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Exponential functionals of conditioned Lvy processes and local time - - PowerPoint PPT Presentation
Exponential functionals of conditioned Lvy processes and local time - - PowerPoint PPT Presentation
Exponential functionals of conditioned Lvy processes and local time of a diffusion in a Lvy environment Conference MADACA Grgoire Vchambre Universit dOrlans June 2016 Introduction Lvy process and exponential functionals
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Lévy process and exponential functionals
Let V be a Lévy process starting from 0. Exponential functional : I(V ) :=
+∞
e−V (t)dt has been intensively studied (Bertoin, Yor, ...). Hypothesis : All the jumps of V are negative (V is spectrally negative). Exponential functional : I(V ↑) :=
+∞
e−V ↑(t)dt. Questions : Finite ? Distribution tails ? Special properties ? Density ? Smoothness of the density ?
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Decomposition of a trajectory of V
L
= V R(V , 0)
I(V ) L = ST + I(V ↑), where S : subordinator, T : exponential distribution, and S, T and I(V ↑) are independent. V → +∞ a.s. ⇒ I(V ) < +∞ a.s. so I(V ↑) < +∞ a.s. P (I(V ) ≤ x) ≤ P
- I(V ↑) ≤ x
- ≤ P (I(V ) ≤ (1 + ǫ)x) /P (ST ≤ ǫx)
so P
- I(V ↑) ≤ x
- ≈ P (I(V ) ≤ x)
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Self-decomposability of I(V ↑)
T(V ,y) R(V , y) = y + V
L
y
Proposition (V. 2015, Self-decomposability)
∀y > 0, I(V ↑) L =
τ(V ↑,y)
e−V ↑(t)dt + Sy
T y + e−yI(V ↑),
where the terms in the right hand side are independent. Sy is a subordinator and T y is an exponential r.v. independent from Sy. as a consequence I(V ↑) admits a density and I(V ↑) L =
- k≥0
e−kyAy
k, avec (Ay k)k≥0 iid and Ay k L
=
τ(V ↑,y)
e−V ↑(t)dt+Sy
T y
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Exponential moments
Hypothesis (H1) : There exists γ > 0 such that E[e−γV (1)] < +∞. The study of the r.v. Ay
0 shows that under hypothesis (H1), Ay
admits exponential moments. Combining with I(V ↑) L =
k≥0 e−kyAy k we get :
Theorem (V. 2015)
(H1) ⇒ ∃λ > 0, E
- eλI(V ↑)
< +∞ Remark :
◮ if V converges toward −∞ a.s., the hypothesis (H1) is not
necessary,
◮ This behavior s different from the one known for I(V ).
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Laplace exponent
The fact that V is spectrally negative implies the existence of a function Ψv such that ∀t, λ ≥ 0, E
- eλV (t)
= etΨV (λ). The expression of ΨV is given by the Lévy-Kintchine formula : ΨV (λ) = Q 2 λ2 − γλ +
−∞
(eλx − 1 − λx1|x|<1)ν(dx), (1) where :
◮ Q > 0 and γ ∈ R are real numbers, ◮ ν is a measure such that
(1 ∧ |x|2)ν(dx) < +∞.
We see that for large λ : cλ ≤ ΨV (λ) ≤ Cλ2.
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Laplace exponent and distribution tail at 0
Hypothesis (H2-α) : ∃c, C > 0 such that cλα ≤ ΨV (λ) ≤ Cλα.
Theorem (V. 2015)
Under hypothesis (H2-α) for α > 1, there are two positive constants, K1, K2 > 0 such that for x small enough exp
- − K2
x
1 α−1
- ≤ P (I(V ) ≤ x) ≤ P
- I(V ↑) ≤ x
- ≤ exp
- − K1
x
1 α−1
- ◮ the proof uses the fact that P
- I(V ↑) ≤ x
- ≈ P (I(V ) ≤ x).
◮ upper bound : We bound the entire moments of 1/I(V )
which are known in term of ΨV .
◮ Lower bound : We study P(Ay 0 ≤ x).
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Density
We already know that the r.v. I(V ) and I(V ↑) are absolutely
- continus. Recall that
I(V ↑) L =
- k≥0
e−kyAy
k,
where Ay
k L
=
τ(V ↑,y)
e−V ↑(t)dt + Sy
T y.
If we show that for some δ > 0, E
- eiξSy
Ty
- =
O
|ξ|→+∞
- |ξ|−δ
, then E[eiξI(V ↑)] converges to 0 faster than any negative power of |ξ|, so the densities of I(V ) and I(V ↑) admit versions of class C∞. E
- eiξSy
T
- =
+∞
petΦSy (ξ)e−ptdt = p p − ΦSy(ξ) = p p −
+∞
(eiξx − 1)µSy(dx).
