Sonia Fourati
email: sonia.fourati@upmc.fr

EXPLICIT SOLUTIONS FOR THE EXIT PROBLEM FOR A CLASS OF L ÉVY PROCESSES. APPLICATIONS TO THE PRICING OF DOUBLE BARRIER OPTIONS

Keywords: 2000 Mathematics Subject Classification. Primary 60G51 Lévy processes, Fluctuation theory, Wiener-Hopf factorization, Exit problems, Options pricing, Inverse problems, Bargmann equations

Lewis and Mordecki have computed the Wiener-Hopf factorization of a Lévy process whose restriction on ]0, +∞[ of their Lévy measure has a rational Laplace transform. That allows to compute the distribution of (Xt, inf 0≤s≤t Xs). For the same class of Lévy processes, we compute the distribution of (Xt, inf 0≤s≤t Xs, sup 0≤s≤t Xs) and also the behavior of this triple at certain stopping time, like the first exit time of an interval containing the origin. Some applications to the pricing of double barrier options with or without rebate are evocated.

Introduction

There are very few examples of Lévy process for which the so-called "exit problem" can be explicitly solved (see [KK05] , [R90], [R72]). We present here this explicit solution for a class of Lévy processes which has been introduced by Lewis A.L. and Mordecki E. [LM08]; that is the class of the Lévy processes whose restriction on ]0, +∞[ of their Lévy measure has a rational Laplace transform. This happens when this restriction is a finite linear combination of exponential or gamma distributions. Lewis A.L and Mordecki E. [LM08] (see also Asmussen, S. Avram, F. and Pistorius M.R. [AAP04]) have computed the so-called Wiener-Hopf factorization of these Lévy process. That brings to closed forms for the distribution of the maximum of the process before an independent exponential time and for the joint distribution of (T x , X T x ) where T x is the first time where the Lévy process X crosses upward a level x and X T x its position at that time. As an application, when adopting the exponential Lévy model, Y t = Y 0 .e Xt , for a financial asset Y t with X t of the preceding form (see for example [CT03]), an immediate consequence of the preceding results is the computation of the (temporal Laplace transform) price of the double barrier option with this underlying asset. That is, an european option which is activated (in) or desactivated (out) when the asset (Y t ) t∈[0,T ] cross up (down) a barrier H before the time of maturity T . This application follows the computation of the price of the simple barrier option, by the Wiener-Hopf factorization (see [AAP04]). In this paper, for the same class of Lévy processes, we solve the exit problem. More precisely, we give closed form of the joint distribution of the minimum and maximum of the process before an independent exponential time and among other behaviors of the process at certain stopping times, we give the Laplace transform of the joint distribution of (T b a , X T b a ) where T b a is the first time that the process leaves a bounded interval [-a, b] containing the origin and X T b a is its position at that time. As an application, we mention how one can deduce the price of the double barrier option, in or out, that means that the option is activated or desactived if the asset crosses up a barrier b or down a barrier a in the exponential Lévy model. The paper is organized as follow : In section 2, we recall the general results on Wiener-Hopf factorization, we give a shortened proof of the results of Lewis-Mordecki and settle few other preliminary results. For that, we use exclusively elementary complex analysis arguments. After introducing few more notations in section 3, we give in section 4, 5, 6, 7 and 8, all the results on the fluctuations of our class of Lévy processes. In section 9, we recall the main result of [F10] which is the main tool of this work. The proofs follow in section 10, 11, 12, 13. These results, in the symmetric case, are related to what is called "Bargmann equations" in the litterature on inverse problems of the spectral theory (see for example [F63]).

The assumption and the Wiener-Hopf factorization

We suppose that X is a real Lévy process possibly killed at an independent exponantial time and we denote ζ the life time of X. Let φ be the Lévy exponent of X, so that the identity E(e -iuXt 1 t<ζ ) = e -tφ (iu) is fulfilled for every time t and every imaginary number iu ∈ iR, φ is continuous on iR and φ(0) is the rate of the exponential distribution of the life time ζ, φ(0) = 0 if and only if ζ = +∞ a.s. ("X does not die").

We now work under Lewis-Mordecki's assumption. It is based on the following rather obvious fact.

Proposition 2.1. The conditions below are equivalent (i) The Lévy measure of X, π, is of the following form

π(dy) = n j=1 c j y nj n j ! e -γjy dy on ]0, +∞[ n ∈ N, n j ∈ N, ℜ(γ j ) > 0, c j ∈ C (j = 1, . . . , n) (ii) The exponent φ is of the form φ(iu) = φ -(iu) + P Q (iu),
where φ -is the exponent of a Lévy process without positive jumps, P and Q are polynomials and Q has all its roots on the complex half plane {λ; ℜ(λ) < 0}.

Proof. Assume (i). Notice that the compound Poisson process with Lévy measure

π(dy) = n j=1 c j y nj n j ! e -γjy 1 y>0 dy has Lévy exponent j∈J c j [ 1 (γ j ) nj +1 - 1 (λ + γ j ) nj +1 ] =: P (λ) Q(λ) ,
The function φ -P Q is then the Lévy exponent of a Lévy process without positive jumps. This establishes (ii). Conservely, assume (ii). Since φ and φ -are exponents of Lévy processes, we know that φ

(iu) = O(u 2 ) and φ -(iu) = O(u 2 ) (see proposition 2 chapter 1 of [B96]), thus P Q (iu), construed as a rational function of u, is O(u 2 ). Consequently deg P ≤ deg Q + 2. Now, write P Q (iu) in its fractional expansion P Q (iu) = au 2 + bu + c + n j=1 k j (iu + γ j ) nj +1 Then P Q (iu)
is the Fourier transform of the Schwartz distribution (here, δ stands for the Dirac mass at 0);

aδ ′′ -ibδ ′ + cδ + n j=1 k j x nj n j ! e -γj x 1 x>0 dx
On the other hand, one can deduce from the Lévy Kinchin formula applied to φ that φ(iu) is the Fourier transform of a distribution whose restriction to ]0, +∞[ is -1 x>0 π(dx) (see chapter 5 of [V02] for example) where π is the Lévy measure.

From this, we see that

1 x>0 π(dx) = - n j=1 k j x nj n j ! e -γjx 1 x>0 dx
This finishes the proof.

The following is a famous result (see proposition 2, chapter I of [B96]) which applies to any Lévy process and which will be of use later .

Lemma 2.2. There exists an exponent of a subordinator ψ and an exponent of the opposite of a subordinator ψ, such that

ψ. ψ = φ on iR Such a couple (ψ, ψ) is unique up to a multiplicative constant.
The functions ψ and ψ will be refered to as the "positive" (for ψ) and "negative" (for ψ) Wiener -Hopf factors of the exponent φ, unlike the ordinary uses which affect these expressions to the functions ψ(0) ψ and ψ(0) ψ . ASSUMPTION : In the rest of this paper, conditions (i) and (ii) of proposition 2.1 will be assumed to be satisfied and this assumption will be call Assumption 2.1 . Property (ii) of proposition 2.1 and the fact that φ -has an holomorphic extension on the half plane {ℜ(λ) < 0} (see chapter 7 of [B96]) imply that φ has a meromorphic extension on this half plane, and we we will still denote by φ this extension.

We will denote

-γ 1 , -γ 2 , ..., -γ n the poles of φ repeated according to their multiplicity. In particular, deg Q = n when the fraction P Q is irreductible, in the expression of condition (ii) of proposition 2.1 (φ = φ -+ P Q ) and we will denote n j := ♯{k < j; γ k = γ j } j = 1, . . . , n Notice that the n j are all zero if and only if all the poles of φ on the half plane {Re(λ) < 0} are simple, or equivalently, if and only if the Lévy measure is a (not necessarily positive) combination of exponential distributions.

Theorem 2.3. ψ is of the form :

ψ(λ) = ψ ∞ Π m 1 (λ + β i ) Π n 1 (λ + γ j )
,

where ψ ∞ is a positive constant. If lim λ→-∞ φ(λ) λ ∈]0, +∞] then m = n + 1. Otherwise, lim λ→-∞ φ(λ) λ ∈] -∞, 0] and m = n.
More over, {-β 1 , . . . , -β n } is the set of the roots of φ lying on the half plane {ℜ(λ) < 0} (repeated according to their multiplicity) together with 0 if φ(0) = 0 and φ ′ (0) ≥ 0.

Proof. Notice first that, thanks to Lévy Kintchin formula, φ(λ) does not vanish for λ on the imaginary axis except possibly for λ = 0, φ is a meromorphic funtion on {ℜ(λ) < 0} and φ is continuous by the left at each point of the imaginary axis iR. On the other hand, the function ψ is an exponent of a subordinator, thus it is analytic on the open half plane {ℜ(λ) > 0}and continuous on the closed half plane {ℜ(λ) ≥ 0} (see [B96] chapter 3 for example). Also, the identity ψ ψ = φ on iR implies that ψ = φ ψ on iR. Thus, since φ is meromorphic and ψ holomorphic on {ℜ(λ) < 0}, we see that ψ has a meromorphic extension on {ℜ(λ) < 0}.We denote again ψ this extension. More over ψ = φ ψ is continuous by the left on the axis iR, except possibly at λ = 0 when ψ(λ) = 0. Then ψ is meromorphic on C except possibly at 0. However, because ψ is the exponent of the opposite of a subordinator, we have (see page 73 of [B96])

lim λ→0;ℜ(λ)<0 λ ψ(λ) →] -∞, 0], then lim λ→0;ℜ(λ)<0 λ. φ(λ) ψ(λ) = lim λ→0;ℜ(λ)<0 λψ(λ) = 0.
Also, ψ is continuous by the right at the point 0, then lim λ→0;ℜ(λ)>0 λψ(λ) = 0 and we deduce that ψ is again holomorphic at 0. In conclusion, ψ is meromorphic on C and its poles, necessarily in the half plane {λ; ℜ(λ) < 0}, are the same as those of φ (because of the identity ψ ψ = φ and because ψ is holomorphic on {λ; ℜ(λ) < 0}). Therefore, these poles are the -γ j . Now, write

ψ(λ) = Ψ(λ) Π n 1 (λ + γ j )
, where Ψ is an entire function. Notice the limit, valid for any exponent of a subordinator ( see proposition 2 of Chapter 1 of [B96])

lim λ→+∞ ψ(λ) λ ∈ [0, +∞[ Then (2.1) Ψ(λ) = O(λ n+1 ) (ℜ(λ) → +∞)
And, since

ψ(λ) = Ψ(λ) Π n 1 (λ + γ j ) = φ -(λ) + P (λ) Q(λ) ψ(λ) , φ -(λ) and P (λ) Q(λ) are O(λ 2 ) on {ℜ(λ) < 0} (see proof of proposition 2.1) and 1 ψ(λ)
is bounded (by 1 ψ(-1) ) on {ℜ(λ) < -1}, because it is the Laplace transform of a measure supported by ] -∞, 0] (see chapter 3 of [B96] for example) and deg

P ≤ deg Q + 2 , we deduce that (2.2) Ψ(λ) = O(λ n+2 ) (ℜ(λ) → -∞)
When joining property (2.1) and (2.2), Ψ is a polynomial with degree at most n+ 1. Moreover, since 1 ψ(λ) is bounded for λ → +∞, then Ψ has a degree at least n. Thus Ψ is a polynomial and its degree is

n or n + 1. If Ψ is a polynomial of degree n + 1 then lim λ→-∞ ψ(λ) λ ∈]0, +∞[ and lim λ→-∞ ψ(λ) ∈]0, +∞]
(this last limit is valid for any exponent of the opposite of a subordinator). Since φ = ψ ψ, we obtain

lim λ→-∞ φ(λ) λ ∈]0, +∞].
And, if Ψ is a polynomial of degree n, then

lim λ→-∞ ψ(λ) ∈]0, +∞[ and lim λ→-∞ ψ(λ) λ ∈] -∞, 0],
(this last limit is valid for any exponent of the opposite of a subordinator) and we obtain lim

λ→-∞ φ(λ) λ ∈] -∞, 0]
Finaly, ψ does not vanish on the left half plane {ℜ(λ) < 0} because it is the exponent of the opposite of a subordinator and so, the roots of ψ, which are necessary on the left half plane {Re(λ) < 0}, except possibly 0 (because the exponent of a subordinator doesn't vanish in the half plane {Re(λ) > 0}), are the one of φ. Let us study when 0 is a root of ψ.

