We deal with a generalization of the risk model with stochastic premiums where dividends are paid according to a constant dividend strategy and consider heuristic approximations for the ruin probability. To be more precise, we construct five- and three-moment analogues to the De Vylder approximation. To this end, we obtain an explicit formula for the ruin probability in the case of exponentially distributed premium and claim sizes. Finally, we analyze the accuracy of the approximations for some typical distributions of premium and claim sizes using statistical estimates obtained by the Monte Carlo methods.
Risk model with stochastic premiumsconstant dividend strategyruin probabilitynet profit conditionDe Vylder approximationMonte Carlo method91B3060G51Introduction
The ruin probability of an insurance company is one of the main risk measures considered in risk theory, and the problems of its calculation and approximation have attracted a lot of attention recently (see, e.g., [2, 16, 22, 25, 27–29] and references therein). Risk models where shareholders receive dividends from their insurance company have been of great interest to researchers since De Finetti first considered dividend strategies in insurance dealing with a binomial model [12]. The classical risk model and its various modifications with different dividend strategies are investigated in a number of papers (see, e.g., [1, 4, 8, 10, 11, 15, 20, 21, 26, 27, 30] and references therein).
It is well known that explicit formulas for the ruin probability can be derived only in a few special cases even for the classical Cramér–Lundberg risk model, so numerous heuristic approximations for this function have been proposed and studied (see [2, 3, 5, 9, 13, 16, 17, 28, 29]). So-called simple approximations, which use only some moments of the distribution of claim sizes and do not take into account its tail behavior, form a special class of approximations for the ruin probabilities.
The De Vylder approximation, which is introduced in [13] for the classical risk model, is supposed to be one of the most successful simple approximations. It is based on the heuristic idea to replace the investigated risk process by a risk process with exponentially distributed claim sizes such that the first three moments coincide (see also [16, 28, 29] for details). Thus, to apply the De Vylder approximation, we need to calculate only the first three moments of the distribution of the claim sizes. Despite its simplicity, the approximation gives surprisingly good results when the initial surplus is not too small, especially when the distribution of claim sizes is light-tailed. This fact was explained later by Grandell [17] after analyzing the simple approximations from a mathematical viewpoint. Analogues to the De Vylder approximation are constructed in the risk model with additional funds [23] and some risk models with reinsurance [7, 19].
The present paper deals with a generalization of the risk model with stochastic premiums where dividends are paid according to a constant dividend strategy. In what follows, we suppose that all stochastic objects we use below are defined on a probability space (Ω,F,P) satisfying the usual conditions. In the risk model with stochastic premiums (see, e.g., [6, 22]), premium sizes form a sequence (Y¯i)i≥1 of non-negative independent and identically distributed (i.i.d.) random variables (r.v.’s) with cumulative distribution function (c.d.f.) FY¯(y)=P[Y¯i≤y], and the number of premiums on the time interval [0,t] is a Poisson process (N¯t)t≥0 with constant intensity λ¯>00$]]>. Similarly, claim sizes form a sequence (Yi)i≥1 of i.i.d. r.v.’s with c.d.f. FY(y)=P[Yi≤y], and the number of claims on the time interval [0,t] is a Poisson process (Nt)t≥0 with constant intensity λ>00$]]>. Thus, the total premiums and claims on [0,t] equal ∑i=1N¯tY¯i and ∑i=1NtYi, respectively. Note that here ∑i=1N¯tY¯i=0 if N¯t=0, and ∑i=1NtYi=0 if Nt=0. In what follows, we also assume that the r.v.’s (Yi)i≥1 and (Y¯i)i≥1 have finite expectations μ>00$]]> and μ¯>00$]]>, respectively, and (Yi)i≥1, (Y¯i)i≥1, (Nt)t≥0 and (N¯t)t≥0 are mutually independent.
Moreover, we make the additional assumption that the insurance company pays dividends to its shareholders according to a constant dividend strategy, which implies that dividends are paid continuously at a rate d>00$]]>. The strategy can be considered as a multi-layer dividend strategy where the number of layers is equal to one (see, e.g., [27])). Next, we denote a non-negative initial surplus of the insurance company by x, and let Xt(x) be its surplus at time t provided that the initial surplus is x. Then the surplus process (Xt(x))t≥0 is defined by the equality
Xt(x)=x+∑i=1N¯tY¯i−∑i=1NtYi−dt,t≥0.
Next, let τ(x)=inf{t≥0:Xt(x)<0} be the ruin time for the risk process (Xt(x))t≥0 defined by (1). For x≥0, the infinite-horizon ruin probability is defined by
ψ(x)=E[𝟙(τ(x)<∞)|X0(x)=x],
where 𝟙(·) is the indicator function. Note that the ruin probability is a special case of the expected discounted penalty function, which is introduced in [14] and also called the Gerber–Shiu function.
Thus, it is easily seen that the risk model described above is a special case of the model with stochastic premiums and a multi-layer dividend strategy investigated in [27], although in that paper it is assumed that the number of layers is more than one. In [27], piecewise integro-differential equations for the Gerber–Shiu function and the expected discounted dividend payments until ruin are derived. In addition, the model is studied in detail in the case of exponentially distributed claim and premium sizes. In particular, explicit formulas for the ruin probability as well as for the expected discounted dividend payments are obtained.
The aim of the present paper is to construct analogues to the De Vylder approximation for the ruin probability in the risk model described above and analyze the accuracy of these approximations. The rest of the paper is organized as follows. In Section 2, we obtain an explicit formula for the ruin probability in the case of exponentially distributed premium and claim sizes. We use this formula in Section 3, where we derive five- and three-moment analogues to the De Vylder approximation. Finally, Section 4 is devoted to numerical illustrations. To be more precise, we deal with some typical distributions of premium and claim sizes and apply the results obtained in Section 3. To analyze the accuracy of the approximations, we use statistical estimates obtained by the Monte Carlo methods.
An explicit formula for the ruin probability in the case of exponentially distributed premium and claim sizes
From now on, we suppose that the net profit condition holds, which in this model means that
λ¯μ¯>λμ+d.\lambda \mu +d.\]]]>
Theorem 1 below is a special case of Theorem 1 in [27], where it is formulated and proved for the Gerber–Shiu function in the model where the number of layers is more than one. It is easy to check that the assertion of the theorem remains true if the number of layers equals one.
