On approximations for the distribution of the time of first level crossing
Vsevolod Malinovskii


Vsevolod K. Malinovskii]Vsevolod K. Malinovskii



Central Economics and Mathematics Institute (CEMI) of Russian Academy of Science,
117418, Nakhimovskiy prosp., 47, Moscow, Russia

Vsevolod.Malinovskii@mail.ru, admin@actlab.ru

ON THE TIME OF FIRST LEVEL CROSSING]ON APPROXIMATIONS FOR THE DISTRIBUTION OF
THE TIME OF FIRST LEVEL CROSSING

[
    [
    December 30, 2023
=====================







§ INTRODUCTION



This paper is an overview of the classical level crossing problem which is
studied extensively in the literature and is fundamental in many branches of
applied probability. We discuss a number of approximations with an emphasis on
their performance, methods of justification and technical conditions which are
required in these methods, including a new approximation called “inverse
Gaussian”. It is derived by a new method, is fruitful for solving related
problems, and is valid under mild regularity conditions. We emphasize its
novelty and boons.




 §.§ Level crossing by a diffusion process



Let s= s+σs, s>0, be a diffusion process with drift
coefficient >0 and diffusion coefficient σ>0, and let c be a
positive constant. The random variable

    u,c=inf{s>0:(-c)s+σs>u},

or +∞, if (-c)s+σs⩽ u for all s>0, is the
first passage time to level u>0 of the shifted diffusion process
s-cs, s>0. It is well known that with 01(x) denoting
compound distribution function (c.d.f.) of a standard Gaussian distribution, we
have

    {u,c⩽ t}={sup_0⩽ s⩽
    t((-c)s+σs)>u}
    
    =1-01(u-(-c)t/σ√(t))
    +exp{2(-c)
    u/σ^2} 01(-u-(-c)
    t/σ√(t)),

or

    {u,c⩽ t}=
    F(t;,,-12) |_=u/-c,=u^2/σ^2,   0<c⩽=,
    
    exp{-2} F(t;,,12) |_=-u/-c,=u^2/σ^2,   c>=,

where

    F(x;,,-12)
    =01(√(/x)(x/-1))
    +exp{2/} 01(-√(/x)(x/+1))

denotes c.d.f. of (proper) inverse Gaussian distribution with parameters
>0 and >0. It is “inverse” in that sense that while the
Gaussian distribution refers to a Brownian motion's position at a fixed time,
the inverse Gaussian distribution refers to the time a diffusion process takes
to reach a fixed level.




 §.§ Level crossing by a compound renewal process



Let us denote by f_1(t) and f_T(t) the probability density
functions (p.d.f.) of a positive random variable 1 and of a set of
positive random variables id=T, i=2,3,…, all
distributed identically. Introducing compound renewal process, the random
variable 1 is the time interval between starting time zero and time of
the first renewal, and the random variables i are the inter-renewal
times. The distribution of 1 may be different from the distribution of
T.

By f_Y(t) we denote p.d.f. of positive random variables
id=Y, i=1,2,…, all distributed identically. The random
variables i are sizes of jumps which occur only in the moments of
renewals. Throughout the entire presentation, p.d.f. f_T(y) and f_Y(y)
are assumed bounded from above by a finite constant.

Having assumed that 1, i.i.d. id=T, i=2,3,…,
i.i.d. id=Y, i=1,2,…, are all mutually independent, we
are within the renewal model, where compound renewal process with time
s⩾ 0 is s=∑_i=1^si, or 0, if
s=0 (or 1>s), where
s=max{n>0:∑_i=1^ni⩽ s}, or 0, if
1>s. The random variable

    u,c=inf{s>0:s-cs>u},

or +∞, as s-cs⩽ u for all s>0, is the first passage
time to level u>0 of the process s-cs, s>0.

Let us denote by {u,c⩽ t|1=v} the distribution of
u,c conditioned by 1=v. It is easily seen that for 0<v<t

    {u,c⩽ t}=∫_0^t{u+cv-Y<0} f_1(v)dv
    
    
    +∫_0^t{v<u,c⩽ t|1=v} f_1(v)dv.

Furthermore (see Theorem 2.1 in <cit.> and references therein),

    {v<u,c⩽
    t|1=v}=∫_v^tu+cvu+cz∑_n=1^∞{M(u+cz)=n} f_T^*n(z-v)dz,

where M(x)=inf{k⩾ 1:∑_i=1^kY_i>x}-1.