The function ψ has a left derivative at 0 (possibly equal to -∞) and the function ψ has a derivative at 0 because it is a rational function and 0 is not a pole. Then, we can write (here, the symbol ′ has to be understood as a left derivative )

φ(0) = ψ(0) ψ(0) φ ′ (0) = ψ′ (0).ψ(0) + ψ(0)ψ ′ (0). Since ψ(0) ∈ [0, +∞[, ψ(0) ∈ [0, +∞[, ψ′ (0) ∈ [-∞, 0[, ψ ′ (0) ∈]0
, +∞] (that are obvious consequences of Lévy-Kinchin formula for exponent of subordinator or opposite of a subordinator), we obtain : φ(0) = 0 =⇒ ψ(0) = 0 and φ ′ (0) ≥ 0 or ψ(0) = 0 and φ ′ (0) ≤ 0 And the proof is finished.

In the sequel we will normalize ψ by setting ψ ∞ = 1. We will always denote -β 1 , -β 2 , . . . , -β m the (possibly equal) roots of ψ and

m i = ♯{k < i; β k = β i } ( m i = 0 for all i's iff the roots -β 1 , . . . , -β m of ψ are simple).
Remark If X is of bounded variations and has a non positive drift then m = n. In all other cases, we have m = n + 1, ψ is then the exponent of a subordinator with positive drift and X "creeps upwards" (see theorem 19 chapter 6 of [B96]). Now, we introduce few notations which will be useful in the sequel, and which are related to the negative Wiener-Hopf factor ψ. First, ψ being the exponent of the opposite of a subordinator, the inverse of ψ, 1 ψ , is the Laplace transform of a measure supported by ] -∞, 0]. We will denote this measure by Ǔ (dy),

1 ψ(λ) =: ]-∞,0] e -λy Ǔ (dy) (ℜ(λ) < 0)
Also, for all x ∈]0, +∞[, we denote by Ǔ[-x,0] (dy) and Ǔ]-∞,-x[ (dy) the measures 1 [-x,0] (y) Ǔ (dy) and 1 ]-∞,-x[ (y) Ǔ (dy) and by Ǔ[-x,0] (λ) and Ǔ]-∞,-x[ (λ) their Laplace transform;

Ǔ[-x,0] (λ) := [-x,0] e -λy Ǔ (dy) (λ ∈ C) Ǔ]-∞,-x[ (λ) := ]-∞,-x[ e -λy Ǔ (dy) (ℜ(λ) < 0)
Using standard facts about the relation between negative Wiener-Hopf factor ψ and the fluctuations of the Lévy process (see chapter 6 of [B96]), we obtain the next proposition.

Proposition 2.4.

1) If φ(0) > 0 or φ ′ (0) > 0 then then the function ψ(0) ψ(λ) is the Laplace transform of the measure ψ(0) Ǔ (dy) which is the distribution of m, the minimum of X (m := inf{X t ; 0 ≤ t < ζ}). 2) ψ(λ) Ǔ]-∞,-x[ (λ) is the Laplace transform of the distribution P(X Tx ∈ dy; T x < ζ), (T x := inf{t; X t < -x}).
Now, using the dual properties settled in this proposition which involves the positive Wiener-Hopf factor ψ instead of the negative one ψ and using the explicit form of ψ given in theorem 2.3, we obtain, (see also Mordecki

[LM08] theorem 2.2), Corollary 2.5. 1) If φ(0) > 0 or φ ′ (0) < 0 then the distribution of the maximum, M := {sup; X t ; t ∈ [0, ζ[} is P(M ∈ dy) = a 0 δ 0 (dy) + m i=1 a i y mi m i ! e -βiy dy,
where the coefficients a i are given by the rational expansion :

Π n j=1 (1 + λ/γ j ) Π m i=1 (1 + λ/β i ) = a 0 + m i=1 a i (λ + β i ) mi+1
2) The distribution of the "over shoot"

X T x -x on T x < ζ (T x = inf{t; X t > x}) is the following P(X T x -x ∈ dy; T x < ζ) = c 0 (x)δ 0 (dy) + n j=1 c j (x) y nj n j ! e -γjy 1 y>0 dy,
where the coefficients c j (x) are given by the rational expension :

c 0 (x) + n j=1 c j (x) (λ + γ j ) nj +1 = Π m i=1 (1 + λ/β i ) Π n j=1 (1 + λ/γ j ) . m i=1 a i m i ! ∂ mi (-∂β) mi e -βx λ + β β=βi
Corollary 2.5 and Proposition 2.4 give us a foretaste of the explicit results that we obtain in the sequel : Most of the explicit distributions that we will obtain will be of the preceding form or of convolution of measures of that form; that is, a combination of exponential or gamma distributions with explicit coefficients, possibly restricted to an interval, and convoluted with (a possible restriction of) the measure Ǔ (dy). The next proposition introduces Laplace transforms of distributions that will be involved in the sequel. The easy proof is left to the reader.

Proposition 2.6. 1)

m i (x, λ) := ∂ mi (-∂β) mi e (λ+β)x Ǔ[-x,0] (-β) -Ǔ[-x,0] (λ) λ + β β=βi is the Laplace transform of the measure (1 y∈[0,x] y mi e -βiy dy) * (1 y∈[-x,0] Ǔ (dy)) 1 y∈[-x,0]
2)

n j (x, λ) := ∂ nj (-∂γ) nj e -λx Ǔ]-∞,-x[ (λ) -e γx Ǔ]-∞,-x[ (-γ) λ + γ γ=γj is the Laplace transform of the measure (y nj e -γjy 1 y>0 ) * ( Ǔ (dy -x)1 y<0 ) 1 y<0 3) i j (λ) = ∂ nj (-∂γ) nj 1 λ + γ γ=γj
is the Laplace transform of the measure y nj e -γj y 1 y>0 dy

Some matrices of coefficients

Recall first all of the notations already introduced.

-γ 1 , . . . , -γ n are the poles of φ (and ψ) located on the half plane {λ, ℜ(λ) < 0}, and

n j = ♯{k < j; γ k = γ j } -β 1 , . . . , -β m
are the roots of φ (and ψ) located on the half plane {λ, ℜ(λ) < 0} together with 0 is φ(0) = 0 and φ ′ (0) > 0. and

m i = ♯{k < i; β k = β i }
We will denote by l and c the line and the column :

l = (1 nj=0 ; 1 ≤ j ≤ n). c = (1 mi=0 ; 1 ≤ i ≤ m) t .
The m i (x, λ) and n j (x, λ) and i j (λ) being defined in the previous proposition 2.6, we denote by m(x, λ) the column

m(x, λ) = (m i (x, λ); 1 ≤ i ≤ m) t and by n(x, λ) the line n(x, λ) = (n j (x, λ); 1 ≤ j ≤ n) and by i(λ) the line i(λ) = (i j (λ); 1 ≤ j ≤ n) Also, define the line v(x), v(x) := - ∂ nj (-∂γ) nj e γx Ǔ]-∞,-x[ (-γ) γ=γj ; 1 ≤ j ≤ n ,
and w(x) be the column,

w(x) := ∂ mi (-∂β) mi e βix Ǔ[-x,0] (-β) β=βi ; 1 ≤ i ≤ m t Proposition 3.1. If m = n + 1, then the limit ǔ(-x) := lim λ→-∞ (-λ)e -λx Ǔ]-∞,-x[ (λ)
exists in [0, +∞[ for all x ∈]0, +∞[. This proposition will be proved incidently here as a consequence of our computations. In fact, it can be shown that the function ǔ is a density of the measure 1 y<0 Ǔ (dy) and we will see that this density is of bounded variations (see remark 4 after corollary 5.2)

If m = n + 1, we will denote w ′ (x) the column w ′ i (x); 1 ≤ i ≤ m t , with w ′ i (x) := ∂ mi (-∂β) mi βe βx Ǔ[-x,0] (β) + ǔ(-x) β=βi Notice that if ǔ is continuous, w ′ i (x)
is the left derivative of w i (x), hence the notation. Let now W(x) be the matrix (m, n) :

W(x) = (W i,j (x); 1 ≤ i ≤ m, 1 ≤ j ≤ n) where, W i,j (x) := ∂ mi ∂ nj (-∂β) mi (-∂γ) nj e βx U [-x,0] (-β) + e γx Ǔ]-∞,-x[ (-γ) β -γ β=βi,γ=γj
When m = n + 1, denote W(x) the square matrix

W(x) = w(x) W(x)
We will show later the next proposition Let W be any matrix and let f be any line, the dimension of f being equal to the number of columns of W, W f i will be the matrix obtained from W by replacing the i-th line by f . And W i will denote the matrix obtained from W by just taking it's i-th line off. Similarly, let e be a column whose dimension is equal to the number of lines of W , W j e will refer to the matrix obtained from W by substituting e to it's j-th column.

The distribution of

(S ζ , X ζ , I ζ ) Theorem 4.1. The potential kernel of the triple (S t , X t , I t ) is characterized by the following identities. For all λ 1 , λ 2 ∈ C, x ∈]0, +∞[, E( ζ 0 e -λ1It 1 St-It≤x e -λ2(Xt-It) dt) = E( ζ 0 e -λ2St 1 St-It≤x e -λ1(Xt-St) dt) = Ǔ[-x,0] (λ 1 ) v(x) m(x, λ 1 ) W(x) |W(x)| 1 l e -λ2x m(x, λ 2 ) W(x) |W(x)| if m = n = Ǔ[-x,0] (λ 1 ) ǔ(-x) v(x) m(x, λ 1 ) w(x) W(x) | W(x)| e -λ2x m(x, λ 2 ) W(x) | W(x)| if m = n + 1 Notations. Let, if m = n, ǎ(x) := Ǔ[-x,0] (0) v(x) m(x, 0) W(x) a(x) := 1 l m(x, 0) W(x) r(x) := |W(x)| And, if m = n + 1, ǎ(x) := Ǔ[-x,0] (0) ǔ(-x) v(x) m(x, 0) w(x) W(x) a(x) := m(x, 0) W(x) r(x) := | W(x)| Corollary 4.2. If φ(0) > 0, then the law of the triple (S ζ , X ζ , I ζ ) is characterized as follow : 1 φ(0) P(S ζ -I ζ ≤ x) = ǎ(x)a(x) r 2 (x) . Given S ζ -I ζ ≤ x, the random variable X ζ -I ζ is independent of I ζ
and has the same distribution as S ζ and

P(I ζ ∈ dy|S ζ -I ζ ≤ x) = 1 y∈[-x,0] ǎ(x) ǎ0 Ǔ (dy)-[1 y∈[-x,0] Ǔ (dy)] * [ m i=1
ǎi (x)y mi e -βiy 1 y∈[0,x] dy]

P(S ζ ∈ dy|S ζ -I ζ ≤ x) = 1 y∈[0,x] a(x) a 0 δ 0 (dy)-[1 y∈[0,x] Ǔ (dy-x)] * [ m i=1 a i (x)y mi e -βiy 1 y∈[0,x] dy] with, if m = n, ǎ0 = a 0 = 1 ǎi (x) = |W v(x) i (x)| |W(x)| a i (x) = |W l i (x)| |W(x)| , i = 1 . . . m And, if m = n + 1, ǎ0 = 1 a 0 = 0 ǎi (x) = | W(x) [ǔ(-x),v(x)] i (x)| | W(x)|| a i (x) = - |W i (x)| | W(x)| i = 1 . . . m
The identities are still true when φ(0) = 0 and when 1 φ(0)

.

P(S ζ -I ζ ≤ x), P(S ζ ∈ dy|S ζ -I ζ ≤ x) P(I ζ ∈ dy|S ζ -I ζ ≤ x)
are replaced by Remark Let φ 0 be a proper (not killed) Lévy exponent and apply the formula of theorem to φ = φ 0 + q when q vary from 0 to +∞, let P q be the corresponding distribution, that is the distribution of the Lévy process X, with Lévy exponent φ 0 , killed at an independent exponential time of rate q. Clearly we have

E q ( ζ 0 e -λ1It 1 St-It≤x e -λ2(Xt-It) dt) = E 0 ( +∞ 0 e -λ1It 1 St-It≤x e -λ2(Xt-It) e -qt dt)
When taking the inverse Laplace transform over q, λ 1 and λ 2 , we find the measure

P 0 (I t ∈ dy, X t -I t ∈ dz, S t -I t ≤ x)dt
Then the previous theorem and corollary give a characterization of the distribution of (S T , X T , I T ) at any time T .

