A Weak Approximation for the Extrema's Distributions of L\'evy Processes

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by  Amir T. Payandeh Najafabadi

A WEAK APPROXIMATION FOR THE EXTREMA’S
DISTRIBUTIONS OF LE´VY PROCESSES
7
1 AMIR T. PAYANDEHNAJAFABADI∗ ANDDAN Z. KUCEROVSKY
0
2

n
a
J Abstract. SupposeXt isaone-dimensionalandreal-valuedL´evypro-
9 cess started from X0 = 0, which (1) its nonnegative jumps measure
1 ν satisfying RRmin{1,x2}ν(dx) < ∞ and (2) its stopping time τ(q)
is either a geometric or an exponential distribution with parameter q

] independent of Xt and τ(0) = ∞. This article employs the Wiener-
R Hopf Factorization (WHF) to find, an Lp∗(R) (where 1/p∗ +1/p = 1
P and 1 < p ≤ 2), approximation for the extrema’s distributions of Xt.
h. Approximating thefinite(infinite)-timeruin probability as a direct ap-
t plication of our findings has been given. Estimation bounds, for such
a
approximation method, along with two approximation procedures and
m
several examples are explored.
[ Keywords: L´evy processes; Positive-definite function; Extrema’s dis-

tributions; theFourier transform; theHilbert transform.
1
MSC(2010): Primary: 60G51;Secondary: 11A55,42A38,60J50,60E10.
v
6
6 1. Introduction
4
5 SupposethatX isaone-dimensionalandreal-valuedL´evyprocessstarted
t
0 from X = 0 and defined by a triple (µ,σ,ν) : the drift µ R, the volatility
. 0 ∈
1 σ 0, and the jumps measure ν which is given by a nonnegative function
0 de≥fined on R 0 satisfying min 1,x2 ν(dx) < . Moreover, suppose
7 \{ } R { } ∞
1 that the stopping time τ(q) isReither a geometric or an exponential distri-
: bution with parameter q independentof the L´evy process X and τ(0) = .
v t
∞
i TheL´evy-Khintchine formulastates thatthecharacteristic exponentψ (i.e.,
X
ψ(ω) = ln(E(exp(iωX ))), ω R) can be represented by
1
r ∈
a 1
(1.1) ψ(ω) = iµω σ2ω2
− 2
+ (eiωx 1 iωxI (x))ν(dx), ω R.
[ 1,1]
ZR − − − ∈
The extrema of the L´evy process X are given by
t
M = sup X : s τ(q) & I = inf X : s τ(q) .
q s q s
{ ≤ } { ≤ }
The Wiener-Hopf Factorization (WHF) is a well known technique to study
thecharacteristicfunctionsoftheextremarandomvariables(see[1]. Namely,
the WHF states that: (i) product of their characteristic functions equal to
Date: Received: ,Accepted: .
∗Corresponding author.
1
2 AMIRT.PAYANDEHNAJAFABADIANDDANZ.KUCEROVSKY
the characteristic function of L´evy process X at its stopping time τ(q),
t
say X and (ii) random variable M (I ) is infinitely divisible, positive
τ(q) q q
(negative), and has zero drift.
In the cases that, the characteristic function of L´evy process X either
t
a rational function or can be decomposed as a product of two sectionally
analyticfunctionsintheclosedupper,i.e.,C+ := λ : λ Cand (λ) 0 ,
and lower half complex planes, i.e., C := λ :{ λ ∈C and ℑ(λ) ≥ 0}.
−
{ ∈ ℑ ≤ }
Then, the characteristic functions of random variables M and I can be
q q
determined explicitly (see [24]). [16] considered a L´evy process X which
t
its negative jumps is distributed according to a mixture-gamma family of
distributions and its positive jumps measure has an arbitrary distribution.
They established that the characteristic function of such a L´evy process can
be decomposed as a product of a rational function in an arbitrary function,
whichareanalyticinC+ andC ,respectively. Theyalsoprovidedananalog
−
result for a L´evy process whose its corresponding positive jumps measure
follows from a mixture-gamma family of distributions while its negative
jumps measure is an arbitrary one, more details can be found in [17].
