Practical approaches to the estimation of the ruin probability in a risk model with additional funds

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1 Modern Stochastics: Theory and Applications (204) DOI: 05559/5-VMSTA8 Practical approaches to the estimation of the ruin probability in a risk model with additional funds Yuliya Mishura a Olena Ragulina a Oleksandr Stroyev b a Taras Shevchenko National University of Kyiv Department of Probability Theory Statistics and Actuarial Mathematics 64 Volodymyrska Str 060 Kyiv Ukraine b Yuriy Fedkovych Chernivtsi National University Department of Mathematical Modelling 2 Kotsjubynskyi Str 5802 Chernivtsi Ukraine myus@univkievua (Yu Mishura) ragulinaolena@gmailcom (O Ragulina) ostroiev@chnueduua (O Stroyev) Received: 4 January 205 Accepted: 9 January 205 Published online: 2 February 205 Abstract 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 Keywords Risk model survival probability exponential bound De Vylder approximation Monte Carlo method 200 MSC 9B30 60G5 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 [0] Corresponding author 204 The Author(s) Published by VTeX Open access article under the CC BY license wwwi-journalsorg/vmsta

2 68 Yu Mishura et al In the classical risk model (see eg [ 7 ]) an insurance company has an initial surplus x 0 and receives premiums with constant intensity c>0 Claim sizes form a sequence (ξ i ) i of nonnegative iid random variables with cdf F (y) = P[ξ i y] and finite expectation E[ξ i ] = μ The number of claims on the time interval [0t] is a homogeneous Poisson process (N t ) t 0 with intensity λ>0 In addition to the classical risk model we suppose that the insurance company gets additional funds η i 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 (η i ) i is a sequence of nonnegative iid random variables with cdf F 2 (y) = P[η i y] and finite expectation E[η i ]=μ 2 The sequences (ξ i ) i (η i ) i and the process (N t ) t 0 are mutually independent Let (F t ) t 0 be the filtration generated by (ξ i ) i (η i ) i and (N t ) t 0 Let X t (x) be the surplus of the insurance company at time t provided that its initial surplus is x Then the surplus process (X t (x)) t 0 is defined as N t X t (x) = x + ct (ξ i η i ) t 0 () Note that we set 0 (ξ i η i ) = 0in()ifN t = 0 The ruin time is defined as τ(x) = inf { t 0: X t (x) < 0 } We suppose that τ(x) = if X t (x) 0 for all t 0 The infinite-horizon ruin probability is given by ψ(x) = P [ inf t 0 X t (x) < 0 ] which is equivalent to ψ(x) = P [ τ(x) < ] The corresponding infinite-horizon survival probability equals ϕ(x) = ψ(x) Note that the ruin never occurs if P[ξ i η i 0] = If P[ξ i η i 0] = then we deal with the classical risk model So in what follows we assume that P[ξ i η i > 0] > 0 and P[ξ i η i < 0] > 0 In this case if c λμ + λμ 2 0 then ϕ(x) = 0 for all x 0; if c λμ + λμ 2 > 0 then lim x + ϕ(x) = (see[0 Lemma 2]) 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 [0] 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

3 Practical approaches to the estimation of the ruin probability 69 Theorem ([0] Theorem 4) Let the surplus process (X t (x)) t 0 follow () under the above assumptions the random variables ξ i and η i i be exponentially distributed with means μ and μ 2 correspondingly and c λμ + λμ 2 > 0 Then ϕ(x) = + λμ ( αμ 2 ) (cα λ)( αμ 2 )(μ + μ 2 ) + λμ 2 e αx (2) for all x 0 where α = λμ μ 2 + cμ cμ 2 c 2 (μ 2 + μ2 2 ) + λ2 μ 2 μ cμ μ 2 (c λμ + λμ 2 ) 2cμ μ 2 Remark ([0] Remark 4) It is justified in the proof of Theorem that α<0 and λμ ( αμ 2 ) < < 0 (cα λ)( αμ 2 )(μ + μ 2 ) + λμ 2 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 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 eg [ ]) 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 ]) 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 2 Exponential bound To get an upper exponential bound for the ruin probability we use the martingale approach introduced by Gerber [6] (seealso[3 7 ])