As an application, one can compute the price of a double barrier knock-out call option under exponential Lévy model for the asset Y , Y t = Y 0 e Xt , when the underlying Lévy process X satisfy the assumption 2.1, that is, an european call option with strike K which is desactivated if the asset Y goes down a barrier value a or up a value b before the time of maturity T . It is clear that the price is given by the expression :

E 0 ((Y 0 e XT -K)1 XT >log K Y 0 1 IT >log a Y 0 1 ST <log a Y 0 )
Then, this price can be deduced from the distribution of the triple (S T , X T , I T ).

5. Behavior of the process (X, S)

at time V x = inf{t, X t -S t < -x} Theorem 5.1. E exp -µ 2 S Vx -µ 1 (X Vx -S Vx ))1 Vx<ζ = ψ(µ 1 ) Ǔ]-∞,-x[ (µ 1 ) e µ1x .n(x, µ 1 ) w(x) W(x) 1 i(µ 2 ) w(x) W(x) if m = n and = ψ(µ 1 ) µ 1 Ǔ]-∞,-x[ (µ 1 ) -Ǔ]-∞,-x[ (µ 1 ) e µ1x .n(x, µ 1 ) w ′ (x) w(x) W(x) µ 2 -1 i(µ 2 ) w ′ (x) w(x) W(x) if m = n + 1 Notations. Denote if m = n, b(x) := 1 -i(0) w(x) W(x) č(x) := lim λ→0 ψ(λ) Ǔ]-∞,-x[ (λ) -n(x, λ) w(x) W(x) . And, if m = n + 1, b(x) := 0 -1 i(0) w ′ (x) w(x) W (x) č(x) := lim λ→0 ψ(λ) λ Ǔ]-∞,-x[ (λ) -Ǔ]-∞,-x[ (λ) n(x, λ) w ′ (x) w(x) W (x)
Corollary 5.2.

P(V x < ζ) = č(x) b(x) ,
The random variables S Vx and X Vx -S Vx are independent conditionally to V x < ζ and the distribution of X Vx -S Vx on V x < ζ is characterized by the identity, Ǔ (dz

) * P(X Vx -S Vx ∈ dz; V x < ζ) = [b(x)] -1 . 1 z<-x Ǔ (dz) * č-1 δ ′ 0 (dz) + č0 (x)δ 0 (dz) - n 1 čj (x)z kj e -γj z 1 z<-x (dz)
And,

P(S Vx ∈ dy; V x < ζ) = č(x) b 0 δ 0 (dy) + m i=1 b i (x)y mi(x) e -βi(x)y 1 y>0 dy .
Where, if m = n,

č-1 = 0 č0 (x) = -1 čj (x) = -W(x) j [w(x)] j = 0, . . . , n And if m = n + 1, then č-1 = 1 čj (x) = W(x) j+1 [w ′ (x)] j = 0, . . . , n
The β i (x), the exponents m i (x) := ♯{k < i; β k (x) = β i (x)} and the coefficients b i (x) are given by the following fractional expansion

b 0 (x) + m i=1 b i (x) (λ + β i (x)) mi (x)+1 = [č -1 (x)λ + č0 (x) - n j=1 čj (x) 1 (λ + γ j ) nj +1 ] -1 = 1 -i(λ) 1 W(x) -1 if m = n = λ -1 i(λ) w ′ (x) w(x) W(x) -1 if m = n + 1
Remark 1) Notice that, when ψ(0) = 0, that means that X does not drift to +∞ (see lemma 2, chapter 6 of [B96]) then

P(X Tx < ζ) = P(X Tx < +∞) = 1, thus ψ(λ) Ǔ]-∞,-x[ (λ) = E(e -λXT x ; T x < +∞) → 1 λ → 0 and we get lim λ→0 ψ(λ)n j (x, λ) = lim λ→0 ∂ nj (-∂γ) nj ψ(λ)e -λx Ǔ]-∞,-x[ (λ) -ψ(λ)e γx Ǔ]-∞,-x[ (-γ) λ + γ γ=γj = ∂ nj (-∂γ) nj 1 γ γ=γj = i j (0)
Thus, we see that b(x) = č(x) when ψ(0) = 0 and, according to the previous corollary, we obtain that P(V x < +∞) = 1 when X does not drift to +∞: this means that the heights of the excursions out of the set {t, X t = S t } are unbounded a.s. This can be obtained by elementary trajectorial arguments.

2) In order to obtain a direct characterization of the distribution P(X Vx -S Vx ∈ dz; V x < ζ) instead of the characterization of the convoluted distribution Ǔ (dy) * P(X Vx -S Vx ∈ dz; V x < ζ), it is necessary to introduce the drift, the Lévy measure and the killing rate associated to the exponent ψ. We leave this part to the interested reader.

3) Due to the lake of memory of exponential distributions and the fact that

X Vx 1 XV x >0 = (S Vx + (X Vx -S Vx )) + ,
the distribution of X Vx given X Vx > 0 is, like the distribution of S Vx as stated in the preceding corollary, a linear combination of the functions y mi(x) e -βi(x)y 1 y>0 . 4) It is clear from the characterisation of the distribution of X Vx -S Vx when m = n + 1 that the convolution 1 z<-x Ǔ (dz) * δ ′ (dz) is a signed measure, that means that the measure Ǔ (dy) has a density, and this density, which is the fonction ǔ appearing in lemma 3.1, is of bounded variations.

6. Behavior of the process (I, X) at time

V x := inf{t; X t -I t > x} Theorem 6.1. E(e -µ1I V x e -µ2(X V x -I V x ) ) = e -µ2x ψ(µ 1 ) 0 i(µ 2 ) c W (x) 1 n(x, µ 1 ) c W (x) if m = n = e -µ2x ψ(µ 1 ) . 0 1 i(µ 2 ) c w(x) W(x) 1 e -µ1x Ǔ]-∞,-x[ (µ 1 ) n(x, µ 1 ) c w(x) W(x) if m = n + 1 Notations If m = n, c(x) = 0 i(0) c W(x) b(x) = lim λ→0 ψ(λ) 1 n(x, λ) c W(x) If m = n + 1, c(x) = 0 1 i(0) c w(x) W(x) b(x) = lim λ→0 ψ(λ) 1 e -λx Ǔ]-∞,-x[ (-λ) n(x, λ) c w(x) W(x)
One can notice again that b(x) = c(x) when ψ(0) = 0.

Corollary 6.2.

P(V x < ζ) = c(x) b(x)
and

I V x and X V x -I V x are independent conditionally to V x < ζ P(X V x -I V x ∈ x+dy|V x < ζ) = [c(x)] -1 . c 0 (x)δ 0 (dy)+ n j=1
c j (x)y nj e -γj y 1 y>0 dy

P(I V x ∈ dz; V x < ζ) = c(x) r(x) Ǔ (dz) * +∞ n=0 µ * n x (dz) with µ x (dz) = 1 z<0 . Ǔ (dz -x) * c 0 (x)δ 0 (dz) +
If m = n + 1, c j (x) = | Wj+1 c (x)| | W(x)|
for j = 0, . . . , n 7. Behavior of the process at time U x = inf{t, S t -I t > x}.

Theorem 7.1.

P(U x < ζ, X Ux = I Ux ) = a(x)č(x) r(x) 2 P(U x < ζ, X Ux = S Ux ) = ǎ(x)c(x) r(x) 2
Given the event {U x < ζ; X Ux = I Ux }, S Ux and X Ux -S Ux are independent. The random variable S Ux has the same distribution as

S ζ given S ζ -I ζ ≤ x if φ(0) > 0,
and has the distribution

∞ 0 P(S t ∈ dz, S t -I t ≤ x)dt ∞ 0 P(S t -I t ≤ x)dt if φ(0) = 0. The random variable X Ux -S Ux has the same distribution as X Vx -S Vx given V x < ζ.
Given the event {U x < ζ; X Ux = S Ux }, I Ux and X Ux -I Ux are independent. The random variable I Ux has the same distribution as

I ζ given S ζ -I ζ ≤ x if φ(0) > 0,
and has the distribution

∞ 0 P(I t ∈ dy, S t -I t ≤ x)dt ∞ 0 P(S t -I t ≤ x)dt if φ(0) = 0. The random variable X Ux -I Ux has the same distribution as X V x -I V x given V x < ζ.
8. Behavior of the process at time

T b a = inf{t; X t ∈ [-a, b]} Theorem 8.1. We have, if φ(0) > 0 P(X T b a = S T b a , T b a < ζ) = [-a,0] c(b -y) a(b -y) P(I ζ ∈ dy, S ζ ≤ b) φ(0) . And P(X T b a = I T b a , T b a < ζ) = [0,b] č(a + x) ǎ(a + x) P(I ζ ≥ -a, S ζ ∈ dx) φ(0)
If φ(0) = 0, we replace in last formulas

P(I ζ ∈dy,S ζ ≤b) φ(0)
and

P(I ζ ≥-a,S ζ ∈dx) φ(0)
by respectively

P(I t ∈ dy, S t ≤ b)dt +∞ 0 P(S t ≤ b, -I t ≥ -a)dt if φ(0) = 0. The distribution of X T b a -I T b a given I T b a = y is the one of X V b+y given V b+y < ζ. Given {X T b a = I T b a , T b a < ζ}, the distribution of S T b a is the distribution of S ζ given {S ζ ≥ -a, I ζ ≤ b} if φ(0) > 0 and the distribution +∞ 0 P(S t ∈ dx, -I t ≥ -a)dt +∞ 0 P(S t ≤ b; -I t ≥ -a)dt if φ(0) = 0. The distribution of X T b a -S T b a given S T b a = x is the one of X Va+x given V a+x < ζ.
Remarks 1) We can deduce from this theorem that the distribution of X T b a -b given X T b a > b is a linear combination of distribution of the distributions y nj e -γjy 1 y>0 dy. That could easily be deduced from the lack of memory of the exponential distribution. The previous result allows also to compute a closed form of the coefficients. 2) As in the remark following theorem 4.1, let φ 0 be a proper (not killed) Lévy exponent and apply the formula of theorem to φ = φ 0 + q when q vary from 0 to +∞, let P = P q be the corresponding distribution, that is the distribution of the Lévy process X, with Lévy exponent φ 0 , killed at an independent exponential time of rate q. The probabilities P q (X T b a = S T b a , T b a < ζ) and P q (X T b a = I T b a , T b a < ζ) are q times the Laplace transforms of the functions of t,

P 0 (X T b a = S T b a , T b a < t) and P 0 (X T b a = S T b a , T b a < t).
On the other hand, the price of a double barrier with rebate, say rebate r + (resp. ) is given by the next formula

E((Y 0 e XT -K)1 XT >log K Y 0 1 IT >-ã;ST < b) + r + P(X T b ã = S T b ã , T b ã < T ) +r -P(X T b ã = I T b ã , T b ã < T )
We have seen that the first term can be computed with the help of theorem 4.1 and now we see that the two other terms can be computed by an inverse Laplace transform in time of

P q (X T b ã = S T b ã , T b ã < ζ) and P q (X T b ã = I T b ã , T b ã < ζ)
which are characterized in the preceding theorem. 9. Recall of the main result of [F10] and its application with the Assumption 2.1

Proposition 9.1. There exist unique functions A(x, λ), Ǎ(x, λ), B(x, λ) and B(x, λ), C(x, λ) and Č(x, λ) such that A(x, λ) and Ǎ(x, λ), are defined for λ ∈ C and entire; B(x, λ) and C(x, λ) are defined, continuous on {ℜ(λ) ≥ 0} and holomorphic on {ℜ(λ) > 0}; B(x, λ) and Č(x, λ) are defined and continuous on {ℜ(λ) ≤ 0}, and holomorphic on {ℜ(λ) < 0}, satisfying the following identities,

P( ζ 0 e -λ1It 1 St-It≤x e -λ2(Xt-It) dt) = P( ζ 0 e -λ2St 1 St-It≤x e -λ1(Xt-St) dt) = Ǎ(x, λ 1 )A(x, λ 2 ) E exp -µ 2 S Vx -µ 1 (X Vx -S Vx )1 Vx<ζ = Č(x, µ 1 ) B(x, µ 2 ) E exp -µ 1 I V x -µ 2 (X V x -I V x )1 V x <ζ = C(x, µ 2 ) B(x, µ 1 ) ψ(λ)A(x, λ) → 1 B(x, λ) ∼ ψ(λ) for λ → +∞ B(x, λ) ∼ ψ(λ) for λ → -∞
Proof. The existence of the 6 functions A, Ǎ, B, B, C, Č has been settled in the general setting of a Lévy process in proposition 6.1 and proposition 6.2 of [F10] , the behavior of the functions is given in theorem 4.2 of [F10]. The uniqueness is obvious.

