists x0 Bb* such that b (x)¾bmax for xÏ [x0, b*].

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3. b*¾G¼ 1(b1) for some b1 Ï (b0, bmax) satisfying H (b1)¾0.

Here we should note that in case 3 the equation H (y)¾0 might not have a uniquesolution in (b0, bmax). If b(x)¾bmax for xÏ [x0, b ] with x0 E0 then it follows fromBuhlmann (1970) that the solution F (x)¾h(xa) where the function h(x) is non-negative, non-decreasing and satis� es

bmaxh (x)¼ch Æ(x)¼ch(x)¾0.

This is, however, easily seen to be a convex function and therefore b*¾x0. Thisshows that case number 2 reduces to b1 ¾bmax. Note that H(b) de� ned by (3.5)does not depend on a. Therefore which of the possible cases we need to treat isdepending only on the functional form of f (b ) and the parameter c. The functionsK(b)¾KÑ (b):a and g(b)¾ag(b), where the º functions does not depend on a.Hence, letting G(u) be given by (3.8) for a¾1. Then we can � nd the solution fora"1 by the relation that ax¾G (b ) for bÏ (b0, b1) or that ba(x)¾b

1(ax). Thisimplies that for any a, Fa(x )¾F 1(ax). Combined with the above it follows that theoptimal return Va(x)¾V 1(ax):a and b*,a¾b*,1:a. Therefore in the followingsubsections we put a¾1. To � nd the maximizer b (x) and the optimal barrier b*,we need to put some explicit functional form on b¾d(h) or equivalently on h (b ).In the following subsections three different cases are considered.

3.1. Decrease of intensity of exponential order

Let d(h)¾k e¼ ah with k, a\0. Then bmax ¾k and h(b)¾ ¼ ln(b:k):a implying

f(b)¾ (1¼ ln(b :k ):a)b, f Æ(b)¾1¼1:a¼ ln(b:k):a, f (b )¾¼1ab

.

Then b0 must satisfy

b0 »c ln(b0:k )¾(a¼1)c,

which has a unique solution less than k if a01 or a\1 but cBk (a¼1)¼ 1. Thefunction H(b ) de� ned by (3.5) is

H(b )¾b»ca»b ln(b :k)a

.

PROPOSITION 3.1. Assume b0 Bk. Let

c¾2k

1» 1»4:aexp a

(1¼ 1»4:a)2

¼1 .

If caE c then b*¾0. If caB c then there exists a unique b0 Bb1 B1 such thatb*¾G¼ 1(b1).

Proof. We have aH Æ(b )¾2» ln(b:k ) and aH (b)¾1:b, hence H attains itsminimum at b*¾k e¼ 2 with value aH (b*)¾ca¼k e¼ 2. Therefore if ca\k e¼ 2

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then H (y)\0 for all y. Assume ca0k e¼ 2. Then aH (ca)¾2ca»ca ln(ca:k)00and aH (k)¾k»ca\0, hence the equation H (b)¾0 has a unique solutionb1 Ï [ca, k ]. We need to ensure that b0 Bb1 and H(y)B0 for all b0 ByBb1. Thelatter follows if we show that caBb0. Assume b0 0ca then by (2.9) c ln(ca:k)»cE0 implying caEk e¼ 1, which contradicts ca0k e¼ 2, so b0 \ca. To ensureb0 Bb1 we then need H (b0)B0 which implies (using that b0 satis� es (2.9))

(c»b0)a¼b 02:cB0

or that

b0 \ca(1 » 1»4:a )

2.

Inserting the latter into (2.9) we get the inequality

a(1 » 1»4:a )

2¼1 »1» ln ca

1» 1»4:a2k

B0

or that acB c. It is easily veri� ed that c is an increasing function of a withlima “Ä c¾e¼ 2, hence acB c implies acBk e¼ 2. The result then followseasily.