Let the surplus process(Xt(x))t≥0be defined by (1) under the above assumptions, and letFY(y)be continuous onR+. Then the functionψ(x)is differentiable onR+and satisfies the integro-differential equationdψ′(x)+(λ¯+λ)ψ(x)=λ¯∫0∞ψ(x+y)dFY¯(y)+λ∫0xψ(x−y)dFY(y)+λ(1−FY(x)),x≥0.
To solve equation (3), we use the following two boundary conditions. Firstly, using standard considerations (see, e.g., [22, 24, 28]) it can be easily shown that limx→∞ψ(x)=0 provided that the net profit condition (2) holds. Secondly, it is obvious that ψ(0)=1 for this risk model. Although equation (3) is not solvable analytically in the general case, we can find explicit expressions for the corresponding ruin probability in some special cases. The uniqueness of the required solution to equation (3) should be justified in each case.
Assume now that the premium and claim sizes are exponentially distributed, i.e. their probability density functions (p.d.f.’s) are
fY¯(y)=1μ¯e−y/μ¯andfY(y)=1μe−y/μ,y≥0,
respectively. In this case, the integro-differential equation (3) can be reduced to a linear differential equation with constant coefficients.
Let the surplus process(Xt(x))t≥0be defined by (1) under the above assumptions, and let the premium and claim sizes be exponentially distributed with meansμ¯and μ, respectively. Then for allx≥0,ψ(x)is a solution to the differential equationdμ¯μψ‴(x)+(d(μ¯−μ)+μ¯μ(λ¯+λ))ψ″(x)+(λ¯μ¯−λμ−d)ψ′(x)=0.
The proof of Lemma 1 is similar to the proof of Lemma 1 in [27]. An explicit formula for the ruin probability is given in Theorem 2 below.
Let the surplus process(Xt(x))t≥0follow (1) under the above assumptions, and let premium and claim sizes be exponentially distributed with meansμ¯and μ, respectively. If the net profit condition (2) holds, thenψ(x)=C1ez1x+C2ez2xfor allx≥0,wherez1=−(d(μ¯−μ)+μ¯μ(λ¯+λ))+D2dμ¯μ,z2=−(d(μ¯−μ)+μ¯μ(λ¯+λ))−D2dμ¯μ,D=(d(μ¯+μ)+μ¯μ(λ−λ¯))2+4λ¯λμ¯2μ2,C1=λ¯μ¯(μ¯+μ)(dz2+λ¯)+dλ¯μ(μ¯z1−1)dλ¯μ¯2(z2−z1)andC2=−λ¯μ¯(μ¯+μ)(dz1+λ¯)+dλ¯μ(μ¯z2−1)dλ¯μ¯2(z2−z1).
By Lemma 1, ψ(x) is a solution to (4) for all x≥0. The characteristic equation corresponding to (4) has the form
dμ¯μz3+(d(μ¯−μ)+μ¯μ(λ¯+λ))z2+(λ¯μ¯−λμ−d)z=0.
The discriminant of the equation
dμ¯μz2+(d(μ¯−μ)+μ¯μ(λ¯+λ))z+(λ¯μ¯−λμ−d)=0
is equal to
(d(μ¯−μ)+μ¯μ(λ¯+λ))2−4dμ¯μ(λ¯μ¯−λμ−d)=(d(μ¯+μ)+μ¯μ(λ−λ¯))2+4λ¯λμ¯2μ2,
which is obviously positive and coincides with the constant D introduced above. Therefore, z1 and z2 defined in the assertion of the theorem are two real roots of equation (7).
By the net profit condition (2), we conclude that λ¯μ¯−λμ−d>00$]]> and
d(μ¯−μ)+μ¯μ(λ¯+λ)=μ(λ¯μ¯−λμ−d)+λμ2+λμ¯μ+dμ¯>0,0,\]]]>
which implies that z1<0 and z2<0 by Vieta’s theorem. Consequently, z1<0, z2<0 and z3=0 are roots of equation (6), from which we deduce that
ψ(x)=C1ez1x+C2ez2x+C3for allx≥0
with some constants C1, C2 and C3. Since the net profit condition (2) holds, by Remark 1, we have limx→∞ψ(x)=0, which yields C3=0. So we obtain (5).
To determine the constants C1 and C2, we use the following two conditions. Firstly, substituting (5) into the equality ψ(0)=1 we get
C1+C2=1.
Secondly, letting x=0 in (3) we obtain
dψ′(0)+(λ¯+λ)ψ(0)=λ¯∫0∞ψ(y)dFY¯(y)+λ.
Since
ψ′(x)=C1z1ez1x+C2z2ez2xfor allx≥0
and
1μ¯∫0∞ψ(u)e−u/μ¯du=−C1μ¯z1−1−C2μ¯z2−1,
from (9) we get
C1(dz1+λ¯μ¯z1−1)+C2(dz2+λ¯μ¯z2−1)=−λ¯.
Taking into account that
(μ¯z1−1)(μ¯z2−1)=μ¯2z1z2−μ¯(z1+z2)+1=μ¯2λ¯μ¯−λμ−ddμ¯μ+μ¯d(μ¯−μ)+μ¯μ(λ¯+λ)dμ¯μ+1=λ¯μ¯(μ¯+μ)dμ,
we find the constants C1 and C2 from the system of linear equations (8) and (10), which always has a unique solution. Applying arguments similar to those in the proof of Theorem 3 in [27] we can show that the function ψ(x) that we found is a unique solution to (3) satisfying the required conditions, which completes the proof. □
Analogues to the De Vylder approximationAn auxiliary result
Let the process (Ut)t≥0 be defined by
Ut=∑i=1N¯tY¯i−∑i=1NtYi−dt,t≥0.
We construct two analogues to the De Vylder approximation replacing the process (Ut)t≥0 by a process (U˜t)t≥0 with exponentially distributed premium and claim sizes. Since in this risk model the process (U˜t)t≥0 is determined by five parameters, which we denote by λ¯0, μ¯0, λ0, μ0 and d0, five equalities are required to determine these parameters. Consequently, we need the first five moments of (Ut)t≥0.
