Asymptotic behaviour of exponential functionals of Lévy...

GW pro esses in random environment Lévy pro esses and CSBP CSBP in random environment Main result

Asymptoti behaviour of exponential fun tionals of

Lévy pro esses with appli ations to random

pro esses in random environment

Charline Smadi, Irstea (with Sandra Palau and Juan Carlos

Pardo, CIMAT)

May 8th, 2017

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Motivations

Aim

Study the asymptoti behaviour of E[F (It

(ξ))] as t goes to in�nity,

where F is non in reasing at in�nity and with at least a polynomial

de ay at in�nity, ξ is a Lévy pro ess, and

I

t

(ξ) =

∫t

0

exp(−ξs

)ds.

Motivations

Appli ation to the long time behaviour of bran hing pro esses in

random environment

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GW pro esses in random environment

Lévy pro esses and CSBP

CSBP in random environment

Main result

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Galton-Watson pro esses

◮Models for population dynami s in dis rete time and spa e.

◮No intera tion between individuals

Z

n

=

Z

n−1∑

i=1

ξ(n)i

where (ξ(k)j

, (j , k) ∈ N2) are iid with law Q.

4 / 40


◮Galton-Watson pro esses in iid random environment:

reprodu tion law hanges at ea h generation

5 / 40


Galton-Watson pro esses in random environment

Z

n

=

Z

n−1∑

i=1

ξ(n)i

, where (ξ(n)j

, j ∈ N) are iid with law Qn

.

◮De�ne the random walk (S

n

, n ∈ N) by

X

n

= logm(Qn

) S

n

=

n∑

k=1

X

i

where

m(Qn

) =

∞∑

y=0

yQn

({y})

◮Then Z

n

e

−S

n

is a martingale =⇒ Properties of (Zn

, n ∈ N)determined by its asso iated random walk (S

n

, n ∈ N)

6 / 40


Asymptoti behaviour of Galton-Watson pro esses in random

environment has been well studied

◮First work: Smith and Wilkinson 69, Athreya and Karlin 71

◮Reviews: Birkner, Geiger and Kersting 05, Dyakonova, Vatutin

and Sagitov 11

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Main result

8 / 40


Lévy pro esses

De�nition

Sto hasti pro ess issued from the origin with iid in rements and

almost sure right ontinuous paths

= CONTINUOUS VERSION OF RANDOM WALKS

9 / 40


Continuous State Bran hing Pro esses (CSBP)

De�nition

◮Non-negative strong Markov pro ess (Y

t

, t ≥ 0) where 0 and

∞ are two absorbing states

◮Bran hing property: if Y

(x)is the pro ess with initial state x ,

Y

(x+y) L= Y

(x) + Y

(y),

where Y

(x)and Y

(y)are independent

(Lamperti 1967)

CSBP ⇔ SCALING LIMITS OF GALTON-WATSON PROCESSES

10 / 40


Properties

The law of Y is ompletely hara terized by its Lapla e transform

Ex

e

−λYt = e

−xu

t

(λ), ∀x > 0, t ≥ 0,

where u is a di�erentiable fun tion in t satisfying

∂ut

(λ)

∂t= −Φ(u

t

(λ)), u

0

(λ) = λ.

Φ: bran hing me hanism. Given by the Lévy-Khin thine formula

Φ(λ) = −aλ+ γ2λ2 +

∫

(0,∞)

(e

−λx − 1+ λx)µ(dx),

where (a, γ) ∈ R2

, µ σ-�nite measure s.t.

∫ (x ∧ x

2

)µ(dx) <∞.

11 / 40


Can be de�ned as the unique strong solution (Fu and Li 10) of the

SDE

Y

t

= Y

0

+

∫t

0

aY

s

dz+

∫t

0

√2γ2Y

s

dB

s

+

∫t

0

∫ ∞

0

∫Y

s

−

0

zN(ds, dz , du),

where B is a standard Brownian motion, N(ds, dz , du) is a Poisson

random measure with intensity dsµ(dz)du independent of B , and

N is the ompensated measure of N.

12 / 40


Self similar CSBP: a parti ular lass

Φ(λ) = βλβ+1

, λ ≥ 0,

for some β ∈ (−1, 0) ∪ (0, 1] and β is su h that

{ β < 0 if β ∈ (−1, 0), β > 0 if β ∈ (0, 1].

Y is the unique non-negative strong solution of (Fu and Li 2010)

Y

t

= Y

0

+1β=1

∫t

0

√2 βYs

dB

s

+1β 6=1

∫t

0

∫ ∞

0

∫Y

s−

0

zN(ds, dz , du)

B Brownian motion, N Poisson random measure independent of B

with intensity ββ(β + 1)dsdzdu/(Γ(1 − β)z2+β), and N is the

ompensated Poisson measure.