We have already argued that if b0 ¾k, which corresponds to cEk(a¼1)¼ 1. Thenalso b*¾0.

From the proof of Proposition 3.1 it is easily seen that H(y)B0 for all yÏ [b0, b1)and the solution F(x ) derived above is well-de� ned.

3.2. Decrease of intensity of power order

Let d(h)¾k (1»h)¼ r, k\0, r\1. Then h(b )¾k 1:rb

¼ 1:r ¼1 and we get

f(b)¾k 1:rb

1 ¼ 1:r.

The equation for b0 is then

b 01 ¼ 1:r ¼(c»b0)(1¼1:r )b 0

¼ 1:r ¾0

or

b0

r¼c(1¼1:r)¾0, (3.11)

which implies that

b0 ¾c(r¼1),

which is less than k if and only if cBk (r¼1)¼ 1. Now b1 must be the root of

H(y)¾y»c¼k 1:r(1¼1:r)y 1 ¼ 1:r ¾0. (3.12)

LEMMA 3.1. The equation (3.12) has a unique solution b1 Eb0 if and only if

c0kr

(1¼1:r)2r ¼ 1. (3.13)

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Proof. The function H has second derivative H (y)¾k 1:r(1¼1:r)21:ry¼ (1:r » 1) \0. Let y* satisfy H Æ(y*)¾0 then it follows from H (0)¾c\0 and H (k)¾c»k:r\0 that (3.12) has a solution if and only if H (y*)00. Simple calculations yieldsy*¾k(1¼1:r)2r and H (y*)¾k (1¼1:r)2r »c¼k(1¼1:r)2r ¼ 1, which is non-posi-tive if and only if (3.13) is satis� ed. Assume now (3.13) is satis� ed with strictinequality. Then convexity of H implies that there exists two roots y1, y2 of (3.12)with y1 By*By2. To show uniqueness of a solution greater than b0 we provey1 Bb0 By*. Then b1 ¾y2. Now

y*¼b0 ¾k (1¼1:r)2r ¼c(r¼1)¾k (1¼1:r)2r ¼cr (1¼1:r)\0

by (3.13). To prove b0 \y1 we show that H (b0)B0.

H(b0)¾c(r¼1) »c¼k 1:r(1¼1:r )c 1 ¼ 1:r(r¼1)1 ¼ 1:r

¾cr¼ck 1:r(1¼1:r)2 ¼ 1:rc¼ 1:r(r)1 ¼ 1:r.

The above expression is negative if

1¼k 1:r(1¼1:r)2 ¼ 1:rc¼ 1:r(r)¼ 1:r B0,

which corresponds to (3.13) being satis� ed with strict inequality. It follows easilyfrom the preceding calculations that if (3.13) is satis� ed with equality then b0 ¾y*¾b1.

The following result is an easy consequence

COROLLARY 3.1. If (3.13) is satis� ed then H (y)B0 for all yÏ [b0, b1) andb*¾G(b1), else b*¾0.

3.3. Linear decrease of intensity

Here d(h)¾k¼ah, k, a\0. Then we only allow hÏ [0, k :a ] and we get bÏ [0, k].With this restriction on h we can invert b(h) and get h(b )¾(k¼b):a implying

f(b)¾ 1»k¼b

ab,

which is strictly concave. The equation for b0 becomes

b 02 »2cb0 ¼c(a»k )¾0

having solution

b0 ¾ ¼c» c 2 »c(a»k),

which is less than k if and only if kEa or kBa and c0k 2:(a¼k ). Now

H(y)¾y»c¼y 1»k¼2y

a(3.14)

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satis� es H (0)¾c\0 and H(k)¾c»k 2:a\0 and therefore the equation H(y)¾0is easily seen to have a unique solution b1 Ï (0, k ) if and only if c0k 2:(8a). Thesolution is

b1 ¾k» k 2 ¼8ac

4. (3.15)

Consider the difference D(c)¾b1 ¼b0. Then D (0)¾1:2\0 and

D(k 2:(8a))¾k

4»

k 2

8a¼

k 4

64»

k 2

8a(a»k )B0

Also

D Æ(c)¾¼a

k 2 ¼8ca»1¼

1»(a»k ):(2c)

1»(a»k):cB0.