Let the process(Ut)t≥0be defined by (11) under the above assumptions,E[Y¯i5]<∞andE[Yi5]<∞. Then for allt≥0, we haveE[Ut]=(λ¯μ¯−λμ−d)t,E[Ut2]=(E[Ut])2+(λ¯E[Y¯2]+λE[Y2])t,E[Ut3]=(E[Ut])3+3E[Ut]·(λ¯E[Y¯2]+λE[Y2])t+(λ¯E[Y¯3]−λE[Y3])t,E[Ut4]=(E[Ut])4+6(E[Ut])2·(λ¯E[Y¯2]+λE[Y2])t+4E[Ut]·(λ¯E[Y¯3]−λE[Y3])t+3(λ¯E[Y¯2]+λE[Y2])t2+(λ¯E[Y¯4]+λE[Y4])t,E[Ut5]=(E[Ut])5+10(E[Ut])3·(λ¯E[Y¯2]+λE[Y2])t+10(E[Ut])2·(λ¯E[Y¯3]−λE[Y3])t+15E[Ut]·(λ¯E[Y¯2]+λE[Y2])2t2+5E[Ut]·(λ¯E[Y¯4]+λE[Y4])t+10(λ¯E[Y¯2]+λE[Y2])(λ¯E[Y¯3]−λE[Y3])t2+(λ¯E[Y¯5]−λE[Y5])t.
Let MY¯(s) and MY(s) be the moment generating functions of the r.v.’s Y¯i and Yi, respectively, provided that they exist in some neighborhood of s=0. Furthermore, we denote the moment generating function of (Ut)t≥0 by M(s). An easy computation shows that
M(s)=E[esUt]=exp{λ¯t(MY¯(s)−1)+λt(MY(−s)−1)−dts}.
Taking the first five derivatives of M(s) yields
M′(s)=(λ¯tMY¯′(s)−λtMY′(−s)−dt)M(s),M″(s)=((λ¯tMY¯′(s)−λtMY′(−s)−dt)2+λ¯tMY¯″(s)+λtMY″(−s))M(s),M‴(s)=((λ¯tMY¯′(s)−λtMY′(−s)−dt)3+3(λ¯tMY¯′(s)−λtMY′(−s)−dt)(λ¯tMY¯″(s)+λtMY″(−s))+λ¯tMY¯‴(s)−λtMY‴(−s))M(s),M(IV)(s)=((λ¯tMY¯′(s)−λtMY′(−s)−dt)4+6(λ¯tMY¯′(s)−λtMY′(−s)−dt)2(λ¯tMY¯″(s)+λtMY″(−s))+4(λ¯tMY¯′(s)−λtMY′(−s)−dt)(λ¯tMY¯‴(s)−λtMY‴(−s))+3(λ¯tMY¯″(s)+λtMY″(−s))2+λ¯tMY¯(IV)(s)+λtMY(IV)(−s))M(s)
and
M(V)(s)=((λ¯tMY¯′(s)−λtMY′(−s)−dt)5+10(λ¯tMY¯′(s)−λtMY′(−s)−dt)3(λ¯tMY¯″(s)+λtMY″(−s))+10(λ¯tMY¯′(s)−λtMY′(−s)−dt)2(λ¯tMY¯‴(s)−λtMY‴(−s))+15(λ¯tMY¯′(s)−λtMY′(−s)−dt)(λ¯tMY¯″(s)+λtMY″(−s))2+5(λ¯tMY¯′(s)−λtMY′(−s)−dt)(λ¯tMY¯(IV)(s)+λtMY(IV)(−s))+10(λ¯tMY¯″(s)+λtMY″(−s))(λ¯tMY¯‴(s)−λtMY‴(−s))+λ¯tMY¯(V)(s)−λtMY(V)(−s))M(s).
Since E[Utk]=M(k)(0) for all integer k≥1, substituting s=0 into the formulas above gives (12)–(16).
If the moment generating functions of Y¯i and Yi do not exist, we can obtain (12)–(16) by a direct computation of the required expectations provided that E[Y¯i5]<∞ and E[Yi5]<∞, which completes the proof. □
In what follows, we use the following constants:
γ2=λ¯E[Y¯2]+λE[Y2],γ3=λ¯E[Y¯3]−λE[Y3],γ4=λ¯E[Y¯4]+λE[Y4],γ5=λ¯E[Y¯5]−λE[Y5].
A five-moment approximation
To construct a five-moment analogue to the De Vylder approximation, we replace the process (Ut)t≥0 by a process (U˜t)t≥0 with exponentially distributed premium and claim sizes such that
E[Utk]=E[U˜tk],k=1,2,3,4,5.(a five-moment analogue to the De Vylder approximation).
Let the surplus process(Xt(x))t≥0be defined by (1) under the above assumptions,E[Y¯i5]<∞,E[Yi5]<∞, and let the net profit condition (2) hold. Then the ruin probability is approximately equal toψDV5(x)=C1ez1x+C2ez2xfor allx≥0,wherez1=−(d0(μ¯0−μ0)+μ¯0μ0(λ¯0+λ0))+D2d0μ¯0μ0,z2=−(d0(μ¯0−μ0)+μ¯0μ0(λ¯0+λ0))−D2d0μ¯0μ0,D=(d0(μ¯0+μ0)+μ¯0μ0(λ0−λ¯0))2+4λ¯0λ0μ¯02μ02,C1=λ¯0μ¯0(μ¯0+μ0)(dz2+λ¯0)+d0λ¯0μ0(μ¯0z1−1)d0λ¯0μ¯02(z2−z1),C2=−λ¯0μ¯0(μ¯0+μ0)(dz1+λ¯0)+d0λ¯0μ0(μ¯0z2−1)d0λ¯0μ¯02(z2−z1),and the constantsλ¯0,μ¯0,λ0,μ0andd0are defined by the following equalities:μ0=−5γ3γ4−3γ2γ540γ32−30γ2γ4+(5γ3γ4−3γ2γ5)2+(4γ3γ5−5γ42)(20γ32−15γ2γ4)|40γ32−30γ2γ4|,μ¯0=5γ3γ4−3γ2γ540γ32−30γ2γ4+(5γ3γ4−3γ2γ5)2+(4γ3γ5−5γ42)(20γ32−15γ2γ4)|40γ32−30γ2γ4|,λ0=3μ¯0γ2−γ36μ02(μ¯0+μ0),λ¯0=γ2−2λ0μ022μ¯02,d0=λ¯0μ¯0−λ0μ0−(λ¯μ¯−λμ−d),provided thatλ¯0>00$]]>,μ¯0>00$]]>,λ0>00$]]>,μ0>00$]]>,d0>00$]]>,4γ32−3γ2γ4≠0and(5γ3γ4−3γ2γ5)2+(4γ3γ5−5γ42)(20γ32−15γ2γ4)>0.0.\]]]>
Taking into account that the kth moments of the r.v.’s that are exponentially distributed with means μ¯0 and μ0 equal k!μ¯0k and k!μ0k, respectively, from Lemma 2 we conclude that (17) is equivalent to the system of equations (19)–(23): λ¯0μ¯0−λ0μ0−d0=λ¯μ¯−λμ−d,2λ¯0μ¯02+2λ0μ02=γ2,6λ¯0μ¯03−6λ0μ03=γ3,24λ¯0μ¯04+24λ0μ04=γ4,120λ¯0μ¯05−120λ0μ05=γ5.