§ EXPLICIT SOLUTIONS IN LEVEL CROSSING



Similarly to diffusion set-up, in the compound renewal framework with T, Y
exponential with parameters , , the distribution of
u,c may be written explicitly. Assuming that 1d=T
and writing
n(x)=∑_k=0^∞1/k! (n+k)! (x/2)^n+2k
for the modified Bessel function of the first kind of order n (see, e.g.,
<cit.>), we have the following equivalent formulas
(<ref>)–(<ref>) which all may be derived[See,
in particular, Theorem 2.2 in <cit.>. The proof of equivalence of Type I
and Type II formulas applies Lommel's formula.] from (<ref>) and
(<ref>): Type I formula

    {u,c⩽
    t}=e^-u∫_0^te^-(
    c+)x(0(2√( x(cx+u)))
    
    
    -cx/cx+u2(2√( x(cx+u)))) dx,

Type II formula

    {u,c⩽
    t}=e^-u√()/√(c)∫_0^c
    te^-(1+/(c))x
    ×∑_n=0^∞u^n/n!(/c)^n/2n+1/xn+1(2x√(/(c)) ) dx,

and Type III formula

    {u,c⩽
    t}={u,c<∞}-1/π∫_0^πf(x,u,t) dx,

where

    {u,c<∞}=cexp{-u(c-)/c},   /(c)<1,
    
    
    1,   /(c)⩾ 1,

and

    f(x,u,t)   =(/(c))(1+/(c)-2√(/(c))cos
    x)^-1
       ×exp{u(√(/(c))cos x-1)-t
    (c/)(1+/(c)-2√(/(c))cos
    x)}
       ×(cos(u√(/(c))sin x)
    -cos(u√(/(c))sin x+2x)).


The equivalent Type I–Type III explicit formulas
(<ref>)–(<ref>), as well as equation (<ref>)
in diffusion set-up, may seem very cumbersome and non-informative for intuitive
understanding of the level crossing phenomenon. However, the formulas
(<ref>)–(<ref>) do comply with the following observation
done in <cit.>, Ch. II,  7:

surprisingly many explicit solutions in diffusion theory, queuing
theory, and other applications involve Bessel functions. It is usually far from
obvious that the solutions represent probability distributions, and the
analytic theory required to derive their Laplace transforms and other relations
is rather complex. Fortunately, the distributions in question (and many more)
may be obtained by simple randomization procedures. In this way many relations
lose their accidental character, and much hard analysis can be avoided.

To clarify this idea, let us show (see as well <cit.>) that Type II formula connects the problem of level crossing with random
walk with random displacements.

Recall that the random walk with random displacements (see, e.g., <cit.>, Chapter 4,  22) is defined as follows. Suppose that a particle
performs a random walkProcess!random walk!with random displacements 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⩽ z<∞} is a Poisson processProcess!Poisson of
density 1/p and that the successive displacements are independent of each
other and independent of the process {ν(z),0⩽ z<∞}. Denote
by ξ_p(z) the position of the particle at time z. In this case
{ξ_p(z),0⩽ z<∞} is a stochastic process having stationary
independent increments, {ξ_p(0)=0}=1 and almost all sample functions
of {ξ_p(z),0⩽ z<∞} are step functions having jumps of
magnitude 1 and -1.

In this random walk model, for y>0 and k=0,± 1,± 2,…, we have (see
equality (3) in <cit.>)

    {ξ_p(y)=k}   =e^-y/p(p/q)^k/2k(2y√(q/p) )
    
       =y/k k(y| p),

where k(y|
p)=(pq)^k/2ky e^-y/pk(2y√(q/p) ).
Direct calculation yields

    ξ_p(y)=(1-(q/p))y,  ξ_p(y)=y/p.

For y>0 and integer k>0, we have (see equalities (8) and (9) in
<cit.>)

    {sup_0⩽ z⩽
    yξ_p(z)<k}   ={ξ_p(y)<k}-(p/q)^k {ξ_p(y)<-k}
       =1-k(p/q)^k/2∫_0^ye^-z/pk(2z√(q/p) ) dz/z
       =1-k∫_0^y{ξ_p(z)=k} dz/z=1-∫_0^yk(z|
    p) dz.

Manipulating with (<ref>) and (<ref>), we have the
equality

    k(p/q)^k/2∫_0^ye^-z/pk(2z√(q/p) ) dz/z
    =e^-y/p∑_i=k^∞(p/q)^i/2i(2y√(q/p) )
    
    
    +e^-y/p∑_i=k+1^∞(q/p)^i/2-ki(2y√(q/p) ),

which proof by the methods of Bessel functions is not at all simple.

