Bridges with Random Length: Gamma Case

In this paper, we generalize the concept of gamma bridge in the sense that the length will be random, that is, the time to reach the given level is random. The main objective of this paper is to show that certain basic properties of gamma bridges with deterministic length stay true also for gamma bridges with random length. We show that the gamma bridge with random length is a pure jump process and that its jumping times are countable and dense in the random interval bounded by 0 and the random length. Moreover, we prove that this process is a Markov process with respect to its completed natural filtration as well as with respect to the usual augmentation of this filtration, which leads us to conclude that its completed natural filtration is right continuous. Finally, we give its canonical decomposition with respect to the usual augmentation of its natural filtration.


Introduction
The gamma process has proven very successful when modelling accumulation processes.Early studies by e.g.Hammersley [13], Moran [22], Gani [12], Kendall [19], Kingman [20] addressed the modelling of water stored in and released from reservoirs and accumulation related to storage in general.Dufresne et al. [8] show how to employ the gamma process to model liabilities of insurance portfolios for continuous claims.For these risk models the fixed budget horizon of one year is assumed.The authors investigate furthermore the gamma process in the setting of ruin theory and supply ruin probabilities in form of tables.The gamma process replaces the compound processes used traditionally.In Eméry and Yor [11] and Yor [28] gamma bridges were studied and their application to stop loss reinsurance and credit risk management were pointed out in Brody et al [7].This work introduces and focusses on random gamma bridges, which model accumulated losses of large credit portfolios in credit risk management.These studies were continued by Hoyle et al. in a series of papers, see e.g.Hoyle and Mengütürk [15].In a further article pricing at an intermediate time is studied Hoyle et al [14].Returning to the starting point accumulation processes for storage, we refer to recent developments in Chan et al [9] for further references.Numerical results may be found in Assmusen and Hobolth [2].In this paper we generalize the concept of a gamma bridge to random times, at which the bridge is pinned, to study amongst others its Markov property and to give its decomposition semimartingale.There are two recent works in which bridges with random length are studied.The first by Bedini et al [4] studies related properties of the Brownian bridge with random length, the second by Erraoui and Louriki [10] studies Gaussian bridges with random length.In both works existence of an explicit expression for the bridge with random length is exploited.Applications for the random gamma bridge suggest itself for accumulation processes in financial mathematics, see the results of Jeanblanc and Le Cam in [16] and [17], if heavier tails are desired, electricity production in river power plants with implications to electricity prices, life insurance portfolios, where the instant of death is random, accumulation storage in the setting of just in time production.The paper is organized as follows.Section 2 begins by recalling the definitions and some properties of gamma processes and gamma bridges of deterministic length, which will be used throughout the paper.In Section 3 we define the gamma bridge with random time τ which will be denoted by ζ and we consider the stopping time property of τ with respect to the right continuous and completed filtration F ζ,c + generated by the process ζ.Moreover, we give the conditional distribution of τ and ζ u given ζ t for u > t > 0. Next we establish the Markov property of the process ζ with respect to its completed natural filtration.As a consequence, we derive Bayesian estimates for the distribution of the default time τ , given the past behaviour of the process ζ up to time t.After that we study the Markov property of the gamma bridge with random length, with respect to F ζ,c + .Finally we give its semimartingale decomposition with respect to F ζ,c + .The following notation will be used throughout the paper: For a complete probability space (Ω, F , P), N p denotes the collection of P-null sets.If θ is a random variable, then P θ denotes the law of θ under P. D denotes the space of right continuous functions with left limits (càdlàg) from R + to R + , endowed with Skorohod's topology, under which the space D is a Pollish space.If E is a topological space, then the Borel σ-algebra over E will be denoted by B(E).The characteristic function of a set A is written I A .The symmetric difference of two sets A and B is denoted by A∆B.Finally for any process Y = (Y t , t ≥ 0) on (Ω, F , P), we define by: ≥ 0 the smallest filtration containing F Y and satisfying the usual hypotheses of right-continuity and completeness.

Gamma and Gamma bridge processes
The purpose of this section is to recall the definition and some properties of the standard gamma process and the gamma bridge with deterministic length.