More over, we know from theorem 4.2 of [F10] (or we can deduce them from proposition 9.1), the behavior of the six functions as settled in next proposition Proposition 9.2. The functions of λ, A(x, λ), e -λx Ǎ(x, λ) , B(x,λ) |λ|+1 , e λx C(x, λ) are bounded on {Re(λ) ≥ 0}. The functions Ǎ(x, λ), e λx A(x, λ), B(x,λ) |λ|+1 , e -λx Č(x, λ) are bounded on {Re(λ) ≤ 0}.

For ℜ(λ) → +∞ e λx C(x, λ) → 0 if ψ(λ) is bounded, and for ℜ(λ) → -∞, e -λx Č(x, λ) → 0 if ψ(λ) is bounded.
Remark The property "ψ(λ) is bounded" means that ψ is the exponent of a compound Poisson process and this is equivalent to the property (see chapter 6 of [B96] that the first time the Lévy process X visits ]0, +∞[ is non zero a.s. (here it is also equivalent to the condition m = n). In the next lemma, we state that the determination of these six functions reduces to the computation of four polynomials.

and we obtain M (x, λ) = R -1 (x, λ).N (x, λ)

= P 1 (x, λ) e -λx Q 1 (x, λ) e λx Q 2 (x, λ) P 2 (x, λ) × 1 0 Ǔ[-x,0] (λ) 1 × 1 Π m 1 (λ+βi) 0 0 1 Π n 1 (λ+γj ) for ℜ(λ) > 0 = P 1 (x, λ) e -λx Q 1 (x, λ) e λx Q 2 (x, λ) P 2 (x, λ) × ψ(λ) 1 -ψ(λ). Ǔ]-∞,-x[ (λ) Ǔ[-x,0] (λ) × 1 Π n 1 (λ+γj ) 0 0 1 Π m 1 (λ+βi)
for ℜ(λ) < 0

Developping each term of this matrix product, we obtain the identities of the theorem. Now, according to proposition 9.2, e -λx C(x, λ) is bounded on {ℜ(λ) > 0} and goes to

0 if ψ is bounded (that is if m = n), that gives that deg Q 1 ≤ m -1.
Futhermore, the equivalence of the proposition 9.1 B(x, λ) ∼ ψ(λ) gives that P 2 (x, λ) = λ m for λ → +∞. Since ψ(λ)A(x, λ) → 1, we deduce that P 1 (λ) ∼ λ n for λ → +∞. Also e -λx Č(x, λ) is bounded for ℜ(λ) → +∞ and goes to 0 if ψ(λ) is bounded. This is equivalent to say that

P 2 (x, λ)e -λx Ǔ]-∞,-x[ (λ) -Q 2 (x, λ) Π n j=1 (λ + γ j ) → 0 if ℜ(λ) → -∞
In other words, (9.2)

P 2 (x, λ)e -λx Ǔ]-∞,-x[ (λ) -Q 2 (x, λ) = o(λ n )
If m = n then P 2 (x, λ) ∼ λ n and we have clearly that e -λx Ǔ]-∞,-x[ (λ) → 0 These two facts put together with property (9.2) give us that Q 2 (x, λ) = o(λ n ) and so,

deg Q 2 (x, λ) ≤ n -1 = m -1 If m = n + 1, then P 2 (x, λ) ∼ λ n+1 and P 2 (x, λ)e -λx Ǔ]-∞,-x[ (λ) = o(λ n+1 ) then,
with property (9.2), we deduce that

Q 2 (x, λ) = o(λ n+1 ), in other words, deg Q 2 (x, λ) ≤ n = m -1
The uniqueness of the 4 polynomials come from the uniqueness of the 6 functions. This finish the proof of proposition 9.3. Now, when m = n + 1 and when looking more precisely the preceding identities, since deg

Q 2 (x, λ) ≤ n, we have that Q2(x,λ) Π n j=1 (λ+γj ) has a finite limit. Since P 2 (x, λ)e -λx Ǔ]-∞,-x[ (λ) -Q 2 (x, λ) Π n j=1 (λ + γ j ) → 0 λ → -∞
we deduce that

P 2 (x, λ)e -λx Ǔ]-∞,-x[ (λ) Π n j=1 (λ + γ j )
has a finite limit when λ → -∞ Since P 2 (x, λ) ∼ λ n+1 and Π n j=1 (λ + γ j ) ∼ λ n , we obtain that λe -λx Ǔ]-∞,-x[ (λ) has a finite limit (-ǔ(-x)) when λ → -∞. That proves proposition 3.1. Now since

P2(x,λ)e -λx Ǔ]-∞,-x[ (λ)-Q2(x,λ) Π n j=1 (λ+γj )
goes to 0( for we deduce that

Q2(x,λ) λ n
→ -ǔ(-x). We state this property in the next lemma.

Lemma 9.4.

lim |λ|→+∞ Q 2 (x, λ) λ n = -ǔ(-x)
10. Computation of A and Ǎ and proof of theorem 4.1 and corollary 4.2

Proposition 10.1.

A(x, λ) = 1 l e -λx m(x, λ) W(x) |W (x)| when m = n, A(x, λ) = e -λx m(x, λ) W(x) | W(x)| when m = n + 1
Proof. Recall the equation of proposition 9.1

A(x, λ) = 1 Π m 1 (λ + β i ) P 1 (x, λ) + Q 1 (x, λ)e -λx Ǔ[-x,0] (λ)
Since A(x, λ) is an entire function and because deg

P 1 = n and deg Q 1 ≤ m -1 (see proposition 9.3), then A(x, λ) is necessarily of the next form A(x, λ) = a 0 (x) - m 1 a i (x) ∂ mi (-∂β) mi e βx Ǔ[-x,0] (-β) -e -λx Ǔ[-x,0] (λ) β + λ β=βi = a 0 (x) - m 1 a i (x)e -λx m i (x, λ)
for some coefficients a i (x), i = 0, 1, . . . , m Thus,

P 1 (x, λ) = Π m 1 (λ + β i ) a 0 (x) - m 1 a i (x) ∂ mi (-∂β) mi e βx Ǔ[-x,0] (-β) λ + β β=βi Q 1 (x, λ) = Π m 1 (λ + β i ) m 1 a i (x) ∂ mi (-∂β) mi 1 λ + β β=βi Now using property P 1 (x, λ) ∼ λ n of lemma 9.3, one obtain (10.1) a 0 (x) = 1 if m = n (10.2) a 0 (x) = 0 and m i=1 a i (x)w i (x) = -1 if m = n + 1
On the other hand, take the identity

B(x, λ) = ψ(λ) Π(λ + γ j ) . P 1 (x, λ) -Q 1 (x, λ)e -λx Ǔ]-∞,-x[ (λ) ,
and replace P 1 and Q 1 by the above expressions, we get

B(x, λ) = ψ(λ) Π m 1 (λ + β i ) Π n 1 (λ + γ j ) a 0 (x)- m i=1 a i (x) ∂ mi (-∂β) mi e βx Ǔ[-x,0] (-β) + e -λx Ǔ]-∞,-x[ (λ) β + λ ) β=βi
The function B(x, λ) is holomorphic on the left half plan, in particular at points λ = -γ j , this implies that for every j = 1, . . . , n,

∂ nj (∂λ) nj m i=1 a i (x) ∂ mi (-∂β) mi e βx Ǔ[-x,0] (-β) + e -λx Ǔ]-∞,-x[ (λ) β + λ ) β=βi,λ=-γj = ∂ nj (∂λ) nj [a 0 (x)] λ=-γj
In other words,

m i=1 a i (x)W i,j (x) = 1 nj =0 if m = n And m i=1 a i (x)W i,j (x) = 0 if m = n + 1
These n equations added to equation ( 10.2) if m = n + 1, form a linear system of m equations with variables a i (x), i = 1, . . . , m. It is easy to be convinced that each solution of this systems brings a new couple of polynomials P 1 (x, λ) and Q 1 (x, λ).

The uniquiness of such a couple leads to the uniquiness of this solution. Thus the system is a Cramer System and the determinant of the matrix W(x) for m = n (resp. of the matrix W (x) for m = n + 1) does not vanish. Finally we obtain, For m = n,

a 0 (x) = 1 a i (x) = |W l i (x)| |W(x)| for i = 1, . . . , n and
A(x, λ) = a 0 (x) - m 1 a i (x)e -λx m i (x, λ) = 1 1 e -λx m(x, λ) W(x) |W(x)| . For m = n + 1, a 0 (x) = 0 a i (x) = - |W i (x)| | W(x)| for i = 1, . . . , m, and
A(x, λ) = - m 1 a i (x)e -λx m i (x, λ) = e -λx m(x, λ) W(x) | W(x)| Proposition 10.2. Ǎ(x, λ) = Ǔ[-x,0] (λ 1 ) v(x) m(x, λ 1 ) W(x) |W(x)| if m = n Ǎ(x, λ) = Ǔ[-x,0] (λ 1 ) ǔ(-x) v(x) m(x, λ 1 ) w(x) W(x) | W(x)| if m = n + 1
Proof. Take the equation of proposition 9.1,

Ǎ(x, λ) = 1 Π m 1 (λ + β i ) Q 2 (x, λ)e λx + P 2 (x, λ) Ǔ[-x,0] (λ) .
Since Ǎ(x, λ) is an entire function of λ and

P 2 (λ) ∼ λ m , deg Q 2 ≤ m -1 according to proposition 9.3, then Ǎ(x, λ) is necessarily of the next form Ǎ(x, λ) = Ǔ[-x,0] (λ) + m 1 ǎi (x) ∂ mi (-∂β) mi Ǔ[-x,0] (λ) -e (λ+β)x Ǔ[-x,0] (-β) λ + β β=βi = Ǔ[-x,0] (λ) - m 1 ǎi (x)m i (x, λ)
For some coefficients ǎi (x), i = 1, . . . , m. Thus,

Q 2 (x, λ) = -Π m 1 (λ + β i ) m 1 ǎi (x) ∂ mi (-∂β) mi e βx Ǔ[-x,0] (-β i ) λ + β β=βi P 2 (x, λ) = Π m 1 (λ + β i ) 1 + m 1 ǎi (x) ∂ mi (-∂β) mi 1 λ + β β=βi
Take now the identity of proposition 9.1, Č(x, λ) = e λx ψ(λ) Π(λ + γ j )

P 2 (x, λ)e -λx Ǔ]-∞,-x[ (λ) -Q 2 (x, λ)
Č(x, λ) is holomorphic on {Re(λ < 0}. Then for all j = 1, . . . , n,

∂ nj (∂λ) nj P 2 (x, λ)e -λx Ǔ]-∞,-x[ (λ) -Q 2 (x, λ) Π m 1 (λ + β i ) λ=-γj = 0
In this last equation, when replacing P 2 (x, λ) and Q 2 (x, λ) by their above expressions, we obtain the next system, for j = 1, . . . , n,

∂ nj (∂λ) nj e -λx Ǔ]-∞,-x[ (λ) λ=-γj + m i=1 ǎi (x) ∂ nj (∂λ) nj ∂ mi (-∂β) mi e -λx Ǔ]-∞,-x[ (λ) + e βx Ǔ[-x,0] (-β) λ + β β=βi,λ=-γj = 0
In other words ,

(10.3) m i=1 ǎi (x)W i,j (x) = v j (x)
This a system of n equations where the variables are the m terms ǎi (x).