Denoting by ς_k(p) the first hitting time of the point k by the
random walk with p∈(0,1) and bearing in mind (<ref>), we have

    {ς_k(p)⩽ y}={sup_0⩽ z⩽
    yξ_p(z)⩾ k}=∫_0^yk(z| p) dz.

Thus, the function k(y| p) introduced in (<ref>) has a
clear probabilistic meaning. It is a probability density function of
ς_k(p), i.e., of the first hitting time of the point k in the
model of random walk with random displacements. It is not a surprise that the
density k(y| p) is defective[See Chapter II, Section 7 and
Chapter XIV, Section 6 in <cit.>.] for p<1/2, i.e., when the
random walk drifts to the left, and proper for p⩾ 1/2, i.e., when the
drift is absent or to the right.

Making the change of variables y=x/(c) in Type II formula
(<ref>), we have

    {u,c⩽
    t}   =e^-u∑_n=0^∞(u)^n/n !∫_0^t(/c)^(n+1)/2n+1/y
    e^-y(c+)/n+1(2y√((c)/)) dy
    
       =e^-u∑_n=0^∞(u)^n/n !∫_0^tn+1(y| p) |_p =/(c+) dy
    
       =e^-u∑_n=0^∞(u)^n/n !{ς_n+1(p)⩽
    t} |_p =/(c+),

which makes a link between the random walk with random displacements and the
problem of level crossing evident.

It is noteworthy that under standard assumptions in the renewal model with T
and Y exponential we have

    {u,c<∞}=
    1,   0<c⩽/,
    
    
    (/(c))exp{-u (1-/(c))},   c>/,

which follows from

    ∫_0^∞n+1(y| p) dy=
    1,   p⩾ 1/2,
    (p/q)^n+1,   p<1/2

and
e^-x∑_n=0^∞x^n/n !(p/q)^n+1=(p/q)exp{-x (1-(p/q))}.




§ APPROXIMATIONS IN LEVEL CROSSING



Among many types of approximations for {u,c⩽ t}, the
most significant[Many other approximations, including Arfwedson's
saddlepoint approximation (see <cit.>, p. 133) are
set aside, but may be also discussed in the similar way.] are “normal” and
“diffusion” which refinement is “corrected diffusion” (see
<cit.>, pp. 37 and 42, and <cit.>). The
former name emphasizes the structure of the approximation, while the latter
name emphasizes the method of its construction.




 §.§ “Normal”



Assuming that 1d=T and writing X=Y-cT, let us introduce

    =T/X,  ^2=(X T-T X)^2/(X)^3.

For 0<c<=Y/T, or for  X=Y-c T>0, we have
>0.


In the renewal model with  0<c<, we assume that 0<^2<∞.
Then

    sup_t>0 | {u,c⩽ t}- u^2
    u(t) | =o(1),   u→∞.

If, in addition, (Y^3)<∞ and (T^3)<∞, then this supremum is
O(u^-1/2), as u→∞.


Let us assume that a positive solution  of the equation
e^rX=1 exists. This assumption entails that the tail of the random
variable Y must decrease exponentially fast[This clue assumption
does not allow extension of this approach to Y with power tail.]. Let us
introduce the associated random variables X, T which joint
distribution is F_XT(dz,dw)=e^ zF_XT(dz,dw), and

    =T/X,
      ^2=(XT-TX)^2/(X)^3,
    
    
    C=1/ Xexp{-∑^∞_n=11/n {n>0}-∑^∞_n=11/n {S_n⩽
    0}},

where S_n=∑_i=1^nX_i, n=∑_i=1^ni. For
c>, we have X>0 and >0.


In the renewal model with  c>, we assume that 0<^2<∞. Then
for C>0 defined in (<ref>)

    sup_t>0 | e^ u {u,c⩽
    t}-C  u^2 u (t) | =o(1),  
    u→∞.

If, in addition, (Y^3)<∞ and (T^3)<∞, then this supremum is
O(u^-1/2), as u→∞.


Developing the classical results by Lundberg and Cramér, the approximation
(<ref>) was first obtained by Segerdahl <cit.>. There
exist different approaches to the proof (see, e.g., Siegmund <cit.>, von Bahr <cit.>). Refinements in terms of Edgeworth expansions
were first discussed[The formula (4.7) in <cit.> which
“invokes the higher cumulants of τ [i.e., time to ruin –
V.M.]… and uses an Edgeworth expansion … to produce
correction terms”, and which was put forth as “first-order correction, which
suggests a formula of the type (4.7)” differs from rigorously constructed
Edgeworth expansion (see <cit.>); the method of proof in
<cit.> is essentially similar to the method in <cit.>,
<cit.>.] in Asmussen <cit.> and then proved in Malinovskii
<cit.>.