Unfortunately, in the most situations, the characteristic function of the
process neither is a rational function nor can bedecomposed as a productof
two sectionally analytic functions in C+ and C . Therefore, the character-
−
istic functions of M and I should be expressed in terms of a Sokhotskyi-
q q
Plemelj integral (see Equation, 2.1). But, this form, also, presents some dif-
ficulties in numerical work due to slow evaluation and numerical problems
caused by singularities near the integral contour (see [11]). To overcome
these difficulties, an appropriate (in some sense) approximation method has
to be considered. It is well known that a L´evy process X which its jumps
t
distribution follows from the phase-type distribution has a rational charac-
teristic function (see [7]). [13] utilized this fact and approximated a jumps
measureν of a ten-parameter L´evy processes (named β family of L´evy pro-
−
cess) by a sequence of the phase-type measures. Then, he determined the
characteristic functions of random variables M and I , approximately. [14]
q q
extended [13]’s findings to class of Meromorophic L´evy processes. More-
over, [15] provided a uniform approximation for the cumulative distribution
function of M whenever X is a symmetric L´evy process. [12] employed
τ(q) t
the Shannon sampling method to find the distributions of the extrema for a
wide class of L´evy processes.
Thisarticlebeginswithanextensionof[11]’sresultsforthemultiplicative
WHF
(1.2) Φ+(ω)Φ (ω) = g(ω) ω R,
−
∈
where g() is a given function with some certain conditions (see below) and
·
Φ () are to be determined. Then, it utilizes such results to approximate
±
·
the extrema’s distributions of a class of L´evy processes. Estimation bounds,
for such approximate method, along with two approximation proceduresare
given.
APPROXIMATION FOR THE EXTREMA’S DISTRIBUTIONS 3
Section 2 collects some useful elements for other sections. Moreover, it
provides an Lp(R), 1 < p 2 approximation technique for solving a multi-
≤
plicative WHF (1.2). Section 3 considers the problem of approximating the
extrema’s density functions for a class of L´evy processes. Then, it develops
twoapproximatetechniquesforsituationswherethosedensityfunctionscan-
not be determined, explicitly. Error bounds for such techniques are given.
Several examples are given in Sections 4. Section 5 provides concluding re-
marks along with some suggestions for other application of our techniques.
2. Preliminaries
The Sokhotskyi-Plemelj integral for s(), which satisfies the H¨older condi-
·
tion, is defined by a principal value integral, as follows
1 s(x)
(2.1) φ (λ) := dx, for λ C.
s
2πiZ−R x λ ∈
−
It is worth mentioning that, the Sokhotskyi-Plemelj integral can be ex-
isted for non-integrable function such as sin(x). Therefore, the Sokhotskyi-
Plemelj integral φ ()shouldbeviewed different fromtheusualintegral over
s
R. ·
The radial limits of the Sokhotskyi-Plemelj integral of s(), are given
·
by φ (ω) = lim φ (λ) and satisfy the following jump formulas: (1)
±s s
λ ω+i0±
φ (ω) = s(ω)→/2+φ (ω),forω Rand(2)φ (ω)= s(ω)/2+H (ω)/(2i),
w±shere H ±(ω) stands fsor the Hilb∈ert transform±sof s() a±nd ω R. s
s
· ∈
ThemultiplicativeWHFistheproblemoffindingananalyticandbounded,
except on the real line, function Φ() where its upperand lower radial limits
·
Φ () satisfy Equation (1.2). Given function g() is a bounded above by
±
· ·
1, zero index1, continuous, and positive function which satisfies the H¨older
condition on R, g(0) = 1, and g(ω) = 0 for all ω R.
6 ∈
The following extends [11]’s results to the multiplicative WHF (1.2). We
begin with what we term the Resolvent Equation for Sokhotskyi-Plemelj
integrals.
Lemma 2.1. The Sokhotskyi-Plemelj integral of a function f() satisfies
·
φ (λ) φ (µ) = (λ µ)φ (µ), where λ and µ are real or complex values.
f f f(x)
− −
x−λ
Proof. In general, (x λ) 1 (x µ) 1 = (λ µ)(x µ) 1(x λ) 1. Then,
− − − −
see [8], we have an eq−uation−of C−auchy integr−als, wh−ere Γ = R−:
1 f(x) 1 f(x) λ µ f(x)
dx dx = − dx.
