VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA18 10.15559/15-VMSTA18 Research Article Practical approaches to the estimation of the ruin probability in a risk model with additional funds MishuraYuliyamyus@univ.kiev.uaa RagulinaOlenaragulina.olena@gmail.coma∗ StroyevOleksandro.stroiev@chnu.edu.uab Taras Shevchenko National University of Kyiv, Department of Probability Theory, Statistics and Actuarial Mathematics, 64 Volodymyrska Str., 01601 Kyiv, Ukraine Yuriy Fedkovych Chernivtsi National University, Department of Mathematical Modelling, 2 Kotsjubynskyi Str., 58012 Chernivtsi, Ukraine Corresponding author. 2014 22201512167180 412015 1912015 © 2014 The Author(s). Published by VTeX2014 Open access article under the CC BY license.

We deal with a generalization of the classical risk model when an insurance company gets additional funds whenever a claim arrives and consider some practical approaches to the estimation of the ruin probability. In particular, we get an upper exponential bound and construct an analogue to the De Vylder approximation for the ruin probability. We compare results of these approaches with statistical estimates obtained by the Monte Carlo method for selected distributions of claim sizes and additional funds.

 Risk model survival probability exponential bound De Vylder approximation Monte Carlo method 91B30 60G51 Introduction

Let (Ω,F,P) be a probability space satisfying the usual conditions, and let all the objects be defined on it. We deal with the risk model that generalizes the classical one and was considered in [10].

In the classical risk model (see, e.g., [1, 7, 11]), an insurance company has an initial surplus x≥0 and receives premiums with constant intensity c>0 0$]]>. Claim sizes form a sequence (ξi)i≥1 of nonnegative i.i.d. random variables with c.d.f.  F1(y)=P[ξi≤y] and finite expectation E[ξi]=μ1 . The number of claims on the time interval [0,t] is a homogeneous Poisson process (Nt)t≥0 with intensity λ>0 0$]]>.

In addition to the classical risk model, we suppose that the insurance company gets additional funds ηi when the ith claim arrives. These funds can be considered, for instance, as additional investment income, which does not depend on the surplus of the company. We assume that (ηi)i≥1 is a sequence of nonnegative i.i.d. random variables with c.d.f. F2(y)=P[ηi≤y] and finite expectation E[ηi]=μ2 . The sequences (ξi)i≥1 , (ηi)i≥1 and the process (Nt)t≥0 are mutually independent. Let (Ft)t≥0 be the filtration generated by (ξi)i≥1 , (ηi)i≥1 , and (Nt)t≥0 .

Let Xt(x) be the surplus of the insurance company at time t, provided that its initial surplus is x. Then the surplus process (Xt(x))t≥0 is defined as Xt(x)=x+ct− ∑i=1Nt(ξi−ηi),t≥0. Note that we set ∑i=10(ξi−ηi)=0 in (1) if Nt=0 .

The ruin time is defined as τ(x)=inf{t≥0:Xt(x)<0}. We suppose that τ(x)=∞ if Xt(x)≥0 for all t≥0 . The infinite-horizon ruin probability is given by ψ(x)=P[inft≥0Xt(x)<0], which is equivalent to ψ(x)=P[τ(x)<∞]. The corresponding infinite-horizon survival probability equals φ(x)=1−ψ(x).

Note that the ruin never occurs if P[ξi−ηi≤0]=1 . If P[ξi−ηi≥0]=1 , then we deal with the classical risk model. So in what follows, we assume that P[ξi−ηi>0]>0 0]>0$]]> and P[ξi−ηi<0]>0 0$]]>. In this case, if c−λμ1+λμ2≤0 , then φ(x)=0 for all x≥0 ; if c−λμ1+λμ2>0 0$]]>, then limx→+∞φ(x)=1 (see [10, Lemma 2.1]).

In this paper, we consider some practical approaches to the estimation of the ruin probability. In particular, we get an upper exponential bound and construct an analogue to the De Vylder approximation for the ruin probability. Moreover, we compare results of these approaches with statistical estimates obtained by the Monte Carlo method for selected distributions of claim sizes and additional funds.

Paper [10], where this risk model is considered, is devoted to the investigation of continuity and differentiability of the infinite-horizon survival probability and derivation of an integro-differential equation for this function. When claim sizes and additional funds are exponentially distributed, a closed-form solution to this equation can be found.