For Y and T exponential with parameters >0 and >0
respectively, easy calculation yields =/,

    C   =/(c),      =(1-/(c)),
    
       =-1/c(1-/(c)),   ^2   =-2(/(c))^2+(1-/(c))^2/
    c(1-/(c))^3,
    
       =(/(c)/c(1-/(c)),   ^2   =2(/(c))/c^2(1-/(c))^3.

The following result follows straightforwardly from Theorems <ref>
and <ref>.






In the renewal model with Y and T exponential with parameters >0
and >0, for 0<c<=/ we have

    sup_t>0 | {u,c⩽ t}-
    u^2 u(t) | =o(1),   u→∞,

and for c> we have

    sup_t>0 | e^ u {u,c⩽
    t}-C  u^2 u (t) | =o(1),  
    u→∞,

with >0, ^2>0, 0<C<1, >0, >0,
^2>0 defined in (<ref>).


The case c= is excluded from consideration in both
Theorems <ref> and <ref>. Poor performance of the
“normal” approximation around  is illustrated in Figs. <ref>
and <ref>.

In applications, the case c= is known under different names. Related is
the term “heavy traffic”. It comes from queueing theory, but has an obvious
interpretation also in risk theory: on the average, the premiums cs within
time s>0 exceed only slightly the expected claims (Y/T)s within the
same time s>0. That is, heavy traffic conditions mean that the safety loading
τ=(T/Y)c-1 is positive but small.

The approximation of Theorem <ref> when positive τ depends on
u and tends to zero, as u→∞, was investigated in <cit.>. In <cit.>, further insight into performance
of Theorems <ref> and <ref>, with rationale of its
poorness in the vicinity of , may be found.




 §.§ “Diffusion”



The idea behind the “diffusion” approximation is to first approximate the
claim surplus process by a Brownian motion with drift by matching the two first
moments, and next to note that such an approximation in particular implies that
the first passage probabilities are close[This and three following
paragraphs are quotations from <cit.>.]
(<cit.>, p. 136).

The idea behind the simple “diffusion” approximation is to replace the risk
process by a Brownian motion (by fitting the two first moments) and use the
Brownian first passage probabilities as approximation for the ruin
probabilities. Since Brownian motion is skip-free, this idea ignores (among
other things) the presence of the overshoot, which we have seen to play an
important role for example for the Cramér–Lundberg approximation. The
objective of the corrected “diffusion” approximation is to take this and
other deficits into consideration (<cit.>, p. 139).

Diffusion approximations of random walks via Donsker's theorem[Called
also Donsker–Prohorov's Invariance Principle, with credits to
<cit.>. — V.M.] is a classical topic of probability
theory. See for example Billingsley <cit.>. The first application
in risk theory is Iglehart <cit.> and two further standard
references in the area are Grandell <cit.>, <cit.>. All material of this section can be found in these references. For
claims with infinite variance, Furrer, Michna, Weron <cit.> suggested an approximation by a stable Lévy process rather than a
Brownian motion. Further relevant references in this direction are Furrer
<cit.> Boxma, Cohen <cit.> and Whitt <cit.>
(<cit.>, p. 139)

A “corrected diffusion” approximations were introduced by Siegmund
<cit.> in a discrete random walk setting, with the translation to
risk processes being carried out by Asmussen <cit.>; this case is in
part simpler than the general random walk case because the ladder height
distribution can be found explicitly which avoids the numerical integration
involving characteristic functions which was used in <cit.> to
determine the constants. In Siegmund's book <cit.> the approach
to the finite horizon case is in part different and uses local central limit
theorems. The adaptation to risk theory has not been carried out. The
“corrected diffusion” approximation was extended to the renewal model in
Asmussen, Højgaard <cit.>, and to the Markov-modulated
model of Chapter VII in Asmussen <cit.>; Fuh <cit.>
considers the closely related case of discrete time Markov additive processes.
Hogan <cit.> considered a variant of the “corrected diffusion”
approximation which does not require exponential moments. His ideas were
adapted by Asmussen, Binswanger <cit.> to derive
approximations for the infinite horizon ruin probability when claims are
heavy-tailed; the analogous analysis of finite horizon ruin probabilities has
not been carried out and seems non-trivial. For “corrected diffusion”
approximations with higher-order terms, see Blanchet, Glynn <cit.> their results also cover some heavy-tailed cases (<cit.>, p. 145).

