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Improved asymptotic upper bounds on the ruin capital in the Lundberg model of risk
Insurance: Mathematics and Economics 55 (2014) 301–309
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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime
Improved asymptotic upper bounds on the ruin capital in the Lundberg model of risk✩ Vsevolod K. Malinovskii ∗ Central Economics and Mathematics Institute (CEMI) of Russian Academy of Science, 117418, Nakhimovskiy prosp., 47, Moscow, Russia Gubkin Russian State University of Oil and Gas, 119991, Leninsky prosp., 65, Moscow, Russia
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This paper deals with ruin capital uα,t (c | λ, µ) in the classical Lundberg model of risk. It is defined as the initial capital needed to keep the probability of ruin within finite time t equal to a predefined value α . Considered as a decreasing function of premium rate c, the ruin capital is shown to be convex (i.e., concave downward) for c > λ/µ and t sufficiently large. This observation is used to construct explicit upper bounds on the ruin capital. © 2014 Elsevier B.V. All rights reserved.
Keywords: Lundberg model of risk Ruin capital Asymptotic bounds
1. Introduction and definitions In this paper, we consider the classical collective Lundberg model, where the risk reserve process is R(s) = u + cs − V (s),
s > 0.
Here u > 0 is the initial risk reserve, c > 0 is the premium rate,
N (s) = max n > 0 :
n
Ti 6 s ,
s > 0,
i=1
or 0, if T1 > s, is the claims arrival process, or the number of claims which have occurred up to time s, V ( s) =
N (s)
Yi ,
s > 0,
i=1
or 0, if N (s) = 0 (or T1 > s), is the claims payout process, or the aggregate claims amount payed out up to time s. By Ti , i = 1, 2, . . . , and Yi , i = 1, 2, . . . , we denote mutually independent sequences of i.i.d. exponentially distributed random variables with intensities λ > 0 and µ > 0 respectively. So, the aggregate claims payout process V (s), s > 0, is compound Poisson, starting at zero. The mean and variance for any s are well known to be EV (s) = (λ/µ)s and DV (s) = 2(λ/µ2 )s respectively. Throughout this paper, we assume that 0 < α < 21 . We denote by
ψt (u, c | λ, µ) = P{ inf R(s) < 0} 0
DOI of original article: http://dx.doi.org/10.1016/j.insmatheco.2013.12.005. ✩ This work was supported by RFBR (grant No. 14-06-00017-a).
∗ Correspondence to: Central Economics and Mathematics Institute (CEMI) of Russian Academy of Science, 117418, Nakhimovskiy prosp., 47, Moscow, Russia. E-mail addresses: [email protected], [email protected]. URL: http://www.actlab.ru. 0167-6687/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.insmatheco.2013.12.004
the probability of ruin within finite time t > 0, and by ψ+∞ (u, c | λ, µ) the probability of ultimate ruin. The solution with respect to u of the equation
ψt (u, c | λ, µ) = α
(1.1)
is called α -level initial capital, or α -level ruin capital, or merely ruin capital. We denote it by uα,t (c | λ, µ). Considered as a function of c, it allows us a choice of the initial capital balanced with premiums, when to achieve a given level of non-ruin is a goal. The solution with respect to c of Eq. (1.1) is called the α -level premium rate. We denote it by cα,t (u | λ, µ). It is easy to see that for any c > 0 cα,t (uα,t (c | λ, µ) | λ, µ) = c
(1.2)
and for any u > 0 uα,t (cα,t (u | λ, µ) | λ, µ) = u.
(1.3)
The inverse functions for both uα,t (c | λ, µ) and cα,t (u | λ, µ) −1 exist. From (1.2) and (1.3) we have uα,t (c | λ, µ) = cα, t (c | λ, µ)
1 for any c > 0 and cα,t (u | λ, µ) = u− α,t (u | λ, µ) for any u > 0. So, enough is to focus on the ruin capital uα,t (c | λ, µ). The ruin capital is paramount in modeling solvency. It lays grounds for the investigation of controlled multi-period models of insurance business (see Malinovskii (2012, 2013b, 2014)). As we will find it below, the shape of the ruin capital considered as a function of premium rate c, as t is sufficiently large, is surprisingly simple. It is (see Fig. 1 in Section 2) monotone decreasing, as c increases, and
• almost linear for 0 < c < λ/µ, • convex for c > λ/µ,
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Theorem 2.1 (Corollary 2.1 in Malinovskii (1998)). In the classical Lundberg model, we have
ψt (u, c | λ, µ) = ψ+∞ (u, c | λ, µ) −
1
π
π
ft (x, u) dx,
(2.1)
0
where2
λ exp{−u(c µ − λ)/c }, ψ+∞ (u, c | λ, µ) = c µ 1, λ/c µ > 1
λ/c µ < 1,
and ft (x, u) = (λ/c µ)(1 + λ/c µ − 2 λ/c µ cos x)−1
Fig. 1. Ruin capital uα,t (c | λ, µ) as a function of c (X -axis) calculated numerically using Eq. (2.1) (solid line) and the results of simulation (dots) for λ = 4/5, µ = 3/5, α = 0.05, t = 200. Horizontal line: uα,t (λ/µ | λ, µ) = 59.9033. Vertical line: λ/µ = 4/3.
• asymptotically independent on t and close to uα (c | λ, µ) for ∗ ∗ c > cα, t , where cα,t is a certain value larger than λ/µ and uα (c | λ, µ) is a solution (w.r.t. u) of the equation1 ψ +∞ (u, c | λ, µ) = α . This paper is a development of that part of Malinovskii (2013a), which refers to the classical Lundberg model of risk. It is a counterpart of Malinovskii (submitted for publication), where the similar results are obtained in a diffusion model of risk. The extension of the results of this paper to Andersen’s model is still an open and difficult problem. In Malinovskii and Kosova (submitted for publication), simulation studies are conducted in order to see to what extent the ruin capital in Andersen’s model preserves the properties listed above. The rest of the paper is arranged as follows. In Section 2, we write the fundamental formula for ψ t (u, c | λ, µ) and discuss some results for uα,t (c | λ, µ) obtained in Malinovskii (2012, 2013a). Our goal is their further development. In Section 3, we formulate Theorem 3.3 which is the main result of the paper. Its proof is based on the monotony and convexity theorems (i.e., on Theorems 3.1 and 3.2). Further on, we formulate Theorem 3.4 which yields a weaker result than the result of Theorem 3.3, but explicit. Since both Theorems 3.3 and 3.4 give asymptotic bounds on the ruin capital, we discuss how to perform calculations for finite t. We outline a straightforward heuristic method and discuss the way of its justification. Section 4 contains the proof of the monotony theorem in the strong form, i.e., for all t. We mention the relationship between the ruin theory and the theory of random walks with random displacements developed, e.g., in Takács (1967). Section 5 contains the proof of the convexity theorem in the weak form, i.e., for t large. The proofs of the monotony and convexity theorems require a series of calculations involving Bessel functions. In Section 6 two auxiliary results from calculus are presented. 2. Rationale and motivation Apparently different methods lead to the same explicit expression for the probability of ruin ψ t (u, c | λ, µ) in the classical risk model (see, e.g., Prabhu (1965), Chapter 5, Section 5.5, Takács (1967), Chapter 7, Section 35). We refer to Malinovskii (1998), where this expression was obtained as a corollary of a more general result.
1 Since (see Theorem 2.1) ψ +∞ (u, c | λ, µ) = (λ/(c µ)) exp{−u(c µ − λ)/c } for λ/c µ < 1 and 1 elsewhere, the explicit expression for uα (c | λ, µ) is straightforward.
× exp uµ λ/c µ cos x − 1 − t λ(c µ/λ) × 1 + λ/c µ − 2 λ/c µ cos x cos uµ λ/c µ sin x − cos uµ λ/c µ sin x + 2x . The cases when there exist explicit formulas like Eq. (2.1) are exceptional.3 In Malinovskii (1998), Eq. (2.1) was obtained by manipulating a formula which yields the probability of ruin within finite time in a model more general than classical. But even this formula which comprises Laplace transform of the probability of ruin can hardly be used per se to extend the analysis of the solution of Eq. (1.1) to the more general model. All values of c in Eqs. (1.1), (2.1) are of a great interest, including those lying around the critical value4 c = λ/µ. This is so because our investigation lays grounds for a number of other endeavors, e.g., for the development of equitable solvent controls in a multiperiod game model of risk (see Malinovskii (2012)). Being focused on the neighborhoods of c = λ/µ, we find useless the asymptotic results like Cramér’s approximations which are of a simpler form than Eq. (2.1). It refers as well to the refinements of these approximations (see, e.g., Malinovskii (1994, 2000)). Being cumbersome, the explicit formula in Eq. (2.1) can be used for the numerical calculation of uα,t (c | λ, µ). In Fig. 1, the ruin capital uα,t (c | λ, µ) as a function of c calculated numerically is shown by a solid line, while the dotted line is a result of simulation study of uα,t (c | λ, µ). This study in the classical case and in the cases more general than classical is described in detail in Malinovskii and Kosova (submitted for publication). The analytical investigation is quite desirable, though not developed yet in Andersen’s model, and simulation seems to be the only effective method to tackle this problem. To conclude this section, we recall several results on the ruin capital uα,t (c | λ, µ) for time t sufficiently large,5 which will be referred to in the sequel. We write κγ = Φ(−0,11) (1 −γ ) for the (1 −γ )quantile (0 < γ < 1) of the standard normal c.d.f. Φ(0,1) (x). Since 0 < α < 1/2, we have 0 < κα < κα/2 < 1. Theorem 2.2 (Theorem 3.1 in Malinovskii (2012)). In the classical Lundberg model, we have
√ uα,t (λ/µ | λ, µ) =
2t λ
µ
κα/2 (1 + o(1)),
t → ∞.
(2.2)
2 Note that the factor λ/(c µ) is the well-known Lundberg’s constant and the factor (c µ − λ)/c is the well-known Lundberg’s exponent. In the framework of the classical model both are explicit. 3 Another exceptional case is provided by the diffusion model. It is considered in Malinovskii (submitted for publication). 4 The case c = λ/µ is usually referred to as the case of zero safety loading.
5 Bearing in mind that time is operational, large t means large portfolio. We use the ‘‘small o notation’’, i.e., we say f (x) = o(g (x)), as x → a, if limx→a f (x)/g (x) = 0.