([<xref ref-type="bibr" rid="j_vmsta18_ref_010">10</xref>], Theorem 4.1).

Let the surplus process (Xt(x))t≥0 follow (1) under the above assumptions, the random variables ξi and ηi , i≥1 , be exponentially distributed with means μ1 and μ2 correspondingly, and c−λμ1+λμ2>0 0$]]>. Then φ(x)=1+λμ1(1−αμ2)(cα−λ)(1−αμ2)(μ1+μ2)+λμ2eαx for all x≥0 , where α=λμ1μ2+cμ1−cμ2−c2(μ12+μ22)+λ2μ12μ22+2cμ1μ2(c−λμ1+λμ2)2cμ1μ2.

([<xref ref-type="bibr" rid="j_vmsta18_ref_010">10</xref>], Remark 4.1).

It is justified in the proof of Theorem 1 that α<0 and −1<λμ1(1−αμ2)(cα−λ)(1−αμ2)(μ1+μ2)+λμ2<0. So the function φ(x) defined by (2) satisfies all the natural properties of the survival probability. In particular, this function is nondecreasing and bounded by 0 from below and by 1 from above.

It is well known that even for the classical risk model, there are only a few cases where an analytic expression for the survival probability can be found. So numerous approximations have been considered and investigated for the classical risk model (see, e.g., [1, 2, 4, 5, 7, 11]). “Simple approximations” form a special class of approximations for the ruin or survival probabilities. They use only some moments of the distribution of claim sizes and do not take into account the detailed tail behavior of that distribution. Such approximations may be based on limit theorems or on heuristic arguments. The most successful “simple approximation” is certainly the De Vylder approximation [5], which is based on the heuristic idea to replace the risk process with a risk process with exponentially distributed claim sizes such that the first three moments coincide (see also [7, 11]). This approximation is known to work extremely well for some distributions of claim sizes. Later, Grandell analyzed the De Vylder approximation and other “simple approximations” from a more mathematical point of view and gave a possible explanation why the De Vylder approximation is so good (see [8]).

We deal with the case where the claim sizes have a light-tailed distribution. The rest of the paper is organized as follows. In Section 2, we get an upper exponential bound for the ruin probability, which is an analogue of the famous Lundberg inequality. Section 3 is devoted to the construction of an analogue of the De Vylder approximation. In Section 4, we give a simple formula that relates the accuracy and reliability of the approximation of the ruin probability by its statistical estimate obtained by the Monte Carlo method. In Section 5, we compare the results of these approaches for some distributions of claim sizes and additional funds. Section 6 concludes the paper.

 Exponential bound

To get an upper exponential bound for the ruin probability, we use the martingale approach introduced by Gerber [6] (see also [3, 7, 11]).

Let Ut=ct− ∑i=1Nt(ξi−ηi),t≥0.

For all R≥0 , we define the exponential process (Vt(R))t≥0 by Vt(R)=e−RUt.

If there is Rˆ>0 0$]]> such that λ(∫0+∞eRˆydF1(y)·∫0+∞e−RˆydF2(y)−1)=cRˆ, then (Vt(Rˆ))t≥0 is an (Ft) -martingale.

For all R>0 0$]]> such that E[eRξi]<∞ , if any, we have E[Vt(R)]=e−cRtE[exp{R ∑i=1Nt(ξi−ηi)}]=e−cRt ∑j=0∞e−λt(λt)jj!(E[eR(ξi−ηi)])j=exp{t(λE[eR(ξi−ηi)]−λ−cR)}.

If there is Rˆ>0 0$]]> such that (3) holds, then E[eRˆξi]<∞ , and for all t2≥t1≥0 , we have E[Vt2(Rˆ)/Ft1]=E[exp{−Rˆ(ct2− ∑i=1Nt2(ξi−ηi))}()/Ft1]=E[exp{−Rˆ(ct1− ∑i=1Nt1(ξi−ηi))}]×E[exp{−Rˆ(c(t2−t1)− ∑i=Nt1Nt2(ξi−ηi))}()/Ft1]=E[Vt1(Rˆ)]·E[exp{−Rˆ(c(t2−t1)− ∑i=1Nt2−t1(ξi−ηi))}]=E[Vt1(Rˆ)]. Here we used the fact that E[exp{−Rˆ(c(t2−t1)− ∑i=1Nt2−t1(ξi−ηi))}]=exp{(t2−t1)(λE[eRˆ(ξi−ηi)]−λ−cRˆ)}=1 by (3) and (4).

