Distributional properties of exponential functionals of Levy processes

0.41 MB·

by  A. Kuznetsov

Distributional properties of exponential
functionals of L´evy processes
A. Kuznetsov , J. C. Pardo , M. Savov
∗ † ‡
2
1
This version: January 26, 2013
0
2

n
a
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Abstract
7
2 ∞
We study the distribution of the exponential functional I(ξ,η) = exp(ξ )dη , where
0 t− t
] ξ and η are independent L´evy processes. In the general setting, using the theory of Markov
R R
processes and Schwartz distributions, we prove that the law of this exponential functional sat-
P
isfies an integral equation, which generalizes Proposition 2.1 in [9]. In the special case when
.
h
η is a Brownian motion with drift, we show that this integral equation leads to an important
t
a functional equation for the Mellin transform of I(ξ,η), which proves to be a very useful tool
m
for studying the distributional properties of this random variable. For general L´evy process ξ
[
(η being Brownian motion with drift) we prove that the exponential functional has a smooth

2 density on R 0 , but surprisingly the second derivative at zero may fail to exist. Under the
v \{ }
additional assumption that ξ has some positive exponential moments we establish an asymp-
5
6 totic behaviour of P(I(ξ,η) > x) as x + , and under similar assumptions on the negative
→ ∞
3 exponential moments of ξ we obtain a precise asymptotic expansion of the density of I(ξ,η) as
6
x 0. Under further assumptions on the L´evy process ξ one is able to prove much stronger
.
5 →
results about the density of the exponential functional and we illustrate some of the ideas and
0
1 techniques for the case when ξ has hyper-exponential jumps.
1
v: Keywords: L´evy processes, exponential functional, integral equations, Mellin transform, asymp-
i totic expansions.
X
r AMS 2000 subject classifications: 60G51.
a
Submitted to EJP on June 30 2011, final version accepted December 1 2011.
∗Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3,
Canada. Email: [email protected]. ResearchsupportedbytheNaturalSciencesandEngineeringResearch
Council of Canada.
†Centro de Investigaci´on en Matema´ticas A.C. Calle Jalisco s/n. 36240 Guanajuato, M´exico. Email: jc-
[email protected]. Research supported by CONACYT.
‡New College, Holywell Street, Oxford, OX1 3BN, UK. Email: [email protected]; [email protected]
1
1 Introduction
In this paper, we are interested in studying distributional properties of the random variable
∞
I(ξ,η) := eξt−dη , (1.1)
t
Z0
where ξ and η are independent real-valued L´evy processes such that ξ drifts to and E[ ξ ] <
1
−∞ | | ∞
and E[ η ] < .
1
| | ∞
The exponential functionals I(ξ,η) appear in various aspects of probability theory. They describe
the stationary measure of generalized Ornstein-Uhlenbeck processes and the entrance law of positive
self-similar Markovprocesses, see[6,9]. Theyalsoplayaroleinthetheoryoffragmentationprocesses
and branching processes, see [4, 22]. Besides their theoretical value, the exponential functionals are
very important objects in Mathematical Finance and Insurance Mathematics. They are related to
Asianoptions, present valuesofcertainperpetuities, etc., see[10,17, 14] forsomeparticular examples
and results.
In general, the distribution of exponential functionals is difficult to study. It is known explicitly
only in some very special cases, see [8, 14, 19]. Properties of the distribution of I(ξ,η) are also of
particular interest. Lindner and Sato [26] show that the density of I(ξ,η) doesn’t always exist, and
in the special case when ξ and η are specific compound Poisson processes, distributional properties
of I(ξ,η) can be related to the problem of absolute continuity of the distribution of Bernoulli con-
volutions, which dates back to Erd˝os, see [12]. The distribution of I(ξ,η), when ξ = s and in
s
−
some other instances, is known to be self-decomposable and hence absolutely continuous, see [5, 18].
When η is a subordinator with a strictly positive drift, the law of the exponential functional I(ξ,η)
is absolutely continuous, see Theorem 3.9 in Bertoin et al. [5]. Some further results are obtained in
[24, 29, 30, 35].