4 70 Yu Mishura et al Let N t U t = ct (ξ i η i ) t 0 For all R 0 we define the exponential process (V t (R)) t 0 by V t (R) = e RU t Lemma If there is ˆR >0 such that ( + + ) λ e ˆRy df (y) e ˆRy df 2 (y) = c ˆR (3) 0 then (V t ( ˆR)) t 0 is an (F t )-martingale Proof For all R>0such that E[e Rξ i ] < ifanywehave E [ V t (R) ] [ { N t }] = e crt E exp R (ξ i η i ) 0 = e crt j=0 λt (λt)j ( [ e E e R(ξ i η i ) ]) j j! = exp { t ( λe [ e R(ξ i η i ) ] λ cr )} If there is ˆR >0 such that (3) holds then E[e ˆRξ i ] < and for all t 2 t 0 we have E [ { ( N ] t2 / ] V t2 ( ˆR)/F t = E [exp ˆR ct 2 (ξ i η i ))} F t [ { ( = E exp ˆR N t )}] ct (ξ i η i ) [ { ( E exp ˆR c(t 2 t ) N t2 )} / (ξ i η i ) F t] i=n t = E [ V t ( ˆR) ] [ { ( N t2 t )}] E exp ˆR c(t 2 t ) (ξ i η i ) = E [ V t ( ˆR) ] Here we used the fact that [ { ( E exp ˆR c(t 2 t ) N t2 t )}] (ξ i η i ) = exp { (t 2 t ) ( λe [ e ˆR(ξ i η i ) ] λ c ˆR )} = by (3) and (4) Thus (V t ( ˆR)) t 0 is an (F t )-martingale which is the desired conclusion (4)

5 Practical approaches to the estimation of the ruin probability 7 Theorem 2 If there is ˆR >0 such that (3) holds then for all x 0 we have ψ(x) e ˆRx (5) Proof It is easily seen that τ(x) is an (F t )-stopping time Hence τ(x) T is a bounded (F t )-stopping time for any fixed T 0 The process (V t ( ˆR)) t 0 is an (F t )- martingale by Lemma Moreover (V t ( ˆR)) t 0 is positive as by its definition Consequently applying the optional stopping theorem yields = V 0 ( ˆR) = E [ V τ(x) T ( ˆR) ] = E [ ] [ ] V τ(x) ( ˆR) I {τ(x)<t} + E VT ( ˆR) I {τ(x) T } E [ ] V τ(x) ( ˆR) I {τ(x)<t} [ { ( N τ(x) )} ] = E exp ˆR cτ(x) (ξ i η i ) I {τ(x)<t} e ˆRx P [ τ(x) < T ] where I { } is the indicator of an event This gives P [ τ(x) < T ] e ˆRx (6) for all T 0 Letting T in (6) yields P [ τ(x) < ] e ˆRx which is our assertion Example Let the random variables ξ i and η i i be exponentially distributed with means μ and μ 2 respectively Then (3) can be rewritten as ( ) λ ( μ ˆR)( + μ 2 ˆR) = c ˆR where ˆR (0 /μ ) This condition is equivalent to cμ μ 2 ˆR 3 + (λμ μ 2 + cμ cμ 2 ) ˆR 2 (c λμ + λμ 2 ) ˆR = 0 (7) If c λμ + λμ 2 > 0 then c 2( μ 2 + μ2 2) + λ 2 μ 2 μ cμ μ 2 (c λμ + λμ 2 )>0 So there are three real solutions to (7) They are ˆR = 0 ˆR 2 = λμ μ 2 + cμ cμ 2 A(c λ μ μ 2 ) 2cμ μ 2