Now our aim is to find the constants λ¯0, μ¯0, λ0, μ0 and d0 from this system. From (20) we have 2λ¯0μ¯02=γ2−2λ0μ02. Substituting this into equations (21)–(23) we get 3γ2μ¯0−6λ0μ02(μ¯0+μ0)=γ3,12γ2μ¯02−24λ0μ02(μ¯02−μ02)=γ4,60γ2μ¯03−120λ0μ02(μ¯03+μ03)=γ5.
Next, from (24) we have 6λ0μ02(μ¯0+μ0)=3γ2μ¯0−γ3. Substituting this into equations (25)–(26) we obtain
12γ2μ¯0μ0+4γ3(μ¯0−μ0)=γ4
and
60γ2μ¯0μ0(μ¯0−μ0)+20γ3((μ¯0−μ0)2+μ¯0μ0)=γ5.
Multiplying (27) by (−5(μ¯0−μ0)) and adding (28) we get
20γ3μ¯0μ0+5γ4(μ¯0−μ0)=γ5.
Note that (27) and (29) form a system of two equations, which are linear with respect to variables μ¯0μ0 and μ¯0−μ0. Solving this system we obtain
μ¯0μ0=4γ3γ5−5γ4280γ32−60γ2γ4
and
μ¯0−μ0=5γ3γ4−3γ2γ520γ32−15γ2γ4
provided that 4γ32−3γ2γ4≠0.
Substituting the expression for μ¯0 from (31) into (30) gives
μ02+5γ3γ4−3γ2γ520γ32−15γ2γ4μ0−4γ3γ5−5γ4280γ32−60γ2γ4=0,
from which we have
μ0=−5γ3γ4−3γ2γ540γ32−30γ2γ4±(5γ3γ4−3γ2γ5)2+(4γ3γ5−5γ42)(20γ32−15γ2γ4)|40γ32−30γ2γ4|
provided that (5γ3γ4−3γ2γ5)2+(4γ3γ5−5γ42)(20γ32−15γ2γ4)>00$]]> and 4γ32−3γ2γ4≠0. Hence, taking into account (31) we get
μ¯0=5γ3γ4−3γ2γ540γ32−30γ2γ4±(5γ3γ4−3γ2γ5)2+(4γ3γ5−5γ42)(20γ32−15γ2γ4)|40γ32−30γ2γ4|.
Since both μ¯0 and μ0 must be positive, from the expressions for μ¯0 and μ0 we deduce that we can take only the values of the parameters given in the assertion of the theorem (otherwise, if we take the values with “−”, at least one of the parameters μ¯0 or μ0 will be negative). Finally, we obtain the corresponding values of the parameters λ0, λ¯0 and d0 from (24), (20) and (19), respectively, provided that all the values are positive.
Thus, we have a new process (U˜t)t≥0 with exponentially distributed premium and claim sizes, which is completely determined by λ¯0, μ¯0, λ0, μ0 and d0. Now the assertion of the theorem follows immediately from Theorem 2. □
A three-moment approximation
From the assertion of Theorem 3 it is clear that its conditions are quite restrictive. In particular, some of the parameters λ¯0, μ¯0, λ0, μ0 and d0 can be negative, and numerical computations show that such situations happen quite often. Hence, it is impossible to construct the five-moment approximation in those cases. Namely for this reason we also consider a simplified three-moment analogue to the De Vylder approximation. To construct it, we replace the process (Ut)t≥0 by the process (U˜t)t≥0 with exponentially distributed premium and claim sizes such that
E[Utk]=E[U˜tk],k=1,2,3,
and the following proportionality conditions hold:
μ¯μ=μ¯0μ0andλ¯λ=λ¯0λ0.
In particular, condition (34) implies that λ¯μ¯/λμ=λ¯0μ¯0/λ0μ0. This means that the ratio between the expected premiums and the expected claims per unit time remains the same, which seems to be natural.
(a three-moment analogue to the De Vylder approximation).
Let the surplus process(Xt(x))t≥0is defined by (1) under the above assumptions,E[Y¯i3]<∞,E[Yi3]<∞, and let the net profit condition (2) hold. Then the ruin probability is approximately equal toψDV3(x)=C1ez1x+C2ez2xfor allx≥0,wherez1,z2,C1andC2are defined as in Theorem3and the constantsλ¯0,μ¯0,λ0,μ0andd0are defined by the following equalities:μ0=γ3μ(λ¯μ¯2+λμ2)3γ2(λ¯μ¯3−λμ3),λ0=9γ23λ(λ¯μ¯3−λμ3)22γ32(λ¯μ¯2+λμ2)3,μ¯0=γ3μ¯(λ¯μ¯2+λμ2)3γ2(λ¯μ¯3−λμ3),λ¯0=9γ23λ¯(λ¯μ¯3−λμ3)22γ32(λ¯μ¯2+λμ2)3,d0=λ¯0μ¯0−λ0μ0−(λ¯μ¯−λμ−d),provided thatγ3(λ¯μ¯3−λμ3)>00$]]>andd0>00$]]>.
From Lemma 2 we conclude that (33) is equivalent to the system of equations (19)–(21), and from (34) we get
λ¯0μ¯02=λ0μ02λ¯μ¯2λμ2andλ¯0μ¯03=λ0μ03λ¯μ¯3λμ3.
Substituting (36) into (20) and (21) we obtain
2λ0μ02(λ¯μ¯2λμ2+1)=γ2
and
6λ0μ03(λ¯μ¯3λμ3−1)=γ3.
Dividing (38) by (37) we easily get the expression for μ0 provided that γ3(λ¯μ¯3−λμ3)>00$]]>. Next, substituting this expression into (37) we obtain the expression for λ0. The constants μ¯0 and λ¯0 are determined from (34). Finally, we can find d0 from (19). Applying Theorem 2 for exponentially distributed premium and claim sizes yields the assertion of the theorem. □
Comparing the assertions of Theorems 3 and 4 we deduce that the conditions of Theorem 4 are much less restrictive.