The asymptotic behaviour P(I(ξ,η) > x), as x , is a question which has attracted the
→ ∞
attention of many researchers. In the general case, but under rather stringent requirements on the
existence of exponential moments for ξ and absolute moments for η, it has been studied in [25]. The
special case when η = t has been considered in [27, 31, 32] and properties of the density of the law
t
of I(ξ,η) at zero and infinity have been studied by [19, 21, 28] and results such as asymptotic and
convergent series expansions for the density have been obtained.
The first objective of this paper is to develop a general integral equation for the law of I(ξ,η)
under the assumptions that E[ ξ ] < , E[ξ ] < 0, E[ η ] < and ξ being independent of η.
1 1 1
| | ∞ | | ∞
Using the fact that in general I(ξ,η) is a stationary law of a generalized Ornstein-Uhlenbeck process,
Carmona et al. [9] show that if ξ has jumps of bounded variation and η = t then the law of
t
I(ξ,η) satisfies a certain integral equation. We refine and strengthen their approach and using both
stationarity properties of I(ξ,η) and Schwartz theory of distributions, we show that in the general
setting the law of I(ξ,η) satisfies a certain integral equation. This equation is important on its own
right, as demonstrated by Corollary 1, but it is also amenable to different useful transformations as
can be seen from the discussion below.
The second mainobjective of thepaper is tostudy someproperties of I := I(ξ,η)in thespecific
µ,σ
case when η = µs + σB , where B is a standard Brownian motion. Quantities of this type have
s s s
already appeared in the literature, see [14], but have not been thoroughly studied. The latter, as it
seems to us, is due to the lack of suitable techniques, which are available in the case when η = s,
s
and in particular due to the lack of any information about the Mellin transform of I(ξ,η), which is
the key tool for studying the properties of I(ξ,η), see [19, 21, 27]. We use the integral equation (2.3)
2
and combine techniques from special functions, complex analysis and probability theory to study the
Mellin transform of I , which is defined as (s) = E (I )s−11 . In particular we derive
µ,σ M µ,σ {Iµ,σ>0}
an important functional equation for (s), see (3.13), and study the decay of (s) as Im(s) .
M (cid:2) (cid:3) M → ∞
These results supply us with quite powerful tools for studying the properties of the density of I via
µ,σ
the Mellin inversion. Furthermore, the functional equation (3.13) allows for a meromorphic extension
of (s) when ξ has some exponential moments. This culminates in very precise asymptotic results
M
for P(I > x), as x , see Theorem 5, and asymptotic expansions for k(x), the density of
µ,σ
→ ∞
I , as x 0, see Theorem 4. The latter results show us that while k(x) C∞(R 0 ), rather
µ,σ
→ ∈ \ { }
unexpectedly k′′(0) may not exist. Finally, we would like to point out that while the behaviour of
P(I > x), as x , might be partially studied via the fact that I solves a random recurrence
µ,σ µ,σ
→ ∞
equation, see for example [25], the behaviour of k(x), as x 0, seems for the moment to be only
→
tractable via our approach based on the Mellin transform.
As another illustration of possible applications of our general results, we study the density of I
µ,σ
when ξ has hyper-exponential jumps (see [7, 8, 20]). This class of processes is quite important for
applications in Mathematical Finance and Insurance Mathematics, and it is particularly well suited
for investigation using our methods due to the rich analytical structure enjoyed by these processes.
In this case we show how to derive complete asymptotic expansions of k(x) both at zero and infinity.
We point out that our methodology is not restricted to this particular case, and can be easily applied
to more general classes of L´evy processes.
The paper is organizedas follows: inSection 2, we study the lawof I(ξ,η)for general independent
L´evyprocesses ξ andη andderiveanintegralequationforthelawofI(ξ,η); inSection3, wespecialize
the results obtained in Section 2 to the case when η = µs+σB and, employing additionally various
s s
techniques fromspecial functions and complex analysis, we study the properties of the density of I .
µ,σ
Section 4 is devoted to some applications of the results derived in the previous section. In particular,
we study the asymptotic behaviour at infinity of the tail of I and of its density at zero, and in
µ,σ
the case of processes with hyper-exponential jumps, we show how these results can be considerably
strengthened.