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Laplace exponent and density
By studying the excursions we can prove E
- eiξSy
Ty
- =
O
|ξ|→+∞
- |ξ|−δ
, under the hypothesis (H2 − α). This implies the smoothness of the density :
Theorem (V. 2015)
Under (H2 − α) for α > 1, the densities of I(V ) and I(V ↑) are C∞, all their derivatives converge to 0 at +∞ and 0. If moreover I(V ↑) admits moments of any positive order, then the density of I(V ↑) belongs to Schwartz’s space.
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Diffusions in random media
We consider the diffusion process (X(t))t≥0 which moves in a random medium given by the potential V . It is the solution of the formal SDE :
- dXt = V ′(Xt)dt + dBt
X0 = 0 (2) In the study of this diffusion, we have two randomness to take into consideration
◮ The randomness of the medium V , ◮ The randomness driving the motion in this medium.
Here, we choose for V a spectrally negative Lévy process which converges a.s. to −∞. In this case the diffusion is transient and converges to +∞. Let κ := inf{λ > 0, ΨV (λ) = 0}.
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Local time
There is a process (LX(t, y), t ≥ 0, y ∈ R) which satisfies the densities of occupation formula : ∀t ≥ 0, ∀f ∈ L∞,
t
f (Xs)ds =
- R
f (y)LX(t, y)dy. (LX(t, y), t ≥ 0, y ∈ R) is continus in time and càd-làg in space. We call it local time of the diffusion X. Supremum of local time : ∀t ≥ 0, L∗
X(t) := sup x∈R
LX(t, x).
Theorem (Andreoletti, Devulder, V. 2015, V. 2016+)
If 0 < κ < 1, V has unbounded variations and V (1) ∈ Lp (for some p > 1), L∗
X(t)/t L
→ I where I is expressed in term of I(V ↑) and I((−V )↑).
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lim sup when 0 < κ < 1
Assume that 0 < κ < 1, V has unbounded variations, V (1) ∈ Lp (for some p > 1) and V admits jumps. We link the behavior of the local time to the left tail of I(V ↑) : P
- I(V ↑) ≤ x
- ≤ exp
- − C
x
1 γ−1
- ⇒ lim sup
t→+∞
L∗
X(t)
t(log(log(t)))γ−1 ≤ C1−γ. P
- I(V ↑) ≤ x
- ≥ exp
- − C
x
1 γ−1
- ⇒ lim sup
t→+∞
L∗
X(t)
t(log(log(t)))γ−1 ≥ C1−γ. Assume that V (t) = W (t) − κ
- 2t. Then the above implications are
true with I(V ↑) replaced by I(V ↑) + ˜ I(V ↑) where ˜ I(V ↑) is an independent copy of I(V ↑).
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lim sup when 0 < κ < 1
Under (H2-α) : ∃c, C > 0 such that cλα ≤ ΨV (λ) ≤ Cλα, for α > 1 we had : ∃K1, K2 > 0, exp
- − K2
x
1 α−1
- ≤ P
- I(V ↑) ≤ x
- ≤ exp
- − K1
x
1 α−1
- Theorem (V. 2016+)
If 0 < κ < 1, V has unbounded variations, V (1) ∈ Lp (for some p > 1) and (H2-α) is satisfied then we have a.s. 0 < lim sup
t→+∞
L∗
X(t)
t(log(log(t)))α−1 < +∞. In particular if V (t) = W (t) − κ
2t then
lim sup
t→+∞
L∗
X(t)
t(log(log(t))) = 1 8.
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lim inf when 0 < κ < 1
Theorem (V. 2016+)
If 0 < κ < 1, V has unbounded variations and V (1) ∈ Lp (for some p > 1) we have a.s. 0 < lim inf
t→+∞
L∗
X(t)
t/ log(log(t)) ≤ 1 − κ κ(E[I(V ↑)] + E[I((−V )↑)]). In particular if V (t) = W (t) − κ
2t then
0 < lim inf
t→+∞
L∗
X(t)
t/ log(log(t)) ≤ (1 − κ2) 4κ .
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Bibliography
Andreoletti, Devulder, Véchambre, Renewal structure and local time for diffusions in random environment (accepted in ALEA), 2015 Véchambre, Exponential functionals of spectrally one-sided Lévy processes conditioned to stay positive (submitted), 2015 Bertoin, Yor, Exponential functionals of Lévy processes, 2005
- K. Sato, Lévy Processes and Infinitely Divisible Distributions, 1999
- J. Bertoin, Lévy processes, 1996