This implies there exists an unique c*Ï (0, k 2:(8a)) satisfying D (c*)¾0 such thatb1 Eb0 if and only if c0c*.

PROPOSITION 3.2. Assume c0c* then H (y)00 for all yÏ [b0, b1].

Proof. Let

y*¾k¼ k 2 ¼8ac

4

then H (y)00 for all yÏ [y*, b1]. Since c0c* implies b0 0b1, it is suf� cient to provethat b0 Ey*. Let D2(c)¾y*¼b0, then D2(0)¾0 and D2(k

2:(8a))¾D(k 2:8a))B0.It is easily veri� ed that D 2(c)00, which implies that D2(c)00 for all cÏ [0, k 2:(8a)]and since c*Bk 2:(8a) the result follows.

We get the following corollary

COROLLARY 3.2. If c0c* then b*¾G(b1) where b1 Eb0 is given by (3.15), elseb*¾0.

3.4. Numerics

Let us look at the behavior of the optimal choices of safety loading and dividendbarriers under different scenarios of costumers reaction on changes in the safetyloading. In all case we let k¾bmax ¾1. In Fig. 1 we consider the case ofexponential decrease with the parameter a varying in [0.5, 2].

In Fig. 2 we consider the case of power order decrease with the parameter rvarying in [2, 4] and the case of linear decrease with a varying in [0.5, 2.5]. Theparameters a and r in all cases describes the rate of reaction on a change in thesafety loading. We see that in all cases the optimal barrier b* is a decreasingfunction of this rate. For � xed x such that x0b* for all values of the parameter

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Fig. 1. The optimal controlh(x), x0b* in the case ofd(h)¾e¼ a h with aÏ [0.5, 2].

Fig. 2. The optimal control h(x), x0b* in the case of d (h)¾ (1»h)¼ r with rÏ [2, 4] and d (h)¾1¼ah with aÏ[0.5, 2.5].

also h(x ) is a decreasing function of the rate. However, the value h(b*) is onlydecreasing in the case of a decrease of power order, while it is increasing in the othercases. We are not able to give any explanation to these phenomena. Finally we shouldnote, that the reason for the optimal safety loadings being unreasonably high is theunrealistically large choice of k¾1.

4. GENERAL CLAIM SIZE DISTRIBUTION

In the case of non-exponential claim size distribution we can not analytically � ndsolutions F of (2.1). To � nd solutions numerically we follow the idea of Hipp & Taksar(2000), Schmidli (2001) to develop numerical schemes for approximations of F.

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To illustrate the idea behind the numerical scheme we start by some heuristictransformations of (2.1). Obviously b¾0 is not optimal to choose so we need onlyconsider bÏ(0, bmax). This implies f(b)\0 for all b and let us then assume that wecan divide by f(b) under the maximization (pure heuristics). Then we can rewrite(2.1) as

F Æ(x )» supbÏ (0,b max]

b

x

0

F(x¼y)G(dy)¼F (x) ¼cF (x)

f(b)¾0 (4.1)

or that

F Æ(x )¾ infbÏ (0,b max]

¼b

x

0

F (x¼y )G (dy )¼F (x ) »cF (x)

f(b). (4.2)

Integration by parts gives

¼x

0

F(x¼y)G(dy)¼F (x) ¾GÉ (x )»x

0

GÉ (x¼y)F Æ(y) dy,

and obviously F (x)¾1» x0 F Æ(y) dy. Therefore (4.2) can be rewritten as

F Æ(x )¾ infbÏ (0,b max]

bGÉ (x)»c»x

0

(bGÉ (x¼y)»c)F Æ(y) dy

f (b ). (4.3)

