Exponential functionals of conditioned Lvy processes and local time - PowerPoint PPT Presentation

Exponential functionals of conditioned Lévy processes and local time of a diffusion in a Lévy environment Conference MADACA Grégoire Véchambre Université d’Orléans June 2016

Introduction Lévy process and exponential functionals Study of the functional Self-decomposability Exponential moments Distribution tail at 0 Density Diffusions in random media Diffusion and local time Local time when 0 < κ < 1

Lévy process and exponential functionals Let V be a Lévy process starting from 0. Exponential functional : � + ∞ e − V ( t ) dt I ( V ) := 0 has been intensively studied (Bertoin, Yor, ...). Hypothesis : All the jumps of V are negative ( V is spectrally negative). Exponential functional : � + ∞ e − V ↑ ( t ) dt . I ( V ↑ ) := 0 Questions : Finite ? Distribution tails ? Special properties ? Density ? Smoothness of the density ?

Decomposition of a trajectory of V L = V R(V , 0) I ( V ) L = S T + 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. � � I ( V ↑ ) ≤ x P ( I ( V ) ≤ x ) ≤ P ≤ P ( I ( V ) ≤ (1 + ǫ ) x ) / P ( S T ≤ ǫ x ) so � � I ( V ↑ ) ≤ x ≈ P ( I ( V ) ≤ x ) P

Self-decomposability of I ( V ↑ ) T(V ,y) R(V , y) L = y + V y Proposition (V. 2015, Self-decomposability) � τ ( V ↑ , y ) ∀ y > 0 , I ( V ↑ ) L e − V ↑ ( t ) dt + S y T y + e − y I ( V ↑ ) , = 0 where the terms in the right hand side are independent. S y is a subordinator and T y is an exponential r.v. independent from S y . as a consequence I ( V ↑ ) admits a density and � τ ( V ↑ , y ) I ( V ↑ ) L L e − ky A y k , avec ( A y k ) k ≥ 0 iid and A y e − V ↑ ( t ) dt + S y � = = T y k 0 k ≥ 0

Exponential moments Hypothesis (H1) : There exists γ > 0 such that E [ e − γ V (1) ] < + ∞ . The study of the r.v. A y 0 shows that under hypothesis (H1), A y 0 admits exponential moments. Combining with I ( V ↑ ) L k ≥ 0 e − ky A y = � k we get : Theorem (V. 2015) � e λ I ( V ↑ ) � ( H 1) ⇒ ∃ λ > 0 , E < + ∞ Remark : ◮ if V converges toward −∞ a.s., the hypothesis (H1) is not necessary, ◮ This behavior s different from the one known for I ( V ).

Laplace exponent The fact that V is spectrally negative implies the existence of a function Ψ v such that � e λ V ( t ) � = e t Ψ V ( λ ) . ∀ t , λ ≥ 0 , E The expression of Ψ V is given by the Lévy-Kintchine formula : � 0 Ψ V ( λ ) = Q 2 λ 2 − γλ + ( e λ x − 1 − λ x 1 | x | < 1 ) ν ( dx ) , (1) −∞ where : ◮ Q > 0 and γ ∈ R are real numbers, � (1 ∧ | x | 2 ) ν ( dx ) < + ∞ . ◮ ν is a measure such that We see that for large λ : c λ ≤ Ψ V ( λ ) ≤ C λ 2 .

Laplace exponent and distribution tail at 0 Hypothesis (H2- α ) : ∃ c , C > 0 such that c λ α ≤ Ψ V ( λ ) ≤ C λ α . Theorem (V. 2015) Under hypothesis (H2- α ) for α > 1 , there are two positive constants, K 1 , K 2 > 0 such that for x small enough � � � � − K 2 − K 1 � � I ( V ↑ ) ≤ x exp ≤ P ( I ( V ) ≤ x ) ≤ P ≤ exp 1 1 x x α − 1 α − 1 � � I ( V ↑ ) ≤ x ◮ the proof uses the fact that P ≈ P ( I ( V ) ≤ x ). ◮ upper bound : We bound the entire moments of 1 / I ( V ) which are known in term of Ψ V . ◮ Lower bound : We study P ( A y 0 ≤ x ).