V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309
Theorem 2.3 (Theorem 3.2 in Malinovskii (2012)). In the classical Lundberg model, we have uα,t (c | λ, µ)
√ √ µ(λ/µ − c ) t λ 2t λ , √ µ − c t + µ Uα,t 2λ = √ √ µ(λ/µ − c ) t 2t λ U , c > λ/µ, √ α,t µ 2λ
c 6 λ/µ,
where the function Uα,t (v) is continuous and monotone increasing, as v increases from −∞ to 0, and monotone decreasing, as v increases from 0 to +∞, and such that lim Uα,t (v) = 0,
v→+∞
and Uα,t (0) = κα/2 (1 + o(1)), as t → ∞. For brevity, set6 u∗α,t = uα,t (λ/µ | λ, µ). The following theorem is a counterpart of Corollary 3.1 in Malinovskii (2013a). Theorem 2.4. In the classical Lundberg model, as t → ∞, we have for c 6 λ/µ
λ κα ∗ λ −c t + u 6 uα,t (c | λ, µ) 6 − c t + u∗α,t , (2.3) µ κα/2 α,t µ
and7 for c > λ/µ
uα,t (c | λ, µ) 6 min
ln(α c µ/λ) u∗α,t , −
µ − λ/c
.
is a convex function of c > λ/µ, it yields the desired upper bound in the interval between the critical point λ/µ and the point of tangency, where h(c ) is an upper bound for the ruin capital. So, we formulate first the monotony and convexity theorems (i.e., Theorems 3.1 and 3.2). After this, we formulate Theorem 3.3 which is the main result of the paper. We formulate Theorem 3.4 which yields upper bounds rougher than in Theorem 3.3, but written explicitly. Further, we outline a heuristic construction applicable for finite t and discuss the ways of its justification. Theorem 3.1 (The Monotony Theorem). In the classical Lundberg model, for c > λ/µ and for all t > 0, the α -level initial capital uα,t (c | λ, µ) is a monotone decreasing function of c. This theorem is quite intuitive. It was proved in Malinovskii (2008a). To make the presentation self-contained, we return to its proof in Section 4. The following theorem, being not at all intuitive, is fundamental. Its proof is given in Section 5.
lim Uα,t (v) = κα
v→−∞
303
Theorem 3.2 (The Convexity Theorem). In the classical Lundberg model, for c > λ/µ and for t > 0 sufficiently large, the α -level initial capital uα,t (c | λ, µ) is a convex (i.e., concave downward) function of c. ∗ ∗ Let cα, t = cα,t (λ, µ) be a solution (w.r.t. c) of the non-linear equation
λ + cµ µc ln α = µu∗α,t − 1. λ − cµ λ
(3.1)
The following theorem improves the upper bounds (2.4). (2.4)
Proof of Theorem 2.4. The proof of (2.3) is straightforward from Theorem 2.3. For the proof of (2.4), we use the trivial inequality ψt (u, c | λ, µ) 6 ψ+∞ (u, c | λ, µ) for c > λ/µ, and note that ψ+∞ (u, c | λ, µ) = α yields as solution u = − ln(α c µ/λ)/(µ − λ/c ). We strive to improve the upper bound (2.4) in Theorem 2.4. 3. Improved asymptotic upper bounds on the ruin capital Outline the desired improvements in Theorem 2.4. Note that √ the accuracy8 of the bounds in (2.3) is o( t ), as t → ∞. It is straightforward from Eq. (2.2) and, bearing in mind Theorem 2.3, looks satisfactory. In contrast to that, the upper bound in (2.4) performs well only for c much larger than λ/µ since uα,t (c | λ, µ) → − ln(α c µ/λ)/(µ − λ/c ), as t → ∞, but is rather trivial, and in some sense unsatisfactory, for c close to λ/µ. If we can prove that uα,t (c | λ, µ) is a convex (i.e., concave downward) function of c > λ/µ, then we can easily improve the upper bound in (2.4) for c close to λ/µ. We do the following. Instead of the horizontal upper bound u∗α,t in (2.4), we take the tangent line to h(c ) = − ln(α c µ/λ)/(µ − λ/c ) starting from the point with abscissa λ/µ and ordinate u∗α,t . Given that uα,t (c | λ, µ)
√
6 We will always bear in mind that u∗ = ( 2t λ/µ) κ (1 + o(1)), as t → ∞, α/2 α,t by Theorem 2.2. 7 Eq. (3.7) in Corollary 3.1 of Malinovskii (2013a) (cf. Theorem 4.2 in Malinovskii (2013b)) contains a technical deficiency which we eliminate here. 8 The bounds, as well as the ruin capital, are of order O(t ), as t → ∞. By accuracy we√mean the difference between the true value and the bounds, which is of order o( t ), as t → ∞. Normalized by dividing by t, this difference becomes o(t −1/2 ), as t → ∞. It corresponds better to the term accuracy, but we will not move to the relative terms.
Theorem 3.3. In the classical Lundberg model, as t → ∞, we have uα,t (c | λ, µ)
λ − µc ∗ + λ ln(α c ∗ µ/λ) α,t α,t ∗ (c − cα, t) ∗ 2 (λ − µ c ) α, t ∗ ∗ cα,t ln(α cα,t µ/λ) λ ∗ 6 < c 6 cα, + , t, ∗ λ − µ c µ α,t − ln(α c µ/λ) , c > c ∗ . α,t µ − λ/c Proof of Theorem 3.3. Bearing in mind Theorem 3.2 and taking t so large that uα,t (c | λ, µ) is convex, we draw a tangent line l(c ) =
∗ ∗ λ − µcα, t + λ ln(α cα,t µ/λ) ∗ (c − cα, t) ∗ 2 (λ − µcα,t ) ∗ ∗ cα, t ln(α cα,t µ/λ) + ∗ λ − µcα, t
to the upper bound h(c ) = −
ln(α c µ/λ)
µ − λ/c
yielded by the evident inequality ψ t (u, c | λ, µ) 6 ψ +∞ (u, c | λ, µ). This line starts from the point with abscissa 1 and ordinate ∗ u∗α,t and is tangent to h(c ) at the point cα, t defined as the solution of Eq. (3.1) which is equivalent to l(λ/µ) = u∗α,t . ∗ While for c > cα, t the examination of accuracy of the upper bound in Theorem 3.3 consists in the study of the rate of convergence9 uα,t (c | λ, µ) → − ln(α c µ/λ)/(µ − λ/c ), as t → ∞, for
9 This is closely linked to the rate of convergence ψ (u, c | λ, µ) → ψ t +∞ (u, c | λ, µ), as t → ∞. We shall not give here a detailed consideration of this well-known problem.
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Fig. 2. Ruin capital uα,t (c | λ, µ) as a function of c (X -axis) calculated numerically using Eq. (2.1) (lower solid line for c > λ/µ and middle solid line for c < λ/µ), the results of simulation (dots), the bounds (2.3) of Theorem 2.4, and the bounds of Theorem 3.4 for λ = 4/5, µ = 3/5, α = 0.05, t = 200. Horizontal line: uα,t (λ/µ | λ, µ) = 59.9033. Vertical line: λ/µ = 4/3.
Fig. 3. Shown are the function uα,t (c | λ, µ) of the variable c (X -axis) calculated numerically using Eqs. (1.1) and (2.1), and the bounds of Theorems 3.3 and 3.4 for c > λ/µ, with λ = µ = 1, α = 0.1, t = 100. Here u∗α,t is taking equal to
√ ( 2t λ/µ) κα/2 = 23.2617, i.e., with the term 1 + o(1) replaced by 1. The value of uα,t (λ/µ | λ, µ) calculated numerically is 23.5722.
∗ λ/µ < c 6 cα, t we cannot say much. Being high for c close to λ/µ ∗ and to cα, by construction, it is unknown for c between these valt ues. Numerical examples show however that the accuracy is surprisingly good. The following theorem yields rougher upper bounds than Theorem 3.3. It applies the upper bound on the right hand side of the inequality
−
ln(α c µ/λ)
µ − λ/c
6−
ln α
µ − λ/c
,
true for c < λ/µ. This is done to find a tangent line to h(c ) = − ln α/(µ − λ/c ) explicitly rather than via the solution of Eq. (3.1). Theorem 3.4. In the classical Lundberg model, as t → ∞, we have uα,t (c | λ, µ)
(µu∗ + ln α)2 α,t c 4λ ln α ∗ (µuα,t − ln α)2 λ λ µu∗α,t − ln α − , . − µ − λ/c µ µu∗α,t + ln α
Fig. 4. The functions shown in Fig. 3, but with u∗α,t (ε) taken equal to 24.0722, i.e., with the term 1 + o(1) replaced by 1.0348. Two other horizontal lines are uα,t (λ/µ |
√ λ, µ) calculated numerically, equal to 23.5722, and ( 2t λ/µ) κα/2 = 23.2617.
Fig. 5. The upper bounds of Theorems 3.3 (above) and 3.4 (middle) shown with uα,t (c | λ, µ) calculated numerically (below). This drawing differs from one in Fig. 3 by that the auxiliary lines are removed.
In Fig. 2, which illustrates the main result of the paper, the graphs shown in Fig. 1 are supplemented by the upper and lower bounds, as c < λ/µ, and by the upper bounds, as c > λ/µ. The former are the bounds (2.3) of Theorem 2.4. The latter are the bounds of Theorem 3.4. 3.1. A heuristic construction of the upper bounds Suppose we are given an arbitrary ε > 0. By Theorems 2.2 and 3.2, there exists t (ε) > 0 such that uα,t (c | λ, µ) is a convex function of c (as c > λ/µ) for all t > t (ϵ) and uα,t (λ/µ | λ, µ) 6 √ u∗α,t (ϵ) = ( 2t λ/µ) κα/2 (1 + ε) for all t > t (ϵ). The constructions of Theorems 3.3 and 3.4 are illustrated in Fig. 3 for ε = 0, which is deficient to garner the upper bound uniformly on c > λ/µ, and in Fig. 4 for ε = 0.0348, which yields a valid upper bound uniformly on c > λ/µ, as t = 100. In Figs. 3 and 4 we show the auxiliary lines illustrating the tangent line construction. In Figs. 5 and 6 these auxiliary lines are removed. We call ε -heuristic the upper bounds which are obtained by a formal replacement of o(1) in Eq. (2.2) by an ε > 0. The heuristic consists in the implicit assumption that t (ε) is chosen so large that it makes the bounds in Theorems 2.2 and 3.2 valid uniformly on c > λ/µ for all t > t (ε). In order to specify t (ε) > 0 for a given ε > 0, further insight into the structure of the term o(1) in Theorem 2.2 and into the phrase ‘‘for t > 0 sufficiently large’’ in
V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309
305
and that
x ∂ n+1 vn+1 (x | p) dx = vn+2 (x | p) dx. (4.5) vn+1 (x | p) − ∂p p2 n+2 4.2. First derivative of α -level initial capital Here we will calculate the first derivative of uα,t (c | λ, µ) given implicitly by Eq. (1.1) with respect to the argument c. For x, t > 0 and 0 < p < 12 , introduce
ℵt (x, p) =
(0,1)
( x, p ) , λ Ht(1,0) (x, p) 1 Ht
where Fig. 6. The upper bounds of Theorems 3.3 (above) and 3.4 (middle) shown with uα,t (c | λ, µ) calculated numerically (below). This drawing differs from one in Fig. 4 by that the auxiliary lines are removed.
