On an exit problem for a jump-di usion model Results …cpss/2012/short/BenAri-short.pdf · ·...

Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

On an exit problem for a jump-diffusion model

Iddo Ben-Ari

University of Connecticut

Eighth Cornell Probability Summer School

2012/07/26

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Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

Model

Brownian Motion with Instantaneous Jumps (BMJ)

I Recently studied in a series of papers by R. Pinsky, N. Arcusin and R. Pinsky.

I “Ladders and chutes”, or dynamics with occasional catastrophes.other ideas ?

Ingredients

I BM on a bounded domain D ⊂ Rd , with

I Instantaneous “events” at spatially dependent rate γV ,

I γ positive parameter,I V jump intensity function, V ∈ C(D), positive, bounded with limits on ∂D.

I Each “event” (jump) process redistributed in the domain according to prescribedprobability measure µ.

Infinitesimal Generator

Lγu =1

2∆u︸︷︷︸

Generator of BM

+ γV

(∫udµ− u

)︸︷︷︸Generator of Jump

, u ∈ C 20 (D).

Note. Reversible if and only if µ = Vdx = 1|V | dx , not typical !

Notation

I X = (X (t) : t ≥ 0), path of BMJ.

I Px , Ex probability and expectation for X conditioned on X (0) = x . 2/ 11

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Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

Objects of interest

Observe

I Each event, starting afresh, with positive probability to exit before next event.

BB (essentially) Geometric number of events before exiting.BB Time to exit is geometric sum of IID.

Problem How fast ?

As explained below, we are actually asking

Problem Behavior of principal eigenvalue for −Lγ ?

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Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

Principal eigenvalue

Let τ denote the exit time of X from D,

τ = inf{t ≥ 0 : X (t) 6∈ D}.

General principles (compactness of resolvents for Lγ , Krein-Rutman theorem) give

I −Lγ has a discrete spectrum consisting of eigenvalues.

I ∃ simple strictly positive e.v., the principal e.v., unique minimizer of real part.

Denote principal e.v. by λc (γ).

Probabilistically,

λc (γ) is the exponential tail for τ

More precisely (positivity of the Markov-semigroup) :

Proposition 1

λc (γ) = − limt→∞

1

tln Px (τ > t) = − lim

t→∞

1

tsupx∈D

ln Px (τ > t)

= sup{λ ∈ R : Ex (eλτ ) <∞} = sup{λ ∈ R : supx∈D

Ex (eλτ ) <∞}

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Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

Statement of problem

Problem Asymptotic behavior of λc (γ) as γ →∞.

Why ?

I γ → λc (γ) analytic. Taylor expansion at 0 with known coefficients in terms of BM.

I When γ large,

Lγ = γ

(1

2γ∆︸︷︷︸

unbounded “perturbation”

+ V

(∫udµ− u

)︸︷︷︸

bounded

).

No standard expansion technique.

Regimes

Recall we assume V ∈ C(D) is strictly positive and has (finite) limits at ∂D.

Asymptotic of λc (γ) strongly related to behavior of V near ∂D.

We identify two regimes :

I Nondegenerate. inf V > 0 (treated by P,PA)

I Degenerate. V |∂D = 0 (treated by B)

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Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

Nondegenerate case, inf V > 0

Notation

I σ surface area measure on ∂D.

I Dε = {x ∈ D : dist(x, ∂D) < ε}.I D smooth.

For ε > 0 small and x ∈ Dε unique “projection” p(x) ∈ ∂D minimizer of distance to ∂D.

Theorem 1Assume inf V > 0 and µ has density on Dε given by

µ(x) ∼x→∂D

dist(x, ∂D)β f (p(x)),

where f ∈ C(∂D) is nonnegative and not identically 0 and β > −1. Then

i. λc (γ) ∼γ→∞

γ1−β

2

∫∂D

(2V )−β+1

2 fdσ∫D

dµV

Γ(β + 1).

ii. λc (γ)τ ⇒γ→∞

Exp(1), with convergence of the moment generating function on the

unit disk.

Remarks

1. Part (i) for integer-valued β proved by Pinsky and Arcusin, and generalized toelliptic diffusions by Pinsky.

2. When µ is uniform, λc (γ) ∼γ→∞

√γ

∫∂D

dσ√2V∫

D1V dx

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Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

Some heuristics for degenerate case, V |∂D = 0

Recall that when inf V > 0 and µ is uniform,

λc (γ) ∼γ→∞

√γ

∫∂D

dσ√2V∫

D1V dx

.

What if V vanishes on ∂D ?

Numerator of RHS does not make sense.

Assume that V (x) ∼ dist(x, ∂D)α, α > 0.

Approximating numerator by volume integrals, formula heuristically predicts :

Iλc (γ)√γ

→γ→∞

∞ α < 2;

C ∈ (0,∞) α = 2;

0 α > 2.

,

but does not predict order sharply, except for α = 2.

I “Phase transition” at α = 1 : denominator converges for α < 1; diverges for α ≥ 1.(equivalently, jump process is positive recurrent for α < 1; null-recurrent for α ≥ 1)

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Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

More heuristics ?