Thus, (Vt(Rˆ))t≥0 is an (Ft) -martingale, which is the desired conclusion.  □

If there is Rˆ>0 0$]]> such that (3) holds, then for all x≥0 , we have ψ(x)≤e−Rˆx.

It is easily seen that τ(x) is an (Ft) -stopping time. Hence, τ(x)∧T is a bounded (Ft) -stopping time for any fixed T≥0 . The process (Vt(Rˆ))t≥0 is an (Ft) -martingale by Lemma 1. Moreover, (Vt(Rˆ))t≥0 is positive a.s. by its definition. Consequently, applying the optional stopping theorem yields 1=V0(Rˆ)=E[Vτ(x)∧T(Rˆ)]=E[Vτ(x)(Rˆ)·I{τ(x)<T}]+E[VT(Rˆ)·I{τ(x)≥T}]≥E[Vτ(x)(Rˆ)·I{τ(x)<T}]=E[exp{−Rˆ(cτ(x)− ∑i=1Nτ(x)(ξi−ηi))}·I{τ(x)<T}]≥eRˆx·P[τ(x)<T], where I{·} is the indicator of an event. This gives P[τ(x)<T]≤e−Rˆx for all T≥0 . Letting T→∞ in (6) yields P[τ(x)<∞]≤e−Rˆx, which is our assertion.  □

Let the random variables ξi and ηi , i≥1 , be exponentially distributed with means μ1 and μ2 , respectively. Then (3) can be rewritten as λ(1(1−μ1Rˆ)(1+μ2Rˆ)−1)=cRˆ, where Rˆ∈(0,1/μ1) . This condition is equivalent to cμ1μ2Rˆ3+(λμ1μ2+cμ1−cμ2)Rˆ2−(c−λμ1+λμ2)Rˆ=0.

If c−λμ1+λμ2>0 0$]]>, then c2(μ12+μ22)+λ2μ12μ22+2cμ1μ2(c−λμ1+λμ2)>0. 0.\]]]> So there are three real solutions to (7). They are Rˆ1=0, Rˆ2=−λμ1μ2+cμ1−cμ2−A(c,λ,μ1,μ2)2cμ1μ2, Rˆ3=−λμ1μ2+cμ1−cμ2+A(c,λ,μ1,μ2)2cμ1μ2, where A(c,λ,μ1,μ2)=c2(μ12+μ22)+λ2μ12μ22+2cμ1μ2(c−λμ1+λμ2).

Furthermore, it is easy to check that, in this case, |λμ1μ2+cμ1−cμ2|<A(c,λ,μ1,μ2). From this we conclude that Rˆ2>0 0$]]> and Rˆ3<0 . Since A(c,λ,μ1,μ2)<(λμ1μ2+cμ1+cμ2)2, we have Rˆ2<(λμ1μ2+cμ1+cμ2)−(λμ1μ2+cμ1−cμ2)2cμ1μ2<1μ1. Hence, Rˆ2 is a unique positive solution to (7), and an exponential bound (5) can be rewritten as follows: ψ(x)≤e−Rˆ2x. Comparing (8) with (2), we see that the exponential bound and the analytic expression for the ruin probability differ in a constant multiplier only.

If c−λμ1+λμ2≤0 , then μ1>μ2 \mu _{2}$]]>, which gives λμ1μ2+cμ1−cμ2>0 0$]]>. Let Rˆ2 and Rˆ3 be two nonzero solutions to (7). Applying Vieta’s formulas yields Rˆ2+Rˆ3<0 and Rˆ2Rˆ3>0 0$]]>. Consequently, if Rˆ2 and Rˆ3 are real, they are negative. Thus, (7) has no positive solution, and Theorem 2 does not give us an exponential bound for the ruin probability. Indeed, in this case, ψ(x)=1 for all x≥0 (see [10, Lemma 2.1]).

 Analogue to the De Vylder approximation

To construct an analogue to the De Vylder approximation, we replace the process (Ut)t≥0 with a process (U˜t)t≥0 with exponentially distributed claim sizes such that E[Utk]=E[U˜tk],k=1,2,3.