Density We already know that the r.v. I ( V ) and I ( V ↑ ) are absolutely continus. Recall that � τ ( V ↑ , y ) I ( V ↑ ) L L e − V ↑ ( t ) dt + S y e − ky A y where A y � = k , = T y . k 0 k ≥ 0 If we show that for some δ > 0, e i ξ S y � � � | ξ | − δ � = O , E Ty | ξ |→ + ∞ then E [ e i ξ I ( 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 ∞ . � + ∞ p e i ξ S y � � pe t Φ Sy ( ξ ) e − pt dt = = E T p − Φ S y ( ξ ) 0 p = ( e i ξ x − 1) µ S y ( dx ) . � + ∞ p − 0

Laplace exponent and density By studying the excursions we can prove e i ξ S y � � � | ξ | − δ � = O , E Ty | ξ |→ + ∞ under the hypothesis ( H 2 − α ). This implies the smoothness of the density : Theorem (V. 2015) Under ( H 2 − 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.

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 : � dX t = V ′ ( X t ) dt + dB t (2) X 0 = 0 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 , Ψ V ( λ ) = 0 } .

Local time There is a process ( L X ( t , y ) , t ≥ 0 , y ∈ R ) which satisfies the densities of occupation formula : � t � ∀ t ≥ 0 , ∀ f ∈ L ∞ , f ( X s ) ds = f ( y ) L X ( t , y ) dy . 0 R ( L X ( 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 L X ( t , x ) . x ∈ R Theorem (Andreoletti, Devulder, V. 2015, V. 2016+) If 0 < κ < 1 , V has unbounded variations and V (1) ∈ L p (for some p > 1 ), L L ∗ X ( t ) / t → I where I is expressed in term of I ( V ↑ ) and I (( − V ) ↑ ) .

lim sup when 0 < κ < 1 Assume that 0 < κ < 1, V has unbounded variations, V (1) ∈ L p (for some p > 1) and V admits jumps. We link the behavior of the local time to the left tail of I ( V ↑ ) : � � L ∗ − C X ( t ) � � I ( V ↑ ) ≤ x t (log(log( t ))) γ − 1 ≤ C 1 − γ . P ≤ exp ⇒ lim sup 1 t → + ∞ x γ − 1 � � L ∗ − C X ( t ) � � I ( V ↑ ) ≤ x t (log(log( t ))) γ − 1 ≥ C 1 − γ . P ≥ exp ⇒ lim sup 1 x t → + ∞ γ − 1 Assume that V ( t ) = W ( t ) − κ 2 t . 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 ↑ ).

lim sup when 0 < κ < 1 Under (H2- α ) : ∃ c , C > 0 such that c λ α ≤ Ψ V ( λ ) ≤ C λ α , for α > 1 we had : � � � � − K 2 − K 1 � � I ( V ↑ ) ≤ x ∃ K 1 , K 2 > 0 , exp ≤ P ≤ exp 1 1 x x α − 1 α − 1 Theorem (V. 2016+) If 0 < κ < 1 , V has unbounded variations, V (1) ∈ L p (for some p > 1 ) and (H2- α ) is satisfied then we have a.s. L ∗ X ( t ) 0 < lim sup t (log(log( t ))) α − 1 < + ∞ . t → + ∞ In particular if V ( t ) = W ( t ) − κ 2 t then L ∗ t (log(log( t ))) = 1 X ( t ) lim sup 8 . t → + ∞

lim inf when 0 < κ < 1 Theorem (V. 2016+) If 0 < κ < 1 , V has unbounded variations and V (1) ∈ L p (for some p > 1 ) we have a.s. L ∗ X ( t ) 1 − κ 0 < lim inf t / log(log( t )) ≤ κ ( E [ I ( V ↑ )] + E [ I (( − V ) ↑ )]) . t → + ∞ In particular if V ( t ) = W ( t ) − κ 2 t then t / log(log( t )) ≤ (1 − κ 2 ) L ∗ X ( t ) 0 < lim inf . 4 κ t → + ∞

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 Véchambre, Path decompostion of a spectrally negative Lévy process, and application to the local time of a diffusion in this environment (preprint arxiv), 2016+ Véchambre, Almost sure behavior for the local time of a diffusion in a spectrally negative Lévy environment (to be submitted), 2016+