6 72 Yu Mishura et al where ˆR 3 = λμ μ 2 + cμ cμ 2 + A(c λ μ μ 2 ) 2cμ μ 2 A(c λ μ μ 2 ) = c 2( μ 2 + 2) μ2 + λ 2 μ 2 μ cμ μ 2 (c λμ + λμ 2 ) Furthermore it is easy to check that in this case λμ μ 2 + cμ cμ 2 < A(c λ μ μ 2 ) From this we conclude that ˆR 2 > 0 and ˆR 3 < 0 Since A(c λ μ μ 2 )<(λμ μ 2 + cμ + cμ 2 ) 2 we have ˆR 2 < (λμ μ 2 + cμ + cμ 2 ) (λμ μ 2 + cμ cμ 2 ) 2cμ μ 2 < μ Hence ˆR 2 is a unique positive solution to (7) and an exponential bound (5) can be rewritten as follows: ψ(x) e ˆR 2 x (8) 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 λμ + λμ 2 0 then μ >μ 2 which gives λμ μ 2 + cμ cμ 2 > 0 Let ˆR 2 and ˆR 3 be two nonzero solutions to (7) Applying Vieta s formulas yields ˆR 2 + ˆR 3 < 0 and ˆR 2 ˆR 3 > 0 Consequently if ˆR 2 and ˆR 3 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) = for all x 0(see[0 Lemma 2]) 3 Analogue to the De Vylder approximation To construct an analogue to the De Vylder approximation we replace the process (U t ) t 0 with a process (Ũ t ) t 0 with exponentially distributed claim sizes such that E [ Ut k ] [Ũ ] = E k t k = 2 3 (9) Since the process (Ũ t ) t 0 in this risk model is determined by the four parameters ( c λ μ μ 2 ) in contrast to the classical risk model where it is determined by three parameters we use the additional condition μ μ 2 = μ μ 2 (0) Note that we could have used the condition E[Ut 4]=E[Ũ t 4 ] instead of (0) but it would have led to tedious calculations and solving polynomial equations of higher degree

7 Practical approaches to the estimation of the ruin probability 73 Let ( ξ i ) i be a sequence of iid random variables exponentially distributed with mean μ Similarly let ( η i ) i be a sequence of iid random variables exponentially distributed with mean μ 2 An easy computation shows that E [ ξ i k ] = k! μ k and E [ η i k ] = k! μ k 2 () Let E[ξi 3] < and E[η3 i ] < Then we have [ Nt ] E[U t ]=ct E (ξ i η i ) E [ Ut 2 ] Nt = (ct) 2 2ct E[ (ξ i η i ) ] = ct λt E[ξ i η i ] [( Nt ) 2 ] + E (ξ i η i ) = (ct) 2 2ct λt E[ξ i η i ]+λt E [ (ξ i η i ) 2] + (λt) 2( E[ξ i η i ] ) 2 = λt E [ (ξ i η i ) 2] + ( ct λt E[ξ i η i ] ) 2 = λt E [ (ξ i η i ) 2] + ( E[U t ] ) 2 E [ Ut 3 ] Nt ] = (ct) 3 3(ct) E[ 2 (ξ i η i ) [( Nt + 3ct E (ξ i η i ) ) 2 ] [( Nt ) 3 ] E (ξ i η i ) = (ct) 3 3(ct) 2 λt E[ξ i η i ]+3ct(λt E [ (ξ i η i ) 2] + (λt) 2( E[ξ i η i ] ) 2 [ λt E (ξi η i ) 3] 3(λt) 2 E [ (ξ i η i ) 2] + λt E [ (ξ i η i ) 2] E [ (ξ i η i ) 2] (λt) 3( E[ξ i η i ] ) 3 = λt E [ (ξ i η i ) 3] + ( ct λt E[ξ i η i ] ) 3 + 3λt E [ (ξ i η i ) 2]( ct λt E[ξ i η i ] ) = λt E [ (ξ i η i ) 3] + ( E[U t ] ) 3 ( [ ] ( + 3 E U 2 t E[Ut ] ) 2) E[Ut ] Applying similar arguments to the process (Ũ t ) t 0 we conclude that (9) is equivalent to ct λt E[ξ i η i ]= ct λt E[ ξ i η i ] λt E [ (ξ i η i ) 2] = λt E [ ( ξ i η i ) 2] λt E [ (ξ i η i ) 3] = λt E [ (2) ( ξ i η i ) 3] By () we can rewrite (2) as c λ(μ μ 2 ) = c λ( μ μ 2 ) λ E [ (ξ i η i ) 2] = 2 λ ( μ 2 μ μ 2 + μ 2 2) λ E [ (ξ i η i ) 3] = 6 λ ( (3) μ 3 μ2 μ 2 + μ μ 2 ) 2 μ3 2