2 Integral equation satisfied by the law of I(ξ,η)
Let us introduce some notation which will be used throughout this paper. The main underlying
objects are two independent L´evy processes ξ and η defined on a probability space (Ω, ,P). As is
F
standard, we assume that both processes are started from zero under the probability measure P.
Assumption 1. Everywhere in this paper we will assume that
E[ ξ ] < , E[ξ ] < 0, E[ η ] < . (2.1)
1 1 1
| | ∞ | | ∞
The characteristics of the L´evy processes ξ and η will be denoted by (b ,σ ,Π ) and (b ,σ ,Π ).
ξ ξ ξ η η η
In particular Π (dx) and Π (dx) are the L´evy measures of ξ and η, respectively. We use the following
ξ η
notation for the double-integrated tail
∞ ∞
(+) (−)
Π (x) = Π ((y, ))dy and Π (x) = Π (( , y))dy,
ξ ξ ∞ ξ ξ −∞ −
Zx Zx
(+) (−)
and similarly for Π and Π . Using the L´evy-Itoˆ decomposition (see Theorem 2.1 in [23]) it is
η η
easy to check that Assumption 1 implies that the above quantities are finite for all x > 0.
3
We define the Laplace exponents ψ (z) = ln E ezξ1 and ψ (z) = ln(E[ezη1]), where without
ξ η
any further assumptions ψ and ψ are defined at least for Re(z) = 0, see [3, Chapter I]. The Laplace
ξ η
(cid:0) (cid:2) (cid:3)(cid:1)
exponent ψ can be expressed in the following two equivalent ways
ξ
σ2
ψ (z) = ξz2 +b z + (ezx 1 zx)Π (dx) (2.2)
ξ ξ ξ
2 − −
R
Z
σ2 ∞ (+) ∞ (−)
= ξz2 +b z +z2 Π (w)exzdx+ Π (x)e−xzdx ,
2 ξ ξ ξ
(cid:18)Z0 Z0 (cid:19)
with a similar expression for ψ . The first equality in (2.2) is essentially the L´evy-Khintchine formula
η
(see Theorem 1 in [3]) with the cutoff function h(x) 1. The standard choice for the cutoff function
≡
in the L´evy-Khintchine formula would be 1 , however it is well-known that if E[ ξ ] < then
{|x|<1} 1
| | ∞
we can take a simpler cutoff function h(x) 1. The second equality in (2.2) follows easily by
≡
repeated integration by parts. Note that according to (2.2), we have b = ψ′(0) = E[ξ ] and similarly
ξ ξ 1
b = ψ′(0) = E[η ].
η η 1
We recall that the exponential functional I(ξ,η) is defined by (1.1), its law will be denoted by
m(dx) := P(I(ξ,η) dx). The density of I(ξ,η), provided it exists, will be denoted by k(x).
∈
Our main result in this section is the derivation of an integral equation for the law of I(ξ,η).
This equation will be very useful later, when we’ll derive the functional equation (3.13) for the Mellin
transform of the exponential functional in the special case when η is a Brownian motion with drift.
The main idea of this Theorem comes from Proposition 2.1 in [9].
Theorem 1. Assume that condition (2.1) is satisfied. Then the exponential functional I(ξ,η) is well
defined and its law satisfies the following integral equation: for v > 0
∞ σ2 ∞ (−) x v (+) v
ξ
b m(dx) dv+ vm(dv)+ Π ln m(dx) dv+ Π ln m(dx) dv
ξ 2 ξ v ξ x
(cid:18) Zv (cid:19) (cid:18)Zv (cid:16) (cid:17) (cid:19) (cid:18)Z0 (cid:16) (cid:17) (cid:19)
∞ m(dx) σ2m(dv) σ2 ∞ m(dx)
η η
+ b dv + dv
η x 2 v − 2 x2
(cid:18) Zv (cid:19) (cid:18) Zv (cid:19)
1 v (+) 1 ∞ (−)
+ Π (v x)m(dx) dv + Π (x v)m(dx) dv (2.3)
v η − v η −
(cid:18) Z0 (cid:19) (cid:18) Zv (cid:19)
∞ 1 w (+) ∞ 1 ∞ (−)
Π (w x)m(dx)dw dv Π (x w)m(dx)dw dv = 0,
− w2 η − − w2 η −
(cid:18)Zv Z0 (cid:19) (cid:18)Zv Zw (cid:19)
where all quantities in (2.3) are a.e. finite. Equation (2.3) for the law of I(ξ, η) on (0, ) describes
− ∞
m(dx) on ( ,0).
