Gabriel Faraud
email: faraud@wias-berlin.de

Stéphane Goutte
email: goutte@math.univ-paris-diderot.fr

Bessel bridges decomposition with varying dimension. Applications to finance

Keywords: Squared Bessel process, Bessel bridges decomposition, Laplace transform, Lévy Ito representation, Financial applications. MSC Classification (2010):60G07 60H35 91B70

We consider a class of stochastic processes containing the classical and well-studied class of Squared Bessel processes. Our model, however, allows the dimension be a function of the time. We first give some classical results in a larger context where a time-varying drift term can be added. Then in the non-drifted case we extend many results already proven in the case of classical Bessel processes to our context. Our deepest result is a decomposition of the Bridge process associated to this generalized squared Bessel process, much similar to the much celebrated result of J. Pitman and M. Yor. On a more practical point of view, we give a methodology to compute the Laplace transform of additive functionals of our process and the associated bridge. This permits in particular to get directly access to the joint distribution of the value at t of the process and its integral. We finally give some financial applications to illustrate the panel of applications of our results.

Introduction

We will consider in this article a family of stochastic processes which contains the classical squared Bessel processes. We recall that for δ ≥ 0, a δ-dimensional squared Bessel process started at x ≥ 0 is defined as the unique strong solution of the stochastic differential equation dX u = δdu + 2 X u dW u and X 0 = x ≥ 0 for all u ∈ R + where (W u ) u≥0 denotes a Brownian motion. There has been a huge number of publications concerning this class of processes, resulting in many tractable tools for its study. We refer in particular to Shiga-Watanabe [START_REF] Shiga | Bessel diffusions as a one-parameter family of diffusion processes[END_REF], Pitman-Yor [START_REF] Pitman | A decomposition of Bessel bridges[END_REF], Revuz-Yor [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] and Deelstra-Debaen [START_REF] Deelstra | Long-term returns in stochastic interest rate models[END_REF], [START_REF] Deelstra | Long-term returns in stochastic interest rate models: Convergence in law[END_REF]. A natural extension of this family is to replace the dimension δ by a function of the time variable. This was done by Carmona in [START_REF] Carmona | Généralisations de la loi de l'arc sinus et entrelacements de processus de Markov[END_REF], extending some of the previously mentioned tools to the class of squared Bessel processes with dimension δ which is a time varying function. A second generalization of this class of processes was done by Deelstra and Delbaen in [START_REF] Deelstra | Long-term returns in stochastic interest rate models[END_REF], [START_REF] Deelstra | Long-term returns in stochastic interest rate models[END_REF] and [START_REF] Deelstra | Long-term returns in stochastic interest rate models: Convergence in law[END_REF]. They introduced a constant drift parameter β and used a dimension δ which may be stochastic. Indeed, they extend results of existence of the solution of an extended stochastic differential equation dX u = (δ u + 2βX u )du + 2 X u dW u and X 0 = x ≥ 0 They proved that the properties of scaling and additivity of the process (X u ) found by Pitman and Yor in [START_REF] Pitman | A decomposition of Bessel bridges[END_REF] for classical squared Bessel processes stay valid for these extended squared Bessel processes. Moreover, Pitman and Yor in [START_REF] Pitman | A decomposition of Bessel bridges[END_REF], using some classical spacetime transformations and Girsanov's theorem, have shown that the study of squared Bessel processes with drift can be reduced to the study of the squared Bessel process without drift (i.e. β = 0). This property is equally true in the case of non-constant dimension. Finally, Delbaen and Shirakawa in [START_REF] Delbean | Squared Bessel processes and their applications to the square root interest rate model[END_REF] applied the class of squared Bessel processes with time varying dimension δ to the square root interest rate model. Hence our contributions are: firstly, we generalize the family of squared Bessel processes to the class of squared Bessel processes with stochastic dimension (δ u ) and with a drift parameter β u which is a function of the time. We will call this kind of process a generalized squared Bessel process (GBESQ) with stochastic dimension (δ u ) and with a drift function β u , defined as the unique solution of the stochastic differential equation dX u = (δ u + 2β u X u )du + 2 X u dW u and X 0 = x ≥ 0.

(0.1)

We establish several classical properties for this generalized process, extending or sometimes overlapping results from the aforementioned articles. This includes results of existence and uniqueness of the solution of (0.1); scaling and additivity properties of this solution. We finish by showing that, as in the two other previously mentioned settings, one can reduce to the situation β = 0 through a change of law. Therefore we make a more precise study of the un-drifted case and give some rather tractable tools in order to compute the law of additive functionals of this process.

We then give a more precise and theoretic version of the last result, in the form of a Lévy-Itô representation of this process, similar to the one exposed in Pitman and Yor [START_REF] Pitman | A decomposition of Bessel bridges[END_REF]. However due to the fact that the dimension parameter belongs to a function space and not, as in the classical setting, to R + , we cannot extend to our case all the results of global monotone coupling that exist in the classical setting. Our tools rely mostly, apart from the probabilistic tool used in the setting of Bessel Processes, on functional analysis in order to extend results on the half line to a functional space. The Fréchet-Riesz representation Theorem is a quite typical example of the theoretical results we use.

Then we introduce the notion of Generalized Squared Bessel Bridges by conditioning our process to its endpoints. We give some extensions of the classical properties of Bessel bridges to our case, including an extension of the Bessel Bridge decomposition of Pitman and Yor [START_REF] Pitman | A decomposition of Bessel bridges[END_REF]. We use this decomposition in order to give a general formula for the Laplace transform of additive functionals of the Bessel bridge. We give an example of how we can use this formula to get an explicit result. The techniques we use are very tractable and can be applied whenever one knows how to handle classical Bessel processes. We give a (if not very tractable) complete formula covering most settings in Appendix A. Our formula contains a power series, the value of which we are unable to compute explicitly, we however give some estimates on the speed of convergence of this series.

We conclude this paper by giving some applications to financial mathematics of our results. Indeed, we will first give some examples of financial models which are GBESQ. Then, we will give three examples of applications: first we will give an explicit way to simulate stochastic volatility model where the volatility process is again a GBESQ, then the evaluation of zero coupon bond where the interest rate is a GBESQ, and finally, we will give an explicit method to simulate default times in credit risk model using a stochastic default intensity which is a GBESQ process.

1 Squared Bessel processes with time varying dimension.

Squared Bessel process

Let Ω, (F t ) t∈[0,T ] , P be a filtered probability space and let (W t ) t∈[0,T ] be a Brownian motion under P. We suppose that the filtration (F t ) t∈[0,T ] is generated by {W s ; 0 ≤ s ≤ t}. We first recall some results of [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], about squared Bessel processes with constant dimension δ ≥ 0.

Definition 1.1. The squared Bessel process of dimension δ ≥ 0, denoted by BESQ δ

x is defined as the unique strong solution of the stochastic differential equation (SDE)

dX u = δdu + 2 X u dW u and X 0 = x ≥ 0. (1.2)
In this case X u ≥ 0, for all u ∈ [0, T ].

We will denote in the sequel by Q δ x , the law of BESQ δ x . We now recall the property of additivity of squared Bessel processes which can be found in Theorem 1.2 Chap XI of [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] Theorem 1.1. For every δ, δ ≥ 0 and x, x ≥ 0, we have

Q δ x ⊕ Q δ x = Q δ+δ x+x . (1.3)

Generalized squared Bessel process (GBESQ)

Let C be the space of continuous functions R + → R + and D the space of continuous by part, measurable and locally bounded functions R + → R + . Definition 1.2. For all u ∈ R + , δ, β ∈ D, we will call the solution in C of the stochastic differential equation

dX u = (2β u X u + δ u ) du + 2 X u dW u (1.4)
with initial value X 0 = x ≥ 0, a generalized squared Bessel process with dimension (δ u ) and drift function (β u ). We will denote X as a GBESQ β,δ x .

We will denote by β Q δ x the law of X and

Q δ x := 0 Q δ x .
Remark 1.1. In the above definition (δ u ) and (β u ) are deterministic functions of the time.

It is obviously possible to extend these definitions to random (δ u ) and (β u ) provided their trajectories belong almost surely to D, by conditioning on the realization of (δ u ) and (β u ). Thus all the following result are true conditionally on (δ u ) and (β u ), or, using the typology of random media, in the "quenched" setting. It is also straightforward by time restriction of the above SDE that the solution in this case is progressively measurable with respect to σ((δ u ) u≤t , (β u ) u≤t , (W u ) u≤t ).

Existence, unicity and positivity of the solution

We will obtain the uniqueness of the existence of the GBESQ and its positivity for a more general volatility structure. Indeed, we take the process X given by

dX u = (2β u X u + δ u ) du + σ(X u )dW u (1.5)
where σ : R + → R + is a function, vanishing at zero and satisfies the Holder condition, namely that for all x, y ∈ R + , there exists a constant c such that

|σ(x) -σ(y)| ≤ c |x -y|. (1.6) Remark 1.2.
1. We can remark that the volatility structure of our GBESQ process (1.4) satisfies this condition.

2. This class of processes was studied by Deelstra in [START_REF] Deelstra | Long-term returns in stochastic interest rate models[END_REF] with a constant drift β and a dimension δ such that

t 0 δ u du < ∞ for all t ∈ R + . 3. If δ is in D then δ satisfies point 2.
Proposition 1.1. Let β and δ be in D and σ a function satisfying (1.6), then for every x ≥ 0 there is a unique solution X t of the stochastic differential equation (1.5) such that X 0 = x.

Proof. The proof follows from Theorem 3.2.1 in [START_REF] Deelstra | Long-term returns in stochastic interest rate models[END_REF]. Whereas as we said in Remark 1.2 point 2. our model has a non constant drift term. But since β is in D, then it is locally bounded. Hence we can work with this new drift in the proof of Theorem 3.2.1 and obtain the same result.

To obtain the positivity of this solution, we will use a comparison result.

Proposition 1.2. Let X 1 and X 2 be two solution of (1.5) with dimension δ 1 ∈ D and δ 2 ∈ D and respectively initial state x 1 > 0 and x 2 > 0. We suppose too, that the processes X 1 and X 2 are associated with the same Brownian motion (W u ) and the same drift β ∈ D.

If for all u ≥ 0 we have x 2 ≥ x 1 and δ 2 u ≥ δ 1 u a.s., then for every u ≥ 0, we have

P X 2 u ≥ X 1 u = 1.
Proof. We will again follow the proof of Theorem 3.2.2 in [START_REF] Deelstra | Long-term returns in stochastic interest rate models[END_REF] with a non constant drift β using the fact that β u is in D for all u ∈ [0, T ].

Corollary 1.1. The process X is always positive.