Gamma process
By a standard gamma process (γ t , t ≥ 0) on (Ω, F , P), we mean a subordinator without drift having the Lévy-Khintchine representation given by where ν(dx) = exp(−x) x I (0,∞) (x) dx is the so-called Lévy measure.We note that the formula (2.2) is obtained from (2.1) using the Frullani formula.
The following properties, inferred from (2.2) by means of standard arguments (see, e.g., Sato [25], ch. 2 and 4), describe the paths of the gamma process.
Proposition 2.1.The gamma process (γ t , t ≥ 0) has the following properties: (i) γ is a purely jump process; (ii) γ is not a compound Poisson process and its jumping times are countable and dense in [0, ∞) a.s.; (iii) the map t → γ t is strictly increasing and not continuous anywhere a.s.; (iv) γ has sample paths of finite variation a.s.; (v) γ t , t > 0, follows a gamma distribution with density where Γ is the gamma function.
The second property means that, for any t > 0, γ has infinite activity, that is, almost all paths have infinitely many jumps along any time interval of finite length.It is a direct consequence of ν(R + ) = +∞, whereas the fourth property arises from x ν(dx) < +∞.
1.It is clear that the process gamma (γ t , t ≥ 0) is a process with paths in D.
2. The process (γ t − γ t − := e t , t ≥ 0) of jumps of the gamma process (γ t , t ≥ 0) is a Poisson point process whose intensity measure is the Lévy measure of (γ t , t ≥ 0), see Bertoin [3].For r > 0, let us denote by (J r 1 ≥ J r 2 ≥ . ..) the sequel of the lengths of jumps of the process (γ t , t ∈ [0, r]) ranked in decreasing order.It is not difficult to see that since the intensity measure of the Poisson point process ((t, e t ), t ≥ 0)) is dt exp(−x) x I (0,∞) (x) dx, then the jump times (U r 1 , U r 2 , . ..) constitute a sequence of i.i.d r.v.'s with uniform law on [0, r] which is independent from the sequence (J r k , k ≥ 1).Thus we have the following representation: We note that : The next proposition gives three other useful properties of of the gamma process.
(ii) For any r > 0, (γ t , 0 ≤ t ≤ r) satisfies the following equation where (iii) (γ t , t ≥ 0) has the Markov property with respect to its natural filtration.
For a deeper investigation on the properties of the gamma process we refer to Kyprianou [21], Sato [25] and Yor [28].

Gamma bridge with deterministic length
A bridge is a stochastic process that is pinned to some fixed point at a fixed future time.In this section we define the gamma bridge with deterministic length and we give some important properties of this process.For fixed r > 0, we define the gamma bridge of length r by setting Definition 2.4.Let r ∈ (0, +∞).The map ζ r : Ω −→ D defined by is the bridge associated with the standard gamma process (γ t , t ≥ 0).Then clearly ζ r 0 = 0 and ζ r r = 1.We refer to ζ r as the standard gamma bridge of length r associated with γ.We note that ζ r is also called the Dirichlet process with parameter r.
We note that the process ζ r is really a function of the variables (r, t, ω) and for technical reasons, it is convenient to have certain joint measurability properties.
Proof.Since the map (r, t) −→ t ∧ r is Lipschitz continuous and t → γ t is càdlàg for all almost ω ∈ Ω, then the map (r, t, ω) −→ ζ r t (ω) can be obtained as the pointwise limit of sequences of measurable functions.So, it is sufficient to use standard results on the passage to the limit of sequences of measurable functions.
As a consequence we have the following corollary.
A number of properties of the gamma bridge ζ r sample paths can be easily deduced from the corresponding properties of the gamma sample paths.Hence, we have Proposition 2.7.The gamma bridge ζ r t , t ≥ 0, has the following properties: (i) ζ r is a purely jump process and its jumping times are countable and dense in [0, r] a.s.; (ii) the map t → ζ r t is strictly increasing and not continuous anywhere in [0, r] a.s.; (iii) ζ r has sample paths of finite variation in [0, +∞) a.s.; (iv) ζ r has the following representation: We now turn to distributional properties of the gamma bridge.
Proposition 2.8.(i) For all 0 < t < r, the random variable ζ r t has a beta distribution β(t, r −t) i.e. its density function is given by with respect to the Lebesgue measure dx 1 . . .dx n−1 (or, as well, dx 2 . . .dx n ) on the simplex (iii) For all t < u < r and x ∈ (0, 1), the regular conditional law of ζ r u given ζ r t = x is given by: In the same spirit as in the Proposition 2.3 we have Proposition 2.9.(i) ζ r is a Markov process with respect to its natural filtration.
(ii) ζ r satisfies the following equation where 3 in Blumenthal and Getoor [5] it suffices to prove that for every bounded measurable function g we have: for all 0 ≤ t 1 < . . .< t n < u ≤ r and for all n ≥ 1.Using Proposition 2.3 (i) we have Hence the formula (2.11) is proved, then ζ r is a Markov process with respect to its natural filtration.
(ii) We have from Proposition 2.3 (ii) that where M r is a martingale with respect to the filtration Then it is easy to see that where t -measurable for all t ≤ r.Moreover, equation (2.12) yields that is the process N r is F ζ r -adapted.In view of these considerations, as well as the fact that M r t is a G (r) t -martingale we obtain Hence the equation (2.10) is satisfied.
Remark 2.10.We can rewrite (2.5) in the form Then we obtain For every t ≥ 0, we set N r t = M r t∧r γ r .We have thus It follows from the above proposition that ( N r t , t ≥ 0) is a F ζ r -martingale stopped at r.