−∞
The proof of Theorem 1 is based on the so-called generalized Ornstein-Uhlenbeck (GOU) process,
which is defined as
t t
U = U (ξ,η) = xeξt +eξt e−ξs−dη =d xeξt + eξs−dη , for t > 0. (2.4)
t t s s
Z0 Z0
Note that the GOU process is a strong Markov process, see [9, Appendix 1]. Lindner and Maller
[25] have shown that the existence of a stationary distribution for the GOU process is closely related
to the a.s. convergence of the stochastic integral teξs−dη , as t . Necessary and sufficient
0 s → ∞
R
4
conditions for the convergence of I(ξ,η) were obtained by Erickson and Maller [13]. More precisely,
they showed that this happens if and only if
log y
lim ξ = a.s. and | | Π (dy) < . (2.5)
t→∞ t −∞ ZR\[−e,e]"1+ 1log|y|∨1Πξ(R\(−z,z))dz# η ∞
R
It is easy to see that Assumption 1 implies (2.5). Hence I(ξ,η) is well-defined and the stationary
d
distribution satisfies U = I(ξ,η). This identity in distribution is the starting point of the proof of
∞
Theorem 1.
As the proof of Theorem 1 is rather long and technical, we will divide it into several steps. We
first compute the generator of U, here denoted by L(U). This result may be of independent interest,
therefore we present it in Proposition 1 below. Then we note that the stationary measure m(dx)
satisfies the equation
∞
L(U)f(x)m(dx) = 0, (2.6)
Z0
where f is any infinitely differentiable function with a compact support in (0, ). Indeed, (2.6)
∞
follows from (2.1) in [9] or from the definition of infinitesimal generator and the observation that, for
all t 0,
≥ ∞ ∞
E[f(U )]m(dx) = f(x)m(dx).
t
Z0 Z0
Finally, an application of Schwartz theory of distributions after rephrasing (2.6) gives (2.3).
We start by working out how the infinitesimal generator of U, i.e. L(U), acts on functions in
C (R), where
0
K ⊂
= f(x) : f(x) C2(R), f(ex) C2(R) C (R)
K ∈ b ∈ b ∩ 0
f(x) = 0, for x 0; f′(0) = f′′(0) = 0 (2.7)
(cid:8) (cid:9)
∩{ ≤ }
and C2(R) stands for two times differentiable, bounded functions with bounded derivatives on R and
b
C (R) is the set of continuous functions vanishing at . Denote by L(ξ) and L(η) ( resp. Dξ and
0
±∞
Dη) the infinitesimal generators (resp. domains) of ξ and η. Note that
σ2
L(ξ)f(x) = b f′(x)+ ξf′′(x)+ (f(x+y) f(x) yf′(x))Π (dy) (2.8)
ξ ξ
2 − −
R
Z
σ2 (+) (−)
= b f′(x)+ ξf′′(x)+ f′′(x+w)Π (w)dw+ f′′(x w)Π (w)dw,
ξ 2 ξ − ξ
R R
Z + Z +
with a similar expression for L(η). The first formula in (2.8) is a trivial modification of the form of the
generator of L´evy processes for the case when the cutoff function is h(x) 1, see [3, p. 24], whereas
≡
the second expression follows easily by integration by parts, the fact that f and E[ ξ ] < .
1
∈ K | | ∞
Finally, we are ready to state our result, which should strictly be seen as an extension of Proposition
5.8 in [9] where the generator L(U) has been derived under very stringent conditions.