Assuming that 1d=T, we focus on the case of T and Y
exponential[It is noteworthy that “diffusion” and “corrected
diffusion” approximations are identical in this case.] with parameters
 and , where the distribution of u,c may be
written explicitly. We follow the idea to use the approximation
(<ref>) and to match the two first moments. We note that for
R_s=(-c)s+σs we have R_s=(-c)s,
R_s=σ^2s and for R_s=∑_i=1^si-cs we have
R_s=(/-c) s, R_s=(2/^2)s. Matching
these two first moments, means taking =/ and
σ^2=2/^2 in (<ref>). It
yields[Notation f(x;z) |_z=a means f(x;a). Notation
f(x) |_x=a^b means the difference f(b)-f(a).]

    {u,c⩽ t}≈
    F(t;,,-12) |_=
    u/-c,=^2
    u^2/2,   0<c⩽,
    
    exp{-2} F(t;,,12) |_=-
    u/-c,=u^2/σ^2,   c>,

where =/, and

    =u/-c=u/(/)-c,  =u^2/σ^2=^2 u^2/2,  =-u/-c=-u/(/)-c.


The approximation (<ref>) illustrated in Fig. <ref> below is
further discussed in Section <ref>.




§ NEW “INVERSE GAUSSIAN” APPROXIMATION IN LEVEL CROSSING





Being in the renewal framework of Section <ref>, let us write
M=T/Y, D^2=((T)^2Y+(Y)^2T)/(Y)^3, and

    Mu,ct| v=∫_0^c(t-v)/u+cv1/1+y cM(1+y)c^2D^2(1+y)/u+cv(y)dy,

which is expressed in terms of c.d.f. of inverse Gaussian distribution
(<ref>) as

    Mu,ct| v=
    F(x;,,-12) |_x=1^c(t-v)/cv+u+1,   0<c⩽,
    
    exp{-2 }
    F(x;,,-12) |_x=1^c(t-v)/cv+u+1,   
    c⩾,

where =1/M, =u/c^2D^2>0,
=1/1-cM, and =1/cM-1. Plainly, >0 for
0<c< and >0 for c>.




 §.§ Approximation for conditional distribution



The following result is formulated and proved as Theorem 1.1 in <cit.>.


 In the renewal model, let p.d.f. f_T(y) and f_Y(y) be
bounded from above by a finite constant, D^2>0, (T^3)<∞,
(Y^3)<∞. Then for fixed c>0 and 0<v<t we have

    sup_t>v | {v<u,c⩽
    t|1=v}-Mu,ct| v | 
    =O (ln(u+cv)/u+cv),

as u+cv→∞.


Since Mu,ct| v in (<ref>) is defined by means of
c.d.f. of inverse Gaussian distribution, let us call[The “diffusion”
approximation also involves the inverse Gaussian distribution. Someone will not
concur with the proposed name, but to us it seems sensible.] the approximation
of Theorem <ref>, as well as its corollaries put forth below,
“inverse Gaussian”.

Bearing in mind easy equality

    {M(u+cv+cy)=n}={∑_i=1^ni⩽
    u+cv+cy<∑_i=1^n+1i}
    
    =∫_0^u+cv+cyf^*n_Y(u+cv+cy-z){n+1>z}dz,

the idea behind the proof of Theorem <ref> consists in application of
the identity (<ref>). After the change of variables y=z-v, it
writes as

    {v<u,c⩽
    t|1=v}   =∫_0^t-vu+cvu+cv+cy  _{∑_i=2^M(u+cv+cy)+1i}(y)dy
    
       =∑_n=1^∞∫_0^t-vu+cv/u+cv+cy∫_0^u+cv+cy{n+1>z}
        90pt× f_Y^*n(u+cv+cy-z)f_T^*n(y)dydz,

and non-uniform central limit theorem applied to approximate n-fold
convolutions f_Y^*n, f_T^*n yields the approximating and residual
terms, both investigated in <cit.> by direct analytical method. This
method is a considerable development of the method used previously in
<cit.>, <cit.>, and <cit.>. It applies identical transformations, approximations of integral sums
by the corresponding integrals, evaluation of elliptic integrals of the third
kind and of sums related to zeta-functions.




 §.§ Approximation for non-conditional distribution



Though approximation for conditional distribution is more convenient to prove
because in this case the identity (<ref>) is a convenient structure,
the traditional object of interest is the unconditional distribution
{u,c⩽ t}. Accordingly, let us formulate some
corollaries of Theorem <ref>.