Gamma bridges with random length
In this section we define and study a process (ζ t , t ≥ 0) which generalizes the gamma bridge in the sense that the time r at which the bridge is pinned is substituted by an independent random time τ .We call it gamma bridge with random length.We prove that the random time τ is a stopping time with respect to the completed filtration F Since ζ is obtained by composition of two maps (r, t, ω) −→ ζ r t (ω) and (t, ω) −→ (τ (ω), t, ω), it's not hard to verify that the map ζ : Ω, F ) −→ (D, B (D) is measurable.The process ζ will be called gamma bridge of random length τ .
As mentioned above, we work under the following standing assumption: Assumption 3.1.The random time τ and the gamma process γ are independent.
Using the fact that the process ζ is obtained by the substitution of r in ζ r by the random time τ allows us to derive a lot of information about its path properties.Hence, we have Proposition 3.2.The gamma bridge ζ t , t ≥ 0, has the following properties: (i) ζ is a purely jump process and its jumping times are countable and dense in [0, τ ] a.s.; (ii) the map t → ζ t is increasing and not continuous anywhere on [0, τ ] a.s.; (iii) ζ has sample paths of finite variation a.s.
(iv) ζ has the following representation: where the jump times (U τ 1 , U τ 2 , . ..) constitute a sequence of r.v.'s identically distributed with the law given by

Stopping time property of τ
The aim of this subsection is to prove that the random time τ is a stopping time with respect to F ζ,c .Proposition 3.3.For all t > 0, we have P ({ζ t = 1} △ {τ ≤ t}) = 0. Then τ is a stopping time with respect to F ζ,c and consequently it is a stopping time with respect to F ζ,c + .
Proof.First we have from the definition of ζ that ζ t = 1 for τ ≤ t.Then {τ ≤ t} ⊆ {ζ t = 1}.On the other hand, using the formula of total probability we obtain The latter equality uses the fact that ζ r t is a random variable has a beta distribution for 0 < t < r.Thus P ({ζ t = 1} △ {τ ≤ t}) = 0.It follows that the event {τ ≤ t} belongs to F ζ t ∨ N P , for all t ≥ 0. Hence τ is a stopping time with respect to F ζ,c and consequently it is also a stopping time with respect to F ζ,c + .In order to determine the conditional law of the random time τ given ζ t we will use the following Proposition 3.4.Let t > 0 such that F (t) > 0. Let g : R + −→ R be a Borel function satisfying E[|g(τ )|] < +∞.Then, P-a.s., we have where the function φ ζ r t is defined on R by: Proof.Let us consider the measure µ defined on B(R) by where δ 1 (dx) and dx are the Dirac measure and the Lebesgue measure on B(R) respectively.Then for any B ∈ B(R) we have where the function q t is a nonnegative and measurable in the two variables jointly given by q t (r, x) = I {x=1} I {r≤t} + ϕ ζ r t (x)I {0<x<1} I {t<r} .It follows from Bayes formula (see [26] p. 272) that P-a.s.: Using (3.7), for t < u < r, we get It follows from (3.5), that P-a.s.
This induces that Hence the formula (3.8) is proved and then the proof of the proposition is completed.

