Elect. Comm. in Probab. 14 2009, 132–142
ELECTRONIC COMMUNICATIONS
in PROBABILITY
SMALL TIME EXPANSIONS FOR TRANSITION PROBABILITIES OF SOME LÉVY PROCESSES
PHILIPPE MARCHAL
1
CNRS and DMA, Ecole Normale Supérieure, 45 rue d’Ulm 75230 Paris Cedex 05, France. email: Philippe.Marchalens.fr.
Submitted December 1, 2008, accepted in final form February 18, 2009 AMS 2000 Subject classification: 60J30
Keywords: Lévy process, transition probability
Abstract We show that there exist Lévy processes X
t
, t ≥ 0 and reals y 0 such that for small t, the
probability PX
t
y has an expansion involving fractional powers or more general functions of t. This constrats with previous results giving polynomial expansions under additional assumptions.
1 The Brownian case
1.1 Main result
Let X
t
, t ≥ 0 be a real-valued Lévy process with Lévy measure Π and let y 0. It is well-known
see for example [B], Chapter 1 that when t → 0,
P X
t
≥ y ∼ tΠ y 1
whenever Π y 0 and Π is is continuous at y, where Π stands for the tail of Π: for every z 0,
Πz = Π[z, ∞ It has been proved that under additional assumptions, which in particular include the smoothness
of Π, one gets more precise expansions of the probability PX
t
≥ y and that these are polynomial in t. See [L, P, RW, FH2] among others.
The problem of relating Π to the marginals of the process have several applications. The paper [RW], as well as [FH1], is concerned with problems of mathematical finance. Applications of
statistical nature can be found in [F]. From a more theoretical point of view, this relation plays an important role when studying small-time behaviour of Lévy processes, which involves fine
properties of the Lévy measure see for instance Section 4 in [BDM]. Our goal is to exhibit some examples where this expansion involves more general functions of t,
such as fractional powers, powers of the logarithm and so on. We shall focus on the case when X
1
WORK UPPORTED BY ANR, GRANT ATLAS
132
Expansions for Lévy processes 133
has the form X
t
= S
t
+ Y
t
where Y
t
, t ≥ 0 is a compound Poisson process with Lévy measure Π
and S
t
, t ≥ 0 is a stable process, S and Y being independent. Assume first that
X
t
= B
t
+ Y
t
where B
t
, t ≥ 0 is a standard Brownian motion. Then we have:
Theorem 1. i Suppose that Π has a continuous density f on [ y −δ, y∪ y, y +δ] for some δ 0.
Suppose that f
+
:= lim
x →0+
f y + x 6= f
−
:= lim
x →0−
f y + x Then as t
→ 0, P
X
t
≥ y − t Πz −
Π{z} 2
∼ λt
3 2
f
−
− f
+
EG 2
where G is the absolute value of a standard gaussian random variable. ii Define the functions g
−
, g
+
: R
+
→ R
+
by g
+
x = Π y, y + x g
−
x = Π y − x, y Suppose that
x
2
= o|g
+
x − g
−
x| as x
→ 0+. Then as t → 0, P
X
t
≥ y − t Πz −
Π{z} 2
∼ tE[g
−
p t G
− g
+
p t G]
2
Remarks i Suppose that for small x
0, g
+
x = a x + c x
α
| log x|
β
+ ox
α
| log x|
β
g
−
x = a x + c
′
x
γ
| log x|
δ
+ ox
γ
| log x|
δ
with the conditions that c, α, β 6= c
′
γ, δ and 1 minα, γ 2. Then x
2
= o|g
+
x − g
−
x| and the conclusion of ii applies. For example, if
α γ, this gives the estimate P
X
t
≥ y − t Πz −
Π{z} 2
∼ − cEG
α
| log G|
β
2 t
1+ α2
| log t|
β
Of course, one could take any slowly varying function instead of the logarithm. On the other hand, if Π has a density that is twice differentiable in the neighbourhood of y, then
|g
+
x − g
−
x| = Ox
2
and ii does not apply. ii For a fixed time t, adding B
t
to Y
t
has a smoothing effect on the probability measure PX
t
∈ d x. In turn, if we fix y and consider the function h
y
: t 7→ PX
t
≥ y, the effect of adding B
t
to Y
t
is counter-regularizing. Indeed, h
y
would be analytic in the absence of Brownian motion while it is not twice differentiable in the presence of Brownian motion. This is not very intuitive in our
view.
Proof of Theorem 1
134 Electronic Communications in Probability
Let λ be the total mass of Π. For every y 0 one can write
P X
t
≥ y = e
−λt
P B
t
≥ y + λte
−λt
P B
t
+ Z
1
≥ y +
λt
2
e
−λt
2 P
B
t
+ Z
1
+ Z
2
≥ y + . . .
2 where the random variables Z
n
are iid with common law λ
−1
Π. As t → 0, for every integer n ≥ 0, P
B
t
≥ y = ot
n
. Moreover, λt e
−λt
P B
t
+ Z
1
≥ y = λtPB
t
+ Z
1
≥ y + Ot
2
Hence, as t → 0,
P X
t
≥ y = λtPB
t
+ Z
1
≥ y + Ot
2
Since PB
t
+ Z
1
≥ y = PZ
1
≥ y − B
t
, we have P
B
t
+ Z
1
≥ y = λ
−1
Π y + PZ
1
∈ [ y − B
t
, y, B
t
−PZ
1
∈ [ y, y + |B
t
|, B
t
The stability property B
t d
= p
t B
1
entails P
B
t
+ Z
1
≥ y − λ
−1
Π y = 1
2
P Z
1
∈ [ y − p
t G, y − PZ
1
∈ [ y, y + p
t G
where G is the absolute value of a standard gaussian random variable. Under the assumptions of i, as t
→ 0, P
Z
1
∈ [ y − p
t G, y = λ
−1
f
−
p tEG + o
p t
and P
Z
1
∈ [ y, y + p
t G = λ
−1
Π{ y} + λ
−1
f
+
p tEG + o
p t
Therefore P
B
t
+ Z
1
≥ y − λ
−1
Πz − Π{z}
2 =
λ
−1
f
−
p tEG
− f
+
p tEG
2
+ o p
t and, together with 2, this entails i. The proof of ii is similar. Remark that proving ii does
not involve the existence of the expectation EG.
1.2 Additional remarks