Theorem 3.2 is necessary.10 It seems that it requires considerable efforts. The same applies to the bounds (2.3) in Theorem 2.4. 4. The proof of monotony Theorem 3.1
t ∞ xn vn+1 (y | p) − vn+2 (y | p) dy, n! 0 n =0 t ∞ xn (0,1) −x Ht ( x, p ) = e y vn+1 (y | p) n! 0 n=0 n+1 vn+2 (y | p) dy. − n+2 (1,0)
Ht
(x, p) = −e−x
In this section, we introduce the key ideas which will be further developed in Section 5 to prove Theorem 3.2. These ideas are essentially the same as in the case of diffusion model of risk considered in Malinovskii (submitted for publication). Basically, it is the standard examination of the sign of the first derivative of a function in order to show its monotony.
Theorem 4.1. For t , λ, µ > 0, 0 < α <
4.1. A formula for ruin probability
ψ(t 0,1) (u, c | λ, µ)|u=uα,t (c |λ,µ) ∂ . uα,t (c | λ, µ) = − (1,0) ∂c ψt (u, c | λ, µ)|u=uα,t (c |λ,µ)
Besides the expression (2.1) for the probability of ruin ψ t (u, c |
λ, µ), we will apply its equivalent,11 ψt (u, c | λ, µ) = Ft λ (x, p)| Ft (x, p) = e−x
x=uµ p=λ/(c µ+λ)
,
t ∞ xn vn+1 (y | p) dy, n! 0 n =0
p
vn+1 (y | p) =
q
Applying (4.5), differentiate (4.1). Since −λµ/(c µ + λ)2 = −µp2 /λ, we have
(4.2)
ψ(t 0,1) (u, c | λ, µ) = −
F∞ (x, p) = e−x
n =0
=
p q
xn
(4.3)
n!
vn+1 (y | p) dy
exp −x 1 −
p q
,
d dc
(λ/(c µ + λ)) =
µ 2 (0,1) p Ft λ (x, p)| x=uµ p=λ/(c µ+λ) λ
(1,0)
x=uµ p=λ/(c µ+λ)
(1,0)
.
(0,1)
(0,1)
Since Ft (x, p) = Ht (x, p) and p2 Ft (x, p) = Ht (x, p), which is checked straightforwardly, the proof is complete.
∂ uα,t (c | λ, µ) < 0. ∂c
0
=
Theorem 4.2 (Weak form Derivative Inequality). For λ, µ > 0, 0 < α < 21 , c > λ/µ and for t > 0 sufficiently large, we have
∞
d p dc
and
Note that ψ ∞ (u, c | λ, µ) = F∞ (x, p)| x=uµ , where14 p=λ/(c µ+λ) ∞
Proof of Theorem 4.1. Bearing in mind Eq. (1.1), apply the first part of Theorem 6.1 which yields15
ψ(t 1,0) (u, c | λ, µ) = µFt(λ1,0) (x, p)| n + 1 −y/p e In+1 (2y q/p). y
and c > λ/µ, we have
∂ uα,t (c | λ, µ) = ℵt λ (uµ, λ/(c µ + λ))|u=uα,t (c |λ,µ) . ∂c
(4.1)
where12 q = 1 − p and13
(n+1)/2
1 2
(4.4)
Proof. Using Eq. (4.4), for x > 0 and 0 < p < (1,0) H∞ ( x, p ) = −
10 This situation is typical for asymptotic analysis, unless the remainder term is presented in an explicit form or estimated by an explicitly given expression. 11 See, e.g., Eq. (13), or Eqs. (17)–(20) in Malinovskii (2008a). See also Malinovskii (2008b). 12 Since in (4.1) we set p = λ/(c µ + λ) and since we will be interested in the case c > λ/µ, we will be concentrated on the case 0 < p < 12 . 13 By I (·), k = 1, 2, . . . , we denote the Bessel functions (see Section 6.2). k
14 It follows from Theorem 6.2 and yields the well-known formula for the probability of ultimate ruin.
(4.6)
(0,1) H∞ ( x, p )
=
p(q − p) q2
p2 (q + px) q3
exp −x 1 −
exp −x 1 −
p q p q
1 2
we have
, .
15 Here and in what follows, the symbols f (i,j) (x , x ) denote ith and jth 1 2 partial derivatives of f (x1 , x2 ) with respect to the first and the second variables, respectively.
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Apply Theorem 4.1 and note that
ℵt (x, p) = ℵ∞ (x, p)(1 + o(1)),
Theorem 4.6 (Eq. (9) in Section 22 of Takács (1967)). In the random walk process model,
t → ∞,
where
ℵ∞ (x, p) =
1
λ
(0,1) H∞ (x, p) (1,0) H∞ (x, p)
The proof is complete.
=−
1 p(q + px)
λ q(q − p)
P{ sup ξp (z ) < k} = P{ξp (y) < k} − 06z 6y
< 0.
(x, p) < 0,
(0,1)
Ht
1 , 2
k(p/q) (4.7)
∂ uα,t (c | λ, µ) < 0. ∂c
Theorem 4.3 was proved in Malinovskii (2008a). It may be proved analytically, using the properties of Bessel functions, or connecting our problem with the following construction. Suppose that a particle performs a random walk on the X -axis. Starting at the origin, in each step the particle moves either a unit distance to the right with probability p or a unit distance to the left with probability q (p + q = 1, 0 < p < 1). Suppose that the displacements of the particle occur at random times in the time interval (0, ∞). Denote by ν(z ) the number of steps taken in the interval (0, z ]. We suppose that {ν(z ), 0 6 z < ∞} is a Poisson process of density 1/p and that the successive displacements are independent of each other and independent of the process {ν(z ), 0 6 z < ∞}. Denote by ξp (z ) the position of the particle at time z. In this case {ξp (z ), 0 6 z < ∞} is a stochastic process having stationary independent increments, P{ξp (0) = 0} = 1 and almost all sample functions of {ξp (z ), 0 6 z < ∞} are step functions having jumps of magnitude 1 and −1. Theorem 4.5 (Eqs. (3), (8) in Section 22 of Takács (1967)). In the random walk process model, P{ξp (z ) = k} = e−z /p (p/q)k/2 Ik (2z
q/p) =
z > 0,
z k
vk (z | p),
P{ sup ξp (z ) < k} = 1 − k(p/q)k/2 06z 6y
y
e−z /p Ik (2z
q/p)
0
dz z
dz
q/p)
vk (z | p)dz ,
=1−
dz z
= e−y/p
∞ (p/q)i/2 Ii (2y q/p) i =k
(q/p)i/2−k Ii (2y q/p).
(4.8)
i=k+1
Denote by ςk (p) the first hitting time of the point k by the random walk process with the parameter p ∈ (0, 1). Evidently,18 y
vk (z | p) dz = P{ςk (p) 6 y},
(4.9)
0
so that vk (z | p), z > 0, is the p.d.f. of the first hitting time of the point k in the random walk process model (see also Feller (1971), Chapter II, Section 7 and Chapter XIV, Section 6). The densities vk (z | p), z > 0, are defective for p < 1/2 (random walk with drift to the left) and proper for p > 1/2 (random walk without drift, or with drift to the right). For any t > 0 and n = 0, 1, 2, . . . , we have from Theorem 4.5 and Eq. (4.9) P{ξp (y) = n + 2} =
y n+2
P{ sup ξp (z ) = n + 1} =
vn+2 (y | p), y
06z 6y
(4.10)
vn+1 (z | p)dz 0 y
vn+2 (z | p)dz .
−
(4.11)
0
Bearing in mind Eq. (4.11), the following theorem yields the first inequality (4.7). Theorem 4.7. For 0 < p <
1 2
, y > 0 and n = 0, 1, 2, . . . , we have
P{ sup ξp (z ) = n + 1} 06z 6y
= P{ςn+1 (p) 6 y} − P{ςn+2 (p) 6 y} > 0.
(4.12)
Bearing in mind Eqs. (4.10) and (4.11), the following theorem yields the second inequality (4.7).
z
y
y > 0,
Theorem 4.8 (Theorem 3.2 in Section 3.4 of Malinovskii (2008a)). For 0 < p < 12 , y > 0 and n = 0, 1, 2, . . . , we have
0
P{ sup ξp (z ) = n + 1} − P{ξp (y) = n + 2} > 0.
for integer k > 0.
(4.13)
06z 6y
16 Recall it: if the first derivative of a function is negative at all points of an interval, then this function is monotonically decreasing on this interval. 17 Note that Eξ (y) = [1 − (q/p)]y, Dξ (y) = y/p, y > 0. p
Proof of Theorem 4.7. It is straightforward from Eqs. (4.9) and (4.11).
for k = 0, ±1, ±2, . . . , and17
0
+ e−y/p
∞
06z 6y
4.3. Proof of Theorem 4.3 and random walks with random displacements
P{ξp (z ) = k}
e−z /p Ik (2z
P{ sup ξp (z ) > k} =
The monotony theorem, i.e., Theorem 3.1, is straightforward from Theorem 4.4 by the well-known criterion.16
y
y
0
(x, p) > 0.
Theorem 4.4 (Strong form Derivative Inequality). For λ, µ > 0, 0 < α < 21 , c > λ/µ and for all t > 0, we have
=1−k
y > 0,
Bearing in mind Theorem 4.5, from Theorem 4.6 we have k/2
we have
Theorem 4.3 proved in Section 4.3 implies the next enhancement of Theorem 4.2, true for all t > 0.
q
for integer k > 0.
Theorem 4.3. For x, t > 0 and 0 < p < (1,0)
p
× P{ξp (y) < −k},
We can refine Theorem 4.2 and prove that the inequality (4.6) holds true for all t > 0. For this, we need the following result.
Ht
k
p
18 By Eq. (4.1), we have ψ (u, c | λ, µ) = e−x ∞ xn P{ς x=uµ . n+1 (p) 6 t λ}| t n= 0 n ! p=λ/(c µ+λ)
V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309
Proof of Theorem 4.8. By ω we mean sample functions of {ξp (z ), 0 6 z < ∞} correspondence between the . Establish one-to-one sets A1 = ω : ξp (y) = n + 2 and A2 = ω : sup06z 6y ξp (z ) =
n + 1, ξp (y) = n . For each ω = {ξp (z ), 0 6 z 6 y} ∈ A1 , reflect with respect to the level n + 1 that part of ω, which lies above the level n + 1. In this way, for each ω ∈ A1 we obtain the unique ω′ = {ξp′ (z ), 0 6 z 6 y}, such that sup06z 6y ξp′ (z ) 6 n + 1 and ξp′ (z ) = n. Furthermore, we have
(0,2)
Ht
(x, p) = e−x
A2 = ω : sup ξp (z ) = n + 1, ξp (y) = n
06z 6y
+
∞ xn n! n =0
n+1 n+3
06z 6y
Obviously, this implication is strict. Indeed, the set A3 contains, e.g. all trajectories of the random walk process which have only one hitting of the point n + 1 within the interval [0, y]. Such trajectories are not produced by the reflection described above since those obtained in such a way must hit the point n + 1 at least twice. We have therefore19 P(A1 ) = P(A2 ) < P(A3 ), and the proof is complete.