2πi Z x λ − 2πi Z x µ 2πi Z (x µ)(x λ)
Γ Γ Γ
− − − −
The above is valid only for λ and µ not on the real line. However, by
Equation (2.1) the values of φ () on the real line are obtained by averaging
f
·
1The index of a complex-valued function f on a smooth oriented curve Γ, such that
f(Γ)isclosed andcompact, isdefinedtobethewindingnumberoff(Γ)abouttheorigin
(see [23], §1), for more technical details.
4 AMIRT.PAYANDEHNAJAFABADIANDDANZ.KUCEROVSKY
the limit from above, φ+(), and the limit from below, φ . We thus obtain
f · −f
the stated equation in all cases. (cid:3)
Lemma 2.2. Suppose Φ () are sectionally analytic functions that satisfy
±
·
the multiplicative WHF (1.2). Moveover, suppose that given function g()
·
is a zero index function which satisfies the H¨older condition and g(0) =
1.2 Then Φ (λ) = exp (φ (λ) φ (0)) , where φ () stands for the
± lng lng lng
{± − } ·
Sokhotskyi-Plemelj integral of lng().
·
Proof. Using the [10]’s suggestion for solving the homogeneous WHF
(1.2) gives, see also [17]:
λ lng(x)/x
Φ (λ) = exp dx .
±
{±2πiZ−R x λ }
−
Lemma (2.1) with f lng gives φ (λ) φ (µ) = (λ µ)φ (µ).
lng lng lng(x)
≡ − −
x−λ
Letting λ goes to zero from the above, in the complex plane, and using
the fact that lng(0) = 0, Equation (2.1) lets us to conclude that φ (0)
lng
−
φ (µ) = µφ (µ). Substituting this into the above equation for Φ ()
lng lng(x) ±
− ·
gives our claimedxresult. (cid:3)
Using the jump formula one can conclude that
i
(2.2) Φ (ω) = g(ω)exp (H (0) H (ω)) ,
± lng lng
{±2 − }
p
where H () stands for the Hilbert transform of lng().
lng
· ·
The Carlemann’s method explores a situation which one may evaluate
solutions of the multiplicative WHF (1.2) directly, rather than using the
Sokhotskyi-Plemelj integrations. The Carlemann’s method states that: if
g() can be decomposed as a product of two sectionally analytic functions
g+·() and g (), respectively in C+ and C . Then, solutions of the multi-
− −
· ·
plicative WHF (1.2) are given by Φ+ g+ and Φ g .
− −
≡ ≡
P(x)
In a situation that g() is a rational function that has no poles or
· Q(x)
zeros on R. Using the Carlemann’s method, we may conclude that the mul-
tiplicative WHF problem can be solved by factoring the polynomial P (Q),
and then let P+ (Q+) be the product of those factors of P (Q) that have
zeros in C , and P (Q ) be the product of those factors that have zeros
− − −
in C+. Then, setting g+(x) = P+(x) and g (x) = P−(x) gives us (up to a
Q+(x) − Q−(x)
scalar multiple) our desired factorization.
The Hausdorff-Young theorem (see [22]) states that: if s() is an Lp(R)
·
function. Then, s() and its corresponding the Fourier transform, say sˆ(),
· ·
satisfy sˆ (2π) 1/p s , where 1 p 2 and 1/p+1/p = 1. From
p∗ − p ∗
|| || ≤ || || ≤ ≤
the Hausdorff-Young Theorem, one can observe that if s () is a sequence
n
of functions converging in Lp(R), 1 p 2, to s(){. Th·e}n, the Fourier
≤ ≤ ·
2Theconditiong(0)=1doesnotalwaysholdinthemultiplicativeWHF,buthappento
ariseinourapplication,andcanleadtocomplications. Lemma(2.2)isusedtosimplifying
thiscase.
APPROXIMATION FOR THE EXTREMA’S DISTRIBUTIONS 5
transforms of the s () converges in Lp∗(R), to the Fourier transform of s(),
n
· ·
where 1/p+1/p = 1. The converse is false.