Under conditions of Theorem <ref>, with p.d.f. f_1(y) bounded
from above by a finite constant and with 1<∞, we have

    sup_t>0 | {u,c⩽
    t}-∫_0^t{Y>u+cv}f_1(v)dv -∫_0^tMu,ct|
    v f_1(v)dv | =O (ln u/u),

as u→∞.


Having information about “regularity” of 1, the insight into the term
∫_0^t{Y>u+cv}f_1(v)dv is easy. To be particular, application
of the Markov's inequality yields {Y>u+cv}⩽Y^3/(u+cv)^3, and
∫_0^t(u+cv)^-3f_1(v)dv=O (u^-3), as
u→∞. This order of magnitude is less than O (ln u/u),
and this term may be omitted.

Concerning the second term, we bear in mind that for each v>0

    Mu,ct|
    v=Mu,ct-v+O (vu),   u→∞,

where Mu,ct-v=Mu,ct-v| 0. It yields the following
result.


Under conditions of Corollary <ref>, we have

    sup_t>0 | {u,c⩽
    t}-∫_0^t{Y>u+cv}f_1(v)dv
    
    
    -∫_0^tMu,ct-v f_1(v)dv | =O (ln
    u/u),

as u→∞.


The term ∫_0^tMu,ct-v f_1(v)dv which corresponds to
the case 1<u,c, is a convolution. It agrees with the
probabilistic intuition about the rôle of the first interval 1 in the
event of first level u crossing: when 1 is fixed and is equal to v,
the time is modified from the whole time t to reduced time t-v.

Until now we did not assume that time t is large, but deemed that it may be
small, moderate, and large. Since the approximation (<ref>) is
formulated uniformly with respect to t>0, the influence of 1 can not be
eliminated for t small and moderate. Though Mu,ct-v is rendered
in a relatively compact form, the integral
∫_0^tMu,ct-v f_1(v)dv hardly can be evaluated as a
compact explicit expression, even for 1 exponential. On the contrary, for
large t and “regular” 1, the first interval does not affect the
main-term approximation, i.e., for t→∞ we have

    ∫_0^tMu,ct-v f_1(v)dv=Mu,ct (1+o(1)).



The point  is excluded from consideration in Theorems <ref>
and <ref>. In this respect, it is special. In this point, the
approximation (<ref>) for {u,c⩽ t},
being neither outstanding, nor especially good, but merely noteworthy, involves
the expression

    Mu,t=2(01(√(u)
    D )-01(u D√( t+u) )).

The approximation for {u,c⩽∞} involves the
expression

    Mu,∞= 2 01(M√(u)/D )-1.






 §.§ Focus of the approximation and related problems



The approximation (<ref>) is informative when
{u,c⩽ t} tends to a positive value, rather than
to zero, as u→∞. It is paramount, e.g., in the problem of
investigating a solution u_α,t(c) of the equation

    {u,c⩽ t}=α,

where 0<α<1. For diffusion process, u_α,t(c) is investigated in
<cit.>, <cit.>, and <cit.>. For compound renewal process with exponential T and Y, it was done
in <cit.>, <cit.>. Analysis of
u_α,t(c) for general compound renewal process was done in
<cit.> and <cit.>. It was found that the main
term in approximation of u_α,t(c), as t→∞, is proportional to
t^1/2.

The focus in (<ref>) with the remainder term O (ln
u/u) differs from the focus in (<ref>) with the remainder term
o (e^- u), as u→∞; the latter is highly
informative when {u,c⩽ t} tends to zero, as
u→∞, while the former may be not.


For approximations (<ref>) and (<ref>), the difference in
focuses is similar to situation known for the normal approximation and large
deviations in the common central limit theorem for sums of i.i.d. summands. The
former concerns itself with the asymptotic behavior around the mean value of
the sum considered, while the latter deals with the exponential decline of
remote tails of the distribution of the sum.





 §.§ Subexponential distributions



Subexponential distributions are a special class of heavy-tailed distributions
prominent in applied probability. First order approximations to ruin
probabilities and waiting time distributions are by now called “folklore found
in the relevant textbooks”, e.g., Embrechts, Klüppelberg and Mikosch
<cit.>.

For some of such distributions (e.g., Pareto inter-renewal times and Pareto
jump sizes), performance of “inverse Gaussian” approximation obtained in
<cit.>, <cit.> was tested against simulation in <cit.>. As we
claimed in Section <ref> above, the interest in <cit.> was the
case when {u,c⩽ t} tends to a non-zero value, as
u→∞, rather than “large deviations” case, i.e., when
{u,c⩽ t} tends to zero, as u→∞.