5.1. Second derivative of the α -level initial capital Here we will calculate the second derivative of uα,t (c | λ, µ) given implicitly by Eq. (1.1) with respect to the argument c. For x, t > 0 and 0 < p < 12 , introduce
×
λ2 Ht(1,0) (x, p) (0,1)
(x, p)
(1,0)
(x, p)
Ht Ht
(0,2)
Ht
(2,0)
+ Ht
( x, p )
(0,1)
(x, p)
(1,0)
( x, p )
Ht Ht
2 ,
t ∞ xn vn+1 (y | p) − 2vn+2 (y | p) n! 0 n =0 + vn+3 (y | p) dy, ∞ xn (1,1) Ht (x, p) = −e−x n! n =0 t n+1 × y vn+1 (y | p) − vn+2 (y | p) n+2 0 n+2 − vn+2 (y | p) − vn+3 (y | p) dy, n+3 (2,0)
(x, p) = e−x
19 The ‘‘infinitesimal probability masses’’ of the original sample path and of its twin obtained by reflection are the same. 20 The basic technical instruments in Malinovskii (submitted for publication) were the inequalities for Mill’s ratio.
n+2
vn+2 (y | p)
vn+3 (y | p) dy.
1 2
and c > λ/µ, we have
∂2 uα,t (c | λ, µ) = i t λ (uµ, λ/(c µ + λ))|u=uα,t (c |λ,µ) . ∂ c2
∂2 u ( c | λ, µ) = − ψ(t 0,2) (u, c | λ, µ)|u=uα,t (c |λ,µ) α, t ∂ c2 2 × ψ(t 1,0) (u, c | λ, µ)|u=uα,t (c |λ,µ) − 2ψ(t 1,1) (u, c | λ, µ)|u=uα,t (c |λ,µ) × ψ(t 1,0) (u, c | λ, µ)|u=uα,t (c |λ,µ) + ψ(t 2,0) (u, c | λ, µ)|u=uα,t (c |λ,µ) 2 × ψ(t 0,1) (u, c | λ, µ)|u=uα,t (c |λ,µ) −3 × ψ(t 1,0) (u, c | λ, µ)|u=uα,t (c |λ,µ) . Applying (4.5), differentiate (4.1). Since
d2 p dc 2
d2 dc 2
=
(2λµ2 )/(c µ + λ)3 = 2µ2 p3 /λ2 , we have ψ(t 0,2) (u, c | λ, µ) =
µ2 4 (0,2) p Ft λ (x, p) λ2 + 2p3 Ft(λ0,1) (x, p) |
x=uµ p=λ/(c µ+λ)
(λ/(c µ + λ)) =
,
µ2 2 (1,1) p Ft λ (x, p) | x=uµ , p=λ/(c µ+λ) λ (2,0) 2 (2,0) ψt (u, c | λ, µ) = µ Ft λ (x, p)| x=uµ . ψ(t 1,1) (u, c | λ, µ) = −
(0,1)
(0,2) x p p4 Ft (1,1) x p Ht
(2,0)
(2,0)
where Ht
0
n+1
p=λ/(c µ+λ)
(x, p) − 2Ht(1,1) (x, p)
y2 vn+1 (y | p) − 2
× ψ(t 0,1) (u, c | λ, µ)|u=uα,t (c |λ,µ)
In this section, we prove the convexity theorem, though only for t large, i.e., in the weak form. It is noteworthy that in Malinovskii (submitted for publication), in the case of diffusion model of risk, this convexity theorem was proved for all t, i.e., in the strong form.20 The extension of Theorem 3.2 to all t is still an open problem. It seems to us difficult and its solution is quite desirable. Basically, the proof below is a standard examination of the sign of the second derivative of a function in order to show its convexity.
i t (x, p) = −
t
Theorem 5.1. For t , λ, µ > 0, 0 < α <
5. The proof of convexity Theorem 3.2
Proof of Theorem 5.1. Bearing in mind Eq. (1.1), apply the second part of Theorem 6.1 which yields
⊂ A3 = ω : sup ξp (z ) = n + 1 .
µ
307
( , ) + 2p3 Ft (x, p) = (x, p) = Ht (x, p), Since Ft (0,2) 2 (1,1) ( , ), which is checked (x, p), and p Ft (x, p) = Ht straightforwardly, the proof is complete. Theorem 5.2 (Weak form Derivative Inequality). For λ, µ > 0, 0 < α < 12 , c > λ/µ and for t > 0 sufficiently large, we have
∂2 uα,t (c | λ, µ) > 0. ∂ c2 Proof. Using Eq. (4.4), for x > 0 and 0 < p < (2,0) F∞ (x, p) =
(1,1) F∞ (x, p)
(0,2) F∞ (x, p)
=
=
p(q − p)2
p
exp −x 1 −
q3
qp − (q − p)(q + px) q4
,
q
exp −x 1 −
=
p(p − q)2 q3
p
exp −x 1 −
exp −x 1 −
p q
,
,
q
It follows that (2,0) H∞ ( x, p )
we have
2q2 + 2(1 + p)qx + px2 q5
1 2
p q
.
308
V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309
(1,1) H∞ ( x, p )
= p
2
qp + (p − q)(q + px)
Theorem 6.1. Let F be a function that possesses partial derivatives up to second order continuous in some neighborhood of some solution, (x0 , y0 ), of the equation F (x, y) = 0. If F2 (x0 , y0 ) ̸= 0, there are an ϵ > 0 and a unique continuously differentiable function φ such that φ(x0 ) = y0 and F (x, φ(x)) = 0 for |x − x0 | < 0. Moreover, when |x − x0 | < 0, we have
q4
p
× exp −x 1 − (0,2) H∞ ( x, p ) = p 3
,
q
q(q + 2px) + (q + px)2
−1 φ ′ (x) = −F (1,0) (x, φ(x)) F (0,1) (x, φ(x))
q5
p
× exp −x 1 −
q
(0,1)
Bearing in mind that
H∞ (x, p)
=−
(1,0) H∞ (x, p)
and
.
φ ′′ (x) = − F (2,0) (x, φ(x))(F (0,1) (x, φ(x)))2
p(q + px) q(q − p)
− 2F (1,1) (x, φ(x))F (1,0) (x, φ(x))F (0,1) (x, φ(x)) −1 + F (0,2) (x, φ(x))(F (1,0) (x, φ(x)))2 (F (0,1) (x, φ(x)))3 .
, we have
(0,1)
H ∞ ( x, p ) (0,2) (1,1) H∞ (x, p) − 2H∞ (x, p) (1,0) H ∞ ( x, p )
(2,0) + H∞ (x, p)
=
p3 (1 + 2px) q 3 ( q − p)
(0,1)
H∞ (x, p)
6.2. Bessel functions
2
(1,0)
For x > 0 and n = 0, 1, 2, . . . , the modified Bessel function of n-th order (see, e.g., Watson (1945), or Whittaker and Watson (1963), Chapter XVII, 17.7) is
H∞ (x, p)
exp −x 1 −
p q
.
In (x) =
Apply Theorem 5.1 and note that
i t (x, p) = i∞ (x, p)(1 + o(1)),
×
k! (n + k)!
2
e−bx In (ax)x−1 dx = 0
(0,1)
H∞ (x, p) (1,1) H (0,2) (x, p) − 2H∞ (x, p) (1,0) ∞ H∞ (x, p)
(2,0) + H∞ (x, p)
∞
(1,0) λ2 H∞ (x, p)
2 (0,1) H ( x, p )
∞ (1,0) H∞ (x, p)
µ p2 (1 + 2px) > 0. λ2 q(q − p)2
The proof is complete.
k =0
x n+2k
.
(6.1)
Theorem 6.2. For b > a > 0 and n = 1, 2, . . . we have
µ
=
1
t → ∞,
where
i∞ (x, p) = −
∞
The convexity theorem, i.e., Theorem 3.2, is straightforward from Theorem 5.2 by the well-known criterion.21 6. Auxiliary results For convenience, we recall here the definition of Bessel functions used in (4.1) and formulate the theorem about differentiation of implicit functions. This is well known in calculus. We recall also a result used to check the second equality in (4.4). 6.1. Differentiation of implicit functions The following implicit function theorem is well known in analysis (see e.g. Chapter I, Sections 5.2 and 5.3 in Widder (1947)).
21 Recall it: if the second derivative of a function is positive at all points of an interval, then this function is strictly convex on this interval.
∞
0
an
n( b +
√
b2 − a2 )n
,
an e−bx In (ax) dx = √ . √ b2 − a2 (b + b2 − a2 )n
Proof. Since In (z ) = exp(−nπ i/2)Jn (iz ), these equalities follow respectively from Eqs. (6.611.4) and (6.623.3) in Gradshtein and Ryzhik (1980). One can also refer to Eqs. (7) and (8) from Section 13.2, Section XIII of Watson (1945). Acknowledgments The author is grateful to the anonymous referee for the comments which helped to improve this paper. References Feller, W., 1971. An Introduction to Probability Theory and its Applications. Vol. II, second ed. John Wiley & Sons, New York. Gradshtein, I.S., Ryzhik, I.M., 1980. Table of Integrals, Series, and Products. Academic Press, New York. Malinovskii, V.K., 1994. Corrected normal approximation for the probability of ruin within finite time. Scand. Actuar. J. 161–174. Malinovskii, V.K., 1998. Non-Poissonian claims’ arrivals and calculation of the probability of ruin. Insurance Math. Econom. 22, 123–138. Malinovskii, V.K., 2000. Probabilities of ruin when the safety loading tends to zero. Adv. Appl. Probab. 32, 885–923. Malinovskii, V.K., 2008a. Adaptive control strategies and dependence of finite time ruin on the premium loading. Insurance Math. Econom. 42, 81–94. Malinovskii, V.K., 2008b. Risk theory insight into a zone-adaptive control strategy. Insurance Math. Econom. 42, 656–667. Malinovskii, V.K., 2012. Equitable solvent controls in a multi-period game model of risk. Insurance Math. Econom. 51, 599–616. Malinovskii, V.K., 2013a. Level premium rates as a function of initial capital. Insurance Math. Econom. 52, 370–380.
V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309 Malinovskii, V.K., 2013b. Rationale of underwriters’ pricing conduct on competitive insurance market. Insurance Math. Econom. 53, 325–333. Malinovskii, V.K., 2014. Annual intrinsic value of a company in a competitive insurance market. Insurance Math. Econom. 55, 310–318. Malinovskii, V.K., 2013. Improved upper bounds on level values in a diffusion model of risk. Ann. Actuarial Sci. (submitted for publication). Malinovskii, V.K., Kosova, K.O., 2013. Simulation analysis of ruin capital in Andersen’s model of risk. Insurance Math. Econom. (submitted for publication).
309
Prabhu, N.U., 1965. Queues and Inventories. John Wiley & Sons, Inc., New York, London. Takács, L., 1967. Combinatorial Methods in the Theory of Stochastic Processes. John Wiley & Sons, Inc., New York, London, Sydney. Watson, G.N., 1945. A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge. Whittaker, E.T., Watson, G.N., 1963. A Course of Modern Analysis, fourth ed. Cambridge University Press, Cambridge. Widder, D.V., 1947. Advanced Calculus. Prentice-Hall, New York.