∗
AsimilarpropertyfortheHilberttransformiswellknownastheTitmarsh-
Riesz lemma (see [22]). The Titmarsh-Riesz lemma says that: if s() is an
Lp(R) function, where 1 < p 2. Then, H tan(π/(2p)) s ,· where
s p p
≤ || || ≤ || ||
H () stands for the Hilbert transform of s(). Using the Titmarsh-Riesz
s
· ·
lemma, one may conclude that if f () , is a sequence of functions which
n
converge, in Lp(R), 1 < p 2, to f{().·T}hen, the Hilbert transforms H(f )
n
converge, in Lp(R), 1 < p≤ 2, to the· Hilbert transform of f().
≤ ·
Thewell known Paley-Wiener theorem states that: if F() is a function in
L2(R). Then, the real-valued function F() vanishes on R ·if and only if the
−
·
Fourier transform F(), say, Fˆ() is holomorphic on C+ and the L2(R)-norm
· ·
of the functions x Fˆ(x+iy ) are uniformly bounded for all y 0.
0 0
7→ ≥
Thefollowing,from[11],recallssomefurtherusefulpropertiesoffunctions
in Lp(R), for 1 < p 2, space.
≤
Lemma 2.3. Suppose s() and r() are two Lp(R), 1 < p 2, functions.
· · ≤
Then,
i): √s √r 1 s r , wheneverboth s()and r() are bounded,
|| − ||p ≤ 2√a|| − ||p · ·
above by a, functions;
ii): lns lnr a 1 s r , whenever both s() and r() are
p − p
|| − || ≤ || − || · ·
positive-valued and bounded, above by a, functions ;
iii): e is/2 e ir/2 1 s r , whenever s() and r() are real-
|| − − − ||p ≤ 2|| − ||p · ·
valued functions.
In many situations, WHF (1.2) cannot be solved explicitly and has to be
solved approximately (see [11]). The following develops an approximation
technique to solve a multiplicative WHF (1.2).
Theorem 2.4. Suppose Φ () are two sectionally analytic functions satis-
±
·
fying the multiplicative WHF (1.2) where
A ): g() is real, positive, bounded above by a, index zero, satisfies the
1
·
H¨older condition, and g(0) =1;
A ): Thereexistasequenceoffunctionsg ()whereconverge, inLp(R),
2 n
·
1 < p 2, to g().
≤ ·
Then, Φ () can be approximated by Φ (), where
± · ±n ·
1 π π 1
Φ Φ tan( ) g g 2 +(tan( )+ ) g g .
|| ±n − ±||p ≤ 2 2p || n − ||p 2p 2 || n − ||p
Proof. Set k(ω) := H (ω) + H (0) and k (ω) := H (ω) +
− lng lng n − lngn
H (0). Now, from Equation (2.2) and Lemma (2.3) observe that Φ
lngn || ±n −
6 AMIRT.PAYANDEHNAJAFABADIANDDANZ.KUCEROVSKY
Φ
± p
||
= √gne±ikn/2 √ge±ik/2 p
|| − ||
[ √gn √g p+ √g p] e±ikn/2 eik/2 p + e±ik/2 √gn √g p
≤ || − || || || || − || | ||| − ||
1
[ √g √g + √g ] H (ω)+H (0)+H (ω) H (0)
≤ 2 || n− ||p || ||p ||− lngn lngn lng − lng ||p
+ e±ik/2 √gn √g p
| ||| − ||
[ √g √g + √g ] H H + √g √g
≤ || n − ||p || ||p || lngn − lng||p || n− ||p
since k and k are real-valued functions
n
π
[ √g √g + √g ]tan( ) ln(g ) ln(g) + √g √g
n p p n p n p
≤ || − || || || 2p || − || || − ||
π 1 1
tan( ) g g +1 g g + g g
n p n p n p
≤ 2p (cid:20)2|| − || (cid:21)|| − || 2|| − ||
1 π π 1
= tan( ) g g 2 +(tan( )+ ) g g . (cid:3)
2 2p || n − ||p 2p 2 || n − ||p
Now, we recall definition of the positive-definite function which plays a vital
roles in the rest of this article.