Proof. This follows directly from the Proposition 1.2. Indeed, taking X 1 0 = 0 and δ 1 ≡ 0, then the unique solution of (1.5) is given by X 1 ≡ 0. Then taking X 2 an other solution of (1.5) such that X 2 0 ≥ X 1 0 = 0 and δ 2 u ≥ δ 1 u ≡ 0 a.s. we obtain that X 2 u ≥ 0 a.s.

Remark 1.3. Hence taking for all u ∈ [0, T ], σ(X u ) = 2 √ X u , then we have that the stochastic differential equation (1.4) admits a unique positive solution.

Results about GBESQ

We come back now to our GBESQ process given as the unique solution of the stochastic differential equation (1.5).

Proposition 1.3. Let (X (δ) u ) u∈[0,T ] and (Y (δ )
u ) u∈[0,T ] be two independent generalized squared Bessel processes with time varying dimension δ u , δ u ≥ 0 with the same drift β u which is a bounded function and initial states x, x ≥ 0. Then

{X (δ) u + Y (δ ) u ; 0 ≤ u, X (δ) 0 = x, Y (δ ) 0 = x } law = {H (δ+δ ) u ; 0 ≤ u, H (δ+δ ) 0 = x + x }, (1.7)
where H is an other GBESQ with stochastic dimension δ u + δ u , with drift equal to β u and initial state x + x . In other words,

β Q δ x ⊕ β Q δ x = β Q δ+δ x+x . (1.8)
Proof. We proceed similarly to the proof of Theorem 1.2 of [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] which is done for simple squared Bessel processes. Let W and W be two independent linear Brownian motion. Let X and Y defined by (1.4) as the corresponding two solutions for (x, β, δ) and (x , β, δ ) and set H=X+Y. Then for any u ∈ [0, T ],

H u = x + x + u 0 2β s (X s + Y s ) ds + u 0 δ s + δ s ds + 2 u 0 X s dW s + Y s dW s .
Let W be a third Brownian motion independent of W and W . The process ζ defined by

ζ u = u 0 1 {Hs>0} √ X s dW s + √ Y s dW s √ H s + u 0 1 {Hs=0} dW s
is a linear Brownian motion since ζ, ζ u = u and we have

H u = (x + x ) + u 0 2β s (X s + Y s ) ds + u 0 δ s + δ s ds + 2 u 0 H s dζ s H u = (x + x ) + u 0 2β s H s + δ s + δ s ds + 2 u 0 H s dζ s ,
which completes the proof.

Proposition 1.4. For all u ∈ [0, T ], If X is a GBESQ β,δ x (i.e. X 0 = x), then for any c > 0, the process 1 c X cu is again a GBESQ cβ ,δ

x c

with δ u = δ cu and β u = β cu . In terms of distribution, this reformulates to

β Q δ x 1 c X c • = cβ(c •) Q δ(c •) x/c . Proof. We know that X u = x + u 0 δ s ds + u 0 2β s X s ds + 2 u 0 √ X s dW s .
So by a change of variable in this stochastic integral, we obtain

X u = x + u c 0 cδ cv dv + u c 0 2cβ cv X cv dv + 2 u c 0 X cv dW cv . Hence 1 c X u = x c + u c 0 δ cv dv + u c 0 2β cv X cv dv + 2 u c 0 1 c X cv dW cv , 1 c X u = x c + u c 0 δ cv dv + u c 0 2β cv X cv dv + 2 u c 0 1 c X cv 1 √ c dW cv , 1 c X cu = x c + u 0 δ cv dv + u 0 2β cv X cv dv + 2 u 0 1 c X cv 1 √ c dW cv .
Moreover, we know that if for all u ∈ [0, T ], W u is a Brownian motion then, for all c > 0, 1 √ c dW cu is again a Brownian motion. Hence this stochastic differential equation becomes

1 c dX cu = (2β cu X cu + δ cu ) du + 2 1 c X cu d Wu = 2cβ cu 1 c X cu + δ cu du + 2 1 c X cu d Wu .
Consequently, the result follows from the uniqueness of the solution to this stochastic differential equation.

Removal of the drift term.

We now give a generalization of the change of law property which can be found in [START_REF] Pitman | A decomposition of Bessel bridges[END_REF]. This proposition allows us to vanish the deterministic drift function. Let X be a GBESQ 0,δ x and for t ∈ R + define the process Y by Y t = t 0 β s √ X s dW s , where the function β is in D and assume moreover that β is differentiable. Then the continuous local martingale defined by

Z t = E(Y t ) = exp Y t - 1 2 Y, Y t is equal to Z t = exp t 0 β s X s dW s - 1 2 t 0 β 2 s X s ds = exp 1 2 t 0 2β s X s dW s + t 0 β s δ s ds - t 0 β s δ s ds - t 0 β 2 s X s ds = exp 1 2 t 0 β s δ s ds + 2 X s dW s - t 0 β s δ s ds - t 0 β 2 s X s ds .
Since X is a GBESQ 0,δ x , we obtain that

Z t = exp 1 2 t 0 β s dX s - t 0 β s δ s ds - t 0 β 2 s X s ds . (1.9)
By integrating by parts, we have

β t X t = β 0 X 0 + t 0 β s dX s + t 0 X s dβ s = β 0 X 0 + t 0 β s dX s + t 0 X s β s ds.
Hence by substituting this expression in (1.9) we obtain

Z t = exp 1 2 β t X t -β 0 X 0 - t 0 X s β s ds - t 0 β s δ s ds - t 0 β 2 s X s ds = exp 1 2 β t X t -β 0 X 0 - t 0 β s δ s ds - t 0 β s + β 2 s X s ds .
If we show that Z t is a true-martingale then we will can apply the change of law formula given by the following proposition Proposition 1.5.

Z t := β Q δ x Q δ x = exp 1 2 β t X t -β 0 X 0 - t 0 β s δ s ds - t 0 β s + β 2 s X s ds . (1.10)
In fact, since X 0,δ s is positive for s ∈ [0, T ] and β ∈ D, so β is locally bounded. We have that β s + β 2 s ≥ 0 then the local-martingale Z t is bounded, therefore it is true-martingale.

Some law results

Starting from now and having in mind the previous result, we will always assume that the drift term is zero. We first recall a result from [START_REF] Carmona | Généralisations de la loi de l'arc sinus et entrelacements de processus de Markov[END_REF], Proposition 3.4

Proposition 1.6. If δ ∈ D and µ is a positive Radon measure on [0, a[, then Q δ x exp - a 0 X u dµ(u) = exp x 2 Φ µ (0) exp 1 2 a 0 Φ µ (u) Φ µ (u) δ u du ,
where Φ µ is the (well and uniquely defined) solution in the distribution sense of

Φ = Φµ. (1.11)
which is positive, non-increasing on [0, ∞[ and follows the initial condition Φ µ (0) = 1.

Remark 1.4.

With straightforward calculation, we can obtain a extended version of this result with non zero drift term, namely

β Q δ x exp - a 0 X u dµ(u) = exp x 2 Φ µ (0) exp 1 2 a 0 Φ µ (u) Φ µ (u) δ u du ,
where Φ µ the unique solution of the differential equation, in the distribution sense,

dµ u = Φ u 2Φ u du + β u Φ u Φ u du (1.12)
which is positive, non-increasing on [0, ∞[ and follows the initial condition Φ µ (0) = 1.

2. All these results could be extended under proper assumptions to measures without compact support. As our goal is to study the Bessel bridge, we feel free to do this restriction.

The previous form gives a precise explicit expression for the Laplace transform of additive functional of the generalized squared Bessel. However in practical cases this explicit form is rather difficult to use. We give a more abstract form which, surprisingly, suits better our goals, and allows us to introduce some notations.

Proposition 1.7. Let X δ be a generalized squared Bessel process with dimension δ and started in x. If δ ∈ D and µ is a positive Radon measure with compact support on R + , then there exist a constant A µ and a function B µ : R + → R + such that

Q δ x exp - ∞ 0 X u dµ(u) = A x µ exp ∞ 0 B µ(u) δ u du .
Proof. It is a trivial corollary of Proposition 1.6.

Remark 1.5. Note that, by taking δ constant, the A µ term is the same as in Corollary 1.3 of page 440 of [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF].

We can apply this formula to measures of the form µ t := λ t , where t stands for the Dirac measure in t. Our strategy to compute A µt and B µt will be to take well-chosen test values and functions for x and δ. Taking δ = 1, A µt was computed in [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], page 441, and is equal to

A µt = e -λ 1+2λt . (1.13)
To get access to B µt , we take x = 0 and use test functions of the form δ = χ [b,∞] where χ denotes the indicator function. For such δ and x = 0, the generalized squared Bessel process is easily described: indeed it is identically equal to 0 for 0 ≤ t ≤ b and then evolutes like a squared Bessel process with dimension 1 started at 0, that is, the square of a standard Brownian motion. Therefore straightforward computations imply

Q χ [b,∞] 0 exp -X u dµ t = E 0 e -λB 2 (t-b) + = 1 1 + 2λ(t -b) + . (1.14)
(for the last equality see for example p441 of [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF].) On the other hand this is also equal to exp ∞ b Bµ(u) . Taking the log and then the derivative with respect to b, then bringing everything together, we get the following result : Corollary 1.2. Under the statement of the Proposition 1.7, we have

Q δ x e -λXt = exp - λx 1 + 2λt exp - t 0 λδ u 1 + 2λ(t -u) du .
This is the Laplace transform of the transition Kernel of Q δ x . In the case of constant δ an inverse to this transform can be computed and therefore an expression of the transition Kernel, it is however not generally possible for any δ. We give in Remark 3.8 a simplification of this formula in the case of piecewise constant δ, and in section 5.1 some hints on how to overcome this issue in numerical simulations.

The Levy-Itô Representation of GBESQ

We now turn to the so-called Levy-Itô representation. For a motivation of such representation we refer to §4 of [START_REF] Pitman | A decomposition of Bessel bridges[END_REF].

Theorem 2.2.

1. There exists a unique measure M on C such that for every random variable I on C of the form I = X t µ(dt), with µ a positive Radon measure with compact support on (0, ∞) and every α > 0, denoting by M u the distribution of (X (t-u) + , t ≥ 0) for (X t , t ≥ 0) distributed according to M we get

Q δ x e -αI = exp xM + ∞ 0 δ u M u du e -αI -1 . (2.

15)

Moreover M corresponds to the 0 M of Theorem 4.1 of [START_REF] Pitman | A decomposition of Bessel bridges[END_REF].

2. For any δ ∈ D, there exists a

R + → C process (Y δ x , x ≥ 0) such that Y δ x = Y 0 x + Y δ 0 =: Y x + Y δ , (2.16)
where (Y x , x ≥ 0) and (Y δ , δ ∈ D) are independent, Y x is as in Theorem 4.1. of [START_REF] Pitman | A decomposition of Bessel bridges[END_REF] and Y δ is distributed according to Q δ 0 .