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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime
Improved asymptotic upper bounds on the ruin capital in the Lundberg model of risk✩ Vsevolod K. Malinovskii ∗ Central Economics and Mathematics Institute (CEMI) of Russian Academy of Science, 117418, Nakhimovskiy prosp., 47, Moscow, Russia Gubkin Russian State University of Oil and Gas, 119991, Leninsky prosp., 65, Moscow, Russia
article
abstract
info
This paper deals with ruin capital uα,t (c | λ, µ) in the classical Lundberg model of risk. It is defined as the initial capital needed to keep the probability of ruin within finite time t equal to a predefined value α . Considered as a decreasing function of premium rate c, the ruin capital is shown to be convex (i.e., concave downward) for c > λ/µ and t sufficiently large. This observation is used to construct explicit upper bounds on the ruin capital. © 2014 Elsevier B.V. All rights reserved.
Keywords: Lundberg model of risk Ruin capital Asymptotic bounds
1. Introduction and definitions In this paper, we consider the classical collective Lundberg model, where the risk reserve process is R(s) = u + cs − V (s),
s > 0.
Here u > 0 is the initial risk reserve, c > 0 is the premium rate,
N (s) = max n > 0 :
n
Ti 6 s ,
s > 0,
i=1
or 0, if T1 > s, is the claims arrival process, or the number of claims which have occurred up to time s, V ( s) =
N (s)
Yi ,
s > 0,
i=1
or 0, if N (s) = 0 (or T1 > s), is the claims payout process, or the aggregate claims amount payed out up to time s. By Ti , i = 1, 2, . . . , and Yi , i = 1, 2, . . . , we denote mutually independent sequences of i.i.d. exponentially distributed random variables with intensities λ > 0 and µ > 0 respectively. So, the aggregate claims payout process V (s), s > 0, is compound Poisson, starting at zero. The mean and variance for any s are well known to be EV (s) = (λ/µ)s and DV (s) = 2(λ/µ2 )s respectively. Throughout this paper, we assume that 0 < α < 21 . We denote by
ψt (u, c | λ, µ) = P{ inf R(s) < 0} 0
DOI of original article: http://dx.doi.org/10.1016/j.insmatheco.2013.12.005. ✩ This work was supported by RFBR (grant No. 14-06-00017-a).
∗ Correspondence to: Central Economics and Mathematics Institute (CEMI) of Russian Academy of Science, 117418, Nakhimovskiy prosp., 47, Moscow, Russia. E-mail addresses: [email protected], [email protected]. URL: http://www.actlab.ru. 0167-6687/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.insmatheco.2013.12.004
the probability of ruin within finite time t > 0, and by ψ+∞ (u, c | λ, µ) the probability of ultimate ruin. The solution with respect to u of the equation
ψt (u, c | λ, µ) = α
(1.1)
is called α -level initial capital, or α -level ruin capital, or merely ruin capital. We denote it by uα,t (c | λ, µ). Considered as a function of c, it allows us a choice of the initial capital balanced with premiums, when to achieve a given level of non-ruin is a goal. The solution with respect to c of Eq. (1.1) is called the α -level premium rate. We denote it by cα,t (u | λ, µ). It is easy to see that for any c > 0 cα,t (uα,t (c | λ, µ) | λ, µ) = c
(1.2)
and for any u > 0 uα,t (cα,t (u | λ, µ) | λ, µ) = u.
(1.3)
The inverse functions for both uα,t (c | λ, µ) and cα,t (u | λ, µ) −1 exist. From (1.2) and (1.3) we have uα,t (c | λ, µ) = cα, t (c | λ, µ)
1 for any c > 0 and cα,t (u | λ, µ) = u− α,t (u | λ, µ) for any u > 0. So, enough is to focus on the ruin capital uα,t (c | λ, µ). The ruin capital is paramount in modeling solvency. It lays grounds for the investigation of controlled multi-period models of insurance business (see Malinovskii (2012, 2013b, 2014)). As we will find it below, the shape of the ruin capital considered as a function of premium rate c, as t is sufficiently large, is surprisingly simple. It is (see Fig. 1 in Section 2) monotone decreasing, as c increases, and
• almost linear for 0 < c < λ/µ, • convex for c > λ/µ,
302
V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309
Theorem 2.1 (Corollary 2.1 in Malinovskii (1998)). In the classical Lundberg model, we have
ψt (u, c | λ, µ) = ψ+∞ (u, c | λ, µ) −
1
π
π
ft (x, u) dx,
(2.1)
0
where2
λ exp{−u(c µ − λ)/c }, ψ+∞ (u, c | λ, µ) = c µ 1, λ/c µ > 1
λ/c µ < 1,
and ft (x, u) = (λ/c µ)(1 + λ/c µ − 2 λ/c µ cos x)−1
Fig. 1. Ruin capital uα,t (c | λ, µ) as a function of c (X -axis) calculated numerically using Eq. (2.1) (solid line) and the results of simulation (dots) for λ = 4/5, µ = 3/5, α = 0.05, t = 200. Horizontal line: uα,t (λ/µ | λ, µ) = 59.9033. Vertical line: λ/µ = 4/3.
• asymptotically independent on t and close to uα (c | λ, µ) for ∗ ∗ c > cα, t , where cα,t is a certain value larger than λ/µ and uα (c | λ, µ) is a solution (w.r.t. u) of the equation1 ψ +∞ (u, c | λ, µ) = α . This paper is a development of that part of Malinovskii (2013a), which refers to the classical Lundberg model of risk. It is a counterpart of Malinovskii (submitted for publication), where the similar results are obtained in a diffusion model of risk. The extension of the results of this paper to Andersen’s model is still an open and difficult problem. In Malinovskii and Kosova (submitted for publication), simulation studies are conducted in order to see to what extent the ruin capital in Andersen’s model preserves the properties listed above. The rest of the paper is arranged as follows. In Section 2, we write the fundamental formula for ψ t (u, c | λ, µ) and discuss some results for uα,t (c | λ, µ) obtained in Malinovskii (2012, 2013a). Our goal is their further development. In Section 3, we formulate Theorem 3.3 which is the main result of the paper. Its proof is based on the monotony and convexity theorems (i.e., on Theorems 3.1 and 3.2). Further on, we formulate Theorem 3.4 which yields a weaker result than the result of Theorem 3.3, but explicit. Since both Theorems 3.3 and 3.4 give asymptotic bounds on the ruin capital, we discuss how to perform calculations for finite t. We outline a straightforward heuristic method and discuss the way of its justification. Section 4 contains the proof of the monotony theorem in the strong form, i.e., for all t. We mention the relationship between the ruin theory and the theory of random walks with random displacements developed, e.g., in Takács (1967). Section 5 contains the proof of the convexity theorem in the weak form, i.e., for t large. The proofs of the monotony and convexity theorems require a series of calculations involving Bessel functions. In Section 6 two auxiliary results from calculus are presented. 2. Rationale and motivation Apparently different methods lead to the same explicit expression for the probability of ruin ψ t (u, c | λ, µ) in the classical risk model (see, e.g., Prabhu (1965), Chapter 5, Section 5.5, Takács (1967), Chapter 7, Section 35). We refer to Malinovskii (1998), where this expression was obtained as a corollary of a more general result.
1 Since (see Theorem 2.1) ψ +∞ (u, c | λ, µ) = (λ/(c µ)) exp{−u(c µ − λ)/c } for λ/c µ < 1 and 1 elsewhere, the explicit expression for uα (c | λ, µ) is straightforward.
× exp uµ λ/c µ cos x − 1 − t λ(c µ/λ) × 1 + λ/c µ − 2 λ/c µ cos x cos uµ λ/c µ sin x − cos uµ λ/c µ sin x + 2x . The cases when there exist explicit formulas like Eq. (2.1) are exceptional.3 In Malinovskii (1998), Eq. (2.1) was obtained by manipulating a formula which yields the probability of ruin within finite time in a model more general than classical. But even this formula which comprises Laplace transform of the probability of ruin can hardly be used per se to extend the analysis of the solution of Eq. (1.1) to the more general model. All values of c in Eqs. (1.1), (2.1) are of a great interest, including those lying around the critical value4 c = λ/µ. This is so because our investigation lays grounds for a number of other endeavors, e.g., for the development of equitable solvent controls in a multiperiod game model of risk (see Malinovskii (2012)). Being focused on the neighborhoods of c = λ/µ, we find useless the asymptotic results like Cramér’s approximations which are of a simpler form than Eq. (2.1). It refers as well to the refinements of these approximations (see, e.g., Malinovskii (1994, 2000)). Being cumbersome, the explicit formula in Eq. (2.1) can be used for the numerical calculation of uα,t (c | λ, µ). In Fig. 1, the ruin capital uα,t (c | λ, µ) as a function of c calculated numerically is shown by a solid line, while the dotted line is a result of simulation study of uα,t (c | λ, µ). This study in the classical case and in the cases more general than classical is described in detail in Malinovskii and Kosova (submitted for publication). The analytical investigation is quite desirable, though not developed yet in Andersen’s model, and simulation seems to be the only effective method to tackle this problem. To conclude this section, we recall several results on the ruin capital uα,t (c | λ, µ) for time t sufficiently large,5 which will be referred to in the sequel. We write κγ = Φ(−0,11) (1 −γ ) for the (1 −γ )quantile (0 < γ < 1) of the standard normal c.d.f. Φ(0,1) (x). Since 0 < α < 1/2, we have 0 < κα < κα/2 < 1. Theorem 2.2 (Theorem 3.1 in Malinovskii (2012)). In the classical Lundberg model, we have
√ uα,t (λ/µ | λ, µ) =
2t λ
µ
κα/2 (1 + o(1)),
t → ∞.
(2.2)
2 Note that the factor λ/(c µ) is the well-known Lundberg’s constant and the factor (c µ − λ)/c is the well-known Lundberg’s exponent. In the framework of the classical model both are explicit. 3 Another exceptional case is provided by the diffusion model. It is considered in Malinovskii (submitted for publication). 4 The case c = λ/µ is usually referred to as the case of zero safety loading.
5 Bearing in mind that time is operational, large t means large portfolio. We use the ‘‘small o notation’’, i.e., we say f (x) = o(g (x)), as x → a, if limx→a f (x)/g (x) = 0.