Definition 2.5. A positive-definite function is a complex-valued function
f : R Csuchthatforanyrealnumbersx , ,x then nsquarematrix
1 n
→ ··· ×
A= (f(x x ))n is a positive semi-definite matrix.
i− j i,j=1
In the theory of the Fourier transform, it is well known that “f() is a
continuous positive-definite function on R if and only if its correspo·nding
the Fourier transform is a (positive) measure”, see [4] for more details.
Lemma 2.6. Suppose φ : R C is a positive-definite function which two
→
equations q φ(ω) = 0 and 1 q exp φ(ω) = 0 have not any solution
1 2
on R, where−q > 0 and q −(0,1). T{−hen, h}(ω) = q /(q φ(ω)) and
1 2 1 1 1
∈ −
h (ω) = (1 q )/(1 q exp φ(ω) ) are positive-definite functions.
2 2 2
− − {− }
Proof. Using the Taylor expansion of q /(q x) and (1 q )/(1
1 1 2
− − −
q exp x ), about zero, one may restated h () and h () as
2 1 2
{− } · ·
∞ φk(ω)
h (ω) =
1 qk
Xk=1 1
q q (q +1) q (q2+4q +1)
h (ω) = 1+ 2 φ(ω)+ 2 2 φ2(ω)+ 2 2 2 φ3(ω)+
2 q 1 2(q 1)2 6(q 1)3 ···
2 2 2
− − −
Now,thedesiredproofarrivesfromthefactthattheproductoftwopositive-
definite functions is again a positive-definite function (see [30]). (cid:3)
Now, we provide two classes of positive-definite rational functions which
play a vital role in numerical section of this article.
APPROXIMATION FOR THE EXTREMA’S DISTRIBUTIONS 7
Lemma 2.7. Consider the following two class of rational functions and
D
.
∗
D
n mk 4
j
: = r(ω); r(ω)= A + C r (ω); A & C 0 ;
D { 0 kj lk 0 kj ≥ }
Xk=1Xj=1Xl=1
n 2
⋆ : = r(ω); r(ω)= A + C r (ω); A & C 0 ,
0 kj lk 0 kj
D { ≥ }
Xk=1Xl=1
where
1
r (ω) = (where β > 0);
1k k
iω+β
k
1
r (ω) = (where β > 0);
2k k
iω+β
k
−
1
r (ω) = (where α , β > 0);
3k k k
(iω+β )(iω+β +α i)(iω+β α i)
k k k k k
−
1
r (ω) = (where α , β > 0).
4k k k
( iω+β )( iω+β +α i)( iω+β α i)
k k k k k
− − − −
Then, (i) the Fourier transform of functions in are nonnegative and real-
D
valued functions. (ii) the Fourier transform of functions in are nonneg-
∗
D
ative, real-valued, and completely monotone functions.
Proof. Nonnegativity of the Fourier transform of functions in (or )
∗
D D
arrives fromthefact thatr (forl = 1, ,4)andtheir powersarepositive-
lk
···
definite rational functions. Now, from the Bernstein’s theorem observe that
a real-valued function defined on R+ is a completely monotone function,
whenever it is a mixture of exponential functions, see [29] for more details.
(cid:3)
The following may be concluded from properties of the WHF given by
[1].
Lemma 2.8. Suppose g() in the multiplicative WHF (1.2) is a positive-
·
definite function. Then, solutions, of the multiplicative WHF (1.2), Φ ()
±
·
are two positive-definite functions.
Proof. First observe that, using the multiplicative WHF (1.2) the char-
acteristic function of L´evy process X at its stopping time τ(q), say X ,
t τ(q)
can be decomposed as a product of the characteristic functions of two ran-
dom variables I and M , see [1] for more details. Moreover, [3]’s theorem
q q
states that “An arbitrary function φ :Rn C is the characteristic function
→
of some random variable if and only if φ() is positive-definite, continuous
·
at the origin, and if φ(0) = 1”. The desired proof arrives using these obser-
vations. (cid:3)
One may readily observe that the characteristic function of the mixed
gamma family of distributions (given below) are belong to .