Remark 2.6.

1. This theorem should be understood as a Lévy-Khinchine formula on C. For more on this we refer to §4 of [START_REF] Pitman | A decomposition of Bessel bridges[END_REF]. We also refer to §3 of [START_REF] Pitman | A decomposition of Bessel bridges[END_REF] for background on the measure M . In short, M is the excursion law associated to the zero diffusion Q 0

x renormalized so that its entrance law is given by

M (X t ∈ dx) = dx (2t) 2 e -x 2t .
In particular the associated transition kernel starting for any strictly positive time is the same as the one of the squared Bessel process of dimension 0.

2. We also recall the expression of Y x from [START_REF] Pitman | A decomposition of Bessel bridges[END_REF]. Namely,

Y x (0) = x, Y x = v≤x ∆v on (0, ∞),
where ∆ v is a Poisson point process on C with Poisson measure M . Note that, due to the form of M , the sum in the last expression is actually infinite, however, for every finite t, the set of ∆ v such that ∆ v (t) > 0 is finite and its cardinality follows a Poisson distribution of parameter x 2t .

3. In [START_REF] Pitman | A decomposition of Bessel bridges[END_REF], the representation in terms of a Poisson point process is also valid for δ. However we do not have such a representation in our setting and believe it is a difficult question. It is however not needed for our purpose.

Proof of Theorem 2.2: First, we would like to prove (2.15). It is clear that both terms are multiplicative in δ and positive. The form in Proposition 1.6 also easily imply that the left hand side is continuous in δ with respect to the infinite norm. The continuity of the right hand side follows easily once one knows that ∞ 0 δ u M u du e -αI -1 is finite, which is easily deduced from Theorem 4.1. of [START_REF] Pitman | A decomposition of Bessel bridges[END_REF]. Hence we just need to prove it for well chosen test function. We take for test functions the functions that are constant on an interval and equal to zero outside. It is easy to see that such function form a generating family of D. Indeed it is rather easy to see that any constant by parts can be constructed as a sum of such functions. The fact that constant by part functions are dense in D is classical.

More precisely, let δ := dχ [a,b] . We can easily see that for such δ, X t is identically zero on [0, a] then evolves like a d-dimensional squared Bessel process on [a, b], then like a 0-dimensional squared Bessel process. It is the clear that by a shifting of time we can reduce to the case a = 0. The Markov property implies:

Q δ x e -αI = Q d x exp -α b 0 X t dµ(t) E 0 X b exp -α ∞ 0 X t dµ(t + b) .
We write

I 1 := b 0 X t dµ(t) and I 2 = ∞ 0 X t dµ(t + b).
Then the previous expression can be written

Q δ x e -αI = Q d x e -αI 1 Q 0 X b e -αI 2 .
(2.17) Theorem 4.1 of [START_REF] Pitman | A decomposition of Bessel bridges[END_REF] implies that the previous expression is equal to

Q d x e -α(I 1 +X b M (e -αI 2 -1)) .
Denoting by I 1 the function on C defined by

I I ((X t ) t≥0 ) = I 1 ((X t ) t≥0 ) + X b M (e -αI 2 -1))
and applying once again Theorem 4.1. of [START_REF] Pitman | A decomposition of Bessel bridges[END_REF], we get

Q δ x e -αI = exp xM + d ∞ 0 M u du (e -αI 1 -1) .
It is clear from the previous form of I 1 that for every u > b, M u (e -αI 1 -1) = 0, therefore the last expression can be reduced to

Q δ x e -αI = exp xM + d b 0 M u du (e -αI 1 -1)
.

We now want to prove that the above expression equals

exp xM + d b 0 M u du (e -αI -1)
.

For this we only need to prove that for every 0 ≤ u < b, M u (e -αI 1 -1) = M u (e -αI -1).

We get this basically by doing the previous computations backward. Indeed

M u (e -αI 1 -1)) = M u exp -α b 0 X t dµ(t) exp{X b M (e -αI 2 -1}} -1 = M u exp -α b 0 X t dµ(t) Q 0 X b exp{-α ∞ 0 X t dµ(t + b)} -1 = M u exp -α ∞ 0 X t dµ(t) -1 ,
where we used in the second line Theorem 4.1 of [START_REF] Pitman | A decomposition of Bessel bridges[END_REF] and in the third line the Markov property for M u and the fact that, by definition of M , for u < b, the transition kernel of M u for times greater than d is equal to the transition kernel of Q 0

x . Now we need to show (2.16): it is trivial, just by taking Y x as in [START_REF] Pitman | A decomposition of Bessel bridges[END_REF] and any independent δ-squared Bessel process started at zero for Y δ . Then Proposition 1.3 gives the result.

Bessel bridge Decomposition

We now turn to our main interest, namely the squared Bessel bridge with non-constant dimension. We define for δ ∈ D the law

Q δ,t x→y := Q δ (.|X 0 = x, X t = y). (3.18)
In all the result we are going to mention, we are only interested in the trajectory up to time t ∈ [0, T ] of the Bessel bridge. So with a slight abuse of notation we will consider Q δ,t x→y as a probability measure on C([0, t], R + ). We will also make use of the time reversed Bessel bridge, namely we call Q the Q distribution of (X(t -s)) 0≤s≤t . In particular it is easily seen that we have Lemma 3.1. Let X be a GBESQ δ,x such that for t ≥ 0, X t = y, then we have that

Qδ,t x→y = Q δ,t y→x ,
where δ(s) = δ(t -s).

We also restrict to the case where δ is not equal to zero excepted on a discrete set of points, as it will be the case in financial applications. This is only to ensure that for every y ≥ 0, the conditioning event in (3.18) has zero probability. It will also ensure that the distribution Q δ,t x→y is continuous with respect to (x, y). We recall that, by definition, Q 0,t 0→y is the time reversal of Q 0,t y→0 . We first give a "bridge" version of the additivity property. Proposition 3.8. Let δ, δ be in D, then

Q δ,t x→0 ⊕ Q δ ,t y→0 = Q δ+δ ,t x+y→0 .
Remark 3.7. In this setting there is no straightforward representation in term of the unconditioned process in the sense of (5.a) of [START_REF] Pitman | A decomposition of Bessel bridges[END_REF], whereas doubtless a careful application of the arguments of [START_REF] Watanabe | On time inversion of one-dimensional diffusion processes[END_REF] should lead to some similar (but not equal) result. This is however not needed for the above proposition to be true.

Proof of Proposition 3.8. This is directly obtained by conditioning the statement of Proposition 1.3, using the (obvious) fact that, for X and Y two R + -valued processes, the events {Y (t) + X(t) = 0} and {X(t) = 0, Y (t) = 0} are equal.

We are aiming at obtaining an extended Bessel bridge decomposition. Firstly, we need some notation and an introductory Lemma. We denote by L -1 f the inverse Laplace transform of f . And we call

F t δ (λ) = exp - λx 1 + 2λt exp - t 0 λδ u 1 + 2λ(t -u) du . (3.19) Lemma 3.2. Let b t δ,x,y (n) := (x/(2t)) n n! L -1 F t δ+2n (y) ∞ k=0 (x/(2t)) k k! L -1 F t δ+2k (y)
, then b t δ,x,y (n) is the distribution of U given

X * + X 1 + • • • + X U = y,
where U ∼ P oisson(x/(2t)), X i ∼ exp(1/(2t)) are i.i.d. random variables and X * is a random variable independent of U and X i and its Laplace transform is given by F t δ (λ).

Proof. By simple computation one gets that the Laplace transform of X 1 is equal to exp -

t 0 2λ 1 + 2λ(t -u) du = 1 1 + 2tλ .
We deduce that, conditionally on U = n, the Laplace transform of

X * + X 1 + • • • + X U is equal to F t δ+2n (λ).
The continuous version of the Bayes Formula then gives the result.

Then we obtain the extended Bessel bridge decomposition

Theorem 3.3. For every δ ∈ D, x ∈ R + , y ∈ R + , Q δ,t x→y = Q 0,t x→0 ⊕ Q 0,t 0→y ⊕ Q δ,t 0→0 ⊕ ∞ n=0 b t δ,x,y (n)Q 4n,t 0→0 ,
where the last term on the right is the mixture of the laws Q 4n,t 0→0 , n = {0, 1, . . . } with weights given by b t δ,x,y (n). Proof. Using point 2. of Remark 2.6, we get that, for

Y δ x ∼ Q δ x , Y δ x = Y 0 x + Y δ 0 ,
where the two processes are independent, Y δ 0 ∼ Q δ 0 and

Y 0 x = v≤x ∆ v ,
where ∆ v is a Poisson point process with measure M . We separate as in [START_REF] Pitman | A decomposition of Bessel bridges[END_REF] this sum in two, keeping on one side the terms which are zero at t and on the other side the terms which are positive at t, we get that

Y δ x = Y 0 x0 + Y 0 x+ + Y δ 0 ,
where the three processes are independent,

Y 0 x0 ∼ Q 0,t x→0 , and
Y 0 x+ = U i=0 Z i ,
where U is a P oisson(x/(2t)) variable independent of everything else and Z i are i.i.d. trajectories whose distribution is given by M conditioned on X t > 0 and renormalized. We condition on the values of Z i and Y δ 0 at time t. Conditionally on U = u, and Z i (t) = z i , ∀i ≤ U, it is shown in [START_REF] Pitman | A decomposition of Bessel bridges[END_REF] that

Z i ∼ Q 4,t 0→z i = Q 0,t 0→z i + Q 4,t 0→0 .
Summing this, we get that, conditionally on U = u, and

Z i (t) = z i , ∀i ≤ U with z i = z, Y 0 x+ ∼ Q 0,t 0→z + Q 4u,t 0→0 .
On the other hand conditionally on U = u, and Z i (t) = z i , ∀i ≤ U, using time reversal and Proposition 3.8,

Y δ x ∼ Q δ,t 0→y-z = Q δ,t 0→0 ⊕ Q 0,t 0→y-z .
Putting everything together, we get that conditionally on U ,

Q δ,t x→y = Q δ,t 0→0 ⊕ Q 0,t x→0 ⊕ Q 0,t 0→y ⊕ Q 4u,t 0→0 .
On the other hand, conditionally on Y , Y δ 0 (t) has Laplace transform F t δ (λ) and Z i (t) are independent random variable with law exp {1/(2t)} Therefore, using Lemma 3.2, we get the result.