5
Proposition 1. Assume that condition (2.1) is satisfied. Let f , g(x) := (xf′(x)) and φ(x) :=
∈ K
f(ex). Then, f Dη, φ Dξ and
∈ ∈
L(U)f(x) = L(ξ)φ(lnx)+L(η)f(x)
σ2 x (−) x ∞ (+) v
= b g(x)+ ξxg′(x)+ g′(v)Π ln dv + g′(v)Π ln dv
ξ 2 ξ v ξ x
Z0 (cid:16) (cid:17) Zx (cid:16) (cid:17)
σ2 ∞ (+) ∞ (−)
+b f′(x)+ ηf′′(x)+ f′′(x+w)Π (w)dw+ f′′(x w)Π (w)dw. (2.9)
η 2 η − η
Z0 Z0
Proof. The main idea is to use the definition of the infinitesimal generator and Itˆo’s formula. Let
f and note that by definition
∈ K
E [f(U )] f(x) 1 t
L(U)f(x) = lim x t − = lim E f xeξt + eξs−dη f(x) .
s
t→0 t t→0 t −
(cid:18) (cid:20) (cid:18) Z0 (cid:19)(cid:21) (cid:19)
Using the fact that (U ) is a semimartingale and f , we apply Itˆo’s formula to f(U ) to obtain
t t≥0 ∈ K t
t 1 t
f(U ) f(x) = f′(U )dU + f′′(U )d[U,U]c+ (f(U ) f(U ) ∆U f′(U )). (2.10)
t − s− s 2 s− s s − s− − s s−
Z0 Z0 s≤t
X
Now, let H := eξt and V := x+ te−ξs−dη , and note that U = H V . Hence by integration by parts
t t 0 s t t t
R t t
U = x+ H dV + V dH +[H,V] .
t s− t s− s t
Z0 Z0
Using the L´evy-Itoˆ decomposition (see Theorem 2.1 in [23]) and Assumption 1, we find that the
L´evy processes ξ and η can be written as follows
ξ = σ B +b t+X , η = σ W +b t+Y , (2.11)
t ξ t ξ t t η t η t
where B and W are Brownian motions, X and Y are pure jump zero mean martingales, and the
processes B,W,X and Y are mutually independent. Then we get
t t
V = x+b e−ξs−ds+σ e−ξs−dW +N ,
t η η s t
Z0 Z0
where N = te−ξs−dY is a pure jump local martingale. On the other hand using Itˆo’s formula, we
t 0 s
have
R
t 1 t
H = eξt = 1+ eξs−dξ + eξs−d[ξ,ξ]c + eξs−(e∆ξs ∆ξ 1)
t s 2 s − s −
Z0 Z0 s≤t
X
σ2 t t
= 1+ b + ξ eξs−ds+σ eξs−dB +N + eξs− e∆ξs ∆ξ 1 ,
ξ ξ s t s
2 − −
(cid:18) (cid:19)Z0 Z0 Xs≤t (cid:16) (cid:17)
e
where N = teξsdX is a pure jump local martingale. Therefore, we conclude that
s 0 s
R t t
e [H,V] = σ e−ξs−dB ,σ e−ξs−dW + ∆V ∆H = 0 a.s.,
t ξ s η s s s
(cid:20) Z0 Z0 (cid:21)t s≤t
X
6
since ∆V = e−ξs−∆η , ∆H = H (e∆ξs 1) and the fact that ξ and η are independent and do not
s s s s−
−
jump simultaneously a.s. This implies that
t t t t
U = x+ H dV + V dH = x+ eξs−dV + V dH .
t s− s s− s s s− s
Z0 Z0 Z0 Z0
Using the expressions of H and V, we deduce that
t σ2 t
U = x+b t+σ W + eξs−dN + b + ξ V eξs−ds
t η η t s ξ s−
2
Z0 (cid:18) (cid:19)Z0
t t
+σ V eξs−dB + V dN + V eξs− e∆ξs ∆ξ 1
ξ s− s s− s s− s
− −
Z0 Z0 Xs≤t (cid:16) (cid:17)
σ2 t e
= x+K +Kc +b t+ b + ξ U ds+ U e∆ξs ∆ξ 1 ,
t t η ξ 2 s− s− − s −
(cid:18) (cid:19)Z0 s≤t
X (cid:0) (cid:1)
where
t t t
K = eξs−dN + V dN = Y + U dX ,
t s s− s t s− s
Z0 Z0 Z0
t
Kc = σ W +σ U dB . e
t η t ξ s− s
Z0
From the definition of K and Kc, and the mutual independence of B, W, N and N, we get for the
continuous part of the quadratic variation of U
e
t
[U,U]c = [Kc,Kc] = σ2t+σ2 U2 ds.