V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309
Theorem 2.3 (Theorem 3.2 in Malinovskii (2012)). In the classical Lundberg model, we have uα,t (c | λ, µ)
√ √ µ(λ/µ − c ) t λ 2t λ , √ µ − c t + µ Uα,t 2λ = √ √ µ(λ/µ − c ) t 2t λ U , c > λ/µ, √ α,t µ 2λ
c 6 λ/µ,
where the function Uα,t (v) is continuous and monotone increasing, as v increases from −∞ to 0, and monotone decreasing, as v increases from 0 to +∞, and such that lim Uα,t (v) = 0,
v→+∞
and Uα,t (0) = κα/2 (1 + o(1)), as t → ∞. For brevity, set6 u∗α,t = uα,t (λ/µ | λ, µ). The following theorem is a counterpart of Corollary 3.1 in Malinovskii (2013a). Theorem 2.4. In the classical Lundberg model, as t → ∞, we have for c 6 λ/µ
λ κα ∗ λ −c t + u 6 uα,t (c | λ, µ) 6 − c t + u∗α,t , (2.3) µ κα/2 α,t µ
and7 for c > λ/µ
uα,t (c | λ, µ) 6 min
ln(α c µ/λ) u∗α,t , −
µ − λ/c
.
is a convex function of c > λ/µ, it yields the desired upper bound in the interval between the critical point λ/µ and the point of tangency, where h(c ) is an upper bound for the ruin capital. So, we formulate first the monotony and convexity theorems (i.e., Theorems 3.1 and 3.2). After this, we formulate Theorem 3.3 which is the main result of the paper. We formulate Theorem 3.4 which yields upper bounds rougher than in Theorem 3.3, but written explicitly. Further, we outline a heuristic construction applicable for finite t and discuss the ways of its justification. Theorem 3.1 (The Monotony Theorem). In the classical Lundberg model, for c > λ/µ and for all t > 0, the α -level initial capital uα,t (c | λ, µ) is a monotone decreasing function of c. This theorem is quite intuitive. It was proved in Malinovskii (2008a). To make the presentation self-contained, we return to its proof in Section 4. The following theorem, being not at all intuitive, is fundamental. Its proof is given in Section 5.
lim Uα,t (v) = κα
v→−∞
303
Theorem 3.2 (The Convexity Theorem). In the classical Lundberg model, for c > λ/µ and for t > 0 sufficiently large, the α -level initial capital uα,t (c | λ, µ) is a convex (i.e., concave downward) function of c. ∗ ∗ Let cα, t = cα,t (λ, µ) be a solution (w.r.t. c) of the non-linear equation
λ + cµ µc ln α = µu∗α,t − 1. λ − cµ λ
(3.1)
The following theorem improves the upper bounds (2.4). (2.4)
Proof of Theorem 2.4. The proof of (2.3) is straightforward from Theorem 2.3. For the proof of (2.4), we use the trivial inequality ψt (u, c | λ, µ) 6 ψ+∞ (u, c | λ, µ) for c > λ/µ, and note that ψ+∞ (u, c | λ, µ) = α yields as solution u = − ln(α c µ/λ)/(µ − λ/c ). We strive to improve the upper bound (2.4) in Theorem 2.4. 3. Improved asymptotic upper bounds on the ruin capital Outline the desired improvements in Theorem 2.4. Note that √ the accuracy8 of the bounds in (2.3) is o( t ), as t → ∞. It is straightforward from Eq. (2.2) and, bearing in mind Theorem 2.3, looks satisfactory. In contrast to that, the upper bound in (2.4) performs well only for c much larger than λ/µ since uα,t (c | λ, µ) → − ln(α c µ/λ)/(µ − λ/c ), as t → ∞, but is rather trivial, and in some sense unsatisfactory, for c close to λ/µ. If we can prove that uα,t (c | λ, µ) is a convex (i.e., concave downward) function of c > λ/µ, then we can easily improve the upper bound in (2.4) for c close to λ/µ. We do the following. Instead of the horizontal upper bound u∗α,t in (2.4), we take the tangent line to h(c ) = − ln(α c µ/λ)/(µ − λ/c ) starting from the point with abscissa λ/µ and ordinate u∗α,t . Given that uα,t (c | λ, µ)
√
6 We will always bear in mind that u∗ = ( 2t λ/µ) κ (1 + o(1)), as t → ∞, α/2 α,t by Theorem 2.2. 7 Eq. (3.7) in Corollary 3.1 of Malinovskii (2013a) (cf. Theorem 4.2 in Malinovskii (2013b)) contains a technical deficiency which we eliminate here. 8 The bounds, as well as the ruin capital, are of order O(t ), as t → ∞. By accuracy we√mean the difference between the true value and the bounds, which is of order o( t ), as t → ∞. Normalized by dividing by t, this difference becomes o(t −1/2 ), as t → ∞. It corresponds better to the term accuracy, but we will not move to the relative terms.
Theorem 3.3. In the classical Lundberg model, as t → ∞, we have uα,t (c | λ, µ)
λ − µc ∗ + λ ln(α c ∗ µ/λ) α,t α,t ∗ (c − cα, t) ∗ 2 (λ − µ c ) α, t ∗ ∗ cα,t ln(α cα,t µ/λ) λ ∗ 6 < c 6 cα, + , t, ∗ λ − µ c µ α,t − ln(α c µ/λ) , c > c ∗ . α,t µ − λ/c Proof of Theorem 3.3. Bearing in mind Theorem 3.2 and taking t so large that uα,t (c | λ, µ) is convex, we draw a tangent line l(c ) =
∗ ∗ λ − µcα, t + λ ln(α cα,t µ/λ) ∗ (c − cα, t) ∗ 2 (λ − µcα,t ) ∗ ∗ cα, t ln(α cα,t µ/λ) + ∗ λ − µcα, t
to the upper bound h(c ) = −
ln(α c µ/λ)
µ − λ/c
yielded by the evident inequality ψ t (u, c | λ, µ) 6 ψ +∞ (u, c | λ, µ). This line starts from the point with abscissa 1 and ordinate ∗ u∗α,t and is tangent to h(c ) at the point cα, t defined as the solution of Eq. (3.1) which is equivalent to l(λ/µ) = u∗α,t . ∗ While for c > cα, t the examination of accuracy of the upper bound in Theorem 3.3 consists in the study of the rate of convergence9 uα,t (c | λ, µ) → − ln(α c µ/λ)/(µ − λ/c ), as t → ∞, for
9 This is closely linked to the rate of convergence ψ (u, c | λ, µ) → ψ t +∞ (u, c | λ, µ), as t → ∞. We shall not give here a detailed consideration of this well-known problem.
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V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309
Fig. 2. Ruin capital uα,t (c | λ, µ) as a function of c (X -axis) calculated numerically using Eq. (2.1) (lower solid line for c > λ/µ and middle solid line for c < λ/µ), the results of simulation (dots), the bounds (2.3) of Theorem 2.4, and the bounds of Theorem 3.4 for λ = 4/5, µ = 3/5, α = 0.05, t = 200. Horizontal line: uα,t (λ/µ | λ, µ) = 59.9033. Vertical line: λ/µ = 4/3.
Fig. 3. Shown are the function uα,t (c | λ, µ) of the variable c (X -axis) calculated numerically using Eqs. (1.1) and (2.1), and the bounds of Theorems 3.3 and 3.4 for c > λ/µ, with λ = µ = 1, α = 0.1, t = 100. Here u∗α,t is taking equal to
√ ( 2t λ/µ) κα/2 = 23.2617, i.e., with the term 1 + o(1) replaced by 1. The value of uα,t (λ/µ | λ, µ) calculated numerically is 23.5722.
∗ λ/µ < c 6 cα, t we cannot say much. Being high for c close to λ/µ ∗ and to cα, by construction, it is unknown for c between these valt ues. Numerical examples show however that the accuracy is surprisingly good. The following theorem yields rougher upper bounds than Theorem 3.3. It applies the upper bound on the right hand side of the inequality
−
ln(α c µ/λ)
µ − λ/c
6−
ln α
µ − λ/c
,
true for c < λ/µ. This is done to find a tangent line to h(c ) = − ln α/(µ − λ/c ) explicitly rather than via the solution of Eq. (3.1). Theorem 3.4. In the classical Lundberg model, as t → ∞, we have uα,t (c | λ, µ)
(µu∗ + ln α)2 α,t c 4λ ln α ∗ (µuα,t − ln α)2 λ λ µu∗α,t − ln α − ,
Fig. 4. The functions shown in Fig. 3, but with u∗α,t (ε) taken equal to 24.0722, i.e., with the term 1 + o(1) replaced by 1.0348. Two other horizontal lines are uα,t (λ/µ |
√ λ, µ) calculated numerically, equal to 23.5722, and ( 2t λ/µ) κα/2 = 23.2617.
Fig. 5. The upper bounds of Theorems 3.3 (above) and 3.4 (middle) shown with uα,t (c | λ, µ) calculated numerically (below). This drawing differs from one in Fig. 3 by that the auxiliary lines are removed.
In Fig. 2, which illustrates the main result of the paper, the graphs shown in Fig. 1 are supplemented by the upper and lower bounds, as c < λ/µ, and by the upper bounds, as c > λ/µ. The former are the bounds (2.3) of Theorem 2.4. The latter are the bounds of Theorem 3.4. 3.1. A heuristic construction of the upper bounds Suppose we are given an arbitrary ε > 0. By Theorems 2.2 and 3.2, there exists t (ε) > 0 such that uα,t (c | λ, µ) is a convex function of c (as c > λ/µ) for all t > t (ϵ) and uα,t (λ/µ | λ, µ) 6 √ u∗α,t (ϵ) = ( 2t λ/µ) κα/2 (1 + ε) for all t > t (ϵ). The constructions of Theorems 3.3 and 3.4 are illustrated in Fig. 3 for ε = 0, which is deficient to garner the upper bound uniformly on c > λ/µ, and in Fig. 4 for ε = 0.0348, which yields a valid upper bound uniformly on c > λ/µ, as t = 100. In Figs. 3 and 4 we show the auxiliary lines illustrating the tangent line construction. In Figs. 5 and 6 these auxiliary lines are removed. We call ε -heuristic the upper bounds which are obtained by a formal replacement of o(1) in Eq. (2.2) by an ε > 0. The heuristic consists in the implicit assumption that t (ε) is chosen so large that it makes the bounds in Theorems 2.2 and 3.2 valid uniformly on c > λ/µ for all t > t (ε). In order to specify t (ε) > 0 for a given ε > 0, further insight into the structure of the term o(1) in Theorem 2.2 and into the phrase ‘‘for t > 0 sufficiently large’’ in
V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309
305
and that
x ∂ n+1 vn+1 (x | p) dx = vn+2 (x | p) dx. (4.5) vn+1 (x | p) − ∂p p2 n+2 4.2. First derivative of α -level initial capital Here we will calculate the first derivative of uα,t (c | λ, µ) given implicitly by Eq. (1.1) with respect to the argument c. For x, t > 0 and 0 < p < 12 , introduce
ℵt (x, p) =
(0,1)
( x, p ) , λ Ht(1,0) (x, p) 1 Ht
where Fig. 6. The upper bounds of Theorems 3.3 (above) and 3.4 (middle) shown with uα,t (c | λ, µ) calculated numerically (below). This drawing differs from one in Fig. 4 by that the auxiliary lines are removed.
Theorem 3.2 is necessary.10 It seems that it requires considerable efforts. The same applies to the bounds (2.3) in Theorem 2.4. 4. The proof of monotony Theorem 3.1
t ∞ xn vn+1 (y | p) − vn+2 (y | p) dy, n! 0 n =0 t ∞ xn (0,1) −x Ht ( x, p ) = e y vn+1 (y | p) n! 0 n=0 n+1 vn+2 (y | p) dy. − n+2 (1,0)
Ht
(x, p) = −e−x
In this section, we introduce the key ideas which will be further developed in Section 5 to prove Theorem 3.2. These ideas are essentially the same as in the case of diffusion model of risk considered in Malinovskii (submitted for publication). Basically, it is the standard examination of the sign of the first derivative of a function in order to show its monotony.