Remark 3.8. The expression in Lemma 3.2 is fairly general, however in most practical applications δ is constant by parts. We thus give a simpler expression in this case. Indeed, let 0 = τ 0 < τ 1 • • • < τ n = t be an increasing sequence of times such that on each [τ i , τ i+1 ), δ is constant equal to δ i , then

F t δ = exp - λx 1 + 2λt n-1 i=0 1 + 2λ(t -τ i+1 ) 1 + 2λ(t -τ i ) δ i /2 .

About the coefficients in the Bessel bridge decomposition

The main difference between the classical setting and ours is the fact that the coefficients b t δ,x,y (n) are not as explicit. However they can be computed for given values of δ. For numerical purposes it is interesting to have a bound on the speed of decay of these terms.

We have the following result: Proposition 3.9. Let δ ∈ D, d = inf [0,t] δ and x, y ∈ R + , then for every increasing function f : N → R + , and with

ν = d/2 -1 and z = √ xy, ∞ k=0 b t δ,x,y (k)f (k) ≤ ∞ k=0 β ν,z (k)f (k), (3.20) where β ν,z (k) = z 2 2n+ν 1 Γ(n + ν + 1)I ν (z)n!
are the coefficients of the Bessel(ν, z) distribution (I ν (z) stands for the modified Bessel function). In particular the power series associated to the b t δ,x,y (k) has an infinite radius of convergence.

Remark 3.9. The philosophy of this proposition is that, when one wants to compute quantities related to the Bessel bridge, the truncation arguments used in the classical setting are also valid in our setting.

Proof. From the representation of Lemma 3.2, it is clear that b t δ,x,y (k) is the distribution of a variable U which is negatively correlated with X * , the distribution of the generalized squared Bessel process at time t. Proposition 1.3 implies that we can couple the generalized squared Bessel process of dimension δ with a classical squared Bessel process of dimension d, the first one being always bigger than the second one. From this we deduce that the distribution of U for δ is dominated by the distribution for d. This distribution is computed in [START_REF] Pitman | A decomposition of Bessel bridges[END_REF], (5.j) and is the Bessel(ν, z) distribution. The result follows easily. In particular the parameter t does not intervene, as it simplifies between α and λ in [START_REF] Pitman | A decomposition of Bessel bridges[END_REF], (5.j).

4 Laplace transform formula associated to extended Bessel bridges.

We are now interested in computing the Laplace transform of some functionals of the Brownian bridge. We first deal with the simpler case of the bridge to zero. In this case, because additivity holds, things are simpler. We have the following: Proposition 4.10. For every Radon measure µ on [0, t], there exist A 0 (α) and a function

B α 0 (u) : [0, t] → R depending on µ such that Q δ,t x→0 exp - t 0 αX u dµ(u) = (A 0 (α)) x exp t 0 B α 0 (u)δ u du .
Remark 4.10. Note that, for t = 1, A 0 (α) is as in in (5.d') of [START_REF] Pitman | A decomposition of Bessel bridges[END_REF]. We will give an example of explicit computation of A 0 (α) and B α 0 (u) in Theorem 4.4.

Proof. We first assume that the function Φ defined by (1.11) is such that

t 0 Φ Φ 2 du < ∞. Using Proposition 3.8, it is easy to check that φ(x, δ) := -log Q δ,t x→0 exp - t 0 αX u dµ(u)
is additive in both terms, therefore φ(x, δ) = φ(x, 0) + φ(0, δ), and φ(x, 0) being linear, it is necessarily of the form φ(x, 0) = a 0 x. We now want to show that

φ(0, δ) = -log Q δ,t 0→0 exp - t 0 αX u dµ(u) is of the form ∞ 0 B α 0 (u)δ u du.
To show this we need to apply the Fréchet-Riesz representation, which states (in our context) that any continuous linear form on the Hilbert space L 2 ([0, t]) is of the above mentioned form. So, keeping in mind that D is a dense subset of L 2 ([0, t]), the only thing we need is to show that φ(0, δ) is continuous with respect to the L 2 norm on [0, t]. As we now it is linear, it suffices to show that it is bounded on the unit ball. By proposition 1.6, and the Cauchy-Schwartz inequality, we now that it is true for the unconditioned process, as by assumption Φ Φ is square integrable. Therefore the result holds as a consequence of the following coupling result, whose interest goes further than the context of this proof. Lemma 4.3. For every δ ∈ D, x ∈ R + , there exist a (R

+ ×R + ) valued process (X s , X 0 s ) s∈[0,t] such that X s ∼ Q δ x , X 0 s ∼ Q δ,t x→0
, and X 0 s ≤ X s a.s., ∀0 ≤ s ≤ t.

Before giving the proof of this lemma we finish the proof of Proposition 4.10. We only need to get rid of the assumption

t 0 Φ Φ 2 du < ∞. First note that by definition of Φ, for all , t- Φ Φ 2 du < ∞.
We deduce from the same arguments as above that for all , there exist B α, 0 such that

Q δ,t x→0 exp - t- αX u dµ(u) = (A 0 (α)) x exp t- B α, 0 (u)δ u du .
From this it is clear that for < , the restriction to [ , t -] of B α, 0 is B α, 0 . We can accordingly define B α 0 on (0, t) such that the restriction to [ , t -] of B α 0 is B α, 0 . One can then easily check that B α 0 verifies the statement of Proposition 4.10.

We now turn to the Proof of Lemma 4.3.

Proof of Lemma 4.3. The proof follows a standard scheme, see for example pages 205-206 of [START_REF] Faraud | Estimates on the speedup and slowdown for a diffusion in a drifted Brownian Potential[END_REF]. The idea is to express the conditioned process as a solution of a SDE whose terms are smaller than the SDE defining the unconditioned process, and then apply the Comparison Theorem of [START_REF] Watanabe | A comparison theorem for solutions of stochastic differential equations and its applications[END_REF] to prove the existence of a coupling. We have first that for s ≤ t, denoting by q t (x, y) the transition kernel associated to Q δ x . Using 1.3, we can write q t (x, y) = q 0 t (x, .) q δ t (0, .) (y), (

where stands for the convolution, and q 0 t is the transition kernel of the 0-dimensional squared Bessel Process. In particular q t (x, 0) = q 0 t (x, 0)q δ t (0, 0). (

We have dQ δ,t x→0 dQ δ

x Fs = q t-s (X s , 0) q t (x, 0) = exp (log q t-s (X s , 0) -log q t (x, 0)) .

We write h(x, s) = log q t-s (x, 0). We now from (4.22) that

h(x, s) = h 0 (x, s) + h δ (0, s), (4.23)
where h δ (x, s) corresponds to the squared Bessel process of dimension δ.

From this form it is obvious that the derivative of h with respect to x does not depend on δ, therefore is equal to its value for the 0-dimensional squared Bessel process, that is 1 2(s-t) (see [START_REF] Faraud | Estimates on the speedup and slowdown for a diffusion in a drifted Brownian Potential[END_REF] for details), and in particular the second derivative with respect to x is 0 . We have thus, applying Itô's formula that

dQ δ,t x→0 dQ δ x Fs = exp - s 0 1 2(t -u) dX u + s 0 dh ds (X u , u)du .
Using once more (4.23) we see that dh ds can be decomposed into a part which only depends on δ and another one which only depends on x. Putting everything together we get

dQ δ,t x→0 dQ δ x Fs = exp - s 0 1 2(t -u) dX u + s 0 X u 2(t -u) 2 du + s 0 dh δ ds (0, u)du) .
We do not need to compute the last term in the exponential, indeed it is a deterministic term, and we now that the Radon derivative is a martingale, therefore its value is imposed by this condition. Recalling that by definition

dX u = δ u du + 2 X u dW u ,
and replacing in the above expression, we have

dQ δ,t x→0 dQ δ x Fs = exp - s 0 √ X u (t -u) dW u , + s 0 X u 2(t -u) 2 du + a deterministic function of s . As X s - s 0 δ u du is a martingale (under Q δ x )
, and keeping in mind that the deterministic term is imposed by the martingale condition, we have:

dQ δ,t x→0 dQ δ x Fs = E - s 0 √ X u (t -u) dW u .
Girsanov's Theorem then implies that, under Q δ,t x→0 ,

dX t = δ u du + 2 X u dW u - 2X u t -u du.
Therefore we obtain the result by an application of the Comparison Theorem of [START_REF] Watanabe | A comparison theorem for solutions of stochastic differential equations and its applications[END_REF].

We deduce from Proposition 4.10 and Theorem 3.3 the following Corollary 4.3. Let for µ a Radon measure on [0, t], I = Xdµ, A 0 (α), B α 0 (u) : R + → R as in Proposition 4.10, and Â0 (α) be the A 0 (α) for the image of µ under the map s → (t-s), let also B 0 (α) = exp t 0 B α 0 (u)du then we have

Q δ,t x→y (e -αI ) = A 0 (α) x Â0 (α) y exp t 0 B α 0 (u)δ u du ∞ n=0 b t δ,x,y (n)B 0 (α) 4n .
We now give explicit computations when µ is the Lebesgues measure on [0, t].

Theorem 4.4. Let X be a GBESQ δ x and let β = √ 2αt, then we have

Q δ,t x→y (e -α t 0 Xsds ) = exp (x + y) 2t (1 -β coth β) exp t 0 1 2(t -u) - √ 2α 2 coth( √ 2α(t -u)) δ u du ∞ n=0 b t δ,x,y (n) β sinh β 2n .
Proof. Here µ is invariant under the map s → (t -s), so that Â0 (α) = A 0 (α). Corollary 3.3 on page 465 of [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] gives for t = 1, b ∈ R + and constant δ that

Q δ,1 x,0 exp - b 2 2 1 0 X s ds = b sinh b δ 2 exp x 2 (1 -b coth b) .
Taking first α := b 2 2 , then although this expression is given for t = 1 we can extend it to [0, t] using the fact that, for δ constant,

Q δ,t x→0 e -α t 0 Xsds = Q δ,t x→0 e -αt 1 0 Xstds = Q δ,1 x/t→0 e -αt 2 1 0 Xsds ,
where first, we made a change of variable and then used the scaling of Bessel processes.

Hence taking y = 0 and δ = 0, we obtain the expression of A 0 (α)

A 0 (α) = exp 1 2t 1 - √ 2αt coth √ 2αt . (4.24)
And taking y = 0 and x = 0 we obtain the expression of B 0 (α)

B 0 (α) = √ 2αt sinh √ 2αt . (4.25)
In order to compute B α 0 (u), we take as before test functions for δ, namely we assume δ = χ [s,t] , with 0 < s < t. We have by the same arguments as before

Q 1,1 0→0 e -α t 0 Xsds = Q 1,(t-s) 0→0 e -α t-s 0 Xudu = Q 1,(t-s) 0→0 e -α(t-s) 1 0 X v(t-s)du .
By scaling, this is equal to B 1 0 (α(t -s) 2 ), where B 1 0 (α) is the value of B 0 (α) for t = 1. On the other hand it is equal to exp ( t s B α 0 (u)du). Then straightforward computations give

B α 0 (u) = 1 2(t -u) - √ 2α 2 coth( √ 2α(t -u)).
This, together with the previous computations, gives the result. Note that B α 0 (u) is not defined for u = t, however it is clearly integrable on [0, t].