Instead of conditions (34), we can consider the following more general conditions:
μ¯μ=ν1μ¯0μ0andλ¯λ=ν2λ¯0λ0,
where ν1>00$]]> and ν2>00$]]>. The corresponding approximation for the ruin probability is calculated using the same formula (35), but the constants λ¯0, μ¯0, λ0, μ0 and d0 are defined by the following equalities:
μ0=γ3μν1(λ¯μ¯2+λμ2ν12ν2)3γ2(λ¯μ¯3−λμ3ν13ν2),λ0=9γ23λν2(λ¯μ¯3−λμ3ν13ν2)22γ32(λ¯μ¯2+λμ2ν13ν2)3,μ¯0=γ3μ¯(λ¯μ¯2+λμ2ν12ν2)3γ2(λ¯μ¯3−λμ3ν13ν2),λ¯0=9γ23λ¯(λ¯μ¯3−λμ3ν13ν2)22γ32(λ¯μ¯2+λμ2ν12ν2)3,d0=λ¯0μ¯0−λ0μ0−(λ¯μ¯−λμ−d),
provided that γ3(λ¯μ¯3−λμ3ν13ν2)>00$]]> and d0>00$]]>. Conditions (39) enable us to consider different cases by changing the values of the coefficients ν1 and ν2 and choose those ones that approximate the ruin probability more accurately. Nevertheless, note that for some values of ν1 and ν2, the corresponding approximations give not so good results, and consequently, should not be applied.
Numerical illustrationsA statistical estimate for the ruin probability
To analyze the accuracy of the approximations proposed in Section 3, we will need a statistical estimate for the ruin probability obtained by the direct simulation of the surplus process (Xt(x))t≥0 using the Monte Carlo methods. To this end, we use the approach described in [23]. Let N be the total number of simulations of (Xt(x))t≥0. To get the corresponding statistical estimate ψˆ(x) for the ruin probability ψ(x), we divide the number of simulations leading to ruin by the total number of simulations N. To find the number of simulations N, which is necessary in order to calculate the ruin probability with the required accuracy and reliability, we apply the following proposition, which follows immediately from Hoeffding’s inequality (see [18]).
Let the surplus process(Xt(x))t≥0be defined by (1) under the above assumptions. Then for anyε>00$]]>, we haveP[|ψ(x)−ψˆ(x)|>ε]≤2e−2ε2N.\varepsilon \big]\le 2{e^{-2{\varepsilon ^{2}}N}}.\]]]>
In all examples below, we set ε=0.005 and 2e−2ε2N=0.005. Therefore, we get N=119830. Moreover, let λ¯=2.3, μ¯=0.2, λ=0.1, μ=3 and d=0.05.
Gamma distributions for the premium and claim sizes
Let the p.d.f. of Y¯i be
fY¯(y)=1Γ(α¯)β¯α¯yα¯−1e−y/β¯,y≥0,
where α¯>00$]]>, β¯>00$]]> and α¯β¯=μ¯, and let the p.d.f. of Yi be
fY(y)=1Γ(α)βαyα−1e−y/β,y≥0,
where α>00$]]>, β>00$]]> and αβ=μ.
Then
E[Y¯i]=α¯β¯=μ¯,E[Y¯i2]=α¯(α¯+1)β¯2,E[Y¯i3]=α¯(α¯+1)(α¯+2)β¯3,E[Y¯i4]=α¯(α¯+1)(α¯+2)(α¯+3)β¯4,E[Y¯i5]=α¯(α¯+1)(α¯+2)(α¯+3)(α¯+4)β¯5,
and analogous formulas hold for the moments of Yi.
Therefore, we get
γ2=λ¯α¯(α¯+1)β¯2+λα(α+1)β2,γ3=λ¯α¯(α¯+1)(α¯+2)β¯3−λα(α+1)(α+2)β3,γ4=λ¯α¯(α¯+1)(α¯+2)(α¯+3)β¯4+λα(α+1)(α+2)(α+3)β4,γ5=λ¯α¯(α¯+1)(α¯+2)(α¯+3)(α¯+4)β¯5−λα(α+1)(α+2)(α+3)(α+4)β5.
A number of numerical examples show that the conditions of Theorem 3 hold provided that α is very close to 1. So the five-moment approximation can be constructed only in those cases. We now consider two examples.
Results of computations: the gamma distributions for the premium and claim sizes, α¯=2, β¯=0.1, α=1 and β=3
x
ψˆ(x)
ψDV5(x)
(ψDV5(x)ψˆ(x)−1)·100%
ψDV3(x)
(ψDV3(x)ψˆ(x)−1)·100%
1
0.6912
0.6832
−1.15%
0.6766
−2.11%
2
0.6205
0.6270
1.05%
0.6210
0.08%
3
0.5681
0.5754
1.29%
0.5700
0.34%
5
0.4870
0.4846
−0.50%
0.4802
−1.41%
7
0.4040
0.4081
1.01%
0.4045
0.12%
10
0.3149
0.3154
0.17%
0.3128
−0.66%
15
0.2024
0.2053
1.42%
0.2037
0.66%
20
0.1374
0.1336
−2.73%
0.1327
−3.38%
30
0.0584
0.0566
−3.09%
0.0563
−3.58%
50
0.0098
0.0102
3.63%
0.0101
3.45%
Values of (ψDV3(x)ψˆ(x)−1)·100% for different ν1 and ν2: the gamma distributions for the premium and claim sizes, α¯=2, β¯=0.1, α=1 and β=3
x
ν1=0.7ν2=1.5
ν1=0.9ν2=1.2
ν1=1.1ν2=0.7
ν1=1.5ν2=0.7
ν1=2ν2=0.2
ν1=5ν2=0.05
ν1=10ν2=0.05
1
−3.48%
−2.36%
−2.21%
−1.00%
−1.41%
−0.35%
0.52%
2
−1.29%
−0.17%
0.00%
1.20%
0.70%
1.84%
2.72%
3
−1.02%
0.09%
0.26%
1.43%
0.97%
2.08%
2.93%
5
−2.69%
−1.65%
−1.46%
−0.38%
−0.76%
0.28%
1.05%
7
−1.13%
−0.12%
0.09%
1.12%
0.80%
1.80%
2.52%
10
−1.82%
−0.89%
−0.67%
0.26%
0.04%
0.95%
1.57%
15
−0.39%
0.44%
0.70%
1.48%
1.42%
2.21%
2.67%
20
−4.26%
−3.58%
−3.29%
−2.70%
−2.60%
−1.97%
−1.68%
30
−4.21%
−3.75%
−3.39%
−3.13%
−2.70%
−2.33%
−2.36%
50
3.31%
3.31%
3.86%
3.45%
4.61%
4.45%
3.76%
Let now α¯=2, β¯=0.1, α=1 and β=3.