13 / 40





Main result

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Self similar CSBP in a Lévy environment

◮CSBP in iid random environment? Catastrophes, ontinuous

variations,...?

◮Boeinghof and Hutzenthaler 12, Palau and Pardo 15:

Brownian motion, Bansaye, Pardo and S. 13: ompound

Poissons pro esses

◮A Lévy pro ess somewhere...

◮See Bansaye and Simatos 2015 for general results on s aling

limits of GW pro esses in random environment

◮See Palau and Pardo 2015 and He, Li and Xu 2016 for

existen e and uni ity of strong solutions

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Self similar CSBP in a Lévy environment

Unique non-negative strong solution of (Palau and Pardo 2015, He,

Li and Xu 2016)

Z

t

=Z0

+ 1β=1

∫t

0

√2 βZsdBs

+ 1β 6=1

∫t

0

∫ ∞

0

∫Z

s−

0

zN(ds, dz , du) +

∫t

0

Z

s−dSs ,

where S independent Lévy pro ess

S

t

= αt+σWt

+

∫t

0

∫

(0,1)

(ev−1)M(ds, dv)+

∫t

0

∫

R\(0,1)

(ev−1)M(ds, dv),

α ∈ R, σ ≥ 0,W Brownian motion, M Poisson random measure in

R+ × R independent of W with intensity dsπ(dy), and π σ-�nitemeasure su h that∫

R

(1 ∧ v

2)π(dv) <∞ and

∫

(−1,0)|ev − 1|π(dv) <∞.

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Random environment

S

t

= αt + σWt

+

∫t

0

∫

(0,1)(ev − 1)M(ds, dv)

+

∫t

0

∫

R\(0,1)(ev − 1)M(ds, dv)

If we de�ne the auxiliary pro ess,

K

t

=

(α− σ2

2

−∫

(0,1)(ev − 1− v)π(dv)

)t

+σWt

+

∫t

0

∫

(0,1)vM(ds, dv) +

∫t

0

∫

R\(0,1)vM(ds, dv),

then Z

t

e

−K

t

martingale =⇒ Will give the long term behaviour of

the pro ess.

17 / 40


Proposition (Bansaye, Pardo and S. 2013, Palau and Pardo

2015, He, Li and Xu 16)

For all z , λ > 0 and t ≥ 0, we have

Ez

[exp

{− λZ

t

e

−K

t

}∣∣∣K]

= exp

{−z(λ−β + β β

∫t

0

e

−βKu

du

)−1/β}.

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Proof.

We introdu e M

t

= Z

t

e

−K

t

and take G (s, x) = exp

−xv

t

(s,λ,K).

From It�'s formula, we observe that (G (s,Ms

), s ≤ t) onditionedon K is a martingale i� v

t

(s, λ,K ) satis�es

∂

∂sv

t

(s, λ,K ) = βvβ+1

t

(s, λ,K )e−βKs , v

t

(t, λ,K ) = λ.

v

t

(s, λ,K ) =

(λ−β + β β

∫t

s

e

−βKu

du

)−1/β

As (G (s,Ms

), s ≤ t) onditioned on K is a martingale,

Ez

[exp{−λMt

}|K ] = Ez

[exp{−M0

v

t

(0, λ,K )}|K ] = e

−zv

t

(0,λ,K).

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Ez

[exp

{− λZ

t

e

−K

t

}]

λ→ ∞ ⇒ P(Zt

e

−K

t = 0) = P(Zt

= 0)

λ→ 0 ⇒ P(Zt

e

−K

t <∞) = P(Zt

<∞)

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Consequen es

Probability of non explosion

Pz

(Z

t

<∞)= 1{β∈(0,1]}+

1{β∈(−1,0)}E

[exp

{−z(β β

∫t

0

e

−βKu

du

)−1/β}]

Probability of non extin tion

Pz

(Z

t

> 0

)= 1{β∈(−1,0)}+

1{β∈(0,1]}E

[1− exp

{−z(β β

∫t

0

e

−βKu

du

)−1/β}]

21 / 40


Logisti population model in a Lévy random environment

Unique strong solution of (Palau and Pardo 2015)

U

t

= U

0

+

∫t

0

U

s

(µ− kU

s

)ds +

∫t

0

U

s−dSs

where µ > 0 is the drift and k > 0 is the ompetition. The pro ess

U satis�es the Markov property and we have

U

t

=U

0

e

K

t

1+ kU

0

∫t

0

e

K

s

ds

, t ≥ 0.

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Di�usion in a Lévy random environment

Informal solution to

dX (t) = dβ(t)− 1

2

V

′(X (t))dt, X (0) = 0,

V two-side Lévy pro ess and β Brownian independent of V . More

rigorously, X di�usion whose onditional generator given V is

1

2

e

V (x) d

dx

(e

−V (x) d

dx

).