Remark 4.11. The previous computations may be easily applied to get formulas for general additive functionals of the GEBSQ. We skip for the moment these tedious computations, but give the complete result in Appendix A

Some applications

As we said in the introduction, our work was motivated by applications to Finance. Indeed, our class of GBESQ contains various financial models. Hence we can give an answer to many financial mathematical problems using our Bessel bridge decomposition and Laplace transform formula results applied to particular GBESQ. We begin thus by giving some financial models which are in the class of GBESQ.

Ornstein-Uhlenbeck (OU): Let (V t ) t∈[0,T ] following an Ornstein-Uhlenbeck model given by dV t = µV t dt + σdW t with V 0 = x.

(5.26)

Then we have the classical representation in term of GBESQ given by

V t = e µt X -σ 2 2µ e -2µt -1 1 2
(5.27) and X is a GBESQ δ,x with dimension δ = 1 and initial value x = x 2 .

Cox Ingersoll Ross (CIR): Let (V t ) t∈[0,T ] be a CIR process then it is the solution of the stochastic differential equation

dV t = (α -βV t )dt + σ V t dW t with V 0 = x.
where α ∈ R + , β ∈ R, σ > 0. Then we have by a time space transformation the classical representation in term of GBESQ given by where µ represents the instantaneous mean, σV ρ t the instantaneous variance of V and ρ is called the elasticity factor. In the limiting case, ρ = 1, the CEV model returns to the classical Black and Scholes model. In the case where ρ = 0, it is the Ornstein-Uhlenbeck model and if ρ = 1 2 , it is a Cox-Ingersoll-Ross (CIR) model without mean reverting. Then we obtain that

V t = e -
V t = e µt X (ρ -1)σ 2 2µ e 2(ρ-1)µt -1 1 2(1-ρ) (5.30)
and X is a GBESQ δ,x with dimension δ = 2ρ-1 ρ-1 and initial value x = x -2(ρ-1) . Extended CIR: Let (V t ) t∈[0,T ] following an extended Cox Ingersoll Ross model which is given by the solution of the stochastic differential equation (SDE)

dV t = (α t -β t V t )dt + σ t V t dW t with V 0 = x.
(5.31) where α t , β t ≥ 0 and σ t > 0 are time-varying functions and σ t is continuously differentiable with respect to t ∈ [0, T ]. Now, we recall Corollary 3.1 of [START_REF] Shirakawa | Squared Bessel Processes and Their Applications to the Square Root Interest Rate Model[END_REF].

Corollary 5.4. We have

V t = exp - t 0 β u du X 1 4 t 0 σ 2 u θ u du , (5.32)
where X is a GBESQ δ,x with time varying dimension δ t = 4(α•ν -1 )t (σ 2 •ν -1 )t with ν t :=

1 4 t 0 σ 2 u θu du, θ t = exp - t 0 β u du and initial value x := x θ 0 = x.
Hence, the class of GBESQ contains a large class of standard (financial) models Remark 5.12. We see from these example that one needs in many occasions to evaluate the law of functionals of the form t 0 g(s)X f (s) ds for various functions g and f (with f increasing), for X being a GEBSQ or a Generalized Bessel bridge. Laplace transform of these quantities can be computed using the results of Remark 1.4 or alternatively Appendix A, using the simple argument that, by change of variable,

t 0 g(s)X f (s) ds = f (t) f (0) g(f -1 (s))) f (f -1 (s))
X s ds.

5.1 Generating values of X t given X 0 and its Laplace transform.

Let X be a GBESQ δ,x . Our aim is to obtain a sample of generating values of X t for t ∈ [0, T ] assuming that with know X 0 = x and that we have an explicit formula of the Laplace transform of X t . A classic way to do this is to inverse the Laplace transform formula. But in our case the Laplace transform formula we obtained in Corollary 1.2 is not easy to inverse. So we need an other method to generate a sample of values of X t . One can use for example the method developed by Ridout in [START_REF] Ridoux | Generating random numbers from a distribution specified by its Laplace transform[END_REF] which allows to generate sample of random numbers from a distribution specified by its Laplace transform (without having to inverse the Laplace transform). Hence, as we can evaluate explicitly the Laplace transform of X assuming X 0 = x with our Corollary 1.2 and simplification given by Remark 3.8, one can apply the method of Ridout in [START_REF] Ridoux | Generating random numbers from a distribution specified by its Laplace transform[END_REF] to obtain a sample of values of X t for any t ∈ [0, T ].

This method also permits to jointly simulate X t (for a fixed t) and any additive functional of X s , 0 ≤ s ≤ t without simulating the whole process. To do this first simulate X t through its Laplace transform, then plug this value into the formula of Appendix A and repeat the process. We give an example in Section 5.2 5.2 Simulation of integrate process conditioning to its starting and ending points.

In this application, we follow results found by Broadie and Kaya in [START_REF] Broadie | Exact Simulation of Stochastic Volatility and other Affine Jump Diffusion[END_REF] and extended by Glasserman and Kim in [START_REF] Glasserman | Gamma Expansion of the Heston Stochastic Volatility Model[END_REF]. We will apply the same idea but using our more general framework. Indeed, assume that we would like to simulate a stochastic volatility model given for all t ∈ [0, T ] , δ ∈ D by

dS t = µ t S t dt + V t ρdW 1 t + 1 -ρ 2 dW 2 t , S 0 = s, (5.33
)

dV t = δ t dt + 2 V t dW 1 t , V 0 = x.
(5.34)

where W 1 and W 2 are two independent standard Brownian motion, X t is a Markov jump process on finite space S := {1, 2, . . . , K} and ρ ∈ [-1, 1]. So the stochastic volatility process is a GBESQ δ,x . Let for all t ∈ [0, T ] the solution of the first stochastic differential equation in (5.33):

S t = S 0 exp t 0 µ s ds - 1 2 t 0 V s ds + ρ t 0 V s dW 1 s + 1 -ρ 2 t 0 V s dW 2 s . Moreover t 0 √ V s dW 1 s = 1 2 V t -V 0 - t 0 δ s ds . Then we have that log St S 0 is conditionally normal given t 0 V s ds, V t and V 0 . Hence log S t S 0 ∼ N t 0 µ s ds - 1 2 t 0 V s ds - ρ 2 V t -V 0 - t 0 δ s ds , (1 -ρ 2 ) 2 t 0 V s ds .
Moreover we can sample V t using again method stated in subsection 5.1. Hence we can obtain a sample of couple (V 0 , V t ). Then the problem of exact simulation of our model (5.33) is reduced to the problem of sampling from

t 0 V s ds|V 0 , V t .
Since V is a GESQ δ x , we know by Theorem 3.3 that for every δ ∈ D, x ∈ R + , y ∈ R + , there exist processes X 1 , X 2 , X 3 and X 4 such that

t 0 V s ds|V 0 = x, V t = y = X 1 + X 2 + X 3 + X 4 , (5.35)
where

X 1 ∼ Q 0,t x→0 , X 2 ∼ Q 0,t 0→y , X 3 ∼ Q δ,t 0→0 and X 4 ∼ ∞ n=0 b t δ,x,y (n)Q 4n,t 0→0
. Moreover, depending on the GBESQ V chosen, we can use Theorem 4.4 or Remark 5.12 and Proposition 3.9 to have a way of sampling from t 0 V s ds|V 0 , V t ; and so an exact simulation of our stochastic model (5.33).

Evaluation of Bond price in a credit notation model with regime switching

Assume that we are under an equivalent risk neutral probability P , then from the general asset pricing theory, we have that the discount bond price B(T ) at time 0 with maturity T > 0 of an interest rate r is given by

B(T ) = Ẽ exp - T 0 r s ds |r 0 = r (5.36)
Let the process r be given by the solution of the stochastic differential equation given by (5.31), we assume that the value of the parameters (α t , σ t ) is function for all t ∈ [0, T ] of the value of an exogenous continuous time on finite state Markov chain (S) t∈[0,T ] . Moreover, we assume that β t ≡ 0. Hence the Markov chain S takes value in S := {1, 2, . . . , N } and could represent the credit notation of a firm A given by a credit notation entity. Then the process r could be the spread of the firm A and we would like to evaluate the price at time t ∈ [0, T ] of a bond of this firm with maturity T > 0. (See Goutte and Ngoupeyou in [START_REF] Goutte | Conditional Laplace formula in regime switching model: Application to defaultable bond[END_REF] for more details about this defaultable regime switching modeling). Hence our model becomes dr t = α(S t )dt + σ(S t ) √ r t dW t , r 0 = r (5.37) and so α, β and σ are piecewise continuous functions.

Remark 5.13.

1. The fact that we assume that S is a exogenous Markov chain implies that S is independent of the Brownian motion W for all t ∈ [0, T ].

2. Assuming that S is exogenous has a economic sense, indeed in representation (5.37) there is two source of randomness, firstly the market risk modeled by the Brownian motion W and secondly an external expertise risk factor S given by an external entity of the firm A. This could be the credit notation representing the wealth at time t ∈ [0, T ] of the firm A or an indicator of the wealth of the global international economy at time t.

Let us denote by S the historical values of S, i.e. S := (S t ; t ∈ [0, T ]). Using (5.32), we have that

r t = X 1 4 t 0 σ 2 s ds
where X is a GBESQ δ,x with time varying dimension δ t = 4(α•ν -1 )t (σ 2 •ν -1 )t with ν t := 1 and g(.) ≡ 1. So we can apply Theorem A.6, which is the extended version of our Theorem 4.4, to evaluate the Laplace transform of t 0 X f (s) ds using our extended Bessel bridge decomposition where we take for Radon measure µ the measure given for all t ≥ 0 by:

dµ(t) = 1 f (f -1 (t))
dt.

Hence to apply this result, we need to have a sample of value of r T = X f (T ) with respect to the starting point r 0 = X 0 := x and the law of the associated GBESQ for r. For this we will use the method stated in subsection 5.1 since we have a Laplace transform formula expression given in Corollary 1.2. Moreover the quantity b t δ,x,y (n) given by Lemma 3.2 is in this case simpler since we have Remark 3.8. Finally these allow us to obtain an evaluation of the Bond price B A (T ) at time zero of the firm A using Monte Carlo.