It is noteworthy that in the paper <cit.> called “Large
deviations results for subexponential tails, with applications to insurance
risk”, Asmussen and Klüppelberg gave asymptotic expressions for
{u,c⩽ t} when t=t(u), such that t(u)/u→
k∈(0,∞), which implies that {u,c⩽ t} is tending
to zero, as u→∞. But this case is just “large deviations”, which
lies outside the focus of <cit.>–<cit.>.




 §.§ Extensions of (<ref>)



In <cit.>, the asymptotic expansions are constructed with the first
correction term given explicitly. It is done by extending the same technique as
was used in <cit.>. A concurrent objective of <cit.> was to
demonstrate the amplitude of the method. The terms in which the correction is
found, are the generalized inverse Gaussian distribution. The results of
<cit.>, <cit.> are illustrated numerically in <cit.>.


There is a difference between asymptotic expansions as a mathematical result,
and as a tool to improve numerical performance. Many examples (in a simpler
problems, e.g., in the central limit theorem for sums) show that involving a
correction term, one not always and not necessary acquire a visible improvement
in terms of proximity to the approximated function.





 §.§ The asymptotic behavior, as t→∞





Assuming that s, s>0, is a Poisson process and that
c>=Y/T, Teugels <cit.> obtained[Quoting
<cit.>, “a relation of this type is highly expected… A
rigorous proof and an explicit evaluation of  was, however, first provided
very recently by Teugels <cit.>” (<cit.>, p. 51).]
an approximation of the form

    {u,c⩽ t}≈{u,c<∞}- 
    u t^-3/2e^-γ_1 u-γ_2 t,   as t→∞,

where , γ_1 and γ_2 are certain positive constants. In
<cit.>, the renewal model was considered under the following
conditions. The i.i.d. r.v. id=T>0, i=1,2,…, are
exponential with parameter , while i.i.d. r.v.
id=Y>0, i=1,2,…, are arbitrarily distributed subject to
the following condition[Called Assumption A(iv) on p. 164 of
<cit.>.]: there exists a positive value v inside the region of
convergence of Λ(s)=cs- (1- e^-sY), such that
Λ^'(-v)=0. This rather restrictive constrain implies that e^-sY converges as an analytical function in a neighborhood of the origin,
and the adjustment coefficient >0 exists.

The proof in <cit.> applies the Laplace transforms method. On
p. 174 of <cit.> it is mentioned that for Y exponential, one
can derive (<ref>) straightforwardly, “through an asymptotic expansion
of the modified Bessel function” in Type II formula (<ref>).

Assuming that t→∞, Corollary <ref> yields “Teugels-type”
approximation, as follows. If u/t→ 0 (u≪ t), we have

    Mu,ct=Mu,c∞- uexp{γ_1 u}
    × t^-3/2exp{-γ_2 t}exp{-γ_3u^2/t} (1+o(1)),  
    t→∞,

where =√(2π)Dc^1/2(1-cM)^2,
γ_1=1-cMc^2D^2, γ_2=(1-cM)^22cD^2,
γ_3=12c^3D^2 and

    Mu,c∞=01(M/D√(u) )
    -exp{2(1-cM)/c^2D^2 u} 01(-2-cM/cD√(u) ),    0⩽
    c⩽,
    
    exp{-2(cM-1)/c^2D^2 u } 01(-cM-2/cD√(u) )
    -01(-M/D√(u) ),    c>,


The approximation (<ref>) is called “Teugels-type” since it differs
from the original Teugels approximation (<ref>). First, conditions on
the model in these two results are obviously different: “Teugels-type” was
obtained under much more general conditions. Second, it is assumed in
<cit.> that u is fixed and t is tending to infinity, while
in Corollary <ref> it is assumed that both u and t are tending
to infinity. However, the structure of both approximations (<ref>) and
(<ref>) is the same, as is expected.

Indeed, quoting Teugels' Remark 8.2 on p. 174 of <cit.>, since
“as a function of time the assumption A(iv) essentially introduces an
exponential decay”, for W(x)={inf_s>0(x+cs-s)⩾ 0} and
W(t,x)={inf_0<s⩽ t(x+cs-s)⩾ 0} with c=1,

the extra t^-γ with γ=1/2 for the compound Poisson
process and with γ=3/2 for its supremum is not surprising in view of an
approximating Wiener process. For let us consider the quantity W(t,x) for a
Wiener process with drift 1. Then (<cit.>, p. 82) …
    W(t,x)-W(x)=e^-2x{1-01(t-x/√(t))}
    -{1-01(t+x/√(t))}

… which can be simplified to

    W(t,x)-W(x)∼√(2/π) e^x-t/2x t^-3/2,   t→∞.