If we construct the five-moment analogue to the De Vylder approximation, by Theorem 3, we get λ¯0≈3.923743, μ¯0≈0.132632, λ0≈0.099996, μ0≈3.000027, d0≈0.110423, and consequently,
ψDV5(x)≈0.255492e−29.147189x+0.744508e−0.085895xfor allx≥0.
For the corresponding three-moment approximation using conditions (34), by Theorem 4, we have λ¯0≈2.129067, μ¯0≈0.205450, λ0≈0.092568, μ0≈3.081744, d0≈0.042145, and hence,
ψDV3(x)≈0.262882e−48.085872x+0.737118e−0.085730xfor allx≥0.
Table 1 presents the results of computations for some values of x. Next, Table 2 shows the values of the relative approximation errors ψDV3(x)ψˆ(x)−1·100% for the three-moment approximations constructed using conditions (39) for different ν1 and ν2. Note that here we chose some values of ν1 and ν2 that give more or less good results. Choosing some other values of the coefficients leads to extremely bad approximations. In addition, analyzing the relative approximation errors in Table 2 we conclude that it is difficult to decide which of the approximations is better: choosing ν1 and ν2 that yield smaller errors for some values of the initial surplus results in larger errors for other values.
Results of computations: the gamma distributions for the premium and claim sizes, α¯=4, β¯=0.05, α=3 and β=1
x
ψˆ(x)
ψDV3(x)
(ψDV3(x)ψˆ(x)−1)·100%
1
0.6690
0.6820
1.94%
2
0.6046
0.5971
−1.24%
3
0.5301
0.5228
−1.38%
5
0.4039
0.4008
−0.77%
7
0.3039
0.3073
1.10%
10
0.2065
0.2062
−0.13%
15
0.1056
0.1061
0.50%
20
0.0535
0.0546
2.17%
30
0.0150
0.0145
−3.60%
Let now α¯=4, β¯=0.05, α=3 and β=1.
In this case, the conditions of Theorem 3 do not hold, so we can construct only the three-moment analogue to the De Vylder approximation. By Theorem 4, we obtain λ¯0≈4.871659, μ¯0≈0.111879, λ0≈0.211811, μ0≈1.678181, d0≈0.079577, and therefore,
ψDV3(x)≈0.221130e−55.405586x+0.778870e−0.132881xfor allx≥0.
Table 3 presents the results of computations for some values of x, whereas Table 4 shows the values of the relative approximation errors ψDV3(x)ψˆ(x)−1·100% for the three-moment approximations constructed using conditions (39) for different ν1 and ν2.
Values of (ψDV3(x)ψˆ(x)−1)·100% for different ν1 and ν2: the gamma distributions for the premium and claim sizes, α¯=4, β¯=0.05, α=3 and β=1
x
ν1=0.7ν2=1.5
ν1=0.9ν2=1.2
ν1=1.1ν2=0.6
ν1=1.5ν2=0.7
ν1=2ν2=0.2
ν1=5ν2=0.05
ν1=10ν2=0.03
1
0.82%
1.73%
1.68%
2.85%
2.51%
3.42%
3.97%
2
−2.30%
−1.44%
−1.47%
−0.39%
−0.68%
0.17%
0.68%
3
−2.41%
−1.58%
−1.58%
−0.55%
−0.81%
0.01%
0.50%
5
−1.75%
−0.96%
−0.93%
0.01%
−0.17%
0.59%
1.02%
7
0.16%
0.91%
0.99%
1.84%
1.73%
2.45%
2.84%
10
−0.97%
−0.31%
−0.17%
0.53%
0.53%
1.15%
1.45%
15
−0.21%
0.33%
0.58%
1.03%
1.22%
1.69%
1.85%
20
1.60%
2.01%
2.37%
2.58%
2.96%
3.30%
3.31%
30
−3.86%
−3.72%
−3.17%
−3.46%
−2.74%
−2.71%
−2.96%
Hyperexponential distributions for the premium and claim sizes
Let
FY¯(y)=p¯1FY¯,1(y)+p¯2FY¯,2(y)+⋯+p¯k¯FY¯,k¯(y),y≥0,
where k¯≥1, p¯j>00$]]>, FY¯,j is the c.d.f. of the exponential distribution with mean μ¯j for all 1≤j≤k¯, ∑j=1k¯p¯j=1, ∑j=1k¯p¯jμ¯j=μ¯, and let
FY(y)=p1FY,1(y)+p2FY,2(y)+⋯+pkFY,k(y),y≥0,
where k≥1, pj>00$]]>, FY,j is the c.d.f. of the exponential distribution with mean μj for all 1≤j≤k, ∑j=1kpj=1, ∑j=1kpjμj=μ.
Then
E[Y¯i]=∑j=1k¯p¯jμ¯j=μ¯,E[Y¯i2]=2∑j=1k¯p¯jμ¯j2,E[Y¯i3]=6∑j=1k¯p¯jμ¯j3,E[Y¯i4]=24∑j=1k¯p¯jμ¯j4,E[Y¯i5]=120∑j=1k¯p¯jμ¯j5,
and analogous formulas hold for the moments of Yi.
Hence, we obtain
γ2=2(λ¯∑j=1k¯p¯jμ¯j2+λ∑j=1kpjμj2),γ3=6(λ¯∑j=1k¯p¯jμ¯j3−λ∑j=1kpjμj3),γ4=24(λ¯∑j=1k¯p¯jμ¯j4+λ∑j=1kpjμj4),γ5=120(λ¯∑j=1k¯p¯jμ¯j5−λ∑j=1kpjμj5).
Examples 3 and 4 below present some numerical results.
Let now k¯=2, p¯1=0.75, p¯2=0.25, μ¯1=0.1, μ¯2=0.5, k=2, p1=0.8, p2=0.2, μ1=2.8 and μ2=3.8.
If we construct the five-moment analogue to the De Vylder approximation, by Theorem 3, we obtain λ¯0≈10.626422, μ¯0≈0.141004, λ0≈0.082185, μ0≈3.245591, d0≈1.121624, and consequently,
ψDV5(x)≈0.236453e−2.683647x+0.763547e−0.079854xfor allx≥0.