When m = n + 1, we have the property of lemma 9.4 Q2(x,λ)

λ n → -ǔ(-x) when λ → -∞ then m i=1 ǎi (x) ∂ mi (-∂β) mi e βx Ǔ[-x,0] (-β) β=βi = ǔ(-x)
In other words (10.4)

m i=1 ǎi (x)w i (x) = ǔ(-x)
We have already seen in the previous proof, that det W(x) = 0 if m = n and det W(x) = 0 if m = n + 1. Then the system (10.3) augmented by the equation ( 10.4) if m = n + 1, is a Cramer system; and the solution is given as follows. If

m = n, ǎi (x) = |W [v(x)] i (x)| |W(x)| and Ǎ(x, λ) = Ǔ[-x,0] (λ) - m 1 ǎi (x)m i (x, λ) = Ǔ[-x,0] (λ) v(x) m(x, λ) W(x) |W(x)| If m = n + 1, ǎi (x) = | W(x)] [ǔ(-x),v(x)] i | | W(x)| Ǎ(x, λ) = Ǔ[-x,0] (λ) - m 1 ǎi (x)e λx m i (x, λ) = Ǔ[-x,0] (λ) ǔ(-x) v(x) e λx m(x, λ) w(x) W(x) | W(x)|
Proof of theorem 4.1 and corollary 4.2 Theorem 4.1 follows from ropositions 9.1, 10.1 and 10.2. After that, corollary 4.2 becomes obvious by the Laplace inversion of the functions m i (x, λ) given in proposition 2.6.

11. Computation of Č(x, λ) and B(x, λ) and proof of theorems 5.1 and corollary 5.2

Proposition 11.1.

Č(x, λ) = ψ(λ) Ǔ]-∞,-x[ (λ) -e λx n(x, λ) w(x) W(x) |W(x)| B(x, λ) = 1 -i(λ) w(x) W(x) |W(x)| if m = n Č(x, λ) = ψ(λ) λ Ǔ]-∞,-x[ (λ) -Ǔ]-∞,-x[ (λ) e λx n(x, λ) w ′ (x) w(x) W(x) | W(x)| B(x, λ) = λ -1 i(λ) w ′ (x) w(x) W(x) | W(x)| if m = n + 1
Proof. We take the expression of Č(x, λ) of proposition 9.3,

Č(x, λ) = ψ(λ) Π n 1 (λ + γ j ) -Q 2 (x, λ)e λx + P 2 (x, λ) Ǔ]-∞,-x[ (λ) with deg P 2 = m and deg Q 2 ≤ m -1. Because Č(x, λ) is holomorphic on {ℜ(λ) < 0}, the function e -λx Č(x, λ) is in the next form, e -λx Č(x, λ) = ψ(λ). č-1 (x)λe -λx Ǔ]-∞,-x[ (λ) + č0 (x)e -λx Ǔ]-∞,-x[ (λ) + ď0 (x) - n j=1 čj (x) ∂ nj (-∂γ) nj e -λx Ǔ]-∞,-x[ (λ) -e γx Ǔ]-∞,-x[ (-γ) λ + γ γ=γj In other words, e -λx Č(x, λ) = ψ(λ) č-1 (x)λ.e -λx Ǔ]-∞,-x[ (λ)+č 0 (x)e -λx Ǔ]-∞,-x[ (λ)+ ď0 (x)- n j=1 čj (x)n j (x, λ)
for some coefficients čj (x), j = -1, 0, 1, . . . , n and a coefficient ď0 (x). Thus

P 2 (x, λ) = Π(λ + γ j ) č-1 (x)λ + č0 (x) - n j=1 čj (x) ∂ nj (-∂γ) nj 1 λ + γ γ=γj And Q 2 (x, λ) = Π(λ + γ j ) -ď0 (x) - n j=1 čj (x) ∂ nj (-∂γ) nj e γx Ǔ]-∞,-x[ (-γ) λ + γ ) γ=γj
Take the equation of proposition 9.1

e -λx Ǎ(x, λ) = 1 Π(λ + β i ) Q 2 (x, λ) + P 2 (x, λ)e -λx Ǔ[-x,0] (λ) .
The function Ǎ(x, λ) is an entire function, then we obtain the equation for every

i = 1, . . . , m, ∂ mi (∂λ) mi Q 2 (x, λ) + P 2 (x, λ)e -λx Ǔ[-x,0] (λ) Π n 1 (λ + γ j ) λ=-βi = 0
Taking the expressions of P 2 and Q 2 upwards, we obtain the next equations,

∂ mi (∂λ) mi λč -1 (x)e -λx Ǔ[-x,0] (λ) + č0 (x)e -λx Ǔ[-x,0] (λ) -ď0 (x) λ=-βi - n j=1 čj (x) ∂ mi (∂λ) mi ∂ nj (-∂γ) nj e -λx Ǔ[-x,0] (λ) + e γx Ǔ]-∞,-x[ (γ) λ + γ λ=-βi,γ=γj = 0
In other words, (11.1)

č0 (x)w i (x) + n j=1 čj (x)W i,j (x) = ∂ mi (∂λ) mi -č -1 (x)λe -λx Ǔ[-x,0] (λ) + ď0 (x) λ=-βi
We can compute the coefficients č-1 (x), č0 (x) and ď0 (x) by using the assertions of lemma 9.3. More precisely, if m = n, we have P 2 (x, λ) ∼ λ n and deg

Q 2 ≤ n -1, thus we obtain (11.2) č-1 (x) = ď0 (x) = 0 č0 (x) = 1 If m = n + 1 then P 2 (x, λ) ∼ λ m and according to proposition 9.4, Q 2 (x, λ) ∼ [-ǔ(-x)]λ n , thus we obtain (11.3) d 0 (x) = ǔ(-x) č-1 (x) = 1
Then, if m = n, using the system (11.1) and equation ( 11.2) we get the new system

n j=1 čj (x)W i,j (x) = -w i (x) for i = 1, . . . , n Thus čj (x) = - |W j [w(x)] (x)| |W(x)| And Č(x, λ) = ψ(λ). Ǔ]-∞,-x[ (λ) -e λx . n j=1 čj (x)n j (x, λ) = ψ(λ) Ǔ]-∞,-x[ (λ) -e λx n(x, λ) w(x) W(x) |W(x)| And B(x, λ) = P 2 (x, λ) Π(λ + γ j ) = 1 - n 1 čj (x) ∂ nj (-∂γ) nj 1 λ + γ γ=γj = 1 - n 1 čj (x)i j (λ) = 1 -i(λ) w(x) W(x) |W(x)|
Similarely if m = n + 1, we obtain from system (11.1) and equation (11.3), the new Cramer system č0 (x)w

i (x) + n j=1 c j (x)W i,j (x) = ∂ mi (-∂β) mi βe βx Ǔ[-x,0] (-β) + ǔ(-x) β=βi = w ′ i (x)
Then (for j = 0, 1, . . . , n),

čj (x) = W(x) j+1 [w ′ (x)] W(x) And Č(x, λ) = ψ(λ). λ. Ǔ]-∞,-x[ (λ)+ǔ(-x)e λx +č 0 (x) Ǔ]-∞,-x[ (λ)- n j=1 čj (x)e λx n j (x, λ) = λ Ǔ]-∞,-x[ (λ) + ǔ(-x)e λx -Ǔ]-∞,-x[ (λ) e λx n(x, λ) w ′ (x) w(x) W(x) W(x) B(x, λ) = P 2 (x, λ) Π(λ + γ j ) = λ + č0 (x) - n 1 čj (x) ∂ nj (-∂γ) nj 1 λ + γ γ=γj = λ + č0 (x) - n j=1 čj (x)i j (λ) = λ -1 i(λ) w ′ (x) w(x) W(x) W(x)
Proof of theorem 5.1 and corollary 5.2. The proof of 5.1 follows from the above proposition and proposition 9.1. Corollary 5.2 follow then with the help the inverse Laplace transform of the functions n j (x, λ) given in proposition 2.6. for some coefficients bj (x), j = -1, 0, 1, . . . , n and a coefficient ď0 (x).

Thus the polynomial P 1 and Q 1 are given by the next expressions

P 1 (x, λ) = Π n 1 (λ + γ j ) 1 + n j=1 bj (x) ∂ nj (-∂γ) nj e γx Ǔ]-∞,-x[ (-γ) λ + γ ) γ=γj Q 1 (x, λ) = Π n 1 (λ + γ j ) -b0 (x) + n j=1 bj (x) ∂ nj (-∂γ) nj 1 λ + γ γ=γj
On the other hand, according to proposition 9.1, we have

A(x, λ) = 1 Π m 1 (λ + β i ) P 1 (x, λ) + Q 1 (x, λ)e -λx Ǔ[-x,0] (λ)
Since A(x, λ) is an entire function of λ, then it is holomorphic at λ = -β i ; Thus we have the equations for i = 1, . . . , m.

∂ mi (∂λ) mi Proof of corollary 6.2. This corollary becomes obvious when using propositions 9.1 and the previous one and when using the Laplace thransform inversion of n j (x, λ) and i j (λ) given in proposition 2.6 13. Proof of theorems 7.1 and 8.1

It has been proven in [F10] (propositions 6.3. and 6.4) that the distributions of the triple (X, I, S) at time U x and at time T b a are also characterized by the 6 functions A, Ǎ, B, B, C, Č. Then, we translate these two propositions with reference to the explicit forms of the functions obtained previously.

  x)z nj e -γjz 1 z>0 dz 1 z<0 , and, if m = n, c 0 (x) = 0 and c j (x) = |W j c (x)| |W(x)| for j = 1, . . . , n

  t ∈ dy, S t ≤ b)dt and +∞ 0 P(S t ∈ dx, -I t ≥ -a)dt Given {X T b a = S T b a , T b a < ζ}, the distribution of I T b a is the distribution of I ζ given {S ζ ≤ b, I ζ ≥ -a} if φ(0) > 0 and the distribution ( +∞ 0

  r -) if the option exceed the value b (resp. goes down the value a) before time T . (Denote ã =log a Y0 , b = log b Y0

12.

  Computation of the functions C(x, λ) and B(x, λ) and proof of theorem 6γ j ). P 1 (x, λ) -Q 1 (x, λ)e -λx Ǔ]-∞,-x[ (λ) with deg P 1 (λ) ∼ λ n and deg Q 1 ≤ m -1 Since B(x, λ) is holomorphic on {ℜ(λ) < 0}, we can write B(x, λ) in the following form, B(x, λ) = ψ(λ) 1+ b0 (x)e -λx Ǔ

P 1

 1 (x, λ) + Q 1 (x, λ)e -λx Ǔ)W i,j (x) = 1 mi=0 Moreover, if m = n, since deg Q 1 (x, λ) ≤ n -1seen that the determinants |W(x)| ( if m = n) and | W(x)| (if m = n + 1), then the system (12.1) augmented with (12.2) if m = n, form a Cramer system whose solutions are given by the next expressions, If m = n, If m = n + 1, we have for j = 0, . . . , n, , λ) = ψ(λ) 1 -b0 (x)e -λx Ǔ]-∞,-x[ (λ) -

Acknowledgment. this work has been supported by the ANR, on the project ANR-09-BLAN-0084-01.

Proposition 9.3. There exist unique polynomials of λ, P 1 (x, λ), Q 1 (x, λ), P 2 (x, λ), Q 2 (x, λ), with

satisfies the identity (9.1)

Moreover, det M (x, λ) = 1, and M is invertible. On the other hand, one can easily check that the matrix

for ℜ(λ) < 0 satisfies the same identity 9.1. Thus, the product N (x, λ)M -1 (x, λ) is entire. One can see easily that, because of the boundary conditions fulfilled by the components of M (x, λ) given in proposition 9.2, that this matrix N (x, λ)M -1 (x, λ) is of the form :

=: R(x, λ)

Where P 1 , Q 1 , P 2 and Q 2 are polynomials of λ. Then det R = det N = Π m 1 (λ + β i )Π n 1 (λ + γ j ),


Lucien Wald
email: lucien.wald@cenerg.cma.fr

Thierry Ranchin

COMMENT LIU "SMOOTHING FILTER-BASED INTENSITY MODULATION: A SPECTRAL PRESERVE IMAGE FUSION TECHNIQUE FOR IMPROVING SPATIAL DETAILS"

We discuss a critical point of the paper. It is shown on the one hand that the author of this paper has mistaken the change of spectral content with the change in spatial resolution, and on the other hand, that the protocol used to establish the advantages of his smoothing filter-based intensity modulation (SFIM) technique over other methods is not appropriate at all. Though the SFIM technique has its merits, the above-mentionned article does not demonstrate its expected qualities.