(Kawazu and Tanaka 1993, Carmona and al. 1997)

P

(max

t≥0

X (t) > x

)= E

[A

A+ B

x

]

where

A =

∫0

−∞e

V (t)dt independent of B

x

=

∫x

0

e

V (t)dt.

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Con lusion

◮ P(Zt

<∞) = E

[exp

{−z(β β

∫t

0

e

−βKu

du

)−1/β}]

, β ∈ (−1, 0)

◮ P(Zt

> 0) = E

[1− exp

{−z(β β

∫t

0

e

−βKu

du

)−1/β}]

, β ∈ (0, 1]

◮ E[Ut

] = E

[U

0

e

K

t/

(1+ kU

0

∫t

0

e

K

s

ds

)]

◮ P (max

x≥0

X (x) > t) = E

[A/(A+

∫t

0

e

V (s)ds

)]

= E

[F

[∫t

0

e

−ξs

ds

]],

where F is non in reasing and goes to 0 at in�nity (at least

polynomially)

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Main result

25 / 40


Lapla e exponent of a Lévy pro ess

◮ ψ(λ) = logE[eλξ1 ]

◮ E[eλξt ] = e

tψ(λ), t ≥ 0

◮Convex fun tion, ψ(0) = 0

◮ ψ′(0) = E[ξ1

] when it exists

Then (see (Kyprianou 2006) for instan e)

◮ ψ′(0) > 0 =⇒ ξt

→ ∞◮ ψ′(0) = 0 =⇒ lim sup ξ

t

= − lim inf ξt

= ∞◮ ψ′(0) < 0 =⇒ ξ

t

→ −∞

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◮Assumption: de�ned on (θ−, θ+), with θ− < 0 < θ+

◮De�nition: τ in (0, θ+) su h that ψ′(τ) = 0

PSfrag repla ements

0 τ

ψ

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Conditions on F

Case (A1)

F is non in reasing at in�nity and satis�es

F (x) = A(x + 1)−p

[1+ (1+ x)−ζh(x)

], 0 < p ≤ τ,

where h is a Lips hitz fun tion whi h is bounded and ζ ≥ 1.

Case (A2)

F is an Hölder fun tion with index α > 0, non in reasing at in�nity,

and satis�es

F (x) ≤ A(x + 1)−p , p > τ.

28 / 40


Theorem (Palau, Pardo and S. 2016)

We have the following �ve regimes for the asymptoti behaviour of

EF

(t) = E

[F

(∫t

0

e

−ξs

ds

)]for large t. (moment ondition

θ+ > p)

i) If ψ′(0+) > 0 and F is a positive and ontinuous fun tion

whi h is bounded, then

lim

t→∞EF

(t) = EF

(∞) > 0.

ii) If ψ′(0+) = 0, θ− < 0, F satis�es (A2),then ∃ 1

su h that

lim

t→∞

√tE

F

(t) =

1

> 0.

29 / 40


PSfrag repla ements

0 p τ

ψ

PSfrag repla ements

0 p = τ

ψ

PSfrag repla ements

0 pτ

ψ

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Theorem (Palau, Pardo and S. 2016)

iii) Suppose that ψ′(0+) < 0:

a) If F satis�es (A1) with ψ′(p) < 0, then ∃ 2

su h that,

lim

t→∞e

−tψ(p)EF

(t) =

2

> 0.

b) If F satis�es (A1), ψ′(p) = 0, ψ′′(p) <∞, then ∃ 3

su h that

lim

t→∞

√te

−tψ(p)EF

(t) =

3

> 0.

) If F satis�es (A2), ψ′(p) > 0 and τ + p < θ+, then ∃ 4

su h

that

lim

t→∞t

3/2e

−tψ(τ )EF

(t) =

4

> 0.

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Remark

Reminis ent of asymptoti behaviour of GWRE

◮Strong sub riti al:

P(Zn

> 0) ∼

1

(E[Z1

])n

◮Intermediate sub riti al:

P(Zn

> 0) ∼

2

1√n

(E[Z1

])n

◮Weak sub riti al:

P(Zn

> 0) ∼

3

1

n

3/2(E[E[Z

1

|f0

]τ ])n ,

where

E[E[Z1

|f0

]τ ] = inf

0≤θ≤1

E[E[Z1

|f0

]θ].