De� ning the operator K by

K Z(x)¾ infbÏ (0,b max]

bGÉ (x)»c»x

0

(bGÉ (x¼y)»c)Z(y) dy

f(b). (4.4)

Then we need to � nd a function Z(x) satisfying

Z¾K Z (4.5)

and

F(x)¾1»x

0

Z (y ) dy. (4.6)

That these heuristic transformations lead to the right solution is proved in Lemma4.2 below. First we must prove that there exists an unique solution Z to (4.5). Letb(x) be the minimizer of the right hand side of (4.4) and de� ne bmin ¾inf{b(x): xE0}. Then we will make the assumption that bmin ¾b0. This means thatwe can restrict the control region to be [b0, bmax].

Remark 4.1. The above assumption is not crucial, since it is intuitively clear, thatb(x) is non-decreasing and therefore bmin ¾b0. It is only introduced to simplify the

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proof of the following lemma, which can be proved in a slightly more complicatedway without this assumption (see e.g. Schmidli (2001) for a line of proof).

LEMMA 4.1. Fix b\0 arbitrary. Then there exists a unique solution Z (x ) to (4.5).

Proof. Assume Z1(x) and Z2(x) are two solutions with minimizers b1(x ) andb2(x) respectively. Then

Z1(x)¼Z2(x)¾b1(x)GÉ (x)»c»

x

0

(b1(x)GÉ (x¼y)»c)Z1(y) dy

f(b1(x ))

¼b2(x)GÉ (x)»c»

x

0

(b2(x)GÉ (x¼y)»c)Z2(y) dy

f(b2(x ))

0

x

0

(b2(x)GÉ (x¼y)»c)[Z1(y)¼Z2(y)] dy

f(b2(x ))

0Kx

0

Z1(y)¼Z2(y ) dy,

where

K¾ maxbÏ [b min,b max]

b»cf(b)

.

Interchanging Z1 and Z2 leads to

Z1(x)¼Z2(x ) 0Kx

0

Z1(y)¼Z2(y) dy. (4.7)

It then follows from Gronwalls inequality (see eg. Fleming & Rishel (1975) thatZ1(x)¼Z2(x ) 00 implying uniqueness. To establish existence, de� ne Z0(x)¾0and Zn (x)¾KZn ¼ 1(x) for nE1 with corresponding minimizers bn (x). LetxÏ [0, b ], then

Z1(x)¾ infbÏ [b 0,b max]

bGÉ (x)»cf (b )

\0¾Z0(x).

Assume Zk (x)BZk ¼ 1(x ) for all 10kBn and all xÏ [0, b ]. Then

Zn (x )¾bn (x)GÉ (x)»c»

x

0

(bn (x)GÉ (x¼y)»c)Zn ¼ 1(y) dy

f(bn (x))

\bn (x)GÉ (x)»c»

x

0

(bn ¼ 1(x)GÉ (x¼y)»c)Zn ¼ 2(y) dy

f(bn ¼ 1(x))

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Ebn ¼ 1(x)GÉ (x)»c»

x

0

(bn ¼ 1(x)GÉ (x¼y)»c)Zn ¼ 2(y) dy

f(bn ¼ 1(x))

¾Zn ¼ 1(x ).

Hence E n (x )ÅZn (x)¼Zn ¼ 1(x )\0. It follows using same arguments as above that00 lim supn “Ä E n (x )00 which implies that Z(x)¾ limn “Ä Zn (x) exists for allx0b and obviously Z(x )¾K Z(x) and is therefore a solution.

LEMMA 4.2. Let F be given by (4.6) for x0b with Z(x) being the unique solutionto (4.5) and de� ne W(x) by W (x)¾0 for xB0 and W (x)¾F(x):F Æ(b) for 00x0b.Let b(x ) be the minimizer of the right hand side of (4.4) and de� ne the optimalcontrol b*(t )¾b (R t

b*,b). Then Vb (x)¾W(x )¾V bb*(x ).