Since the process (U˜t)t≥0 in this risk model is determined by the four parameters (c˜,λ˜,μ˜1,μ˜2) in contrast to the classical risk model, where it is determined by three parameters, we use the additional condition μ1μ2=μ˜1μ˜2. Note that we could have used the condition E[Ut4]=E[U˜t4] instead of (10), but it would have led to tedious calculations and solving polynomial equations of higher degree.

Let (ξ˜i)i≥1 be a sequence of i.i.d. random variables exponentially distributed with mean μ˜1 . Similarly, let (η˜i)i≥1 be a sequence of i.i.d. random variables exponentially distributed with mean μ˜2 . An easy computation shows that E[ξ˜ik]=k!μ˜1kandE[η˜ik]=k!μ˜2k.

Let E[ξi3]<∞ and E[ηi3]<∞ . Then we have E[Ut]=ct−E[ ∑i=1Nt(ξi−ηi)]=ct−λtE[ξi−ηi], E[Ut2]=(ct)2−2ctE[ ∑i=1Nt(ξi−ηi)]+E[( ∑i=1Nt(ξi−ηi))2]=(ct)2−2ct·λtE[ξi−ηi]+λtE[(ξi−ηi)2]+(λt)2(E[ξi−ηi])2=λtE[(ξi−ηi)2]+(ct−λtE[ξi−ηi])2=λtE[(ξi−ηi)2]+(E[Ut])2, E[Ut3]=(ct)3−3(ct)2E[ ∑i=1Nt(ξi−ηi)]+3ctE[( ∑i=1Nt(ξi−ηi))2]−E[( ∑i=1Nt(ξi−ηi))3]=(ct)3−3(ct)2·λtE[ξi−ηi]+3ct(λtE[(ξi−ηi)2]+(λt)2(E[ξi−ηi])2−λtE[(ξi−ηi)3]−3(λt)2E[(ξi−ηi)2]+λtE[(ξi−ηi)2]E[(ξi−ηi)2]−(λt)3(E[ξi−ηi])3=−λtE[(ξi−ηi)3]+(ct−λtE[ξi−ηi])3+3λtE[(ξi−ηi)2](ct−λtE[ξi−ηi])=−λtE[(ξi−ηi)3]+(E[Ut])3+3(E[Ut2]−(E[Ut])2)E[Ut].

Applying similar arguments to the process (U˜t)t≥0 , we conclude that (9) is equivalent to ct−λtE[ξi−ηi]=c˜t−λ˜tE[ξ˜i−η˜i],λtE[(ξi−ηi)2]=λ˜tE[(ξ˜i−η˜i)2],−λtE[(ξi−ηi)3]=−λ˜tE[(ξ˜i−η˜i)3],

By (11) we can rewrite (12) as c−λ(μ1−μ2)=c˜−λ˜(μ˜1−μ˜2),λE[(ξi−ηi)2]=2λ˜(μ˜12−μ˜1μ˜2+μ˜22),λE[(ξi−ηi)3]=6λ˜(μ˜13−μ˜12μ˜2+μ˜1μ˜22−μ˜23).

Substituting μ˜2=μ2μ˜1/μ1 into the second and third equations of system (13) yields λE[(ξi−ηi)2]=2λ˜μ˜12(1−μ2μ1+μ22μ12), λE[(ξi−ηi)3]=6λ˜μ˜13(1−μ2μ1+μ22μ12−μ23μ13).

Dividing (15) by (14) gives μ˜1=μ1(μ12−μ1μ2+μ22)E[(ξi−ηi)3]3(μ13−μ12μ2+μ1μ22−μ23)E[(ξi−ηi)2].

Consequently, we have μ˜2=μ2(μ12−μ1μ2+μ22)E[(ξi−ηi)3]3(μ13−μ12μ2+μ1μ22−μ23)E[(ξi−ηi)2].

Substituting (16) into (14), we get λ˜=9λ(μ13−μ12μ2+μ1μ22−μ23)2(E[(ξi−ηi)2])32(μ12−μ1μ2+μ22)3(E[(ξi−ηi)3])2.

Substituting (16)–(18) into the first equation of system (13), we obtain c˜=c−λ(μ1−μ2)(1−3(μ13−μ12μ2+μ1μ22−μ23)(E[(ξi−ηi)2])22(μ12−μ1μ2+μ22)2E[(ξi−ηi)3]).