8 74 Yu Mishura et al Substituting μ 2 = μ 2 μ /μ into the second and third equations of system (3) yields λ E [ (ξ i η i ) 2] ( = 2 λ μ 2 μ ) 2 + μ2 2 μ μ 2 (4) λ E [ (ξ i η i ) 3] = 6 λ μ 3 Dividing (5)by(4) gives ( μ 2 + μ2 2 μ μ 2 μ3 2 μ 3 ) (5) μ (μ 2 μ = μ μ 2 + μ 2 2 ) E[(ξ i η i ) 3 ] 3(μ 3 μ2 μ 2 + μ μ 2 2 μ3 2 ) E[(ξ i η i ) 2 ] (6) Consequently we have μ 2 (μ 2 μ 2 = μ μ 2 + μ 2 2 ) E[(ξ i η i ) 3 ] 3(μ 3 μ2 μ 2 + μ μ 2 2 μ3 2 ) E[(ξ i η i ) 2 ] (7) Substituting (6)into(4) we get λ = 9λ(μ3 μ2 μ 2 + μ μ 2 2 μ3 2 )2 (E[(ξ i η i ) 2 ]) 3 2(μ 2 μ μ 2 + μ 2 2 )3 (E[(ξ i η i ) 3 ]) 2 (8) Substituting (6) (8) into the first equation of system (3) we obtain c = c λ(μ μ 2 ) ( 3(μ3 μ2 μ 2 + μ μ 2 2 μ3 2 )(E[(ξ i η i ) 2 ]) 2 ) 2(μ 2 μ μ 2 + μ 2 2 )2 E[(ξ i η i ) 3 (9) ] Note that since F (y) and F 2 (y) are known it is easy to find E[(ξ i η i ) 2 ] and E[(ξ i η i ) 3 ] if E[ξi 3] < and E[η3 i ] < By (6) and (7) μ and μ 2 are positive provided that ( μ 3 μ 2 μ 2 + μ μ 2 2 ) [ μ3 2 E (ξi η i ) 3] > 0 (20) If (20) holds then μ μ 2 So λ is also positive Moreover c is positive provided that c λ(μ μ 2 ) ( 3(μ3 μ2 μ 2 + μ μ 2 2 μ3 2 )(E[(ξ i η i ) 2 ]) 2 ) 2(μ 2 μ μ 2 + μ 2 2 )2 E[(ξ i η i ) 3 > 0 (2) ] Thus we get the following result Proposition (An analogue to the De Vylder approximation) Let the surplus process (X t (x)) t 0 follow () under the above assumptions c λμ + λμ 2 > 0 E[ξi 3] < E[η3 i ] < and let conditions (20) and (2) hold Then the ruin probability is approximately equal to for all x 0 where ψ DV (x) = λ μ ( α μ 2 ) ( c α λ)( α μ 2 )( μ + μ 2 ) + λ μ 2 e αx