t t η ξ s−
Z0
Putting all the pieces together in identity (2.10), we have
t σ2 t
f(U ) f(x) = M +b f′(U )ds+ b + ξ f′(U )U ds
t t η s− ξ s− s−
− 2
Z0 (cid:18) (cid:19)Z0
σ2 t σ2 t
+ f′(U )U e∆ξs ∆ξ 1 + η f′′(U )ds+ ξ f′′(U )U2 ds
s− s− − s − 2 s− 2 s− s−
Xs≤t (cid:16) (cid:17) Z0 Z0
+ (f(U ) f(U ) ∆U f′(U ))
s s− s s−
− −
s≤t
X
where M is a local martingale starting from 0 and M describes the integration with respect to K
and Kc in the expressions above. Using the fact that f implies f(x) = 0 for x < 0 and
∈ K
x f′(x) + x2 f′′(x) < C(f) < , we deduce that M is a proper martingale as all other terms in
t
| | | | ∞
the expression above have a finite absolute first moment. Furthermore applying the compensation
formula to the jump part of f(U ) we get
t
t
E f′(U )U e∆ξs ∆ξ 1 = E f′(U )U (ey y 1)Π (dy) ds .
s− s− s s− s− ξ
− − − −
"s≤t # (cid:20)Z0 (cid:18)Zy∈R (cid:19) (cid:21)
X (cid:0) (cid:1)
7
Similarly, using the fact that ∆U = ∆η when ∆η = 0 and ∆U = U (e∆ξs 1) when ∆ξ = 0
s s s s s− s
6 − 6
(see the definition of U) we get
t
E (f(U ) f(U ) ∆U f′(U )) = E (f(U +z) f(U ) zf′(U ))Π (dz)ds
s s− s− s− s− s− s− η
− − − −
"s≤t # (cid:20)Z0 Zz∈R (cid:21)
X
t
+E f(U ey) f(U ) ey 1 f′(U )U Π (dy)ds .
s− s− s− s− ξ
− − −
(cid:20)Z0 Zy∈R(cid:16) (cid:16) (cid:17) (cid:17) (cid:21)
Finally, as f , we derive
∈ K
t σ2 t σ2 t
E f(U ) f(x) = b E f′(U )ds + b + ξ E f′(U )U ds + ηE f′′(U )ds
t η s− ξ s− s− s−
− 2 2
h i (cid:20)Z0 (cid:21) (cid:18) (cid:19) (cid:20)Z0 (cid:21) (cid:20)Z0 (cid:21)
σ2 t t
+ ξE f′′(U )U2 ds +E f(U +z) f(U ) zf′(U ) Π (dz)ds
2 s− s− s− − s− − s− η
(cid:20)Z0 (cid:21) (cid:20)Z0 Zz∈R(cid:16) (cid:17) (cid:21)
t
+E f(U ey) f(U ) yf′(U )U Π (dy)ds .
s− s− s− s− ξ
− −
(cid:20)Z0 Zy∈R(cid:16) (cid:17) (cid:21)
˜
and dividing by t, letting t go to 0 and recalling that U = x a.s., we obtain for f the identity
0
∈ K
σ2 σ2 σ2
L(U)f(x) = b f′(x)+ b + ξ xf′(x)+ ηf′′(x)+ ξf′′(x)x2
η ξ
2 2 2
(cid:18) (cid:19)
+ f(x+z) f(x) zf′(x) Π (dz)+ f(xey) f(x) yxf′(x) Π (dy),(2.12)
η ξ
− − − −
Zz∈R(cid:16) (cid:17) Zy∈R(cid:16) (cid:17)
and therefore the infinitesimal generator of U satisfies
L(U)f(x) = L(ξ)φ(lnx)+L(η)f(x).