Note that since F1(y) and F2(y) are known, it is easy to find E[(ξi−ηi)2] and E[(ξi−ηi)3] if E[ξi3]<∞ and E[ηi3]<∞ .

By (16) and (17), μ˜1 and μ˜2 are positive, provided that (μ13−μ12μ2+μ1μ22−μ23)E[(ξi−ηi)3]>0. 0.\]]]> If (20) holds, then μ1≠μ2 . So λ˜ is also positive. Moreover, c˜ is positive, provided that c−λ(μ1−μ2)(1−3(μ13−μ12μ2+μ1μ22−μ23)(E[(ξi−ηi)2])22(μ12−μ1μ2+μ22)2E[(ξi−ηi)3])>0. 0.\]]]>

Thus, we get the following result. (An analogue to the De Vylder approximation).

Let the surplus process (Xt(x))t≥0 follow (1) under the above assumptions, c−λμ1+λμ2>0 0$]]>, E[ξi3]<∞ , E[ηi3]<∞ , and let conditions (20) and (21) hold. Then the ruin probability is approximately equal to ψDV(x)=λ˜μ˜1(α˜μ˜2−1)(c˜α˜−λ˜)(1−α˜μ˜2)(μ˜1+μ˜2)+λ˜μ˜2eα˜x for all x≥0 , where α˜=λ˜μ˜1μ˜2+c˜μ˜1−c˜μ˜2−c˜2(μ˜12+μ˜22)+λ˜2μ˜12μ˜22+2c˜μ˜1μ˜2(c˜−λ˜μ˜1+λ˜μ˜2)2c˜μ˜1μ˜2 and the parameters (c˜,λ˜,μ˜1,μ˜2) are defined by (16)–(19).

Note that α˜<0 and λ˜μ˜1(α˜μ˜2−1)(c˜α˜−λ˜)(1−α˜μ˜2)(μ˜1+μ˜2)+λ˜μ˜2>0 0\]]]> in Proposition 1. Indeed, the parameters (c˜,λ˜,μ˜1,μ˜2) are positive. Moreover, since c−λμ1+λμ2>0 0$]]>, we have c˜−λ˜μ˜1+λ˜μ˜2>0 0$]]> by the first equation of system (13). Hence, Theorem 1 and Remark 1 give us the desired conclusion.

If claim sizes and additional funds are exponentially distributed, then it is easily seen from (16)–(19) that ψ(x)=ψDV(x) .

 Statistical estimate obtained by the Monte Carlo method

Let N be the total number of simulations of the surplus process Xt(x) , and let ψˆ(x) be the corresponding statistical estimate obtained by the Monte Carlo method. To get it, we divide the number of simulations that lead to the ruin by the total number of simulations.

Let the surplus process (Xt(x))t≥0 follow (1) under the above assumptions. Then for any ε>0 0$]]>, we have P[|ψ(x)−ψˆ(x)|>ε]≤2e−2ε2N. \varepsilon \big]\le 2{e}^{-2{\varepsilon }^{2}N}.\]]]>

The assertion of Proposition 2 follows immediately from Hoeffding’s inequality (see [9]).

Formula (22) relates the accuracy and reliability of the approximation of the ruin probability by its statistical estimate obtained by the Monte Carlo method. It enables us to find the number of simulations N, which is necessary in order to calculate the ruin probability with the required accuracy and reliability. An obvious shortcoming of the Monte Carlo method is a too large number of simulations N. In all examples in Section 5, we assume that ε=0.001 and 2e−2ε2N=0.001 . Consequently, N=3800452 .

 Comparison of results Erlang distributions for claim sizes and additional funds

Let the probability density functions of ξi and ηi be f1(y)=k1k1yk1−1e−k1y/μ1μ1k1(k1−1)!andf2(y)=k2k2yk2−1e−k2y/μ2μ2k2(k2−1)! for y≥0 , respectively, where k1 and k2 are positive integers.

In what follows, h1(R) and h2(R) , where R≥0 , denote the moment generating functions of ξi and ηi , respectively, that is, h1(R)=E[eRξi]andh2(R)=E[eRηi].