Theorem 4.1. For t , λ, µ > 0, 0 < α <
4.1. A formula for ruin probability
ψ(t 0,1) (u, c | λ, µ)|u=uα,t (c |λ,µ) ∂ . uα,t (c | λ, µ) = − (1,0) ∂c ψt (u, c | λ, µ)|u=uα,t (c |λ,µ)
Besides the expression (2.1) for the probability of ruin ψ t (u, c |
λ, µ), we will apply its equivalent,11 ψt (u, c | λ, µ) = Ft λ (x, p)| Ft (x, p) = e−x
x=uµ p=λ/(c µ+λ)
,
t ∞ xn vn+1 (y | p) dy, n! 0 n =0
p
vn+1 (y | p) =
q
Applying (4.5), differentiate (4.1). Since −λµ/(c µ + λ)2 = −µp2 /λ, we have
(4.2)
ψ(t 0,1) (u, c | λ, µ) = −
F∞ (x, p) = e−x
n =0
=
p q
xn
(4.3)
n!
vn+1 (y | p) dy
exp −x 1 −
p q
,
d dc
(λ/(c µ + λ)) =
µ 2 (0,1) p Ft λ (x, p)| x=uµ p=λ/(c µ+λ) λ
(1,0)
x=uµ p=λ/(c µ+λ)
(1,0)
.
(0,1)
(0,1)
Since Ft (x, p) = Ht (x, p) and p2 Ft (x, p) = Ht (x, p), which is checked straightforwardly, the proof is complete.
∂ uα,t (c | λ, µ) < 0. ∂c
0
=
Theorem 4.2 (Weak form Derivative Inequality). For λ, µ > 0, 0 < α < 21 , c > λ/µ and for t > 0 sufficiently large, we have
∞
d p dc
and
Note that ψ ∞ (u, c | λ, µ) = F∞ (x, p)| x=uµ , where14 p=λ/(c µ+λ) ∞
Proof of Theorem 4.1. Bearing in mind Eq. (1.1), apply the first part of Theorem 6.1 which yields15
ψ(t 1,0) (u, c | λ, µ) = µFt(λ1,0) (x, p)| n + 1 −y/p e In+1 (2y q/p). y
and c > λ/µ, we have
∂ uα,t (c | λ, µ) = ℵt λ (uµ, λ/(c µ + λ))|u=uα,t (c |λ,µ) . ∂c
(4.1)
where12 q = 1 − p and13
(n+1)/2
1 2
(4.4)
Proof. Using Eq. (4.4), for x > 0 and 0 < p < (1,0) H∞ ( x, p ) = −
10 This situation is typical for asymptotic analysis, unless the remainder term is presented in an explicit form or estimated by an explicitly given expression. 11 See, e.g., Eq. (13), or Eqs. (17)–(20) in Malinovskii (2008a). See also Malinovskii (2008b). 12 Since in (4.1) we set p = λ/(c µ + λ) and since we will be interested in the case c > λ/µ, we will be concentrated on the case 0 < p < 12 . 13 By I (·), k = 1, 2, . . . , we denote the Bessel functions (see Section 6.2). k
14 It follows from Theorem 6.2 and yields the well-known formula for the probability of ultimate ruin.
(4.6)
(0,1) H∞ ( x, p )
=
p(q − p) q2
p2 (q + px) q3
exp −x 1 −
exp −x 1 −
p q p q
1 2
we have
, .
15 Here and in what follows, the symbols f (i,j) (x , x ) denote ith and jth 1 2 partial derivatives of f (x1 , x2 ) with respect to the first and the second variables, respectively.
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V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309
Apply Theorem 4.1 and note that
ℵt (x, p) = ℵ∞ (x, p)(1 + o(1)),
Theorem 4.6 (Eq. (9) in Section 22 of Takács (1967)). In the random walk process model,
t → ∞,
where
ℵ∞ (x, p) =
1
λ
(0,1) H∞ (x, p) (1,0) H∞ (x, p)
The proof is complete.
=−
1 p(q + px)
λ q(q − p)
P{ sup ξp (z ) < k} = P{ξp (y) < k} − 06z 6y
< 0.
(x, p) < 0,
(0,1)
Ht
1 , 2
k(p/q) (4.7)
∂ uα,t (c | λ, µ) < 0. ∂c
Theorem 4.3 was proved in Malinovskii (2008a). It may be proved analytically, using the properties of Bessel functions, or connecting our problem with the following construction. Suppose that a particle performs a random walk on the X -axis. Starting at the origin, in each step the particle moves either a unit distance to the right with probability p or a unit distance to the left with probability q (p + q = 1, 0 < p < 1). Suppose that the displacements of the particle occur at random times in the time interval (0, ∞). Denote by ν(z ) the number of steps taken in the interval (0, z ]. We suppose that {ν(z ), 0 6 z < ∞} is a Poisson process of density 1/p and that the successive displacements are independent of each other and independent of the process {ν(z ), 0 6 z < ∞}. Denote by ξp (z ) the position of the particle at time z. In this case {ξp (z ), 0 6 z < ∞} is a stochastic process having stationary independent increments, P{ξp (0) = 0} = 1 and almost all sample functions of {ξp (z ), 0 6 z < ∞} are step functions having jumps of magnitude 1 and −1. Theorem 4.5 (Eqs. (3), (8) in Section 22 of Takács (1967)). In the random walk process model, P{ξp (z ) = k} = e−z /p (p/q)k/2 Ik (2z
q/p) =
z > 0,
z k
vk (z | p),
P{ sup ξp (z ) < k} = 1 − k(p/q)k/2 06z 6y
y
e−z /p Ik (2z
q/p)
0
dz z
dz
q/p)
vk (z | p)dz ,
=1−
dz z
= e−y/p
∞ (p/q)i/2 Ii (2y q/p) i =k
(q/p)i/2−k Ii (2y q/p).
(4.8)
i=k+1
Denote by ςk (p) the first hitting time of the point k by the random walk process with the parameter p ∈ (0, 1). Evidently,18 y
vk (z | p) dz = P{ςk (p) 6 y},
(4.9)
0
so that vk (z | p), z > 0, is the p.d.f. of the first hitting time of the point k in the random walk process model (see also Feller (1971), Chapter II, Section 7 and Chapter XIV, Section 6). The densities vk (z | p), z > 0, are defective for p < 1/2 (random walk with drift to the left) and proper for p > 1/2 (random walk without drift, or with drift to the right). For any t > 0 and n = 0, 1, 2, . . . , we have from Theorem 4.5 and Eq. (4.9) P{ξp (y) = n + 2} =
y n+2
P{ sup ξp (z ) = n + 1} =
vn+2 (y | p), y
06z 6y
(4.10)
vn+1 (z | p)dz 0 y
vn+2 (z | p)dz .
−
(4.11)
0
Bearing in mind Eq. (4.11), the following theorem yields the first inequality (4.7). Theorem 4.7. For 0 < p <
1 2
, y > 0 and n = 0, 1, 2, . . . , we have
P{ sup ξp (z ) = n + 1} 06z 6y
= P{ςn+1 (p) 6 y} − P{ςn+2 (p) 6 y} > 0.
(4.12)
Bearing in mind Eqs. (4.10) and (4.11), the following theorem yields the second inequality (4.7).
z
y
y > 0,
Theorem 4.8 (Theorem 3.2 in Section 3.4 of Malinovskii (2008a)). For 0 < p < 12 , y > 0 and n = 0, 1, 2, . . . , we have
0
P{ sup ξp (z ) = n + 1} − P{ξp (y) = n + 2} > 0.
for integer k > 0.
(4.13)
06z 6y
16 Recall it: if the first derivative of a function is negative at all points of an interval, then this function is monotonically decreasing on this interval. 17 Note that Eξ (y) = [1 − (q/p)]y, Dξ (y) = y/p, y > 0. p
Proof of Theorem 4.7. It is straightforward from Eqs. (4.9) and (4.11).
for k = 0, ±1, ±2, . . . , and17
0
+ e−y/p
∞
06z 6y
4.3. Proof of Theorem 4.3 and random walks with random displacements
P{ξp (z ) = k}
e−z /p Ik (2z
P{ sup ξp (z ) > k} =
The monotony theorem, i.e., Theorem 3.1, is straightforward from Theorem 4.4 by the well-known criterion.16
y
y
0
(x, p) > 0.
Theorem 4.4 (Strong form Derivative Inequality). For λ, µ > 0, 0 < α < 21 , c > λ/µ and for all t > 0, we have
=1−k
y > 0,
Bearing in mind Theorem 4.5, from Theorem 4.6 we have k/2
we have
Theorem 4.3 proved in Section 4.3 implies the next enhancement of Theorem 4.2, true for all t > 0.
q
for integer k > 0.
Theorem 4.3. For x, t > 0 and 0 < p < (1,0)
p
× P{ξp (y) < −k},
We can refine Theorem 4.2 and prove that the inequality (4.6) holds true for all t > 0. For this, we need the following result.
Ht
k
p
18 By Eq. (4.1), we have ψ (u, c | λ, µ) = e−x ∞ xn P{ς x=uµ . n+1 (p) 6 t λ}| t n= 0 n ! p=λ/(c µ+λ)
V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309
Proof of Theorem 4.8. By ω we mean sample functions of {ξp (z ), 0 6 z < ∞} correspondence between the . Establish one-to-one sets A1 = ω : ξp (y) = n + 2 and A2 = ω : sup06z 6y ξp (z ) =
n + 1, ξp (y) = n . For each ω = {ξp (z ), 0 6 z 6 y} ∈ A1 , reflect with respect to the level n + 1 that part of ω, which lies above the level n + 1. In this way, for each ω ∈ A1 we obtain the unique ω′ = {ξp′ (z ), 0 6 z 6 y}, such that sup06z 6y ξp′ (z ) 6 n + 1 and ξp′ (z ) = n. Furthermore, we have
(0,2)
Ht
(x, p) = e−x
A2 = ω : sup ξp (z ) = n + 1, ξp (y) = n
06z 6y
+
∞ xn n! n =0
n+1 n+3
06z 6y
Obviously, this implication is strict. Indeed, the set A3 contains, e.g. all trajectories of the random walk process which have only one hitting of the point n + 1 within the interval [0, y]. Such trajectories are not produced by the reflection described above since those obtained in such a way must hit the point n + 1 at least twice. We have therefore19 P(A1 ) = P(A2 ) < P(A3 ), and the proof is complete.