D
8 AMIRT.PAYANDEHNAJAFABADIANDDANZ.KUCEROVSKY
Definition 2.9. (Mixed gamma family of distributions)Anonnegative ran-
dom variable X is said to be distributed according to a mixed gamma dis-
tribution if its density function is given by
ν nν αjxj 1 ν∗ nν∗ βj( x)j 1
p(x) = Xk=1Xj=1ckj(jk−1−)!e−αkxI[0,∞)(x)+Xk=1Xj=1c∗kj k(j−−1)−! eβkxI(−∞,0](x)
where c and α are positive value which satisfy ν nν c = 1.
kj k k=1 j=1 kj
P P
We now from [2] recall some useful properties of the characteristic func-
tion, which plays an important role for the next sections.
Lemma 2.10. Suppose pˆ() stands for the characteristic function of a dis-
·
tribution. Then,
i): pˆ() is a positive-definite function;
·
ii): pˆ() is a positive-definite rational function whenever its character-
·
istic function belongs to given by Lemma (2.7);
D
iii): pˆ(0) = 1; and the norm of pˆ() is bounded by 1.
·
Thenextsection provides an application of Theorem (2.4) to theproblem
offindingthedistributionsoftheextremaofL´evyprocessX ,approximately.
t
3. Main results
The following lemma restates the characteristic function of L´evy process
X at its stopping time τ(q), say X .
t τ(q)
Lemma 3.1. Suppose X represents L´evy process X at its stopping time
τ(q) t
τ(q). Then, the characteristic function of X can be restated as:
τ(q)
(i): q/(q ψ(ω)), for an exponential stopping time τ(q) with parameter
−
q > 0;
(ii): (1 q)/(1 qexp ψ(ω) ), for a geometric stopping time τ(q)
− − {− }
with parameter q (0,1).
∈
Proof. Conditioning on stopping time τ(q), one may restates he charac-
teristic function of X as:
τ(q)
For part (i):
E(eiωXτ(q)) = ∞E(eiωXτ(q) τ(q) = t)fτ(q)(t)dt = ∞E(eiωXt)qe−qtdt
Z | Z
0 0
q
= ∞eψ(ω)tqe qtdt = ;
−
Z q ψ(ω)
0
−
For part (ii):
∞ ∞
E(eiωXτ(q)) = E(eiωXτ(q) τ(q) = n)P(τ(q) = n)= E(eiωXn)(1 q)qn
| −
nX=0 nX=0
∞ 1 q
= eψ(ω)n(1 q)qn = − ,
− 1 qexp ψ(ω)
nX=0 − {− }
APPROXIMATION FOR THE EXTREMA’S DISTRIBUTIONS 9
where for both cases, the second equality arrives from the fact that X and
t
τ(q) are independent and the third equality obtains from definition of the
characteristic exponent ψ and infinitely divisibility of L´evy process X . (cid:3)
t
The following theorem represents an error bound for approximating the
density functions of extrema of a L´evy process.
Theorem 3.2. Suppose X is a L´evy process defined by a triple (µ,σ,ν).
t
Moreover, suppose that:
A ): The stopping time τ(q) is either a geometric or an exponential
1
distribution with parameter q independent of X and τ(0) = ;
t
∞
A ): The r (dx) are a sequence of positive-definite rational functions
2 n
which converge, in Lp∗(R) (where 1/p +1/p = 1 and 1 < p 2), to
∗
≤
characteristic exponent q/(q ψ(dx)) (or (1 q)/(1 qexp ψ(dx) )
− − − {− }
for geometric stoping time)
Then, the density function of the suprema and infima of the L´evy process
X , denoted f+ and f , respectively, can be approximated, in Lp∗(R) sense,
t q q−
by a sequence of the density functions f+ and f where:
q,n q−,n
i): For exponentially distributed stopping time τ(q), for q > 0,
1 π q π 1 q
f f tan( ) r 2 +(tan( )+ ) r ;
|| q±− q±,n||p∗ ≤ 2 2p || n−q ψ||p∗ 2p 2 || n−q ψ||p∗
∗ ∗
− −
ii): For geometric stopping time τ(q), for q (0,1),
∈
1 π 1 q π 1 1 q
||fq±−fq±,n||p∗ ≤ 2tan(2p )||rn−1 −qe ψ||2p∗+(tan(2p )+2)||rn−1 −qe ψ||p∗.