Proof. The proof follows the line of the proof of Lemma 2.1. We only need thefollowing observation, that since W satis� es (4.1)

t ‚teb,b

0

e¼ cs[A b(s)W(R sb,b)¼cW (R s

b,b)] ds¾t ‚te

b,b

0

e¼ csf(b(s))

½ W Æ(R sb,b)»

b(s )R s

b,b

0

W (R sb,b ¼y)G(dy)¼W(R s

b,b) ¼cW(R sb,b)

f (b (s))ds00

for all b (s ) and with equality for b*(s).

Finally we must � nd b*¾arg min{F Æ(b), bE0} and then the optimal return is

V(x)¾F(x):F Æ(b*) xBb*

x¼b* »F(b*):F Æ(b*) x\b*.(4.8)

Lemma 4.1 also shows how to approximate a solution F (x) to (2.1) by approximat-ing Z (x )¾F Æ(x) via a Dynamic Programming approach (or the method of succes-sive approximation) . This means to use the iterative scheme Zn (x )¾K Zn ¼ 1(x) forn¾0, . . . , N and then use ZN (x) as an approximation of Z(x) and b* is approxi-mate by arg inf{ZN (x), xE0}. In the following calculations we have used thecontrol region [e, bmax] with eÒb0 to verify the assumption that b(x)Ï [b0, bmax].We have used exponential decrease of intensity, meaning h(b)¾ ¼ ln(b). In Fig. 3the convergence rates of the functions ZN (x ) and hN (x ) are depicted. It indicates arather fast convergence and it should be noticed that the value bN* changessigni� cantly as function of N for N small. Note, that in the case of Gammadistributed claim sizes, the optimal return function is not convex as is the case withexponential claim size distributions and this implies that the optimal controlb(x)¾b0 for xÏ [0, x0] for some x0 \0.

In Fig. 4 comparisons are made of Z15(x), h15(x) with the exact values found inthe previous section (here we have smoothed h15(x) by simple moving averagemethods). These plots indicate that the numerical scheme works very well.

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Fig. 3. Convergence of ZN (x) and of hN (x)¾ ¼ ln(bN (x)), x0bN* (here for N¾2, 5, 8, 12, 15, 16), inthe case of UºGamma(3, 0.6).

Fig. 4. Comparisons of Z15(x), h15(x) with the exact values in case of exponential claim size distribution.

In Fig. 5 plots of the functions in case of uniform claim size distributions aregiven. These plots indicate that we can not in general expect F (x) to be twicecontinuously differentiable as in the case of exponential claim size distribution.

5. ‘‘DE VYLDER’’ APPROXIMATIONS

In the paper de Vylder (1978) a simple but ingenious idea for approximations ofultimate ruin probabilities was introduced. The idea is to approximate the riskprocess {Rt } with a risk process {RÑ

t } with exponential claim size distributions,such that the moments

limt “Ä

[R tn]

t¾ lim

t “Ä

[RÑtn]

t(5.1)

for n03 (under the restriction that the limits on both sides exist and are � nite).The probability of ultimate ruin can be calculated exactly for the process {RÑ

t } andthe

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Fig. 5. Plots of Z20(x) and h20(x), x0b20* in cases of UºUniform[0, 1].

idea is then to use this as an approximation of the ruin probability for the process{Rt }. This, in general, turned out to be a very good approximation (see e.g.Grandell (1990)). The process {RÑ

t } with claim sizes being exponentially distributedwith parameter a is characterized by the three parameters (n, bÑ , a ). Let mi ¾ [U i]then a choice of (n, bÑ , a ), such that (5.1) is satis� ed, is

a¾3m2

m2

, (5.2)

bÑ ¾9m 23

2m 32 b (5.3)

and

h¾2m1m3

3m 22 h. (5.4)