1.

Introduction

 [START_REF] Liu | Smoothing filter-based intensity modulation: a spectral preserve image fusion technique for improving spatial details[END_REF]recently published a paper entitled "Smoothing filter-based intensity modulation: a spectral preserve image fusion technique for improving spatial details". This paper presents a technique called smoothing filter-based intensity modulation (SFIM) that permits the fusion of images having a given spatial resolution with images having a higher spatial resolution. The SFIM aims at producing synthetic images (fused products) having the highest spatial resolution in the same multispectral bands of the original low resolution images. The example given in the paper is the fusion of a Landsat TM multispectral image with a resolution of 30 m with a SPOT panchromatic image with a resolution of 10 m. The result is supposed to be a multispectral image with the same bands than the Landsat original image but with a spatial resolution close to 10 m. The SFIM technique is based upon the extraction by a filtering technique of the high frequencies of the SPOT image and their injection into the Landsat imagery. The SFIM technique is a refinement of the method of [START_REF] Pradines | Improving SPOT image size and multispectral resolution[END_REF] and of that of [START_REF] Liu | Smoothing filter-based intensity modulation: a spectral preserve image fusion technique for improving spatial details[END_REF]Moore (1998, improperly referred to as Guo andMoore in Ranchin, Wald 2000a). All belong to the group of methods known as the ARSIS concept (Ranchin, Wald, 2000a). Liu indicates that the IHS and Brovey techniques cause spectral distortion and that "preserving the original spectral properties is very important for most remote sensing applications". Liu makes efforts to establish the advantages of the SFIM technique. The colour composites are convincingly demonstrating that point. However we cannot agree with the following content of his article where he pretends to analyse better the spectral preservation property of the SFIM technique based on statistics and demonstrate the superiority of the SFIM technique over the IHS and Brovey techniques when spectral content is at stake. The first point is that many articles have been published that demonstrate that the spectral content of an image changes as the spatial resolution changes. Hence it cannot be demonstrated that the fused multispectral images should have the same spectral content than the original Landsat images (preservation of the original spectral properties). Ranchin, Wald (2000a) note that several authors dealing with the methods under concern (including ourselves) improperly write "preservation of spectral content", an inappropriate shortcoming for "high-quality transformation of the multispectral content when increasing the spatial resolution". But here, we do not believe that this is only a problem of expression because the protocol used by Liu for assessing the quality of the transformation is very inappropriate.

The three properties of the fused images

Let denote the acquired images of lowest spatial resolution by B l , and the images of highest spatial resolution by A h . The subscripts l and h denote the spatial resolution of images B or A, i.e., low and high resolution, respectively. B interp h denotes the result of the interpolation (resampling) of B l from resolution l to h. Here the images B l are the Landsat images of resolution 30 m, the B interp h the Landsat images resampled to 10 m superimposable to the image SPOT panchromatic (image A h ) of resolution 10 m. Within the set of images B, B kl denotes the image acquired in the spectral band k. The fusion methods aim at constructing synthetic images B* h , which are close to reality. The methods should perform a high-quality transformation of the multispectral content of B l , when increasing the spatial resolution from l to h. [START_REF] Wald | Fusion of satellite images of different spatial resolutions: assessing the quality of resulting images[END_REF] establish that these synthetic images B* must respect the three following properties. First property. Any synthetic image B* h once degraded to its original resolution l, should be as identical as possible to the original image B l . Approximation induced by the resampling of B* kh into B* kl should be taken into account: typically the root of the mean of the squared differences (B kl -B* kl ) on a pixel basis should not exceed 0.05 times the mean value of B kl . Second property. Any synthetic image B* h should be as identical as possible to the image B h that the corresponding sensor would observe with the highest spatial resolution h, if existent. The second property does not imply an accurate synthesis of the multispectral properties of the set B when increasing the spatial resolution. This should be an additional property. Third property. The multispectral set of synthetic images B* h should be as identical as possible to the multispectral set of images B h that the corresponding sensor would observe with the highest spatial resolution h, if existent. Liu does not check any of these properties.

The protocol for quality assessment

Only one protocol has been proposed up to now, which permits to check these three properties in case the images B h are not available, like in the case of the article of Liu. The first mention of such a protocol originates from [START_REF] Munechika | Resolution enhancement of multispectral image data to improve classification accuracy[END_REF], [START_REF] Mangolini | Evaluation de la qualit€ des images multispectrales • haute r€solution spatiale d€riv€es de SPOT[END_REF] extended it.

It has been strongly refined and justified by [START_REF] Wald | Fusion of satellite images of different spatial resolutions: assessing the quality of resulting images[END_REF]. [START_REF] Vrabel | Multispectral imagery advanced band sharpening study[END_REF] presents a detailed protocol in use in the US Defence for the assessment of fused products for visual analysis. [START_REF] Wald | Fundamentals in data fusion and image fusion techniques[END_REF] integrates both protocols in a general frame. It permits to alleviate the need for a reference image if not available and offers a complete checking of the three properties.

In this particular case, the protocol is as follows:

 the fusion method is applied to the original sets of images  the second and third properties are tested with the synthetic images B* kl . The quality observed for the fused products B* kl is assumed to be close to the quality that would be observed for the fused products B* kh if a reference at resolution h were present. This point has been largely discussed by [START_REF] Wald | Fusion of satellite images of different spatial resolutions: assessing the quality of resulting images[END_REF]. A comparison is performed between B kl and B* kl by the means of visual analysis and analysis of the similarities and discrepancies.

For the third property, the emphasis is put on the spectral similarities. Depending upon the objectives of the assessment and of the available resources, the task of visual analysis will be more or less sophisticated and the computer analysis of the similarities and discrepancies will be more or less extensive. It is also recommended to apply the technique under examination on images of urban areas since these landscapes offer a very large diversity of both spectral signatures and high spatial frequencies. The protocol followed by Liu is based upon the comparison between the synthetic images B* kh and the interpolated original images B interp kh (interpolated Landsat images at 10 m). Except in the case of landscapes with no high frequency, the interpolated images B interp kh are different from the images B kh that should be observed if a sensor of resolution h were existent. This has been proven several times. Accordingly, even if the sets of images B* h and B interp h are very similar, it does not prove the high-quality transformation of the multispectral content when increasing the spatial resolution performed by the SFIM method because the reference image B interp h selected by Liu is not at all the appropriate one. A further comment is the choice made by Liu of the linear correlation coefficient. This coefficient is not sufficient to summarize a comparison. It is notably insensitive to a linear transformation, except for the limitations due to the accuracy of the computers. It follows that if an image B* kh offers a bias and / or a much different signal / radiance dynamics than the reference image, it will not appear in the correlation coefficient. Bias and root mean square error (or standard-deviation) are better representing the errors. Furthermore, the correlation coefficient is sensitive to the very high frequencies. This is a surprising choice knowing that interpolated images do not contain such frequencies.

Conclusion

The protocol used by Liu to demonstrate the quality of the products fused by the SFIM technique is fully inadequate for its part based on statistics. In theory, this is also true for the visual part but the effects are dampened by the combined effects of the computer display and the visual perception. It results that there is no demonstration of the superiority of the SFIM technique on any other technique based on statistics. We encourage Dr. Liu to adopt the protocol discussed here. It has been debated within a joint working group of the European Association of Remote Sensing Laboratories (EARSeL), the European branch of the International Society for Photogrammetry and Remote Sensing (ISPRS) and of the Soci€t€ des Electriciens et Electroniciens, the French arm of the IEEE. The work of [START_REF] Wald | Fusion of satellite images of different spatial resolutions: assessing the quality of resulting images[END_REF] on this protocol has been distinguished by the American Society of Photogrammetry and Remote Sensing. This protocol is already in operation in several companies and agencies. By using this protocol, the demonstration of the advantages of the SFIM technique can be made and comparisons can be made with already published works [START_REF] Ph | Using iterated rational filter banks within the ARSIS concept for producing 10 m Landsat multispectral images[END_REF][START_REF] Mangolini | Evaluation de la qualit€ des images multispectrales • haute r€solution spatiale d€riv€es de SPOT[END_REF][START_REF] Munechika | Resolution enhancement of multispectral image data to improve classification accuracy[END_REF]Ranchin, Wald, 1998, 2000b;[START_REF] Raptis | Assessment of different data fusion methods for the classification of an urban environment[END_REF][START_REF] Vrabel | Multispectral imagery advanced band sharpening study[END_REF][START_REF] Yang | Influence of landscape changes on the results of the fusion of P and XS images by different methods[END_REF].

  A h and B kl . It results into a new set of synthesized images B* kh at resolution h, any synthetic image B* kh , once degraded to its original resolution l, should be as identical as possible to the original image B* kh . To achieve this, the synthetic image B* kh is spatially degraded to an approximate solution (B* kh ) l of B kl . If the first property is true, then (B* kh ) l is very close to B kl . The difference between both images is computed on a per-pixel basis. The fused products together with the difference images are visually compared to the original images B kl in order to detect trends of error, if any, possibly related to the objects in the scene. Then some statistical quantities are computed to quantitatively express the similarities and discrepancies between both images. scale is performed for the second and third properties. Two sets of images A l and B kv are created from the original sets of images A h and B kl . The image A h is degraded to the low resolution l (A l , i.e. a SPOT panchromatic image at 20 m) and the images B kl to a lower resolution v (B kv , i.e. a Landsat multispectral image at 60 m), where v=l(l/h),



testing the first property:  a change of  the fusion method is applied to these two sets of images, resulting into a set of synthesized images B* kl at resolution l. The original images B kl (the Landsat original image) serve now as references.


published or not. The documents may come

The ARSIS method: a general solution for improving spatial resolution of images by the means of sensor fusion

Thierry Ranchin, Lucien Wald, Marc Mangolini


Thierry Ranchin

Lucien Wald

Marc Mangolini

Claude Penicand

On the assessment of merging processes for the improvement of the spatial resolution of multispectral SPOT XS images


Philippe Blanc

Thierry Ranchin

Thierry Blu

Lucien Wald

Roberto Aloisi

Using iterated rational filter banks within

come   L'archive ouverte pluridisciplinaire

USING ITERATED RATIONAL FILTER BANKS WITHIN THE ARSIS METHOD FOR PRODUCING 10 M LANDSAT MULTISPECTRAL IMAGES.

(1,3) Philippe Blanc, (1) Thierry Ranchin, (2) Thierry Blu, (1) Lucien Wald, (3) Roberto Aloisi.

(1) Ecole des Mines de Paris, Groupe Télédétection & Modélisation, B.P. 207, F-06904 Sophia Antipolis cedex.

(2) France Telecom, CNET PAB/STC, Rue du général Leclerc, F-92191 Issy-les-Moulineaux.

(3) Aérospatiale, Service performances et applications de l'imagerie, SE/TNA, B.P. 99, F-06322 Cannes la Bocca cedex.

ABSTRACT:

The ARSIS concept makes use of wavelet transforms and multiresolution analysis to improve the spatial resolutions of images from a set made of a low resolution image in the same spectral band and a high resolution image in an other spectral band. The paper deals with the use of rational filter banks in ARSIS. Indeed, first this concept was operationally applied with dyadic wavelet transforms that limit the merging of images with a ratio between spatial resolutions equals to a power of two. Provided some conditions, rational filter banks can be seen as a good approximation of rational wavelet transforms and, thus, enable a more general merging of images with ARSIS. The advantages of those rational filter banks compared to other methods are discussed and illustrated by an example of fusion of a 10 m SPOT Panchromatic image and a 30 m Landsat Thematic Mapper (TM) multispectral image into a synthetic 10 m multispectral image called hereafter TM-HR.