Markov property of ζ and Bayes estimate of τ
In this part we prove that the gamma bridge with random length ζ is an inhomogeneous Markov process with respect to its completed natural filtration F ζ,c .
Theorem 3.7.The process (ζ t , t ≥ 0) is an F ζ -Markov process.That is, for any t ≥ 0, we have for all t, h ≥ 0 and for every bounded measurable function f .
Proof.First, we would like to mention that since ζ 0 = 0 almost surely then it is easy to see that So it remains to show that To do this it is enough to verify that for all A ∈ F ζ t .We start by remarking that, for t > 0, F ζ t is generated by Using Proposition 2.3 (i), then for t < r the vectors (β 1 , . . ., β n ) and (ζ r t , ζ r t+h ) are independent.Now taking into account all the above considerations, we have Hence (3.11) is proved and this ends the proof.
Corollary 3.8.The Markov property can be extended to the completed filtration F ζ,c .
The aim of this proposition is to provide, using the Markov property, that the observation of ζ t would be sufficient to give estimates of the time τ based on the observation of the information process ζ up to time t.Proposition 3.9.Let 0 < t < u.
Proof.(i) Obviously, we have t -measurable then, P-a.s, one has On the other hand due to the facts that g(τ ∨ t) ζ is a Markov process with respect to its completed natural filtration we obtain P-a.s.
The result is deduced from (3.2).
(ii) The equation (3.13) is an immediate consequence of (i).Concerning the equation (3.14) we use the same method which we used to prove (i).
Remark 3.10.The process ζ cannot be an homogeneous F ζ z -Markov process.Indeed, Proposition 3.9 enables us to see that, for A ∈ B(R) and t < u, we have P-a.s., which is clear that it doesn't depend only on u − t.
First let us remark that the function defined on {(t, r) ∈ (0, +∞) 2 , t < r} × (0, 1) is continuous.Using the facts that ζ tn is decreasing to ζ t and P [ζ t = 0] = 0 then, P-a.s on {t < τ }, we have On the other hand since the function x −→ (1 − x) r is decreasing on (0, 1) for all r > 0 and for large enough r, see [1], p. 257, 6.1.46,then for any compact subset K of (0, +∞) × (0, 1) it yields sup Hence, P-a.s on {t < τ }, we have We conclude assertion (3.21) from the Lebesgue dominated convergence theorem.Now let us prove (3.22).Recall that the function K tn,u (r, ζ tn ) is given by Next, we investigate the second part of the proof, that is the case t = 0.It will be carried out in two steps.In the first one we assume that there exists ε > 0 such that On the other hand we have Then in order to show (3.27) it is sufficient to prove, P-a.s, the following First, for r > ε, we have Since the gamma function is increasing on [2, ∞) then, for r ≥ 2 + t 1 , we obtain is P τ -integrable on (ε, +∞).
Hence (3.28) follows from a simple application of the Lebesgue dominated convergence theorem.
In the same way as in the first case (t > 0) we obtain from the weak convergence that then also (3.29) follows from a simple application of the Lebesgue dominated convergence theorem.Finally, we have to consider the general case, that is P(τ > 0) = 1.In order to prove the Markov property of ζ with respect to F ζ,c + at t = 0 it is sufficient to show that F ζ,c 0+ is P-trivial.This amounts to prove that To do so, let ε > 0 be fixed and consider the stopping time τ ε = τ ∨ ε.We define the process ζ τε t by The first remark is that the sets (τ ε > ε) = (τ > ε) are equal and therefore the following equality of processes holds As P(τ ε > ε/2) = 1 then according to the previous case we have that F ζ τε 0+ is P-trivial.That is P(B) = 0 or 1. Consequently we obtain Now if P(A) > 0, then there exists ε > 0 such that P(A ∩ {τ > ε}) > 0. Therefore for all 0 < ε ′ ≤ ε we have Passing to the limit as ε ′ goes to 0 yields P(A ∩ (τ > 0)) = P(τ > 0) = 1.It follows that P(A) = 1, which ends the proof.
Corollary 3.12.The filtration F ζ,c satisfies the usual conditions of rightcontinuity and completeness.

