Ald´ericJoulinUniversit´edeLaRochelle(France)Workshop“Stochasticandharmonicanalysisofprocesseswithjumps”2-9May2006-Angers Onmaximalinequalitiesforstablestochasticintegrals

On maximal inequalities for stable stochastic integrals

Ald´eric Joulin

Universit´e de La Rochelle (France)

Workshop

“Stochastic and harmonic analysis of processes with jumps”

2-9 May 2006 - Angers

Content

1 Motivations

2 Pruitt’s truncation method

3 Large range estimates

4 Small range estimates

5 Application to first crossing problems

Motivations

Z α-stable process, H predictable integrable.

Our aim: to understand the behavior of H · Z as α is “close to 2”, i.e. to give upper bounds on:

- Large range: P sup

t∈[0,1]

Z t 0

HsdZs

≥ x

!

, x large.

- Small range: P sup

t∈[0,1]

Z t 0

HsdZs ≥ x

!

, x small.

Main difficulties:

- Absence of close formulae for stable densities.

- Infinite variance ⇒ classical maximal inequalities fail.

Motivations

Pruitt ’81. Regularity of paths of L´evy processes. In the SαS case:

P sup

t∈[0,1]

|Z_t| ≥ x

!

≤ D₁

α(2 − α)x^α, x^α> D₁ α(2 − α). Optimal in x since behavior similar to ν ({|x |, ∞}).

Linear explosion as α → 2.

Proof: Truncation of the jumps + Doob’s maximal inequalities.

Motivations

Gin´e-Marcus ’83. To derive a CLT for stable stochastic integrals. Generalization of Pruitt’s result: for any

x^α > D₂kHk^α_α α(2 − α)²,

P sup

t∈[0,1]

Z t 0

H_sdZ_s    

≥ x

!

≤ D₂

α(2 − α)²x^α Z 1

0

E|Ht|^αdt.

Optimal in x .

H ∈ L^α(Ω × [0, 1]) is sufficient.

Explosion of quadratic order as α → 2.

Motivations

Hult-Lindskog ’06. Extremal behavior of L´evy-type stochastic integrals:

P sup

t∈[0,1]

Z _t

0

HsdZs

≥ x

!

∼_{x →∞} Dα

αx^α Z ₁

0

E|Ht|^αdt.

Uniform integrability assumption E supt∈[0,1]|H_t|^α+p < ∞.

D_α bounded in α ⇒ No explosion phenomenon !

⇒ loss of information from asymptotic to non-asymptotic estimates.

Motivations

Natural question: is it possible to avoid the explosion of the upper bound as α → 2 and to involve at the same time optimal bounds in terms of the deviation level x and the L^α-norm of H ? I do not know.

Our purposes: to conserve the optimal rate x^−α and:

- to weaken the speed of explosion in Gin´e-Marcus’ estimate, with the same optimal L^α-norm.

- to avoid the explosion as α is close to 2, at the price of stronger integrability assumptions on H.

Pruitt’s truncation method

Z α-stable with characteristics (b, 0, ν), where ν stable L´evy measure:

ν(dy ) = c−1_{{y <0}}+ c+1_{{y >0}}
dy

|y |^α+1. L´evy-Itˆo decomposition with truncation level R:

Zt = b_Rt + Z t

0

Z

|y |≤R

y ˜µ(dy , dt) + Z t

0

Z

|y |>R

y µ(dy , dt)

= drift + L²-martingale Z_t^(R−) + compound Poisson Z_t^(R+).

Pruitt’s truncation method

Construction of stable stochastic integrals:

H ∈ L²(Ω × [0, 1]).

X_t := A^(R)_t + X_t^(R−)+ X_t^(R+), with A^(R)_t := bR

Z t 0

Hsds, X_t^(R−):= H·Z_t^(R−), X_t^(R+) := H·Z_t^(R+).

Pruitt’s truncation method

Crucial point: the truncation level R is chosen at the end of the proofs, depending on:

the deviation level x . a suitable L^p-norm kHk_p.

Large range estimates (1)

Proposition

Underlying driving Z SαS. Then

P sup

t∈[0,1]

|X_t| ≥ x

!