INTRODUCTION

The ARSIS concept from its French acronym "Amélioration de la Résolution Spatiale par Injection de Structures", enables the fusion of images having different spatial and spectral resolutions. More precisely, it is meant to synthesize a high spatial resolution image by merging a low resolution image in the same spectral band and a high resolution image in an other spectral band, thanks to wavelet transforms and multiresolution analysis. In other words, the aim of ARSIS is to increase the spatial resolution of a multispectral image without modification of its spectral content by using the structures extracted from a higher resolution image provided a sufficient level of local spatial correlation between the images. For more information about ARSIS, see Ranchin et al. (1993). In the past, the ARSIS concept was operationally applied with dyadic wavelet transforms that limit the merging of images with a resolution ratio equals to a power of two. We present an extension of the use of ARSIS by using rational filter banks as non dyadic rational wavelet transforms. First a short overview is made on rational filter banks and their link with rational wavelet transforms. Then, to assess the efficiency of those filter banks applied in ARSIS, we present an example of merging of a 10 m SPOT Panchromatic image and a 30 m Landsat TM multispectral image to synthesize a 10 m multispectral TM-HR image. Two approaches were conducted which both used a 2/3 filter bank in ARSIS to synthesize the 10 m multispectral TM-HR image. Then these two methods were compared to other methods that made use of a bicubic interpolation with resampling rate 2/3 and 3/2. Our study is illustrated by an extract of a Landsat TM multispectral image and SPOT Panchromatic image of the region of Berre l'Etang close to Marseille in the southeast of France (Figure 1) Figure 1. Original Landsat TM 1 (region of Berre l'Etang close to Marseille).

RATIONAL FILTER BANKS AND DISCRETE RATIONAL WAVELET TRANSFORMS

The purpose of this section is to present a short overview of the rational filter banks structure and their link with discrete wavelet transforms. Blu (1993 a) showed that, under some conditions, twoband iterated filter banks with rational rate change p/q can be used to approximate very closely samples of discrete wavelet transform with the same rational dilation factor. More precisely, the structure of rational filter banks is similar to the dyadic filter banks one which provides wavelet analysis of the input signal: this general structure is shown Figure 2. Those filter banks are composed of an iterated low-pass branch and a high-pass branch between two consecutive low-pass branches. The description of the design procedure of the rational lossless finite impulse response (F.I.R.) low-pass and high-pass filters is given in Blu (1993 b). The dyadic case is obtained with p=1 and q=2. In this case, the high-pass branch outputs are samples of discrete dyadic wavelet transform. In the rational case, this is no longer true. It still could be seen as a kind of multiresolution analysis but the decomposition of the signal is not plainly made on dilated and shifted versions of a single function (the wavelet). Nevertheless, when p=q-1, Blu (1993a) proved that the difference between the high-pass outputs and the samples of a corresponding rational discrete wavelet transform can be reduced by suitable parameters in the filters design, so that the iterated rational filter bank can generate a good approximation of a wavelet transform.

Low-pass filter

As a conclusion, we presently have an approximation of a discrete wavelet transform with a rational dilation factor (q-1)/q at our disposal. The extension of the use of those iterated rational filter banks to images is easily done following Mallat (1989). We have to notice that the suitable parameters for the filters design increase the number of coefficients of the filters and, therefore, are usually "longer" than in the dyadic case. For example, the low-pass filter and the high-pass filter selected here to be applied in ARSIS method have impulsional responses comprising respectively 21 and 11 coefficients.

PRODUCING 10 M LANDSAT TM MULTISPECTRAL IMAGE

THE ISSUE

Our aim is to synthesize a 10 m multispectral TM-HR image from the merging of a 10 m SPOT Panchromatic image and a 30 m Landsat TM multispectral image of the same geographical area.

Both images are superimposable. The ratio of their resolution is 1/3. It is the reason why ARSIS with dyadic wavelet transform can not be strictly applied in this case. Moreover, we assume, for the synthesis of a filter bank with a rational rate change p/q, that p=q-1. Therefore, two ways exist to synthesize the TM-HR multispectral image with ARSIS.

ARSIS WITH A 2/3 FILTER BANK

The first way consists in using ARSIS with a 2/3 filter bank. There are two possibilities:

• The method ARSIS #2 is illustrated in Figure 3. Note that the 15 m SPOT Panchromatic image is generated with the 2/3 filter bank but it could have been done by a bicubic interpolation with a resampling rate 2/3.

ARSIS WITHOUT A 2/3 FILTER BANK

This way consists in using a bicubic interpolation in order to circumvent the fact that the ratio of resolutions (i.e. 10 m and 30 m) is not a power of two. There are also two possibilities:

• ARSIS #3: the 30 m Landsat TM image is enhanced to 20 m by the use of a bicubic interpolation with resampling rate 2/3. Then the 10 m TM-HR image is synthesized from the 10 m SPOT Panchromatic and the synthesized 20 m Landsat TM image through ARSIS with a dyadic wavelet transform.

• ARSIS #4: the 30 m Landsat TM image is resampled using a bicubic interpolation with a resampling rate of 2 to generate a 40 m Landsat TM image. Then ARSIS with a dyadic wavelet transform is applied two times to synthesize the 10 m TM-HR.

In both approaches, a bicubic interpolation is used: the Landsat TM images are changed into new images having a pseudo spatial resolution of either 20 m (ARSIS #3) or 40 m (ARSIS #4). As a consequence, the merging between SPOT and Landsat is not as complete as ARSIS #1 or ARSIS #2 because there is no injection of structures from the SPOT image between 30 m and 20 m in the case of ARSIS #3 and a degradation of the Landsat TM image resolution before the merging in the case of ARSIS #4.

QUALITY ASSESSMENT OF THE SYNTHESIZED IMAGES

These different approaches generate four sets of synthetic enhanced multispectral images. The quality of these sets is assessed by the mean of the approach described in Mangolini et al. (1995). Indeed, the following approach was used to quantify the quality of the merging of the SPOT Panchromatic image and the Landsat TM multispectral image. The SPOT and the Landsat images were interpolated to a lower resolution of respectively 30 and 90 m by using bicubic interpolation. Then, the four methods were applied to synthesize Landsat TM multispectral images at 30 m called hereafter TM 1 * , ..., TM 4 * . Those images are compared visually and on a pixel basis to the original TM images. Only the methods ARSIS #2 and ARSIS #3 are presented here for discussion. The two other methods are worse as far as visual and quantitative assessment are concerned.

VISUAL ASSESSMENT

The quality of the synthesized images can be analyzed by a simple visual inspection. For example, the visual inspection is based on the comparison between the original 30 m TM-1 (band 1) in the Figure 1 Fusion of Earth Data, Cannes, France, 6-8 February 1996

QUANTITATIVE ASSESSMENT

Quantitative assessments are based upon usual criteria for comparison (bias, correlation coefficient, variance, etc.) computed in several sub-regions of the multispectral Landsat TM images (the original TM and the synthesized TM * ). This study confirms the visual inspection: basically ARSIS #2 method provides better results than the ARSIS #3 method. But, in the regions with strong radiometric discontinuities, ARSIS #2 approach generates artifacts that locally decrease the quality of the synthesized image. Moreover, generally, we notice that the ARSIS merging is all the more efficient as there is locally a sufficient level of local spatial correlation between the images.

CONCLUSION

As a conclusion, rational filter banks can be useful for ARSIS assuming a certain regularity of the images to merge. More precisely, the ARSIS #2 method using the 2/3 filter bank provides better results as far as visual quality of the resolution and criteria used for comparison (bias, correlation coefficient, variance, etc.) are concerned assuming that the images to merge do not have large discontinuities. However, discontinuities in images introduce artifacts that decrease the quality of the fusion. Moreover, the quality of the ARSIS merging of Landsat TM multispectral with SPOT Panchromatic is less satisfactory than the ARSIS merging of a 20 m SPOT XS multispectral image with a 10 m SPOT Panchromatic image. This can be partially explained by the fact that the resolution gap "to fill in" is more important in the first case than in the second one.

Figure 2 .

 2 Figure 2. General scheme of iterated rational filter banks.

  ARSIS #1: the 30 m Landsat TM image is enhanced to 20 m by the use of ARSIS with a 2/3 filter bank and the SPOT Panchromatic resampled at the resolution 20 m. Then the 10 m TM-HR image is synthesized from the 10 m SPOT Panchromatic and the synthesized 20 m Landsat TM image using ARSIS with a dyadic wavelet transform. • ARSIS #2: the 30 m Landsat TM image is enhanced to 15 m by the use of ARSIS with a dyadic wavelet transform and the SPOT Panchromatic resampled at the resolution 15 m Then the 10 m TM-HR image is synthesized from the 10 m SPOT Panchromatic and the synthesized 15 m Landsat TM image using ARSIS with a 2/3 filter bank.

Figure 3 .

 3 Figure 3. Illustration of the different stages of the method ARSIS #2

  Figure 4. (a) 90 m resampled Landsat TM-1. (b) 30 m Landsat TM 3 * 1 with ARSIS #3. (c) 30 m Landsat TM 2 * -1 with ARSIS #2.

  , 1993 a. Iterated filter banks with rational rate changes. Connection with discrete wavelet transforms. IEEE Transactions on Signal Processing, 41, 12, 3232-3244. Blu T., 1993 b. Lossless filter design in two-band rational filter banks: a new algorithm. In Comptesrendus du Quatorzième colloque GRETSI, Juan-Les-Pins, France, 69-72. Mallat S.G., 1989. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 7, 674-693. Mangolini M., Ranchin T., Wald L., 1995. Evaluation de la qualité des images multispectrales à haute résolution spatiale dérivées de SPOT. Bulletin de la Société Française de Photogrammétrie et Télédétection, 137, 24-29. Ranchin T., Wald L., Mangolini M., 1993. Application de la transformée en ondelettes à la simulation d'images SPOT multispectrales de résolution 10 m In Comptes-rendus du Quatorzième colloque GRETSI, Juan-Les-Pins, France, 1387-1390.


Thierry Ranchin

Isabelle Couloigner

Lucien Wald

come


Serguei Barannikov

Solving the noncommutative Batalin-Vilkovisky equation

I show that summation over ribbon graphs give the construction of the solutions to the noncommutative Batalin-Vilkovisky equation, including the equivariant version, introduced in my previous papers. This generalizes the known construction of A1 algebra via summation over ribbon trees. In particular, the quadratic Maurer-Cartan equation is replaced by the linear equation ( ) = 0. Such solutions are naturally the matrix action functionals giving the equivariantly closed di¤erential forms on the matrix spaces, described in my previous paper.

I explain that the summation over ribbon graphs with legs produces solutions to the noncommutative Batalin-Vilkovisky equation, introduced in [B06a]. This generalizes the known construction of A 1 structure via summation over trees, see [M].

Notations.I work in the tensor category of super vector spaces, over an algebraically closed …eld k, char(k) = 0. Let V = V 0 V 1 be a super vector space. I denote by the parity of an element and by V the super vector space with inversed parity. Element (a 1 a 2 : : : a n ) of A n is denoted by (a 1 ; a 2 ; : : : ; a n ). I denote by V _ the dual vector space Hom(V; k).

1 Algebra with odd di¤erentiation.

I consider a Z=2Z graded algebra A, dim k A < 1 , with multiplication denoted by m 2 : A 2 ! A and an odd di¤ erentiation I : A ! A Im 2 (a; b) = m 2 (Ia; b) + ( 1) a m 2 (a; Ib); in particular, if I 2 = 0 then this is a d(Z=2Z)g-algebra. I assume, at the beginning, that the algebra is cyclic with odd scalar product and that is preserved by I:

(Ia; b) + ( 1) a (a; Ib) = 0:
The modi…cation for the variant with an even scalar product are described below.

Below I consider also the variant for general d(Z=2Z)g-algebra without scalar product. It is reduced to the case with even/odd scalar product by putting which is the associativity of the multiplication m.

Let H be an odd selfadjoint operator

H : A ! A; H _ = H such that Id [I; H] = P (3)
is an idempotent operator P : A ! A;

P 2 = P:

I assume also that H commutes with I 2 , this is automatic if I 2 = 0. I denote by B the subspace which is the image of the idempotent P . Let be a tri-valent ribbon graph, i.e. the trivalent graph with …xed cyclic orders on the sets of the three ‡ags attached to every vertex. Let be the corresponding two-dimensional surface. Then I put: the three-tensors

m v 2 (( A) F lag(v) ) _
on every vertex v and de…ne

= N v2V ert( ) m v the two tensors _;e H 2 ( A) ff;f 0 g ; _;e H = _ (H _ u f ; v f 0 ) = ( 1) u f v f 0 _ (H _ v f 0 ; u f )
for any interieur edge e = (f f 0 ) element a l 2 B, for any exterier leg l; this gives a partition of the set of elements fa l g l2Leg( ) to the subsets corresponding to the components of the boundary @ and cyclic orders on these subsets.