Semimartingale Decomposition of ζ
Our purpose is to derive the semimartingale property of ζ with respect to its own filtration F ζ,c .Firstly, we obtain from the representation (2.15) that where the processes N and Z are defined as follows:  (ii) By assertion (i), the process (Z t , t ≥ 0) is integrable with respect to the Lebesgue measure, hence N is well-defined.It is clear that the process N is H-adapted and N t = N τ , P-a.s, on the set {t ≥ τ }.Now since ( N r t , t ≥ 0) is a F ζ r -martingale stopped at r we obtain, for any 0 < t 1 < t 2 < ... < t n = t, n ∈ N * , h ≥ 0 and g a bounded Borel function, that The desired result follows by a standard monotone class argument.This completes the proof.
Therefore, it follows from Stricker's Theorem [27] that ζ is a semimartingale relative to its natural filtration F ζ,c .A natural question is: what is the explicit form of its canonical decomposition?That is the problem we want to discuss.The method consists in applying the stochastic fltering theory.
ζ,c and we give the regular conditional distribution of τ and (τ, ζ .) given ζ . .Moreover, we prove that the gamma bridge with random length ζ is an inhomogeneous Markov process with respect to its completed natural filtration F ζ,c as well as with respect to F ζ,c + .The last property allows us to deduce an interesting consequence that is the filtration F ζ,c satisfies the usual conditions of completeness and right-continuity.Finally we give the semimartingale decomposition of ζ with respect to F ζ,c + .Now we give precise definition of the process (ζ t , t ≥ 0).Due to Corollary 2.6 we could substitute r by a random time τ in (2.6).Thus we obtain Definition 3.1.Let τ : (Ω, F , P) −→ (0, +∞) be a strictly positive random time, with distribution function F (t) := P(τ ≤ t), t ≥ 0. The map ζ : Ω, F ) −→ (D, B (D) is defined by

3. 3
Markov property with respect to F ζ,c + We have established, in the previous section, the Markov property of ζ with respect to its completed natural filtration F ζ,c .In this section we are interested in the the Markov property of ζ with respect to F ζ,c + .It has an interesting consequence which is none other than the filtration F ζ,c satisfies the usual conditions of completeness and right-continuity.However, we need the following condition of on the integrability of τ .Since lim n−→+∞ I {tn<τ } = I {t<τ } then assertion (3.
is a H t -martingale stopped at τ .Proof.(i) We first note that Z is a nonnegative process.Since, for s ≤ r, ζ r s has a beta distribution β(s, r − s) then E (ζ r s ) = s/r.So, we can see, for any t ≥ 0, that .32) which is equal to the initial enlargement of the filtration F ζ,c by the σ-algebra σ(τ ).Since the processes ζ and Z are H-adaped it follows from equation (3.31) that N is H-adapted.Moreover, τ is a stopping time with respect to H.The next proposition will play a very important role in forthcoming developments, since it shows the semimartingale property of ζ with respect to H t .Proposition 3.13.(i)Wehave E t 0 |Z s | ds < +∞, ∀t ≥ 0.(ii) The process N = ( N t , t ≥ 0) defined byN t = ζ t − E ( N t+h − N t )g(ζ t 1 , . . ., ζ tn , τ ) = E[( N r t+h − N r t )g(ζ r t 1 , . . ., ζ r tn , r)]P τ (dr) = 0.