In order to finish the proof one only has to apply integration by parts.
The following Lemma will also be needed for our proof of Theorem 1.
Lemma 1. Assume that condition (2.1) is satisfied. Let ν(dv) denote the measure in the left-hand
side of formula (2.3). Then ν (dv) and hence ν(dv) define finite measures on any compact subset of
| |
(0, ) and for any a > 0
∞
lim z−1 ν ((a,z)) = 0. (2.13)
z→∞ | |
Proof. We only need to prove (2.13), as the finiteness of ν (dv) on compact subsets of (0, ) follows
| | ∞
from (2.13). It is sufficient to show the claims for 1 a > 0. We integrate every term on the
≥
left-hand side of (2.3) from a to z and then divide by z. This shows that the limit goes to zero, as
z . We first note that
→ ∞
z z m(dx) ∞
lim z−1 xm(dx) = 0 and lim z−1 lim(az)−1 m(dx) = 0.
z→∞ z→∞ x ≤ z→∞
Za Za Za
Hence,
z ∞ z ∞
lim z−1 m(dx)dv lim z−1 xm(dx)+ m(dx) = 0,
z→∞ ≤ z→∞
Za Zv (cid:18) Za Zz (cid:19)
8
z ∞ m(dx) z ∞ m(dx)
lim z−1 dv = 0 and lim z−1 dv = 0.
z→∞ x z→∞ x2
Za Zv Za Zv
So far, we have checked that the terms in (2.3) that do not depend on the tail of the L´evy measure
vanish under the transformation we made, as z . Now, we turn our attention to the terms that
→ ∞
involve the L´evy measure of ξ. When we’ll be dealing with these integrals, the main trick that we
will use is to change the order of integration. First, we check that
z ∞ (−) x
limsupz−1 Π ln m(dx)dv
ξ v
z→∞ Za Zv z ev (cid:16)(−) (cid:17) x (−) z
limsupz−1 Π ln m(dx)dv +limsupz−1 Π (1) m(ev, )dv
≤ ξ v ξ ∞
z→∞ (cid:18)Zza Zevv (−) (cid:16) x (cid:17) (cid:19) z→∞ ez(cid:18) x (−)Za x (cid:19)
= limsupz−1 Π ln m(dx)dv limsupz−1 Π ln dvm(dx)
ξ v ≤ ξ v
z→∞ Za Zv (cid:16) (cid:17) z→∞ Za Zx/e (cid:16) (cid:17)
1 ez
(−)
= Π (w)e−wdw limsupz−1 xm(dx) = 0
ξ ×
(cid:20)Z0 (cid:21) z→∞ Za
where we have applied Fubini’s Theorem, a change of variables w = ln(x/v) and we have used
(−)
the finiteness of E[ ξ ] and henceforth the finiteness of the quantities 1Π w exp( w)dw and
| 1| 0 ξ −
(+)
Π (1). R (cid:0) (cid:1)
ξ
(+)
Next using Fubini’s Theorem and the monotonicity of Π , we note that for any positive number
ξ
b,
z v (+) v
limsupz−1 Π ln m(dx)dv
ξ x
z→∞ Za Z0 (cid:18) (cid:19)
z v (+) v z ln(z/x) (+)
limsupz−1 Π ln m(dx)dv = limsupz−1 x Π w ewdwm(dx)
≤ ξ x ξ
z→∞ Z0 Z0 (cid:16) (cid:17) z→∞ Z0 Z0
b z z ln(z/x)∨b (cid:0) (cid:1)
(+) (+) (+)
limsupz−1 Π w ewdw xm(dx)+ x Π w ewdwm(dx) Π b .
≤ ξ ξ ≤ ξ
z→∞ Z0 Z0 Z0 Zb !