∗ − ∗ −
− −
Proof. From [1] and Lemma (3.1), one can observe that the Fourier
transform of the density functions of random variables M and I , say re-
q q
spectivelyΦ+ andΦ ,satisfyeither themultiplicativeWHFΦ+(ω)Φ (ω) =
− −
q/(q ψ(ω)), where ω R (for exponentially distributed stopping time) or
− ∈
the multiplicative WHF Φ+(ω)Φ (ω) = (1 q)/(1 qexp ψ(ω) ), where
−
ω R(forgeometricstoppingtime). Now,f−romthe−facttha{t−theex}pressions
∈
q/(q ψ()) and (1 q)/(1 qexp ψ() ) are the characteristic function
− · − − {− · }
of the L´evy process X , at exponential and geometric stopping time, respec-
t
tively, we observe that both expressions are bounded above by 1 because of
the property of the characteristic function given by Lemma (2.10, part ii).
For part (i), from Theorem (2.4) observe that
1 π q π 1 q
Φ Φ tan( ) r 2+(tan( )+ ) r .
|| ±n − ±||p ≤ 2 2p || n − q ψ||p 2p 2 || n − q ψ||p
− −
The rest of proof arrives from an application of the Hausdorff-Young Theo-
rem. The proof of part (ii) is quite similar. (cid:3)
Remark 3.3. In case that the distribution of I or M has an atom at x =
q q
0. Then, it corresponding probability mass function at zero can be found,
approximately, by
P(I = 0) = lim Φ ( iω) & P(M = 0) = lim Φ+(iω).
q − q
ω − ω
→∞ →∞
10 AMIRT.PAYANDEHNAJAFABADIANDDANZ.KUCEROVSKY
Using the fact that the Compound Poisson has bounded characteristic
exponent ψ(). The following formulates result of the above theorem in
·
terms of the jumps measure ν(dx).
Theorem 3.4. (Compound Poisson) Suppose X is a Compound Poisson
t
process defined by a triple (µ,σ,ν). Moreover, suppose that
A ): the stopping time τ(q) is either a geometric or an exponential
1
distribution with parameter q independent of X and τ(0) = ;
t
∞
A ): the ν (dx) are a sequence of the density functions which converge
2 n
in L2(R), to jumps measure ν and 1 xν (dx) = 1 xν(dx).
1 n 1
Then, the density functions of the supremRa−and infima Ro−f the Compound
Poisson process X , denoted by f+() and f (), respectively, can be approx-
t q · q− ·
imated by a sequence of the density functions f+ () and f () where:
q,n · q−,n ·
i): For exponentially distributed stopping time τ(q),
1 3
f f ν ν 2+ ν ν ;
|| q±− q±,n||2 ≤ q2√8π|| n − ||2 2q|| n− ||2
ii): For geometric stopping time τ(q),
(1 q)2 3(1 q)
||fq±−fq±,n||2 ≤ q2−√8π ||νn −ν||22+ 2−q ||νn −ν||2.
Proof. Suppose ψ () are sequence of the characteristic exponent cor-
n
·
responding to ν (dx). For part (i) using result of Theorem (3.2), one may
n
conclude that
1 q q 3 q q
Φ Φ 2+
|| ±n − ±||2 ≤ 2||q ψ − q ψ||2 2||q ψ − q ψ||2
n n
− − − −
1 3
ψ ψ 2 + ψ ψ
≤ 2q2|| n− ||2 2q|| n − ||2
1 3
ν ν 2+ ν ν .
≤ 4πq2|| n− ||2 q√8π|| n − ||2
The second inequality arrives from the fact that the characteristic function
q/(q ψ()) is bounded above by 1, while the third inequality comes from
− ·
the Levy-Khintchine representation (Equation, 1.1) along with conditions
A and the Hausdorff-Young Theorem. The rest of proof arrives from an
2
application of the Hausdorff-Young Theorem. The proof of part (ii) is quite
similar. (cid:3)
4. Application to the finite (infinite)-time ruin probability
Suppose surplus process of an insurance company can be restated as
(4.1) U = u+X ,
t t
where L´evy process X and u > 0 stands for initial wealth/reserve of the
t
process.