32 / 40


Remark

The same week as our paper, preprint of Li and Xu on the same

subje t. Their approa h:

◮Flu tuation theory for Lévy pro esses

◮Lévy pro esses onditioned to stay positive

Our approa h

◮Dis retization of the exponential fun tionals

◮Results on asymptoti behaviour of fun tionals of semi-dire t

produ ts of random variables

33 / 40


Proof of the strong sub riti al ase (p < τ)

Change of measure

dP(β)

dP

∣∣∣∣F

t

= e

βξt

−ψ(β)tfor β ∈ (θ−, θ+),

I

t

(ξ) =

∫t

0

e

−ξt−s

ds = e

−ξt

∫t

0

e

ξt

−ξt−s

ds

(d)= e

−ξt

∫t

0

e

ξs

ds = e

−ξt

I

t

(−ξ)

Hen e using the hange of measure with β = p, we have

E[I

t

(ξ)−p

]= E

[e

pξt

I

t

(−ξ)−p

]= e

tψ(p)E(p)[I

t

(−ξ)−p

].

E(p)[−ξ

1

] = −ψ′(p) > 0 ⇒ E(p)[I

t

(−ξ)−p

]→ E

(p)[I∞(−ξ)−p

]> 0

34 / 40


∣∣F (I

t

(ξ)) − AI

t

(ξ)−p

∣∣ ≤ MI

t

(ξ)−(1+ε)p .

So, it is enough to show

E

[I

t

(ξ)−(1+ε)p]= o(etψ(p)), as t → ∞.

Again, from the hange of measure with β = (1+ ε)p, we dedu e

E

[I

t

(ξ)−(1+ε)p]= E

[e

p(1+ε)ξs

I

t

(−ξ)−(1+ε)p]

= e

tψ(p(1+ε))E((1+ε)p)

[I

t

(−ξ)−(1+ε)p].

For ε small enough

E(p(1+ε))[−ξ

1

] = −ψ′(p(1 + ε)) > 0

⇒ E(p(1+ε))

[I

t

(−ξ)−p(1+ε)]→ E

(p(1+ε))[I∞(−ξ)−p(1+ε)

]> 0

35 / 40


Key idea

Theorem (Le Page and Peigné 1997, Guivar 'h and Liu 2010)

Let (an

, bn

)n≥0

be a R2

+-valued sequen e of i.i.d. r.v. su h that

E[ln a0

] = 0. Assume that b

0

/(1− a

0

) not onstant a.s. and de�ne

A

0

:= 1, A

n

:=

n−1∏

k=0

a

k

and B

n

:=

n−1∑

k=0

A

k

b

k

, for n ≥ 1.

Let η, κ, ϑ > 0 su h that κ < ϑ, and φ and ψ be two positive

ontinuous fun tions on R+ su h that they do not vanish and for a

onstant C > 0 and for every a > 0, b ≥ 0, b

′ ≥ 0, we have

φ(a) ≤ Ca

κ, ψ(b) ≤ C

(1+ b)ϑ, and |ψ(b)−ψ(b′)| ≤ C |b−b′|η.

36 / 40


Key idea

Theorem (Guivar 'h and Liu 2010)

Moreover, assume that

E[a

κ0

]<∞, E

[a

−η0

]<∞, E

[b

η0

]<∞ and E

[a

−η0

b

−ϑ0

]<∞.

Then there exist two positive onstants (φ, ψ) and (ψ) su h that

lim

n→∞n

3/2E

[φ(A

n

)ψ(Bn

)]= (φ, ψ) and lim

n→∞n

1/2E

[ψ(B

n

)]= (ψ).

37 / 40


Dis retization

(q, n) ∈ N2

. For k ≥ 0, we also de�ne

a

k

= e

−(ξ(k+1)/q−ξk/q)and b

k

=

∫ (k+1)/q−k/q

0

e

−(ξu+k/q−ξk/q)

du.

∫ (i+1)/q

i/qe

−ξu

du = e

−ξi/qb

i

=

i−1∏

k=0

a

k

b

i

:= A

i

b

i

,

whi h implies

∫n/q

0

e

−ξs

ds =

n−1∑

i=0

∫ (i+1)/q

i/qe

−ξu

du =

n−1∑

i=0

A

i

b

i

:= B

n

.

38 / 40


Example: riti al ase (ψ′(0) = 0)

An appli ation of Guivar 'h and Liu gives

√nE

F

(n/q) ∼ (q), as n → ∞.

Let t > 0. Sin e the mapping s 7→ EF

(s) is non in reasing for large

s, we get

√tE

F

(t) ≤√tE

F

(⌊qt⌋/q) =√

t

⌊qt⌋√⌊qt⌋E

F

(⌊qt⌋/q).

√tE

F

(t) ≥√tE

F

((⌊qt⌋+1)/q) =

√t

⌊qt⌋+ 1

√⌊qt⌋+ 1E

F

((⌊qt⌋+1)/q).

Both ∼ (q)/√q for large t.

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THANK YOU FOR YOUR ATTENTION

40 / 40

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