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
(+)
Since Π (b) decreases to zero as b increases, we see that
z v (+) v
lim z−1 Π ln m(dx)dv = 0.
z→∞ ξ x
Za Z0
(cid:16) (cid:17)
Since η has a finite mean and m is a finite measure
z 1 v (+)
limsupz−1 Π (v x)m(dx)dv
v η −
z→∞ Za Z0
(+) z z 1 v (+)
limsupz−1 Π (a)ln + Π (v x)m(dx)dv
≤ η a v η −
z→∞ (cid:18) (cid:16) (cid:17) Za Zv−a (cid:19)
z (x+a)∧z (+) dv
= limsupz−1 m(dx) Π (v x)
η − v
z→∞ Z0 Za∨x !
a z
(+)
Π (s)ds lim(az)−1 m(dx) = 0.
≤ η ×z→∞
(cid:20)Z0 (cid:21) Z0
9
Similarly, we estimate the following integral
z ∞ 1 w (+)
limsupz−1 Π (w x)m(dx)dwdv
w2 η −
z→∞ Za Zv Z0
(+) z z ∞ 1 w (+)
limsupz−1 Π (a)ln + Π (w x)m(dx)dwdv
≤ η a w2 η −
z→∞ (cid:18) (cid:16) (cid:17) Za Zv Zw−a (cid:19)
z ∞ x+a 1 (+)
= limsupz−1 Π (w x)dwm(dx)dv
w2 η −
z→∞ Za Zv−aZv∨x
a (+) z 1
Π (s)ds limsupz−1 m(v a, )dv = 0.
≤ η × v2 − ∞
(cid:20)Z0 (cid:21) z→∞ Za
As for the remaining two integrals, we split the innermost integrals at the point x = v+a so that
(−) (−)
Π (x v) = Π (a) and similarly estimate the resulting two terms to get
η − η
z 1 ∞ (−) z ∞ 1 ∞ (−)
limsupz−1 Π (x v)m(dx)dv = limsupz−1 Π (x w)m(dx)dwdv = 0.
v η − w2 η −
z→∞ Za Zv z→∞ Za Zv Zw
Thus, we verify (2.13) and conclude the proof of Lemma 1.
Now that we have established Proposition 1 and Lemma 1, we are ready to complete the proof of
Theorem 1.
Proof of Theorem 1. Take an infinitely differentiable function f with compact support in (0, ) and
let g(x) := xf′(x). We use (2.6), (2.9), and the identity g(x) = xg′(v)dv to get, ∞
0
∞ ∞ σ2 ∞ R
L(ξ)φ(lnx)m(dx) = b g(x)m(dx)+ ξ xg′(x)m(dx)
ξ
2
Z0 Z0 Z0
∞ x (−) x ∞ ∞ (+) v
+ g′(v)Π ln dvm(dx)+ g′(v)Π ln dvm(dx)
ξ v ξ x
Z0 Z0 (cid:16) (cid:17) Z0 Zx (cid:16) (cid:17)
∞ ∞ ∞ σ2
= g′(v) b m(dx) dv+ g′(v) ξvm(dv)
ξ
2
Z0 (cid:18) Zv (cid:19) Z0 (cid:18) (cid:19)
∞ ∞ (−) x
+ g′(v) Π ln m(dx) dv
ξ v
Z0 (cid:18)Zv (cid:16) (cid:17) (cid:19)
∞ v (+) v
+ g′(v) Π ln m(dx) dv =: (g′,F ),
ξ x 1
Z0 (cid:18)Z0 (cid:19)
(cid:16) (cid:17)
where the interchange of integrals is permitted due to claims of Lemma 1.
Next, substituting f′(x) = g(x)/x and f′′(x) = g′(x)/x g(x)/x2, we get
−
∞ ∞ g(x) σ2 ∞ g′(x) g(x)
L(η)f(x)m(dx) = b m(dx)+ η m(dx)
η
x 2 x − x2
Z0 Z0 Z0 (cid:18) (cid:19)
∞ ∞ g′(x+w) g(x+w) (+)
+ Π (w)dwm(dx)
x+w − (x+w)2 η
Z0 Z0 (cid:18) (cid:19)
∞ ∞ g′(x w) g(x w) (−)
+ − − Π (w)dwm(dx).
x w − (x w)2 η
Z0 Z0 (cid:18) − − (cid:19)
10