Notice that both m v and _;e H are even elements, so that the products

N v2V ert( ) m v and N e2Edge( ) _;e
H are canonically de…ned.

De…nition 1 I de…ne the tensor W as the contraction

W ( N l2Leg( ) a l ) = D N v2V ert( ) m v ; N e2Edge( ) _;e H N l2Leg( ) a l E (4)
Notice that W is cyclically invariant on every subset of fa l g l2Leg( ) corresponging to a component of the boundary of . Moreover the cyclic orders on ‡ags at vertices induce the orientation on the ribbon graph , whose more detailed analysis, see e.g. [B09b], shows that W lies in

W 2 Symm( 1 j=1 ( B j ) Z=jZ ) _
Let ( ) denotes the genus of ,

( ) = 2 2g( ) i( );
where g( ), i( ) are the genus and the number of boundary components of . I put

S = P f g ~ ( ) W (5)
where the sum is over isomorphism classes of connected trivalent graphs with nonempty subsets of legs on every boundary component of . One can include the graphs with empty subsets of legs on boundary components by adding the constant extra term to the Batalin-Vilkovisky operator below, I leave the details to the interested reader.

I've described in [B06a], [B06b] the Batalin-Vilkovisky formalism associated with a Z=2Z graded vector space with odd scalar product and in particular I've de…ned the second order Batalin-Vilkovisky operator and the odd Poisson bracket f ;

g on Symm( 1 j=1 ( B j ) Z=jZ ) _ .
Theorem 2 The sum over graphs S de…ned in (5) satisfy the equivariant noncommutative Batalin-Vilkovisky equation:

~ S + 1 2 fS; Sg + I _ S = 0; in particular if Ij B is zero then S is the solution of the non-commutative Batalin- Vilkovisky equation from [B06a],[B06b] ~ S + 1 2 fS; Sg = 0 Remark 3 If Ij B 6 = 0, but I 2 j B = 0, then S + S 0;
2 is also a solution to the non-commutative Batalin-Vilkovisky equation , where S 0;2 = (I ; )j B is the quadratic term corresponding to the di¤ erential Ij B .

Proof. The proof is straightforward, see e.g. [B09b]. For a trivalent graph and an internal edge e 2 Edge( ) consider the three tensors

W [I;H] ;e ; W Id ;e ; W P ;e 2 Symm( 1 j=1 (( B) j ) Z=jZ ) _
which are de…ned by the same contraction as W except that at the edge e 2 Edge( ) I put the tensors

_ ([I _ ; H _ ]u f ; v f 0 ); _ (u f ; v f 0 ); _ (P _ u f ; v f 0 )
correspondingly instead of _;e H . Then, from (3)

W P ;e = W Id ;e

W [I;H]

;e :

By summing over v 2 V ert( ) of the Leibnitz rule (1) and noticing that over all internal edges and over the set of trivalent graphs, having the same with same distribution of Leg( ) over the boundary components, is zero:

_ (H _ I _ u f ; v f 0 ) + _ (u f ; H _ I _ v f 0 ) = _ ([I _ ; H _ ]u f ; v f 0 ) I get I _ W P e W [I
P f g; = ;e2Edge( ) W Id ;e = 0:
Notice that P 2 = P implies that

_ (P _ u f ; v f 0 ) = _ (P _ u f ; P _ v f 0 ):
Then, from the de…nition of the Batalin-Vilkovisky operator and the odd Poisson bracket on B it follows that

P f g ~ ( )+1 W + 1 2 f P f g ~ ( ) W ; P f 0 g ~ ( 0 ) W 0 g = = P f e g;e2Edge( e ) ~ ( ) W P e ;e
where each term on left hand side corresponds precisely to the right hand side term ~ ( ) W P e ;e , where e is obtained by gluing two legs to form the edge e from either the single surface or the two surfaces . Notice that the condition, that does not get contributions from the neighboring points on the same circle, corresponds precisely to the fact that the resulting surface e has always nonempty subsets of Legs( e ) on the boundary components.

Remark 4 In general, for in…nite dimensional cyclic Z=2Z graded algebra A the tensor _ (H _ ; ) belongs to some completion of A k A, and extra work is needed in the in…nite dimensinal situation in order to verify that the tensors W are well-de…ned. One can either apply the technique of analysis and construct the propagators as forms with singularity on the diagonal and then verify that the resulting integrals de…ning W converge, or the technique of algebraic geometry by working in the Z graded category and with dg-algebras whose resolution of the diagonal is constructed from modules with …nite-dimensional graded pieces, for example, for smooth dg-algebras, and this implies that the tensors W are well-de…ned, see ??.

Corollary 5 Given a Z=2Z graded cyclic A 1 algebra B, if B has a cyclic d(Z=2Z)g associative model A, such that for A the contractions (4) over trivalent ribbon graphs are well-de…ned, then the summation over such graphs gives an extension of the cyclic A 1 algebra on B to the quantum A 1 algebra, i.e. to the solution of the non-commutative Batalin-Vilkovisky equation.

1.2 Even scalar product.

Assume now that the scalar product on A is even:

: A 2 ! A:
Then given an odd di¤ erentiation I : A ! A and an odd selfadjoint operator H : A ! A; satisfying (3), I construct the tensors W for any ribbon trivalent graph by the same contraction (4). The only di¤erence is that in this case, both the three-tensors m v attached to the vertices and the two-tensors _;e H are odd and an analysis of the corresponding orientation on , similar to [B09b], shows that W belongs to the exterieur power of the space of cyclic tensors

W 2 Symm( 1 j=1 ( B j ) Z=jZ ) _
For the case of even scalar product on Z=2Z graded vector space V I've also described a variant of noncommutative Batalin-Vilkovisky formalism in [B06a].

Theorem 6 The sum over trivalent ribbon graphs S satisfy the equivariant noncommutative Batalin-Vilkovisky equation:

~ S + 1 2 fS; Sg + I _ S = 0;
in particular if Ij B = 0 then S is the solution of the non-commutative Batalin-Vilkovisky equation from [B06a], [B06b] ~ S + 1 2 fS; Sg = 0

General algebras.

Let now A be an arbitrary Z=2Z graded algebra, dim k A < 1 , with an odd di¤ erentiation I : A ! A. This case is reduced to the two previous cases by putting e A = A ( A) _ with odd scalar product given by natural odd pairing between A and ( A) _ , or by putting e A = A A _ with even scalar product given by natural even pairing between A and A _ . Then e A is naturally an associative algebra with odd, respectively even, scalar product and the odd di¤ erentiation I, whose action extends naturally to e A. Consider the case of odd scalar product. Suppose that H is an odd operator

H : A ! A; such that Id [I; H] = P (6)
is an idempotent operator P : A ! A; P 2 = P:

Then both H and P act naturally on e A as self-adjoint operators and I apply to this situation the construction of tensors W for ribbon graphs described above. The tensors

W B 2 Symm( 1 j=1 (( B B _ ) j ) Z=jZ ) _
de…ned by the contraction (4) are given by the sum over markings

F lag( ) ! f A; A _ g
such that for any edge, its two ‡ags are marked di¤erently, and for any vertex there is exactly one ‡ag which is marked by A, with no other extra restrictions.

In particular such marking gives an orientation for every edge, from A _ to A, and there must be exactly one edge exiting every vertex. There are no other restrictions, in particular the edges can in principle form cycles. The legs of , which correspond to the points sitting on the boundary of the surface , are also marked as either entries (B _ ) or exits ( B). And I de…ne S B by the summation as above S B = P f g ~ ( ) W B where the sum is over isomorphism classes of connected trivalent graphs with such markings and with nonempty subsets of legs on every boundary component of . Similarly I de…ne the tensors W in the case of e A = A A _ with its even pairing . These tensors are from the space of exterieur powers of linear functionals on cyclic words consisting of elements of B and

B _ W B 2 Symm( 1 j=1 (( B B _ ) j ) Z=jZ ) _
and I de…ne S B as their sum over ribbon graphs as above.

Theorem 7 Let A an arbitrary Z=2Z graded algebra, dim k A < 1 , with an odd di¤erentiation I : A ! A and a homotopy H, such that the operator ( 6) is idempotent. The sums over ribbon graphs S B and S B give the solutions to the two variants of the equivariant noncommutative Batalin-Vilkovisky equation in the spaces of symmetric, respectfully exterieur powers, of cyclic words, consisting of elements from B _ and B, respectfully from B _ and B:

~ S B + 1 2 fS B ; S B g + I _ S B = 0 ~ S B + 1 2 fS B ; S B g + I _ S B = 0:
2 Graphs with the insertion of A 1 tensors.

Assume now that A is a Z=2Z graded A 1 algebra, dim k A < 1 , as above I relax the condition d 2 = 0, and assume that it is simply an odd operator I : A ! A, which together with other structure maps m n 2 (( A) n ) _ A, n 2, satisfy the standard A 1 constrains, except the very …rst, so that I 2 is in general nonzero: for any n 2

Im n (v 1 ; : : : ; v n ) P l ( 1) m n (v 1 ; : : : ; Iv l ; : : :

v n ) = = P i+j=n+1 (
1) m i (v 1 ; : : : ; m j (: : :); : : : v n )

I assume …rst that A has also an invariant odd scalar product so that all tensors m n 2 (( A) n+1 ) _ ; ( m n (v 1 ; : : : ; v n ); v n+1 ) are cyclic invariant, the variant without scalar product is reduced as above to this case by taking e A = A ( A) _ . Let as above H be an odd selfadjoint operator

H : A ! A; H _ = H such that Id [I; H] = P
is an idempotent operator P : A ! A, whose image I denote by B. Now I de…ne the tensors W , by inserting the cyclyc tensors m n(v) 2 (( A) F lag(v) ) _ at vertices, as above, where is now a ribbon graph, with valency n(v) for any vertice at least three:

W ( N l2Leg

:

  A 2 ! A; so that the three tensor m 2 (( A) 3 ) _ m( a; b; c) = ( 1) b (m 2 (a; b); c) is cyclically invariant m( a; b; c) = ( 1) (c+1)(a+b) m( c; a; b)

  e A = A A _ , or e A = A A _ with their natural scalar products. I have m(Ia; b; c) + ( 1) a m(a; Ib; c) + ( 1) a+b m(a; b; Ic) = 0 (1) which re ‡ect the Leibnitz rule for the di¤erentiation I. Denote by _ 2 ( A) 2 the tensor of the scalar product on the dual vector space, then for any a; b; c; d 2 A, hm( a; b; )m( ; c; d); _ i = ( 1) " hm( d; a; )m( ; b; c); _ i (2)

  (2) to substitute in W Id ;e the contraction hm( a; b; )m( ; c; d); _ i corresponding to the internal edge e 2 Edge( ) by ( 1) " hm( d; a; )m( ; b; c); _ i : This corresponds to passing from the trivalent ribon graph to the trivalent ribbon graph 0 obtained by the standard transformation on the edge e, preserving the overal cyclic order of the ‡ags corresponding to a, b, c, d. This transformation preserves the surface and the distribution of elements of Leg( ) over the boundary components of . Therefore the sum of W Id ;e

  ( ) a l ) = D Nv2V ert( ) m n(v) ; Symm( 1 j=1 ( B j ) Z=jZ ) _ Next however, looking carefully at the proof of the equation for S above, one sees that one immediately runs into a problem because of tadpoles, unless the important condition m n = 0 is imposed, which I assume from now on. I de…ne now, similarly to above, is over isomorphism classes of connected ribbon graphs with vertices of valency n(v) 3, and with nonempty subsets of legs on every boundary component of .Theorem 8 Let the odd operator I and the cyclically invariant tensors m n 2 (( A) n+1 ) _ , n 2, satisfyI _ m + fm; mg = 0 m = 0Then the sum over ribbon graphs S satisfy the equivariant noncommutative Batalin-Vilkovisky equation associated with (B; j B ): Sg + I _ S = 0;in particular if Ij B = 0 then S is the solution of the non-commutative Batalin-Vilkovisky equation from[B06a],[B06b]

Proof. The proof is parallel to the above.

The same result holds starting from the arbitrary solution to the noncommutative Batalin-Vilkovisky equation and the tensors W constructed similarly for stable ribbon graphs. Details will appear elsewhere.

Analogous result holds for arbitrary modular operad.