5.1. Second derivative of the α -level initial capital Here we will calculate the second derivative of uα,t (c | λ, µ) given implicitly by Eq. (1.1) with respect to the argument c. For x, t > 0 and 0 < p < 12 , introduce
×
λ2 Ht(1,0) (x, p) (0,1)
(x, p)
(1,0)
(x, p)
Ht Ht
(0,2)
Ht
(2,0)
+ Ht
( x, p )
(0,1)
(x, p)
(1,0)
( x, p )
Ht Ht
2 ,
t ∞ xn vn+1 (y | p) − 2vn+2 (y | p) n! 0 n =0 + vn+3 (y | p) dy, ∞ xn (1,1) Ht (x, p) = −e−x n! n =0 t n+1 × y vn+1 (y | p) − vn+2 (y | p) n+2 0 n+2 − vn+2 (y | p) − vn+3 (y | p) dy, n+3 (2,0)
(x, p) = e−x
19 The ‘‘infinitesimal probability masses’’ of the original sample path and of its twin obtained by reflection are the same. 20 The basic technical instruments in Malinovskii (submitted for publication) were the inequalities for Mill’s ratio.
n+2
vn+2 (y | p)
vn+3 (y | p) dy.
1 2
and c > λ/µ, we have
∂2 uα,t (c | λ, µ) = i t λ (uµ, λ/(c µ + λ))|u=uα,t (c |λ,µ) . ∂ c2
∂2 u ( c | λ, µ) = − ψ(t 0,2) (u, c | λ, µ)|u=uα,t (c |λ,µ) α, t ∂ c2 2 × ψ(t 1,0) (u, c | λ, µ)|u=uα,t (c |λ,µ) − 2ψ(t 1,1) (u, c | λ, µ)|u=uα,t (c |λ,µ) × ψ(t 1,0) (u, c | λ, µ)|u=uα,t (c |λ,µ) + ψ(t 2,0) (u, c | λ, µ)|u=uα,t (c |λ,µ) 2 × ψ(t 0,1) (u, c | λ, µ)|u=uα,t (c |λ,µ) −3 × ψ(t 1,0) (u, c | λ, µ)|u=uα,t (c |λ,µ) . Applying (4.5), differentiate (4.1). Since
d2 p dc 2
d2 dc 2
=
(2λµ2 )/(c µ + λ)3 = 2µ2 p3 /λ2 , we have ψ(t 0,2) (u, c | λ, µ) =
µ2 4 (0,2) p Ft λ (x, p) λ2 + 2p3 Ft(λ0,1) (x, p) |
x=uµ p=λ/(c µ+λ)
(λ/(c µ + λ)) =
,
µ2 2 (1,1) p Ft λ (x, p) | x=uµ , p=λ/(c µ+λ) λ (2,0) 2 (2,0) ψt (u, c | λ, µ) = µ Ft λ (x, p)| x=uµ . ψ(t 1,1) (u, c | λ, µ) = −
(0,1)
(0,2) x p p4 Ft (1,1) x p Ht
(2,0)
(2,0)
where Ht
0
n+1
p=λ/(c µ+λ)
(x, p) − 2Ht(1,1) (x, p)
y2 vn+1 (y | p) − 2
× ψ(t 0,1) (u, c | λ, µ)|u=uα,t (c |λ,µ)
In this section, we prove the convexity theorem, though only for t large, i.e., in the weak form. It is noteworthy that in Malinovskii (submitted for publication), in the case of diffusion model of risk, this convexity theorem was proved for all t, i.e., in the strong form.20 The extension of Theorem 3.2 to all t is still an open problem. It seems to us difficult and its solution is quite desirable. Basically, the proof below is a standard examination of the sign of the second derivative of a function in order to show its convexity.
i t (x, p) = −
t
Theorem 5.1. For t , λ, µ > 0, 0 < α <
5. The proof of convexity Theorem 3.2
Proof of Theorem 5.1. Bearing in mind Eq. (1.1), apply the second part of Theorem 6.1 which yields
⊂ A3 = ω : sup ξp (z ) = n + 1 .
µ
307
( , ) + 2p3 Ft (x, p) = (x, p) = Ht (x, p), Since Ft (0,2) 2 (1,1) ( , ), which is checked (x, p), and p Ft (x, p) = Ht straightforwardly, the proof is complete. Theorem 5.2 (Weak form Derivative Inequality). For λ, µ > 0, 0 < α < 12 , c > λ/µ and for t > 0 sufficiently large, we have
∂2 uα,t (c | λ, µ) > 0. ∂ c2 Proof. Using Eq. (4.4), for x > 0 and 0 < p < (2,0) F∞ (x, p) =
(1,1) F∞ (x, p)
(0,2) F∞ (x, p)
=
=
p(q − p)2
p
exp −x 1 −
q3
qp − (q − p)(q + px) q4
,
q
exp −x 1 −
=
p(p − q)2 q3
p
exp −x 1 −
exp −x 1 −
p q
,
,
q
It follows that (2,0) H∞ ( x, p )
we have
2q2 + 2(1 + p)qx + px2 q5
1 2
p q
.
308
V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309
(1,1) H∞ ( x, p )
= p
2
qp + (p − q)(q + px)
Theorem 6.1. Let F be a function that possesses partial derivatives up to second order continuous in some neighborhood of some solution, (x0 , y0 ), of the equation F (x, y) = 0. If F2 (x0 , y0 ) ̸= 0, there are an ϵ > 0 and a unique continuously differentiable function φ such that φ(x0 ) = y0 and F (x, φ(x)) = 0 for |x − x0 | < 0. Moreover, when |x − x0 | < 0, we have
q4
p
× exp −x 1 − (0,2) H∞ ( x, p ) = p 3
,
q
q(q + 2px) + (q + px)2
−1 φ ′ (x) = −F (1,0) (x, φ(x)) F (0,1) (x, φ(x))
q5
p
× exp −x 1 −
q
(0,1)
Bearing in mind that
H∞ (x, p)
=−
(1,0) H∞ (x, p)
and
.
φ ′′ (x) = − F (2,0) (x, φ(x))(F (0,1) (x, φ(x)))2
p(q + px) q(q − p)
− 2F (1,1) (x, φ(x))F (1,0) (x, φ(x))F (0,1) (x, φ(x)) −1 + F (0,2) (x, φ(x))(F (1,0) (x, φ(x)))2 (F (0,1) (x, φ(x)))3 .
, we have
(0,1)
H ∞ ( x, p ) (0,2) (1,1) H∞ (x, p) − 2H∞ (x, p) (1,0) H ∞ ( x, p )
(2,0) + H∞ (x, p)
=
p3 (1 + 2px) q 3 ( q − p)
(0,1)
H∞ (x, p)
6.2. Bessel functions
2
(1,0)
For x > 0 and n = 0, 1, 2, . . . , the modified Bessel function of n-th order (see, e.g., Watson (1945), or Whittaker and Watson (1963), Chapter XVII, 17.7) is
H∞ (x, p)
exp −x 1 −
p q
.
In (x) =
Apply Theorem 5.1 and note that
i t (x, p) = i∞ (x, p)(1 + o(1)),
×
k! (n + k)!
2
e−bx In (ax)x−1 dx = 0
(0,1)
H∞ (x, p) (1,1) H (0,2) (x, p) − 2H∞ (x, p) (1,0) ∞ H∞ (x, p)
(2,0) + H∞ (x, p)
∞
(1,0) λ2 H∞ (x, p)
2 (0,1) H ( x, p )
∞ (1,0) H∞ (x, p)
µ p2 (1 + 2px) > 0. λ2 q(q − p)2
The proof is complete.
k =0
x n+2k
.
(6.1)
Theorem 6.2. For b > a > 0 and n = 1, 2, . . . we have
µ
=
1
t → ∞,
where
i∞ (x, p) = −
∞
The convexity theorem, i.e., Theorem 3.2, is straightforward from Theorem 5.2 by the well-known criterion.21 6. Auxiliary results For convenience, we recall here the definition of Bessel functions used in (4.1) and formulate the theorem about differentiation of implicit functions. This is well known in calculus. We recall also a result used to check the second equality in (4.4). 6.1. Differentiation of implicit functions The following implicit function theorem is well known in analysis (see e.g. Chapter I, Sections 5.2 and 5.3 in Widder (1947)).
21 Recall it: if the second derivative of a function is positive at all points of an interval, then this function is strictly convex on this interval.
∞
0
an
n( b +
√
b2 − a2 )n
,
an e−bx In (ax) dx = √ . √ b2 − a2 (b + b2 − a2 )n
Proof. Since In (z ) = exp(−nπ i/2)Jn (iz ), these equalities follow respectively from Eqs. (6.611.4) and (6.623.3) in Gradshtein and Ryzhik (1980). One can also refer to Eqs. (7) and (8) from Section 13.2, Section XIII of Watson (1945). Acknowledgments The author is grateful to the anonymous referee for the comments which helped to improve this paper. References Feller, W., 1971. An Introduction to Probability Theory and its Applications. Vol. II, second ed. John Wiley & Sons, New York. Gradshtein, I.S., Ryzhik, I.M., 1980. Table of Integrals, Series, and Products. Academic Press, New York. Malinovskii, V.K., 1994. Corrected normal approximation for the probability of ruin within finite time. Scand. Actuar. J. 161–174. Malinovskii, V.K., 1998. Non-Poissonian claims’ arrivals and calculation of the probability of ruin. Insurance Math. Econom. 22, 123–138. Malinovskii, V.K., 2000. Probabilities of ruin when the safety loading tends to zero. Adv. Appl. Probab. 32, 885–923. Malinovskii, V.K., 2008a. Adaptive control strategies and dependence of finite time ruin on the premium loading. Insurance Math. Econom. 42, 81–94. Malinovskii, V.K., 2008b. Risk theory insight into a zone-adaptive control strategy. Insurance Math. Econom. 42, 656–667. Malinovskii, V.K., 2012. Equitable solvent controls in a multi-period game model of risk. Insurance Math. Econom. 51, 599–616. Malinovskii, V.K., 2013a. Level premium rates as a function of initial capital. Insurance Math. Econom. 52, 370–380.
V.K. Malinovskii / Insurance: Mathematics and Economics 55 (2014) 301–309 Malinovskii, V.K., 2013b. Rationale of underwriters’ pricing conduct on competitive insurance market. Insurance Math. Econom. 53, 325–333. Malinovskii, V.K., 2014. Annual intrinsic value of a company in a competitive insurance market. Insurance Math. Econom. 55, 310–318. Malinovskii, V.K., 2013. Improved upper bounds on level values in a diffusion model of risk. Ann. Actuarial Sci. (submitted for publication). Malinovskii, V.K., Kosova, K.O., 2013. Simulation analysis of ruin capital in Andersen’s model of risk. Insurance Math. Econom. (submitted for publication).
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Prabhu, N.U., 1965. Queues and Inventories. John Wiley & Sons, Inc., New York, London. Takács, L., 1967. Combinatorial Methods in the Theory of Stochastic Processes. John Wiley & Sons, Inc., New York, London, Sydney. Watson, G.N., 1945. A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge. Whittaker, E.T., Watson, G.N., 1963. A Course of Modern Analysis, fourth ed. Cambridge University Press, Cambridge. Widder, D.V., 1947. Advanced Calculus. Prentice-Hall, New York.