For the corresponding three-moment approximation using conditions (34), by Theorem 4, we have λ¯0≈2.738661, μ¯0≈0.190975, λ0≈0.119072, μ0≈2.864627, d0≈0.071919 (here we use conditions (34)), and therefore,
ψDV3(x)≈0.228569e−34.768023x+0.771431e−0.080413xfor allx≥0.
Table 5 presents the results of computations for some values of x, whereas Table 6 shows the values of ψDV3(x)ψˆ(x)−1·100% for the three-moment approximations constructed using conditions (39) for different ν1 and ν2.
Results of computations: the hyperexponential distributions for the premium and claim sizes, k¯=2, p¯1=0.75, p¯2=0.25, μ¯1=0.1, μ¯2=0.5, k=2, p1=0.8, p2=0.2, μ1=2.8 and μ2=3.8
x
ψˆ(x)
ψDV5(x)
(ψDV5(x)ψˆ(x)−1)·100%
ψDV3(x)
(ψDV3(x)ψˆ(x)−1)·100%
1
0.6988
0.7211
3.20%
0.7118
1.87%
2
0.6365
0.6519
2.43%
0.6568
3.19%
3
0.5909
0.6010
1.71%
0.6061
2.58%
5
0.5023
0.5122
1.97%
0.5160
2.74%
7
0.4253
0.4366
2.67%
0.4394
3.32%
10
0.3367
0.3436
2.06%
0.3452
2.54%
15
0.2210
0.2305
4.31%
0.2309
4.51%
20
0.1542
0.1546
0.26%
0.1545
0.17%
30
0.0670
0.0696
3.84%
0.0691
3.17%
50
0.0141
0.0141
−0.09%
0.0138
−1.84%
Values of (ψDV3(x)ψˆ(x)−1)·100% for different ν1 and ν2: the hyperexponential distributions for the premium and claim sizes, k¯=2, p¯1=0.75, p¯2=0.25, μ¯1=0.1, μ¯2=0.5, k=2, p1=0.8, p2=0.2, μ1=2.8 and μ2=3.8
x
ν1=0.4ν2=2.7
ν1=0.5ν2=2.1
ν1=0.7ν2=0.7
ν1=0.9ν2=0.6
ν1=1.1ν2=0.5
ν1=1.5ν2=0.15
ν1=2ν2=0.1
1
−2.81%
−1.14%
0.28%
0.73%
1.37%
4.60%
4.25%
2
−1.47%
0.19%
−0.01%
1.72%
2.59%
2.18%
2.77%
3
−1.98%
−0.36%
−0.60%
1.16%
2.01%
1.12%
1.93%
5
−1.67%
−0.10%
−0.26%
1.41%
2.22%
1.35%
2.17%
7
−0.95%
0.57%
0.49%
2.08%
2.86%
2.09%
2.87%
10
−1.47%
−0.04%
0.01%
1.45%
2.17%
1.57%
2.27%
15
0.83%
2.14%
2.41%
3.65%
4.27%
3.94%
4.54%
20
−2.97%
−1.85%
−1.38%
−0.42%
0.08%
0.03%
0.50%
30
0.73%
1.59%
2.51%
3.04%
3.35%
3.86%
4.11%
50
−2.62%
−2.36%
−0.62%
−1.02%
−1.12%
0.43%
0.22%
Let now k¯=3, p¯1=0.2, p¯2=0.5, p¯3=0.3, μ¯1=0.1, μ¯2=0.15, μ¯3=0.35, k=3, p1=0.1, p2=0.4, p3=0.5, μ1=1, μ2=2.7 and μ3=3.64.
In this case, the conditions of Theorem 3 do not hold, so we can construct only the three-moment analogue to the De Vylder approximation. By Theorem 4, we get λ¯0≈2.112044, μ¯0≈0.217677, λ0≈0.091828, μ0≈3.265162, d0≈0.049911 (here we use conditions (34)), and therefore,
ψDV3(x)≈0.252988e−39.790359x+0.747012e−0.077929xfor allx≥0.
Table 7 presents the results of computations for some values of x, whereas Table 8 shows the values of ψDV3(x)ψˆ(x)−1·100% for the three-moment approximations constructed using conditions (39) for different ν1 and ν2.
Results of computations: the hyperexponential distributions for the premium and claim sizes, k¯=3, p¯1=0.2, p¯2=0.5, p¯3=0.3, μ¯1=0.1, μ¯2=0.15, μ¯3=0.35, k=3, p1=0.1, p2=0.4, p3=0.5, μ1=1, μ2=2.7 and μ3=3.64
x
ψˆ(x)
ψDV3(x)
(ψDV3(x)ψˆ(x)−1)·100%
1
0.6949
0.6910
−0.56%
2
0.6368
0.6392
0.38%
3
0.5861
0.5913
0.88%
5
0.5035
0.5059
0.49%
7
0.4306
0.4329
0.55%
10
0.3446
0.3427
−0.56%
15
0.2299
0.2321
0.95%
20
0.1594
0.1572
−1.35%
30
0.0694
0.0721
3.91%
Values of (ψDV3(x)ψˆ(x)−1)·100% for different ν1 and ν2: the hyperexponential distributions for the premium and claim sizes, k¯=3, p¯1=0.2, p¯2=0.5, p¯3=0.3, μ¯1=0.1, μ¯2=0.15, μ¯3=0.35, k=3, p1=0.1, p2=0.4, p3=0.5, μ1=1, μ2=2.7 and μ3=3.64
x
ν1=0.5ν2=2.1
ν1=0.7ν2=1.5
ν1=0.9ν2=1.1
ν1=1.1ν2=1
ν1=1.5ν2=0.5
ν1=2ν2=0.2
ν1=3ν2=0.1
1
−3.93%
−1.89%
−0.94%
−0.16%
0.19%
0.19%
0.69%
2
−2.96%
−0.94%
0.00%
0.77%
1.12%
0.98%
1.47%
3
−2.42%
−0.42%
0.52%
1.27%
1.63%
1.50%
1.98%
5
−2.69%
−0.77%
0.13%
0.86%
1.21%
1.12%
1.58%
7
−2.51%
−0.66%
0.21%
0.90%
1.26%
1.20%
1.65%
10
−3.41%
−1.69%
−0.87%
−0.24%
0.12%
0.11%
0.53%
15
−1.65%
−0.09%
0.67%
1.23%
1.61%
1.68%
2.06%
20
−3.61%
−2.26%
−1.59%
−1.12%
−0.75%
−0.60%
−0.26%
30
2.14%
3.18%
3.72%
4.05%
4.47%
4.79%
5.07%
Lomax distributions for the premium and claim sizes
Let
FY¯(y)=1−(β¯y+β¯)α¯,y≥0,
where α¯>11$]]>, β¯>00$]]> and β¯/(α¯−1)=μ¯, and let
FY(y)=1−(βy+β)α,y≥0,
where α>11$]]>, β>00$]]> and β/(α−1)=μ. In what follows, we assume that α¯>55$]]>, α>55$]]> and both α¯ and α are integer. Then
E[Y¯i]=β¯α¯−1=μ¯,E[Y¯i2]=2β¯2(α¯−2)(α¯−1),E[Y¯i3]=6β¯3(α¯−3)(α¯−2)(α¯−1),E[Y¯i4]=24β¯4(α¯−4)(α¯−3)(α¯−2)(α¯−1),E[Y¯i5]=120β¯5(α¯−5)(α¯−4)(α¯−3)(α¯−2)(α¯−1),
and analogous formulas hold for the moments of Yi.