9 Practical approaches to the estimation of the ruin probability 75 λ μ μ 2 + c μ c μ 2 c 2 ( μ 2 α = + μ2 2 ) + λ 2 μ 2 μ c μ μ 2 ( c λ μ + λ μ 2 ) 2 c μ μ 2 and the parameters ( c λ μ μ 2 ) are defined by (6) (9) Remark 2 Note that α <0 and λ μ ( α μ 2 ) > 0 ( c α λ)( α μ 2 )( μ + μ 2 ) + λ μ 2 in Proposition Indeed the parameters ( c λ μ μ 2 ) are positive Moreover since c λμ + λμ 2 > 0 we have c λ μ + λ μ 2 > 0 by the first equation of system (3) Hence Theorem and Remark give us the desired conclusion Remark 3 If claim sizes and additional funds are exponentially distributed then it is easily seen from (6) (9) that ψ(x) = ψ DV (x) 4 Statistical estimate obtained by the Monte Carlo method Let N be the total number of simulations of the surplus process X t (x) and let ˆψ(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 Proposition 2 Let the surplus process (X t (x)) t 0 follow () under the above assumptions Then for any ε>0 we have P [ ψ(x) ˆψ(x) >ε ] 2e 2ε 2N (22) The assertion of Proposition 2 follows immediately from Hoeffding s inequality (see [9]) Remark 4 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 = 000 and 2e 2ε2N = 000 Consequently N = Comparison of results 5 Erlang distributions for claim sizes and additional funds Let the probability density functions of ξ i and η i be f (y) = kk yk e k y/μ μ k (k )! for y 0 respectively where k and k 2 are positive integers and f 2 (y) = kk 2 2 yk 2 e k 2y/μ 2 μ k 2 2 (k 2 )!

10 76 Yu Mishura et al In what follows h (R) and h 2 (R) where R 0 denote the moment generating functions of ξ i and η i respectively that is h (R) = E [ e Rξ ] i and h 2 (R) = E [ e Rη ] i An easy computation shows that + ( ) h (R) = e Ry k k df (y) = k μ R 0 R< k μ h 2 (R) = ( e Ry df 2 (y) = k 2 k 2 μ 2 R ) k2 0 R< k 2 μ 2 Moreover for all R 0 we have + ( ) e Ry k k2 2 df 2 (y) = 0 k 2 + μ 2 R Thus condition (3) can be rewritten as ( ) k k ( ) k k2 2 λ = λ + c ˆR (23) k μ ˆR k 2 + μ 2 ˆR where 0 < ˆR <k /μ Furthermore we have E[ξ i ]=h (0) = μ E[η i ]=h 2 (0) = μ 2 E [ ξ 2 i E [ ξi 3 ] = h ] = h (0) = (k + )μ 2 E [ η 2 ] i = h 2 k (0) = (k 2 + )μ 2 2 k 2 (0) = (k + )(k + 2)μ 3 k 2 Hence we get E [ (ξ i η i ) 2] = E [ ξ 2 i E [ ηi 3 ] = h ] 2E[ξi ]E[η i ]+E [ ηi 2 ] 2 (0) = (k 2 + )(k 2 + 2)μ 3 2 k 2 2 = (k + )μ 2 2μ μ 2 + (k 2 + )μ 2 2 k k 2 E [ (ξ i η i ) 3] = E [ ξi 3 ] [ ] 3E ξ 2 i E[ηi ]+3E[ξ i ]E [ ηi 2 ] [ ] E η 3 i = (k + )(k + 2)μ 3 k 2 3(k + )μ 2 μ 2 k + 3(k 2 + )μ 2 2 μ (k 2 + )(k 2 + 2)μ 3 2 k 2 k2 2 Substituting E[(ξ i η i ) 2 ] and E[(ξ i η i ) 3 ] into (6) (9) we obtain the parameters ( c λ μ μ 2 ) Example 2 Let c = 0 λ = 4 μ = 2 μ 2 = 05 k = 3 k 2 = 2 Then ˆR which may not be an unique positive solution to (23) and ψ DV (x) = e x The results of computations are given in Table

11 Practical approaches to the estimation of the ruin probability 77 Table Results of computations: Erlang distributions for claim sizes and additional funds x ˆψ(x) ψ DV (x) ( ψdv (x) ˆψ(x) ) 00% e ˆRx % % % % % % % % % % ( e ˆRx ˆψ(x) ) 00% 52 Hyperexponential distributions for claim sizes and additional funds Let F (y) = p F (y) + p 2 F 2 (y) + +p k F k (y) y 0 where k p j > 0 k j= p j = k j= p j μ j = μ and F j is the cdf of the exponential distribution with mean μ j ; F 2 (y) = p 2 F 2 (y) + p 22 F 22 (y) + +p 2k2 F 2k2 (y) y 0 where k 2 p 2j > 0 k 2 j= p 2j = k 2 j= p 2j μ 2j = μ 2 and F 2j is the cdf of the exponential distribution with mean μ 2j It is easy to check that h (R) = h 2 (R) = k j= k 2 j= { p j μ j R 0 R<min μ { p 2j μ 2j R 0 R<min μ 2 Furthermore for all R 0 we have + 0 e Ry df 2 (y) = Hence condition (3) can be rewritten as ( k λ j= p j μ j ˆR k 2 j= k 2 j= p 2j + μ 2j R μ 2 μ 22 μ k μ 2k2 } } ) p 2j = λ + c ˆR (24) + μ 2j ˆR where 0 < ˆR <min{/μ /μ 2 /μ k } Moreover we have k E[ξ i ]=h (0) = k 2 p j μ j = μ E[η i ]=h 2 (0) = p 2j μ 2j = μ 2 j= j=