An easy computation shows that h1(R)=∫0+∞eRydF1(y)=(k1k1−μ1R)k1,0≤R<k1μ1, h2(R)=∫0+∞eRydF2(y)=(k2k2−μ2R)k2,0≤R<k2μ2.

Moreover, for all R≥0 , we have ∫0+∞e−RydF2(y)=(k2k2+μ2R)k2.

Thus, condition (3) can be rewritten as λ(k1k1−μ1Rˆ)k1(k2k2+μ2Rˆ)k2=λ+cRˆ, where 0<Rˆ<k1/μ1 . Furthermore, we have E[ξi]=h1′(0)=μ1,E[ηi]=h2′(0)=μ2, E[ξi2]=h1″(0)=(k1+1)μ12k1,E[ηi2]=h2″(0)=(k2+1)μ22k2, E[ξi3]=h1‴(0)=(k1+1)(k1+2)μ13k12,E[ηi3]=h2‴(0)=(k2+1)(k2+2)μ23k22.

Hence, we get E[(ξi−ηi)2]=E[ξi2]−2E[ξi]E[ηi]+E[ηi2]=(k1+1)μ12k1−2μ1μ2+(k2+1)μ22k2, E[(ξi−ηi)3]=E[ξi3]−3E[ξi2]E[ηi]+3E[ξi]E[ηi2]−E[ηi3]=(k1+1)(k1+2)μ13k12−3(k1+1)μ12μ2k1+3(k2+1)μ22μ1k2−(k2+1)(k2+2)μ23k22. Substituting E[(ξi−ηi)2] and E[(ξi−ηi)3] into (16)–(19), we obtain the parameters (c˜,λ˜,μ˜1,μ˜2) .

Let c=10 , λ=4 , μ1=2 , μ2=0.5 , k1=3 , k2=2 . Then Rˆ≈0.349093 , which may not be an unique positive solution to (23), and ψDV(x)=0.612268e−0.332472x.

The results of computations are given in Table 1.

Results of computations: Erlang distributions for claim sizes and additional funds

x ψˆ(x) ψDV(x) (ψDV(x)ψˆ(x)−1)·100% e−Rˆx (e−Rˆxψˆ(x)−1)·100%
0 0.634149 0.612268 −3.45% 1.000000 57.69%
1 0.492768 0.439087 −10.89% 0.705327 43.14%
2 0.355769 0.314891 −11.49% 0.497487 39.83%
5 0.137737 0.116142 −15.68% 0.174564 26.74%
10 0.023224 0.022031 −5.14% 0.030473 31.21%

Hyperexponential distributions for claim sizes and additional funds

Let F1(y)=p1,1F1,1(y)+p1,2F1,2(y)+⋯+p1,k1F1,k1(y),y≥0, where k1≥1 , p1,j>0 0$]]>, ∑j=1k1p1,j=1 , ∑j=1k1p1,jμ1,j=μ1 , and F1,j is the c.d.f. of the exponential distribution with mean μ1,j ; F2(y)=p2,1F2,1(y)+p2,2F2,2(y)+⋯+p2,k2F2,k2(y),y≥0, where k2≥1 , p2,j>0 0$]]>, ∑j=1k2p2,j=1 , ∑j=1k2p2,jμ2,j=μ2 , and F2,j is the c.d.f. of the exponential distribution with mean μ2,j .

It is easy to check that h1(R)= ∑j=1k1p1,j1−μ1,jR,0≤R<min{1μ1,1,1μ1,2,…,1μ1,k1}, h2(R)= ∑j=1k2p2,j1−μ2,jR,0≤R<min{1μ2,1,1μ2,2,…,1μ2,k2}.

Furthermore, for all R≥0 , we have ∫0+∞e−RydF2(y)= ∑j=1k2p2,j1+μ2,jR.

Hence, condition (3) can be rewritten as λ( ∑j=1k1p1,j1−μ1,jRˆ· ∑j=1k2p2,j1+μ2,jRˆ)=λ+cRˆ, where 0<Rˆ<min{1/μ1,1,1/μ1,2,…,1/μ1,k1} . Moreover, we have E[ξi]=h1′(0)= ∑j=1k1p1,jμ1,j=μ1,E[ηi]=h2′(0)= ∑j=1k2p2,jμ2,j=μ2, E[ξi2]=h1″(0)= ∑j=1k12p1,jμ1,j2,E[ηi2]=h2″(0)= ∑j=1k22p2,jμ2,j2, E[ξi3]=h1‴(0)= ∑j=1k16p1,jμ1,j3,E[ηi3]=h2‴(0)= ∑j=1k26p2,jμ2,j3.