Consequently, we have
γ2=2(λ¯β¯2(α¯−2)(α¯−1)+λβ2(α−2)(α−1)),γ3=6(λ¯β¯3(α¯−3)(α¯−2)(α¯−1)−λβ3(α−3)(α−2)(α−1)),γ4=24(λ¯β¯4(α¯−4)(α¯−3)(α¯−2)(α¯−1)+λβ4(α−4)(α−3)(α−2)(α−1)),γ5=120(λ¯β¯5(α¯−5)(α¯−4)(α¯−3)(α¯−2)(α¯−1)−λβ5(α−5)(α−4)(α−3)(α−2)(α−1)).
Note that the Lomax distribution, which is also called the Pareto type II distribution, is a heavy-tailed distribution in contrast to the gamma and hyperexponential distributions. This can be the reason why the conditions of Theorem 3 do not hold for this distribution, at least as a number of numerical examples indicate. However, the three-moment approximation can be applied provided that the conditions of Theorem 4 hold.
Let now α¯=6, β¯=1, α=6 and β=15.
In this case, we can construct only the three-moment analogue to the De Vylder approximation. By Theorem 4, we get λ¯0=1.035, μ¯0≈0.333333, λ0=0.045, μ0=5, d0=0.01 (here we use conditions (34)), and therefore,
ψDV3(x)≈0.313466e−105.137225x+0.686534e−0.062775xfor allx≥0.
Table 9 presents the results of computations for some values of x, whereas Table 10 shows the values of ψDV3(x)ψˆ(x)−1·100% for the three-moment approximations constructed using conditions (39) for different ν1 and ν2.
Results of computations: the Lomax distributions for premium and claim sizes, α¯=6, β¯=1, α=6 and β=15
x
ψˆ(x)
ψDV3(x)
(ψDV3(x)ψˆ(x)−1)·100%
1
0.6881
0.6448
−6.30%
2
0.6391
0.6055
−5.25%
3
0.5899
0.5687
−3.59%
5
0.5086
0.5016
−1.37%
7
0.4429
0.4424
−0.10%
10
0.3638
0.3665
0.73%
15
0.2643
0.2677
1.32%
20
0.1887
0.1956
3.67%
30
0.1025
0.1044
1.92%
50
0.0301
0.0298
−1.16%
Values of (ψDV3(x)ψˆ(x)−1)·100% for different ν1 and ν2: the Lomax distributions for the premium and claim sizes, α¯=6, β¯=1, α=6 and β=15
x
ν1=0.9ν2=1.1
ν1=1.1ν2=0.9
ν1=1.5ν2=0.6
ν1=2ν2=0.4
ν1=5ν2=0.2
ν1=7ν2=0.15
ν1=10ν2=0.1
1
−6.78%
−5.94%
−5.11%
−4.56%
−3.24%
−3.03%
−2.89%
2
−5.73%
−4.90%
−4.07%
−3.52%
−2.21%
−2.00%
−1.86%
3
−4.07%
−3.24%
−2.40%
−1.85%
−0.55%
−0.33%
−0.20%
5
−1.84%
−1.02%
−0.19%
0.36%
1.64%
1.85%
1.99%
7
−0.56%
0.24%
1.06%
1.60%
2.85%
3.05%
3.19%
10
0.29%
1.06%
1.85%
2.37%
3.55%
3.75%
3.88%
15
0.92%
1.62%
2.36%
2.84%
3.90%
4.08%
4.20%
20
3.30%
3.94%
4.63%
5.09%
6.04%
6.20%
6.32%
30
1.65%
2.14%
2.69%
3.06%
3.74%
3.86%
3.95%
50
−1.26%
−1.06%
−0.76%
−0.55%
−0.38%
−0.34%
−0.29%
Conclusion
The results of computations presented in Tables 1, 3, 5, 7 and 9 indicate that both approximations yield very small relative errors. Although the existence of exponential moments of the distributions of the premium and claim sizes is not required to construct the approximations, it is easily seen that the relative errors are smaller when those distributions do not have heavy tails.
The construction of the five-moment approximation is based on the classical approach, where we only require that the first five moments of the processes coincide without any additional assumptions. The numerical illustrations show that this approach gives very good results, but unfortunately, the conditions that are necessary for its construction are too restrictive. A definite advantage of the three-moment approximation is that the corresponding conditions are much less restrictive, but the construction of this approximation requires two additional conditions, which are also based on some heuristic assumptions. Nevertheless, there is no reason to assert that one of the approximations is more accurate than the other one: the corresponding relative errors vary for different values of the initial surplus.
The relative errors for some three-moment approximations using more general conditions (39) are given in Tables 2, 4, 6, 8 and 10. The analysis of the errors shows that the accuracy of those approximations is more or less the same: choosing ν1 and ν2 that yield smaller errors for some values of the initial surplus leads to larger errors for other values. Nonetheless, for some other values of ν1 and ν2, the corresponding approximations can give not so good results. Therefore, the choice of coefficients ν1 and ν2 should be controlled using other methods that enable us to approximate the ruin probability.
Finally, note that although the numerical examples considered above are not sufficient to make conclusions about the accuracy of the suggested approximations in general and it would be highly desirable to have a tool to control the accuracy in terms of parameter values, those illustrations enable us to outline some general tendencies.
Acknowledgement
The author is deeply grateful to the anonymous referees for careful reading and valuable comments and suggestions, which helped to improve the earlier version of the paper.
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