In this case, however, b is the control parameter of the original problem and bÑ thecontrol of the approximative problem. Therefore we cannot ensure both (5.3) and(5.4) to be satis� ed. On the other hand, we known that h¾h(b) and if we in theapproximating problem let h¾hÑ (bÑ ), which means we adjust the behavior of thecostumers to the approximation, then from (5.3) and (5.4) we � nd that we mustchoose

hÑ (bÑ )¾2m1m3

3m 22 h

2m 32

9m 23 bÑ . (5.5)

Let b(x) and bÑ (x) be the optimal control functions for the two problems respec-tively. Then to have a good approximation we need the functions h(b) and hÑ (bÑ ) aswell as the optimal barriers b* and bÑ* to be close. In Figs. 6–8, comparisons of theoptimal control functions and barriers are made for different claim size distribu-tions. Here we have also only considered h(b)¾ ¼ ln(b). We refer to h20(x) and b 20*as the exact values, that is that the non-smooth lines in the plots are regarded as

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Fig. 6. Exact and approximate values of h(x), x0b* in case of UºPareto(4) and Pareto(7) respec-tively.

Fig. 7. Exact and approximate values of h(x), x0b* in case UºPareto(10) and Pareto(12) respectively.

exact values. In Figs. 6 and 7 we consider Pareto(n) claim size distributions, i.e.GÉ (x)¾ (1 »x )¼ n. It is known that for heavy-tailed distributions the De Vylderapproximation of ruin probabilities does not work too well (see Grandell (1990)). InFig. 6 we have n¾4, 7 and also in this case the conclusion is that the heavier a tailthe worse is the approximation. In Fig. 7 we have n¾10, 12 and we get areasonable � t, although the tail is still subexponential .

In Fig. 8 we plot the approximations in case of Gamma(3, 0.6) and Uniform[0, 5]and also in these cases the � ts are reasonable. Especially, as is also the case in Fig.7, one should notice that the approximation of b* is very close. In the case ofGamma distributed claim sizes the approximation cannot catch the phenomenonthat b(x)¾b0 in a right neighborhood of 0, otherwise is has the right behavior. Inthe other cases, the behavior of the approximations also more or less look right,however, there is a systematic under-(respectively over-) estimation.

If we compare the approximating return function we see in Fig. 9 that all errorsin the approximating b(x) is accumulated such that there are rather large dis-crepancies in the return functions. However, again the behavior is rather similar.

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Fig. 8. Exact and approximate values of h(x), x0b* in case UºGamma(3, 0.6) and Uniform[0, 5]respectively.

Fig. 9. Exact and approximate values of V(x), x0b* in case of UºGamma(3, 0.6) and Pareto(10)respectively.

Considering the rather fast convergence of the numerical scheme developed inSection 4 and the rather larger errors on the approximating return functions, onemight ask, what is the use of these approximations based on the idea of De Vylder.One answer to that question is that the De Vylder approximation is a solution givenin closed form with a behavior much similar to the exact solution. Therefore onepossibility is to use the approximation for sensitivity analysis such as analysis ofdependence of the optimal controls on the free parameters in a given class of claimsize distributions. For example in the case of claim size distributions beingUniform[a, b], there a and the function hÑ will depend on a, b and (although beinga dif� cult task) the De Vylder approximation has the opportunity to give someanalytic results on the dependence of the optimal control on theses parameters.Another example is to investigate how large the discount factor c should be to getb*¾0. Let us consider an example:

EXAMPLE 5.1. Let UºGamma(g, b) and let h (b )¾ ¼ ln(b ). Then hÑ (bÑ )¾¼ln(bÑ :k ):a, where

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k¾9m 2

3

2m 32¾ g(g»1)

2(g»2)2

and

a¾3m 2

2

2m1m3

¾3(g»1)2(g»2)

.