≤ D

(2 − α)x^α Z 1

0

E|Ht|^αdt, x^α ≥ DkHk^α_α 2 − α .

Large range estimates (1)

Remark

Optimal in the deviation level x and the L^α-norm of H.

Linear explosion (instead of quadratic) as α → 2:

⇒ generalization of Gin´e-Marcus’ estimate.

Proof: adaptation to stable stochastic integrals of

Bass-Levin-Vondracek’s method + choice of truncation level R = x .

Large range estimates (2)

For non-explosion as α is close to 2, we need the tricky inequality:

P sup

t∈[0,1]

|X_t| ≥ x

!

≤ P sup

t∈[0,1]

|A^(R)_t | ≥ x/2

! (1)

+P sup

t∈[0,1]

|X_t^(R−)| ≥ x/2

! (2)

+P sup

t∈[0,1]

|X_t^(R+)| > 0

!

≤ (1) + (2) + K αR^α.

⇒ Reduced to bound suitably (2), avoidingR

|y |≤Ry²ν(dy )

Large range estimates (2)

Theorem

H ∈ L^α+p(Ω × [0, 1]), p ∼ 2 − α. Then for x^α ≥ K kHk^α_α+p

(2 − α)^β , β := βα+p > 2 :

P sup

t∈[0,1]

|X_t| ≥ x

!

≤ M x^α

Z 1 0

E|Ht|^α+pdt

α/(α+p)

.

Large range estimates (2)

Remark

Optimal in x and no longer explosion as α → 2.

Price to pay compared to Gin´e-Marcus’ estimate:

- H ∈ L^α+p and not only in L^α.

- Restriction on the range of validity of x (β > 2).

Proof: moment estimates + choice of truncation level R = x /kHkα+p.

Small range estimates

Theorem

Underlying driving Z SαS, H bounded. Then for any  > 0 and any x ∈ (0, x_α,),

P sup

t∈[0,1]

Xt ≥ x

!

≤ (1 + ) exp −K_α

 x

kHk_∞,α

α/(α−1)! .

Small range estimates

Remark

Different behaviors as x is small or large. What about intermediate range ?

As α → 2, recover the Gaussian case ?

Small range estimates

Corollary

Yes: choosing a suitable weight in the stable L´evy measure and taking limit as α → 2 yields

P sup

t∈[0,1]

B_t≥ x

!

≤ exp −x²/2
, x > 0.

Application to first crossing problems

Z SαS. Stable (recurrent) O.U. process of parameter λ > 0:

X_t^(λ):=

Z t 0

e^{−λ (t−s)}dZ_s, t ≥ 0, X₀^{(λ )}= 0.

First passage times: if x > 0, define

Tx^{(λ )}:= inf{t ≥ 0 : |X_t^(λ)| ≥ x}, T^curv := inf{t ≥ 0 : |Zt| ≥ x(1 + λαt)^1/α}.

By Alili-Patie’s transform: Tx^{(λ )}

(d )= _{λ α}¹ log (1 + λαT^curv).

Maximal inequalities for X^{(λ )} ⇒ Estimates on T_x^{(λ )}

⇒ Estimates on T^curv.

Application to first crossing problems

Theorem

Estimates on first crossing times:

P (T^curv > r ) ≤ exp



− K1

λx^α log (1 + λαr )



, r > 0, x > 0,

P (T^curv≤ r ) ≤ K₂log (1 + λαr )

λx^α , 0 < r < r₀(α, x , λ).

L. Alili and P. Patie.

On the first crossing times of a Brownian motion and a family of continuous curves.

C. R. Math. Acad. Sci. Paris, 340(3):225–228, 2005.

E. Gin´e and M.B. Marcus.

The central limit theorem for stochastic integrals with respect to L´evy processes.

Ann. Probab., 11(1):58–77, 1983.

H. Hult and F. Lindskog.

Extremal behavior of stochastic integrals driven by regularly varying L´evy processes.

To appear in Ann. Probab., 2006.

W.E. Pruitt.

The growth of random walks and L´evy processes.

Ann. Probab., 9(6):948–956, 1981.

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