I Fact. When V is compactly supported, limγ→∞ λc (γ) <∞.

Proof. Suppose x is outside K , the support of V .Then τ is greater than the exit time of BM from D\K .Hence tail uniformly bounded above. �

BB Expect λc (γ) = O(√γ).

I Intuition (?). When V is small near ∂D, then X has better chance of exiting beforebeing redistributed.

BB Expect√γ = O(λc (γ)).

( “? ” because less jumps lower chance of approaching boundary quickly throughfrequent jumps...)

Proposition 2 (Arcusin and Pinsky)

Assume D = (0, 1), µ is uniform and V (x) = x(1− x). Then

cγ13 ≤ λc (γ) ≤ Cγ

23 , as γ →∞.

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Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

Degenerate case, V |∂D = 0

Theorem 2Let µ be uniform on D = (0, 1). Let 0 ≤ α′ ≤ α, and assume that

V (x) =x→0+

Θ(xα), V (x) =x→1−

Θ((1− x)α′).

Let δ(α) = α∧1+1α+2 . Then

λc (γ) =γ→∞

Θ(γδ(α))×{

1 α 6= 1;1

ln γ α = 1.

Figure: Graph of δ(α)

Remarks

1. Asymptotic behavior determined by Vnear ∂D, and there where it is smaller.

2. Maximal asymptotic λc (γ) = Θ( γ2/3

ln γ )

when α = 1.

3. Extends to regular varying V , µ and alldimensions.

4. Problems

I Constant ?I Convergence of λc (γ)τ ?

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Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

An upper bound

We will prove

λc (γ) ≤ cγ2

2+α (optimal when α > 1).

1. Observe. If for some x , Ex (eητ ) =∞, then η ≥ λc (γ).

2. Starting from (small) x , let

σ=Exit time from interval (0, 2x).J=Time of first jump.

Then τ ≥ J ∧ σ.

3. Observe. Maximal rate of jump before σ is ρ = ρ(γ, x) = c1γxα.

Thus, τ ≥ J ∧ σ ≥ J ∧ σ, where J ∼ Exp(ρ(γ, x)).

4. Ex (eητ ) ≥ηEx

(e(η−ρ)σ

)− ρ

η − ρ.

5. Recall. Ex (eθσ) =∞ if and only if θ >π2

8x2= c2x

−2.

6. Optimize. By 5,4 and 1, if

η ≥ ρ + c2x−2 = c1γx

α + c2x−2 (?)

then η ≥ λc (γ).

Minimum (over x) of RHS of (?) is c3γ2

2+α (attained at c4γ− 1α+2 ). �

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Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

An upper bound

We will prove

λc (γ) ≤ cγ2



This is because Px (τ > t) ≈ e−λc (γ)t .







(e(η−ρ)σ

)− ρ

η − ρ.


8x2= c2x

−2.


η ≥ ρ + c2x−2 = c1γx

α + c2x−2 (?)




10/ 11

reset

Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

An upper bound

We will prove

λc (γ) ≤ cγ2









(e(η−ρ)σ

)− ρ

η − ρ.


8x2= c2x

−2.


η ≥ ρ + c2x−2 = c1γx

α + c2x−2 (?)




10/ 11

reset

Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

An upper bound

We will prove

λc (γ) ≤ cγ2









(e(η−ρ)σ

)− ρ

η − ρ.

Indeed,

Ex (eητ ) ≥ Ex (eη(J∧σ)) (by 2. and 3.)


8x2= c2x

−2.


η ≥ ρ + c2x−2 = c1γx

α + c2x−2 (?)




10/ 11

reset

Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

An upper bound

We will prove

λc (γ) ≤ cγ2









(e(η−ρ)σ

)− ρ

η − ρ.

Indeed,


=

∫ ∞0

E(eη(J∧t)

)︸︷︷︸

MGF for truncated Exp RV

dP(σ ≤ t) (condition on σ)


8x2= c2x

−2.


η ≥ ρ + c2x−2 = c1γx

α + c2x−2 (?)




10/ 11

reset

Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

An upper bound

We will prove

λc (γ) ≤ cγ2









(e(η−ρ)σ

)− ρ

η − ρ.

Indeed,


=

∫ ∞0

E(eη(J∧t)

)︸︷︷︸

MGF for truncated Exp RV

dP(σ ≤ t) (condition on σ)

=ηEx

(e(η−ρ)σ

)− ρ

η − ρ(compute MGF) �


8x2= c2x

−2.


η ≥ ρ + c2x−2 = c1γx

α + c2x−2 (?)




10/ 11

reset

Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

An upper bound

We will prove

λc (γ) ≤ cγ2









(e(η−ρ)σ

)− ρ

η − ρ.

Thus, Ex (eητ ) =∞ if Ex (e(η−ρ)σ) =∞.


8x2= c2x

−2.


η ≥ ρ + c2x−2 = c1γx

α + c2x−2 (?)




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Introduction

Model

Problem

Results

Nondegenerate

Degenerate

Pf. of upper bd

Thank you !

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On an exit problem for a jump-di usion model Results …cpss/2012/short/BenAri-short.pdf · ·...

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