Simulation of the default time in credit risk model

We work in the model of construction of Cox processes with a given stochastic intensity λ. Hence we denote by λ a stochastic default intensity of a firm or country A. We assume for all t ∈ [0, T ] that t 0 λ s ds < ∞ and that ∞ 0 λ s ds = ∞. Then the default time τ of the firm or country A is a real random variable which is given by the first time when the increasing process t 0 λ s ds is below a uniform random variable U independent with λ. Hence we get τ := inf t ≥ 0; exp -t 0 λ s ds ≤ U .

(5.39) So to simulate the default time requires to simulate the stochastic default intensity λ together with its time integral. Hence assuming that λ is a GBESQ δ,x , we can use again our Bessel bridge decomposition given in Theorem 3.3, our Laplace transform formula given in Corollary 1.2 and the method stated in subsection 5.1, to do this.

A Laplace transform of additive functionnals of the GEBSQ

In this appendix we give an explicit formula for the Laplace transform of additive functionals of the Generalized Bessel bridge. We first recall Corollary 4.3:

4 t 0 σ 2 4 t 0 σ 2 s

 4242 s ds and initial value x = r. Hence we have two different ways to evaluateB A (T ) = Ẽ exp -T 0 r s ds |r 0 = r; SFirstly, we can use the Proposition 1.6 and solve the corresponding equation(1.11). Or we can use our Bessel Bridge decomposition to jointly sample the final value of the process X and the Laplace transform. Indeed, by Remark 5.12, we have in this model that f (t) = ν t := 1 ds and r t = X f (t)(5.38)

  Constant elasticity of variance model (CEV): Let (V t ) t∈[0,T ] following a Constant elasticity of variance model (CEV) given by dV t = µV t dt + σV ρ t dW t with 0 ≤ ρ < 1 and V 0 = x.

	βt X	σ 2 4β	e βt -1		(5.28)
	and X is a GBESQ δ,x such that
	dX t = δdt + 2 X t d Wt
	with a constant dimension δ = 4α σ 2 and W σ 2 4β e βt -1 =	t 0	σ 2 e	βs 2 dW s .
					(5.29)

Corollary 4.3. Let for µ a Radon measure on [0, t], I µ = Xdµ, A 0 (α), B α 0 (u) : R + → R as in Proposition 4.10, and Â0 (α) be the A 0 (α) for the image of µ under the map s → (t-s), let also B 0 (α) = exp t 0 B α 0 (u)du then we have

The idea, used before, is to take test functions for δ to get the explicit value of A 0 (α) and B α 0 (u). We recall the result in the case of Standard Bessel Process, as given for example on page 465 of [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], in the proof of Theorem 3.2., taking y = 0 and recalling that lim x→0 Iν (xy)

Theorem A.5. In the case of constant dimension we have

, where, as usual, φ µ is the unique solution of φ = µφ which is positive, non increasing and such that φ µ (0) = 1; and

We first extend this formula to bridges of arbitrary length (keeping for the moment δ constant). Let µ be a positive Radon measure on [0, t]. We have by change of variable, then scaling,

x/t→0 (e -1 0 Xst 2 dµ(ts) ).

By Theorem A.5, we deduce that

, where φ µ is the unique solution of φ = t 2 µ(t•)φ. It is clear that φ µ = φ t µ (t•), where φ t µ is the unique solution of φ = µφ which is positive, non increasing and such that φ t µ (0) = 1. Expressing the previous result in term of φ t µ we have, with obvious notations,

.

We deduce from this expression that A 0 (1) = exp 1 2 F t µ (0) + 1 t -

. We also

, where φt µ is the unique solution of φ = µ(t -•)φ which is positive, non increasing and such that φt µ (0) = 1; and

We now turn to the computation of B 1 0 (u). For this take δ = χ [b,t] . We have as previously explained,

, where φ t,b µ is the unique solution of φ = µ(b + •)φ which is positive, non increasing and such that φ t,b µ (0

We take the logarithm and differentiate with respect to b to get:

Finally, using (A.40), we have

Finally, putting everything together, we have:

where φ t µ is the unique solution of φ = µφ which is positive, non increasing and such that φ t µ (0) = 1, φt µ is the unique solution of φ = µ(t -•)φ which is positive, non increasing and such that φt µ (0) = 1; and


Jean Michel Chapuis
email: jean-michel.chapuis@upf.pf.

Basics of Dynamic Programming for Revenue Management

Keywords: Revenue Management, Dynamic Programming, Booking limits, Time thresholds, Bid prices. JEL classification: M30, M11

The Revenue Management (RM), namely the pricing and the inventory control of a perishable product, is usually used to improve services marketing efficiency. While booking a flight, the manager has to allocate seats to various fare classes. Then, he has to assess the consequence of a current decision on the future stream of revenue, i.e. accept an certain incoming reservation or wait for a possible higher fare demand, but later. Since its practice becomes omnipresent this last decade, this paper presents some basics of Dynamic Programming (DP) through the most common model, the dynamic discrete allocation of a resource to n fare classes. The properties of the opportunity cost of using a unit of a given capacity, the key of any RM optimizations, are studied in details.

INTRODUCTION

For decades, we have learned to revere the letters of Revenue Management (RM). Its spirit of pricing and of inventory control is widely shared among various industries, major companies and academic researchers. Revenue maximization is usually reached by comparing a current discount fare request with an expected higher fare request in the future. The well-known EMRS model [START_REF] Belobaba | Application Of A Probabilistic Decision Model To Airline Seat Inventory Control[END_REF], which is an extension of [START_REF] Littlewood | Forecasting and Control of Passengers Bookings[END_REF] model, gives an heuristic solution but assume that the demand is static and will not evolve during the booking period. However, new low cost carrier entrants, fare transparency from Internet fare search engines and changing customer purchasing patterns are the major forces pushing airline revenue managers to adapt their optimization models. In addition, using traditional forecasting and optimization models that assume fare class independence causes continuous erosion of yield. Specially when there is excess capacity, the surplus seats flows down the fare structure, allowing high yield customers to buy down. Later, this behavior affects forecast, ie expecting lower demand for high fare class in the future, and leads revenue managers to protect less seats for it.

Almost RM situations can usefully be modeled as Dynamic Programming problems (hereafter DP) because those models take into account future possible booking decisions in assessing that current decision. Moreover, DP allows to relax the early birds hypothesis (low contribution customers book in advance of the high contribution ones). The optimal controls are then time-dependent as a function of the remaining capacity. Dynamic pricing is the special case where price is the control used by the managers.

Nonetheless, the practicing of Bellman equations ( 1957) is relatively new in RM since none of the pioneering papers use DP 1 . McGill and Van Ryzin (1999) conclude 1 The basics of Revenue Management can be found in [START_REF] Pfeifer | The Airline Discount Fare Allocation Problem[END_REF], [START_REF] Belobaba | Application Of A Probabilistic Decision Model To Airline Seat Inventory Control[END_REF] or [START_REF] Kimes | Yield Management: A Tool for Capacity-Constrained Service Firms[END_REF] among others. For an overview, check [START_REF] Michel | Revenue Management: Research Overview and Prospects[END_REF] or [START_REF] Michel Chapuis | An Overview of Pricing Models for Revenue Management[END_REF] and more recently [START_REF] Chiang | An Overview of Research on Revenue Current Issues and Future Research[END_REF]. [START_REF] Smith | Yield Management at American Airlines[END_REF] report the experience of American Airlines while implementing Revenue Management in a worldwide scale.

-3 -Jean Michel Chapuis that: "Dynamic formulations of the revenue management problem are required to properly model real world factors like cancellations, overbooking, batch bookings, and interspersed arrivals. Unfortunately, DP formulations, particularly stochastic ones, are well known for their unmanageable growth in size when real-world implementations are attempted. Usually, the only hope for dynamic optimisation in these settings lies in identification and exploitation of structural properties of optimal or near optimal solutions. Knowledge that an optimal solution must be of a control-limit type or be represented by a monotonic threshold curve can be invaluable in development of implementable systems. The existing literature has already identified such structures in special cases of the revenue management problem; however, there are difficult areas still requiring work." (emphasis added).

The goal of this research is to review some Dynamic Programming models dedicated to Revenue Management in order to provide a solid basis for future work. The structure is the following. The first part, §1, formulates both the RM problematic and its DP methodology. Then, the problem of setting booking limits and prices using DP approach is addressed in the second part, §2. In order to maximize operating revenue, RM is the process by which a manager controls the availability of a product or service, marketed with a differential and dynamic pricing. Controls can be effective by varying prices, setting booking limits and managing fences2 . This approach is in line with [START_REF] Talluri | The Theory and Practice of Revenue Management[END_REF], who supported both price and quantity-based models of RM. Most authors often define RM as the application of control and pricing strategies to sell the right capacity to the right customer, in the right place, at the right time and at the right price [START_REF] Kimes | Yield Management: A Tool for Capacity-Constrained Service Firms[END_REF]. RM has proven its potential impact on profitability in the past (Smith, Leimkulher and Darrow, 1992).

DYNAMIC PROGRAMMING OF REVENUE MANAGEMENT OPTIMIZATION

Literature review of DP models to RM optimization.

Dynamic programming addresses how to make optimal decisions over time under uncertain conditions and to control a system. Most RM situations can be analyze assuming a discrete-state and a discrete time over a finite-horizon modeling. The methodological reference is the one of [START_REF] Bertsekas | Dynamic Programming and Optimal Control[END_REF]. [START_REF] Hersh M | A model for dynamic seat inventory control with multiple seat bookings[END_REF] is the most popular reference with their model for dynamic seat inventory control. [START_REF] Gallego | Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons[END_REF] are commonly cited as the reference in dynamic pricing model using dynamic programming. The table 1 summarizes the literature according to four criteria. (i) A paper could consider a single product (at various price) or multiple products (depending on purchase restrictions or independent demands for example). (ii)

A paper could consider a static policy (assuming a strict order of booking arrivals) or allow for a dynamic policy (not assuming the early birds hypothesis). (iii) A paper could Jean Michel Chapuis consider various forms of demand process. (iv) A paper could consider either a single resource for 1 to n products or multiple resources (such as an airline network of hubs and spokes). [START_REF] Brumelle | Airline Seat Allocation with Multiple Nested Fare Classes[END_REF] Lee and Hersh (1993); [START_REF] Gallego | Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons[END_REF]; [START_REF] Robinson | Optimal and Approximate Control Policies for Airline Booking with Sequential Fare Classes[END_REF]; [START_REF] Gallego | A Multiproduct Dynamic Pricing Problem and its Applications to Network Yield Management[END_REF]; [START_REF] Talluri | An aAnalysis of Bid-Price Controls for Network RM[END_REF]; Liang (1999); You (1999); Lautenbacher and Stidam (1999); [START_REF] Feng | Perishable asset revenue management with Markovian time dependent demand intensities[END_REF]; [START_REF] Feng | A dynamic airline seat inventory control model and its optimal policy[END_REF]; [START_REF] Kleywegt | An Optimal Control Problem of Dynamic Pricing[END_REF]; [START_REF] You | Airline Seat Management with rejection-for-possible-upgrade decision[END_REF]; [START_REF] Popescu | Revenue Management in a Dynamic Network Environment[END_REF]; El-Haber and El-Taha ( 2004) [START_REF] Bertsimas | Dynamic pricing and inventory control for multiple products[END_REF]