12 78 Yu Mishura et al Table 2 Results of computations: hyperexponential distributions for claim sizes and additional funds x ˆψ(x) ψ DV (x) ( ψdv (x) ˆψ(x) ) 00% e ˆRx % % % % % % % % % % % % % % ( e ˆRx ˆψ(x) ) 00% E [ k ξi 2 ] = h (0) = 2p j μ 2 j j= E [ k ξi 3 ] = h (0) = 6p j μ 3 j Consequently we get E [ (ξ i η i ) 3] = 6 j= j= k E[ ηi 2 ] 2 = h 2 (0) = 2p 2j μ 2 2j j= k E[ ηi 3 ] 2 = h 2 (0) = 6p 2j μ 3 2j j= j= E [ (ξ i η i ) 2] ( k = 2 p j μ 2 j μ k 2 ) μ 2 + p 2j μ 2 2j p j μ 3 j μ k 2 p j μ 2 j + μ k 2 k 2 p 2j μ 2 2j ) p 2j μ 3 2j ( k j= j= Example 3 Let c = 0 λ = 4 μ = 2 μ 2 = 05 k = 3 k 2 = 2 p = 04 μ = 05 p 2 = 03 μ 2 = 2 p 3 = 03 μ 3 = 4 p 2 = 075 μ 2 = 04 p 22 = 025 μ 22 = 08 Then ˆR which may not be a unique positive solution to (24) and ψ DV (x) = e x The results of computations are given in Table 2 53 Exponential distribution for claim sizes and degenerate distribution for additional funds Let ξ i be exponentially distributed with mean μ and E[η i = μ 2 ]= Then condition (3) can be rewritten as λe μ 2 ˆR j= μ ˆR = λ + c ˆR where ˆR (0 /μ ) which is equivalent to j= λe μ 2 ˆR = cμ ˆR 2 + (c λμ ) ˆR + λ (25)

13 Practical approaches to the estimation of the ruin probability 79 Table 3 Results of computations: exponential distribution for claim sizes and degenerate distribution additional funds x ˆψ(x) ψ DV (x) ( ψdv (x) ˆψ(x) ) 00% e ˆRx % % % % % % % % % % ( e ˆRx ˆψ(x) ) 00% If c λμ + λμ 2 > 0 then it is easy to check that (25) has a unique solution ˆR (0 /μ ) Since E[ξ i ]=μ E [ ξi 2 ] = 2μ 2 E [ ξi 3 ] = 6μ 3 we get E[η i ]=μ 2 E [ η 2 i ] = μ 2 2 E [ ηi 3 ] = μ 3 2 E [ (ξ i η i ) 2] = 2μ 2 2μ μ 2 + μ 2 2 E [ (ξ i η i ) 3] = 6μ 3 6μ2 μ 2 + 3μ μ 2 2 μ3 2 Example 4 Let c = 0 λ = 4 μ = 2 μ 2 = 05 Then ˆR and ψ DV (x) = e x The results of computations are given in Table 3 6 Conclusion Tables 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 References [] Asmussen S: Ruin Probabilities World Scientific Singapore (2000) MR doi:042/ [2] Beekman JA: A ruin function approximation Trans Soc Actuar (969) [3] Boikov AV: The Cramér Lundberg model with stochastic premium process Theory Probab Appl (2002) MR doi:037/s x

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