Then the critical value for c is c:a with c de� ned by Proposition 3.1. Here weobserve that the critical value only depends on g and it is easily veri� ed that it is anincreasing function of g. By choosing c close to this critical value both smaller andlarger in the numerical scheme of the previous section we have checked for thevalidity of this choice of a critical value. In all the cases we investigated it seemedto be a very good approximation.

6. CLOSING REMARKS

First we would like to emphasize, that to get any empirical knowledge about thefunctional form of b¾d(h) is an extremely dif� cult task. On the other hand thismodel can be used to put up different scenarios, where it is to � nd the optimalcontrols and even more interesting to see how sensitive the optimal controls are tochanges in the modeling of costumers behavior.

As mentioned in the Introduction a series of dividend distribution:dynamiccontrol problems has been treated in the diffusion model. Also in these cases closedform solutions are found, but whether these results can be used for conclusions inthe classical model is still unknown. The only result of that kind is in Schmidli(2001), Hipp & Plum (1999), who consider optimal reinsurance and optimalinvestment strategies respectively. In both papers the objective is minimization ofthe probability of ruin and it is shown that the optimal control in the diffusionmodel and in the classical model are of a completely different form. This is,however, not the case with the de Vylder approximations in this paper. Althoughthe approximations are not close we still believe that they can be used for someanalysis of the original problem.

ACKNOWLEDGEMENTS

The author would like to thank Søren Asmussen, who introduced a problem leading to the present paperand Hanspeter Schmidli for valuable discussions.

REFERENCES

Asmussen, S., Højgaard, B. & Taksar, M. (2000). Optimal risk control and dividend distributionpolicies. Example of excess-of-loss reinsurance for an insurance corporation. Finance & Stochastics 4(3), 299–324.

Buhlmann, H. (1970). Mathematical methods in risk theory. Springer Verlag, Berlin.Buzzi, R. (1974). Optimale Dividendestrategien fur den Risikoprozess mit austauschbaren Zuwachsen. PhD

thesis, ETH Zurich.

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de Finetti, B. (1957). Su un’ impostazione alternativa dell teoria collectiva del rischio. Transactions of the15th international Congress of Actuaries, New York 23, 433–443.

de Vylder, F. (1978). A practical solution to the problem of ultimate ruin probability. Scand. Act. J.1978, 114–119.

Fleming, W. & Rishel, R. (1975). Deterministic and stochastic optimal control. Springer Verlag, NewYork.

Gerber, H. (1969). Entschedungskriterien fur den zusammengesetzten poisson-process. Mitt. Ver. Scwiez.Versich. 69, 185–228.

Gerber, H. (1979). An introduction to mathematical risk theory, S.S. Huebner Foundation Monographs,University of Pennsylvania.

Grandell, J. (1990). Aspects of risk theory. Springer Verlag, Berlin.Hipp, C. & Plum, M. (2000). Optimal investments for insurers. Insurance: Mathematics and Economics

27, 215–228.Hipp, C. & Taksar, M. (2000). Stochastic control for optimal new business. Insurance: Mathematics and

Economics 26, 185– 192.Højgaard, B. & Taksar, M. (1998a). Optimal proportional reinsurance policies for diffusion models.

Scand. Act. J. 1998 (2), 166–180.Højgaard, B. & Taksar, M. (1998b). Optimal proportional reinsurance policies for diffusion models with

transaction costs. Insurance: Mathematics and Economics 22, 41–51.Højgaard, B. & Taksar, M. (1999). Controlling risk exposure and dividends pay-out schemes: Insurance

company example. Mathematical Finance 9 (2), 153–182.Paulsen, J. & Gjessing, H. (1997). Optimal choice of dividend barriers for a risk process with stochastic

return of investment. Insurance: Mathematics and Economics 20, 215–223.Schmidli, H. (2001). Optimal proportional reinsurance policies in a dynamic setting. Scand. Act. J. 2001

(1), 51–68.

Address for correspondence:Bjarne HøjgaardDep. of Mathematical SciencesAalborg UniversityDenmark

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