Demand

Deterministic

Stochastic

Both

Non homogeneous [START_REF] Curry R | Optimal Seat Allocation with Fare Classes Nested by Origins and Destinations[END_REF]; Wollmer (1992); [START_REF] Brumelle | Airline Seat Allocation with Multiple Nested Fare Classes[END_REF]; [START_REF] Robinson | Optimal and Approximate Control Policies for Airline Booking with Sequential Fare Classes[END_REF]; [START_REF] Kleywegt | An Optimal Control Problem of Dynamic Pricing[END_REF] Lee and Hersh (1993); [START_REF] Gallego | Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons[END_REF]; [START_REF] Gallego | A Multiproduct Dynamic Pricing Problem and its Applications to Network Yield Management[END_REF]; [START_REF] Talluri | An aAnalysis of Bid-Price Controls for Network RM[END_REF]; [START_REF] Feng | Perishable asset revenue management with Markovian time dependent demand intensities[END_REF]; [START_REF] Feng | A dynamic airline seat inventory control model and its optimal policy[END_REF]; [START_REF] You | Airline Seat Management with rejection-for-possible-upgrade decision[END_REF]; [START_REF] Popescu | Revenue Management in a Dynamic Network Environment[END_REF]; El-Haber and El-Taha ( 2004) [START_REF] Bertsimas | Dynamic pricing and inventory control for multiple products[END_REF] Zhao and Zheng (2000) Jean Michel Chapuis

Resources

Single Network Wollmer (1992); [START_REF] Brumelle | Airline Seat Allocation with Multiple Nested Fare Classes[END_REF]; Lee and Hersh (1993); [START_REF] Gallego | Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons[END_REF]; [START_REF] Robinson | Optimal and Approximate Control Policies for Airline Booking with Sequential Fare Classes[END_REF]; Lautenbacher and Stidam (1999); [START_REF] Feng | Perishable asset revenue management with Markovian time dependent demand intensities[END_REF]; Zhao and Zheng (2000); [START_REF] Kleywegt | An Optimal Control Problem of Dynamic Pricing[END_REF]; [START_REF] You | Airline Seat Management with rejection-for-possible-upgrade decision[END_REF]; [START_REF] Bertsimas | Dynamic pricing and inventory control for multiple products[END_REF] Curry (1990); Bitran and Mondshein (1995); [START_REF] Gallego | A Multiproduct Dynamic Pricing Problem and its Applications to Network Yield Management[END_REF]; [START_REF] Talluri | An aAnalysis of Bid-Price Controls for Network RM[END_REF]; You (1999); [START_REF] Feng | A dynamic airline seat inventory control model and its optimal policy[END_REF]; [START_REF] Popescu | Revenue Management in a Dynamic Network Environment[END_REF]; El-Haber and El-Taha ( 2004)

The way the behavior of customer is incorporated in the optimization process is the next challenge. The following of this part almost borrows to Talluri and Van Ryzin book (2004;p. 651).

A simple formulation of RM with a Dynamic Program

This paper choose a traditional RM problem by considering n > 2 fare classes.

Demand for a single product is then considered discrete as well as capacity (assuming a single resource to simplify). Dynamic models do not assume the traditional hypothesis of early birds. Demand can arrive in a non-strict increasing order of revenue values.

Demand is assumed independent between classes and over time and also independent of the capacity controls. The n classes are indexed by j such that p 1 > p 2 >…> p j >… > p n .

Multiple booking is not considered3 .

Over time, this system evolves as a function of both control decisions and random disturbances according to a system equation. The system generates rewards that are a function of both the state and the control decisions. The objective is to find a Jean Michel Chapuis control policy that maximizes the total expected revenues from the selling period. There are T time-periods. Time, indexed by t, runs reverse so that t = 1 is the last period and t = T is the first period. By the way, t indicates the time remaining before the product is consumed or loses its value. Applying basics of DP to RM gives the following:

• x(t) is, as the state of a system, the remaining capacity constrained between 0 and the full capacity C (both cancellation and overbooking are not considered).

• w(t) is the random disturbance representing the consumer's demand to the firm for product j. The probability of an arrival of class j in the period t is

j t
λ . The horizon is divided into decision periods that are small enough so that no more than one customer arrives during each period, thus 1

1 ≤ ∑ = n j j t λ .
• u(t) is the control decision, assumed discrete and constrained to a finite set that may depend on time t and the current state x(t). u(t) is then the quantity u of demand to accept. In the simplest statement, u is constrained to be either 0 or 1. The amount accepted must be no more than the capacity remaining, so u≤x.

Few remarks must be written down. First, as the manager waits for the demand to realize to make his or her decision -accept or reject an incoming booking requestthe random disturbance is observable. In other words, s/he can build the control action on perfect knowledge of the disturbance4 .

Second, passengers are allowed to arrive in any order (interspersed arrivals), in contrast to static models which assume sequential booking classes or low-before-high fares. Accepting this early-birds hypothesis leads to think in terms of the number of -8 -Jean Michel Chapuis booking asked per period (i.e. the booking limit) and only require the total demand for each class, D j . However, the optimal control would be "static", meaning constant over time (how much to protect or when to reject booking) as the manager never came back on his or her decision. Another approach would have been to reason for only one booking request. The incoming requests are not ordered by price but one needs to model the demand process and then the optimal control changes every period according to the remaining capacity -so "dynamic" means real time adjustment.

• g t (x(t), u(t), w(t)) is a real-valued reward function, specifying the revenue in period t as a function of the parameters. The total revenue is additive and terminal one is assumed to be zero whatever happens; namely that the remaining capacity after the last stage is no value.

The objective is to maximize the total expected revenue for T periods

( )       ∑ = T t t t w t u t x g E 1 ) ( ), ( ), (
by choosing T control decisions u T , …, u 2 , u 1 . This is a

Markovian control because the control depends only on the current state and the current time, and no other information is needed, such as the history of the process up to time.

The collection {u T , …, u 2 , u 1 } is called a policy.

For a given initial state x(T) = x, the expected (total) revenue of a policy is:

( )       = ∑ = T t t t T t w t x u t x g E x V 1 ) ( )), ( ( ), ( ) ( µ .
In other words, this is the total expected revenue that can be generated when there are T decision periods (remaining) and x products to sell. E[] means the expectation is estimated over j, the fare classes or products. The optimal policy, denoted u*, is one that maximizes V T u* (x) and is simply written V T (x). The principle of optimality, due to Bellman (1957), lies at the heart of DP. It says that if a policy is optimal for the original problem, then it must be optimal for any subproblem of this original problem as well. Let say that if {u t *, u t-1 *, …, u 1 *} is not optimal for the tsubproblem and another policy, û, yields a greater expected reward, then the optimality -9 -Jean Michel Chapuis of u* is contradicted. Because the policy {u T *,…, u t+1 *, û t ,…, û 1 } would produce a strictly greater expected revenue than does the policy u*.

Applying this principle leads to use a recursive procedure for finding the optimal policy. The following value function V t (x) is the unique solution to the recursion shown in equation 1.1., for all t and all x.

Once the value of the demand is observed, the value of u is chosen to maximize the current period t revenue plus the revenue to go, or p j .u + V t-1 (x -u) subject to the constraint u ∈ {0,1}. This sentence leads to think in terms of V(x) = E[max{}] instead of V(x) = max{E[]} as in traditional DP, due to the previous assumption about the random disturbance. The value function entering period t, V t (x), is then the expected value of this optimization (namely the maximum) with respect of the demand and is given in the form of the Bellman equation: Equation 1.1.:

{ } ( ) { }       - + = - ∈ u x V u p E x V t j u t 1 . 1 , 0 max ) (
With the boundary V 0 (x) = 0, whatever is x, and V t (0) = 0, whatever is t. This means that the unsold inventory left the last day is useless and a sunk cost.

The motivation of Equation 1.1. uses the Bellman's principle of optimality. Since V t-1 (x -u) is the optimal expected value of the future revenue given the state (x -u) in the next period, t -1, the optimal value of the t-subproblem should be the result of maximizing the sum of the current expected reward E[p j u] and the expected reward for the t -1 subproblem, E[V t-1 (x -u)]. This is the optimal solution for the period t and the process can be restarted until reaching T.

-10 -Jean Michel Chapuis

APPLICATION OF DP TO FIND OUT BOOKING LIMITS AND PRICING

ISSUES.

The RM problem is basically formulated as a marginal one, such that the opportunity cost or expected marginal value (hereafter EMV) becomes the cornerstone of any model. The properties that any control must hold to be optimal are shown in §2.1. Nevertheless, there exists various controls, presented in §2.2, as managers in practice figure out few control types such as a control-limit type or threshold type.

There is also a close correspondence between DP model and Belobaba's EMSR model (see appendix A).

The expected marginal value as the key solution

The values u* that maximize the right-hand side of equation 1.1. for each t and x form an optimal control policy for this model. The solution can be found by evaluating the EMV at period t of the x th unit of capacity.

Equation 1.2.:

) 1 ( ) ( ) ( - - = ∆ x V x V x V t t t
The most important concerns how this marginal value behaves with changes in the capacity left x and the t remaining periods. The expected marginal value of V t (x) satisfies5 ∀x, t :

(i) ∆V t (x+1) ≤ ∆V t (x) Jean Michel Chapuis
EMV is decreasing in x, meaning that the opportunity cost of using one more unit of the capacity increases when there are less products available. In other words, the fewer products on hand the more the chance to loose a sale.

(ii)

∆V t+1 (x)≥ ∆V t (x)
EMV is increasing in time period left or is decreasing as time elapses, meaning that the opportunity cost of using one more unit of the capacity is higher when the time to go is long than when it's short. In other words, the longer the remaining time the more the chance to use the capacity efficiently (the probability of a customer requesting a full fare is high). The same conclusions apply in continuous models 6 .

These two properties are intuitive and greatly simplify the control decisions because the equation 1.1. can be rewritten with equation 1.2. (proof in appendix B).