Consequently, we get E[(ξi−ηi)2]=2( ∑j=1k1p1,jμ1,j2−μ1μ2+ ∑j=1k2p2,jμ2,j2), E[(ξi−ηi)3]=6( ∑j=1k1p1,jμ1,j3−μ2 ∑j=1k1p1,jμ1,j2+μ1 ∑j=1k2p2,jμ2,j2− ∑j=1k2p2,jμ2,j3).

Let c=10 , λ=4 , μ1=2 , μ2=0.5 , k1=3 , k2=2 , p1,1=0.4 , μ1,1=0.5 , p1,2=0.3 , μ1,2=2 , p1,3=0.3 , μ1,3=4 , p2,1=0.75 , μ2,1=0.4 , p2,2=0.25 , μ2,2=0.8 . Then Rˆ≈0.110607 , which may not be a unique positive solution to (24), and ψDV(x)=0.581428e−0.108865x.

The results of computations are given in Table 2.

Results of computations: hyperexponential distributions for claim sizes and additional funds

x ψˆ(x) ψDV(x) (ψDV(x)ψˆ(x)−1)·100% e−Rˆx (e−Rˆxψˆ(x)−1)·100%
0 0.647560 0.581428 −10.21% 1.000000 54.43%
1 0.540924 0.521454 −3.60% 0.895291 65.51%
2 0.488597 0.467667 −4.28% 0.801545 64.05%
5 0.346390 0.337363 −2.61% 0.575201 66.06%
10 0.202323 0.195749 −3.25% 0.330856 63.53%
20 0.067802 0.065903 −2.80% 0.109466 61.45%
25 0.038194 0.038239 0.12% 0.062965 64.86%

Exponential distribution for claim sizes and degenerate distribution for additional funds

Let ξi be exponentially distributed with mean μ1 and E[ηi=μ2]=1 . Then condition (3) can be rewritten as λe−μ2Rˆ1−μ1Rˆ=λ+cRˆ, where Rˆ∈(0,1/μ1) , which is equivalent to λe−μ2Rˆ=−cμ1Rˆ2+(c−λμ1)Rˆ+λ.

If c−λμ1+λμ2>0 0$]]>, then it is easy to check that (25) has a unique solution Rˆ∈(0,1/μ1) .

Since E[ξi]=μ1,E[ξi2]=2μ12,E[ξi3]=6μ13, E[ηi]=μ2,E[ηi2]=μ22,E[ηi3]=μ23, we get E[(ξi−ηi)2]=2μ12−2μ1μ2+μ22, E[(ξi−ηi)3]=6μ13−6μ12μ2+3μ1μ22−μ23.

Let c=10 , λ=4 , μ1=2 , μ2=0.5 . Then Rˆ≈0.195273 and ψDV(x)=0.582498e−0.187764x.

The results of computations are given in Table 3.

Results of computations: exponential distribution for claim sizes and degenerate distribution additional funds

x ψˆ(x) ψDV(x) (ψDV(x)ψˆ(x)−1)·100% e−Rˆx (e−Rˆxψˆ(x)−1)·100%
0 0.637998 0.582498 −8.70% 1.000000 56.74%
1 0.549737 0.482780 −12.18% 0.822610 49.64%
2 0.465171 0.400133 −13.98% 0.676687 45.47%
5 0.277026 0.227808 −17.77% 0.376678 37.97%
10 0.113399 0.089093 −21.43% 0.141886 25.12%

Conclusion

Tables 1–3 provide results of computations when the initial surplus is not too large. In this case, the statistical estimates obtained by the Monte Carlo method can be used instead of the exact ruin probabilities to compare an accuracy of the exponential bound and the analogue to the De Vylder approximation. To get appropriate statistical estimates by the Monte Carlo method for large initial surpluses, the number of simulations must be exceeding. The results of computations show that the exponential bound is very rough. The analogue to the De Vylder approximation gives much more accurate estimations, especially in the case of hyperexponential distributions for claim sizes and additional funds. Nevertheless, it is heuristic, and its real accuracy is unknown.

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