To have an illustration of (<ref>) stated in a general renewal
framework, let us consider T and Y exponential with positive parameters
 and . We have

    M=^-1,   D^2=2^-2,

and

    =
    2/√(π)√()/√(c)(1-c/)^2,  γ_1=(-c)/2c^2,  γ_2=(-c)^2/4
    c,  γ_3=^2/4c^3.


Performance of Teugels-type approximation (<ref>) in this case is
illustrated in Fig. <ref>. It is poor in the vicinity of ; the
same is known for the original Teugels' approximation in <cit.>.
It is no surprise since =
2/√(π)√()/√(c)(1-c/)^2
turns into infinity for c=.




§ “DIFFUSION” AND “INVERSE GAUSSIAN” APPROXIMATIONS VS. EXACT FORMULA WHEN T, Y
ARE EXPONENTIAL





Bearing in mind (<ref>) and (<ref>), equation
(<ref>) for large t>0 yields

    {u,c⩽ t}≈
    F(ct/u+1;,,-12)-F(1;,,-12),
       0<c⩽,
    
    exp{-2 }(F(ct/u+1;,,-12)    
     120pt-F(1;,,-12)),     c>,

where =1/M, =1/1-cM, =u/c^2D^2,
=1/cM-1.

For T, Y exponential with parameters , , we have
M=^-1, D^2=2^-2, and in (<ref>) we
have =/, and

    =1/1-c(/)=(/)/(/)-c,
      =^2u/2c^2>0,  =-1/1-c(/).


Having derived explicit expressions (<ref>), (<ref>) for
“diffusion” approximation and (<ref>), (<ref>) for
“inverse Gaussian” approximation, we are able to compare these results with
each other. Having Type I–Type III explicit formulas for
{u,c⩽ t}, we can further compare them with the
exact formula.

Comparing of (<ref>), (<ref>) and (<ref>),
(<ref>) with each other clearly shows that the structure of
“inverse Gaussian” approximation and the structure of “diffusion”
approximation[Which coincides in this case with “corrected
diffusion” approximation.] are different, though both (<ref>) and
(<ref>) are written in terms of c.d.f. of inverse Gaussian
distribution (<ref>).

When the values of {u,c⩽ t} are significantly
greater than zero (which happens in a region including ), performance of
both “inverse Gaussian” and “diffusion” approximations is compared to exact
values of {u,c⩽ t} in Fig. <ref>.




§ CONCLUSIONS



The “inverse Gaussian” approximation for {u,c⩽
t} considered in Section <ref> is structurally
different[Though “inverse Gaussian”, “diffusion”, and “corrected
diffusion” approximations may be written in terms of c.d.f. of inverse
Gaussian distribution (<ref>).] from “normal”, “diffusion”, and
“corrected diffusion” approximations (see in particular illustration in
Section <ref>).

The “inverse Gaussian” approximation and its refinements in terms of
Edgeworth expansions were rigorously proved by a technique based on an idea
different from the ideas lying behind the “normal”, “diffusion”, and
“corrected diffusion” approximations (see Sections <ref>
and <ref>). The methodology is said to be creative, and cannot be
directly translated to “standard” techniques[Though develops the
technique in <cit.> and <cit.>.]; it is
not very demanding in terms of background.

The conditions which are required in “inverse Gaussian” approximation are
weaker, and even look minimal, in comparison with conditions in “normal”,
“diffusion”, and “corrected diffusion”; these substantially more
restrictive conditions seem immanent to the methods of proof for “normal”,
“diffusion”, and “corrected diffusion” approximations[Quoting
again <cit.> (see quotation in full in Section
<ref>), note that “the analogous analysis of finite horizon ruin
probabilities ψ(u,T) has not been carried out and seems non-trivial.”].

The “Teugels-type” approximation, as expected, is a corollary of the
“inverse Gaussian” approximation; the form of the “Teugels-type”
approximation is straightforward under very broad assumptions on the renewal
model.

The “inverse Gaussian” approximation is important in many applications, in
particular (see Section <ref>) in the problem of finding a solution of
non-linear equation (<ref>); in the risk context this solution is
called “non-ruin capital”; it is used to construct different controls in the
models of long-term insurance solvency regulation (see, e.g.,
<cit.>–<cit.>).

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