Equation 1.3.:

{ } { }       - ∆ - + = - ∈ - 0 ); ( max ) ( ) ( 1 1 , 0 1 u x V u p E x V x V t j u t t
The meaning of this equation is the following. The value of the revenue to go today (in t) is equal to the revenue to go tomorrow (that is already optimized using a recursive method) plus the expected maximum revenue of the decision to make (that is the fare price net of the opportunity cost if positive). The problem is formulated as a backward-recursion dynamic program. In other words, an optimal booking policy is reached by the assessment of accepting a booking request relative to the decrease in expected total revenue associated with removing one product from the available inventory.

6 Equation 1.2. becomes

x x V x V j j ∂ ∂ = ∆ ) ( ) ( and (i) 0 ) ( 2 2 < ∂ ∂ x x V t and (ii) 0 ) ( 2 2 > ∂ ∂ t x V t .

Figure 2-1: critical booking capacity

Legend: for a given t, the decision to accept or reject a request with fare j+1 depends on x the level of capacity left. The critical booking capacity separates the accept zone and the reject zone.

The protection level y j * is the number of units to save for customers who request, or will request in the future, class j or higher (i.e. j, j -1, …, 1). It is the maximum quantity x such that the fare level is too low to compensate the opportunity cost, justifying the reject decision.

Equation 2-1: y j * = max {x : p j+1 < ∆V t (x)}, j = 1,…, n-1.

The optimal control at the period t + 1 is Equation 2-2: u*(t+1 ; x ; D j+1 ) = min{(x -y j *) + ; D j+1 }.

j u -∆V t-1 (x -u) C x y* 2 b* 3 p 3 -∆V t (x) p 2 -∆V t (x) p 1 -∆V t (x) y* 1 b* 2 b* 1 Jean Michel Chapuis -u) is decreasing in t.
Thus, it is optimal to keep refusing incoming requests (not increase u in a static formulation) until the previous term becomes positive.

Figure 2-3: critical time threshold

Legend: For a given booking capacity x, a request for a seat of fare class j in decision period t is accepted if time remaining is short (less than Y j *, the critical time threshold) and rejected otherwise. The caps denote that the control is expressed on a time axis instead of a quantity axis.

This proposition is also easy to understand with the figure 2-1. Since ∆V t (x) is increasing in t, one should accept more booking for a given j when remaining time lessens. In other words, as time elapses, the EMV and the line lifts up. Then the optimal protection level y* j will shift to the left (decrease

Bid prices

Finally, the optimal control can also be implemented through a table of bid prices, defined as π j+1 (x) = ∆V t (x). A bid price is the minimum amount of money to in exchange of a unit of capacity. Under mild assumptions [START_REF] Gallego | Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons[END_REF][START_REF] Talluri | An aAnalysis of Bid-Price Controls for Network RM[END_REF], the optimal price path π j (x) is decreasing in

x and increasing in t. Dynamic pricing is becoming a real challenge for airlines since low cost carriers are introducing less restricted fares. For example, Ryanair or Easyjet are not able to be sure that the more price sensitive passengers (low fare) book first.

Then they apply a dynamic pricing strategy for a single product without price discrimination between their customers.

DISCUSSION AND CONCLUSION

The use of Dynamic Programming in Revenue Management helps to decide whether to accept or reject an incoming booking reservation with more realism than older methods. There are two main points. One proposition of DP in RM is to relax the low-before-high fare order of arrival bookings. In practice, the DP provides the optimal policy for the RM problem, by evaluating the whole tree of possibilities and making at each point in time the decision that would imply higher future expected revenues, processing backward recursion. The dark side is the increase in the computation difficulties according to the dimension of the problem. This means that for a single product with 100 units to sell over a 200 time periods, the number of iterations is 100 x 200 = 20 000. But for three products using the same resource, this number becomes 100

x 200 = 800 millions. [START_REF] Saranathan | Revenue Management in the New Management of the New Fare Environment[END_REF] had related the implementation of DP in United Airlines and that the company expects a 1 to 2 % increase in revenue ($

P

Table 1 : classification of literature Product

 1

Single

Multiple

[START_REF] Curry R | Optimal Seat Allocation with Fare Classes Nested by Origins and Destinations[END_REF]

;

[START_REF] Wollmer R | An Airline Seat Management Model for a Single Leg Route when Lower Fare Classes Book First[END_REF]

;

[START_REF] Brumelle | Airline Seat Allocation with Multiple Nested Fare Classes[END_REF]

;

[START_REF] Hersh M | A model for dynamic seat inventory control with multiple seat bookings[END_REF]

;

[START_REF] Gallego | Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons[END_REF]

;

Bitran and Mondshein (1995)

;

[START_REF] Robinson | Optimal and Approximate Control Policies for Airline Booking with Sequential Fare Classes[END_REF]

;

Lautenbacher and Stidam (1999)

;

Zhao and Zheng (2000)

;

[START_REF] You | Airline Seat Management with rejection-for-possible-upgrade decision[END_REF]

Gallego and Van Ryzin (1997);

[START_REF] Talluri | An aAnalysis of Bid-Price Controls for Network RM[END_REF]

;

[START_REF] Feng | A dynamic airline seat inventory control model and its optimal policy[END_REF]

;

[START_REF] Kleywegt | An Optimal Control Problem of Dynamic Pricing[END_REF]

;

[START_REF] Popescu | Revenue Management in a Dynamic Network Environment[END_REF]

; El-Haber and El-Taha (

2004

); Bertsimas and De Boer (2005) Policy Static Dynamic Both Curry (1990); Wollmer (1992);

  The booking limit b* j+1 is the quantity is excess of the protection level (x -y j *) if positive or zero otherwise. This is the maximum space available for class j + 1 booking request. Those booking limits are nested, meaning that a class j request can be withdrawn in any of the j class and above (to n). ∆V t (x) Jean Michel Chapuis capacity threshold b* j is reached or the time ends, whichever come first. For an illustration, consider the case where n = 3 and p 1 > p 2 > p 3 . and the figure 2.2. Because it's too late to expect high fare requests to come (low opportunity cost), in such a quantity to fulfill the capacity.

	p j u -∆V t-1 (x -u)
	p j+1 -Figure 2-2 : Case with 3 fare classes
		x
		C
	Reject zone	Accept zone
	y j *	b* j+1
	2.2.2 The control by time thresholds
	Second, consider controlling the revenue through t, the time left, to accept (u =
	1) or reject (u = 0) an incoming request. Based on previous properties of equation 1.2.,
	one can show that the optimal policy is characterized by time thresholds. During the
	booking horizon, they are points in time before which requests are rejected 7 and after
	which requests are accepted 8 . Since ∆V t (x) is increasing in t, it follows that p j u -∆V t-1 (x
		By the way, this set of protected
	classes (j, j -1, …, 1) are proposed or "open" to customers being denied a request fare 7 Because one expects to receive booking orders from clients willing to pay high fare (high
	class below j. The policy then is "simply accept requests first come, first serve until the opportunity cost), justifying rejection.

8

In the previous example, one can easily see that the first come first serve rule is not optimal for the hotel. The fence of "stay at least two nights", that can justify the discount rate and help to increase revenue, is not totally efficient. Only a booking limit of one "Two nights" package or a bid price for each night are optimal.

Usually, researchers consider that the time can be divided in small enough intervals such that only one booking arrives.[START_REF] You | Airline Seat Management with rejection-for-possible-upgrade decision[END_REF] develops a model allowing for multiple booking.

This assumption allows to greatly simplify both the mathematical formulation and the optimal control policies because the optimization problem could be spread in multiple subperiods optimization problems that are independents between them. SeeTalluri and Van Ryzin (2004, p. 654) to understand the full consequence of this assumption.

There are different proofs of these statements. However, theTalluri and Van Ryzin (2004, p. 38) proof is one of the clever. Hence, the seat allocation problem and the dynamic pricing problem satisfy well-known sufficient conditions for an optimal policy to be monotonic. These problems translate to the existence of time-dependent controls. In other words, the optimality value function is concave and nonincreasing, from which it follows that an optimal admission policy is monotonic in the state.

Jean Michel Chapuis

Types of controls

Three types of controls can be derived, studying the sign of value of the max function. Needless to say that relaxing the hypothesis of early birds implies that the protection levels, the booking limits and the bid prices are time-dependent because the random disturbance associated with the value function depends on the probability of an arrival of class j in the period t. In order to an efficient optimization, the manager must know the process of demand for each fare class over time (i.e. the booking curve).

Protect levels and nested booking limits

First consider controlling the revenue through u, the accept (u = 1) or reject (u = 0) decision of the current demand. Since ∆V t (x) is decreasing in x, it follows that p j u -∆V t (x -u) is increasing in x. For a given t, it is optimal to keep accepting incoming requests until the previous term becomes negative or the upper bound min{D j ; x} is reached, whichever comes first. Jean Michel Chapuis

The second interest of DP approach, and the main avenue for future research, is that it allows RM to incorporate the consumer choice in the optimization process. El-Haber and El-Taha ( 2004) formulate a dynamic programming model to solve the seat inventory control problem for a two-leg airline with realistic elements of consumer behavior. Ahead of the Origin and Destination formulation, they consider cancellation, no-shows and overbooking. Following Talluri andVan Ryzin (2003) work, Van Ryzin and[START_REF] Van Ryzin | Computing Virtual Nesting Controls for Network Revenue Management Under Customer Choice Behavior[END_REF] consider a revenue management, network capacity control problem in a setting where heterogeneous customers choose among the various products offered by a firm (e.g., different fight times, fare classes and/or routings). Customers may therefore substitute if their preferred products are not offered, even buy up. Their choice model is very general, simply specifying the probability of purchase for each fare product as a function of the set of fare products offered. Overall, the value of our paper is to facilitate the understanding of more complex, and probably more realistic, models of Revenue Management.

-20 -Jean Michel Chapuis

APPENDICES

Correspondence between DP and EMSR models

The recommendation about booking acceptance of DP models is similar to the one of EMSR models. For example, suppose there is one seat left to sell, and a potential customer has just called and asked to make a reservation for a discount fare. The flight will take off tomorrow. Thus T = 1, n = 2 (j = 1 -full fare -to 2 -discount fare -), and

is the probability of an arrival of class 1 in the period 1 before departure.

Since the potential discount customer hold the line,

∑

Then, the manager has to reject while ) 0 ( ) 1 (

. In other words, the policy is close fare class j = 1 when

or when the probability of spoilage is lower than the discount rate offered to customers :

. This is the EMSR rule assuming there is only one seat to sell next period. Jean Michel Chapuis

Appendix B

The last term depends neither on the probability of an arrival in period t or the decision u made in t. In other words, this means that the optimization of the revenue from t-1 to 1 depends on neither what have been done before nor what happens today.

Moreover, this is the true reason why the model is built assuming the demand is known in t before the decision to make. This term is then equal to V t-1 (x) and can be written out of the E[] and the max functions.