ELECTRONIC COMMUNICATIONS in PROBABILITY FURTHER EXPONENTIAL GENERALIZATION OF PITMAN’S 2M-X THEOREM

We present a class of processes which enjoy an exponential analogue of Pitman's $2M-X$ theorem, improving hence some works of H. Matsumoto and M. Yor.

In this work, we show that theorem 1 can be extended to a large class of processes which are constructed from the geometric Brownian motion by a simple transformation and which satisfy a generalized Dufresne identity (originally discussed in [5] and [10]).This paper is essentially self-contained and contains a new proof of theorem 1.
2 Diffusions for which (X −1 t t 0 X 2 s ds) t≥0 is also a diffusion Let (Ω, (F t ) t≥0 , P) a filtered probability space on which a standard Brownian motion (B t ) t≥0 and Z a positive F ∞ -measurable variable independent of the process (B t ) t≥0 are defined.
In [6], C. Donati-Martin, H. Matsumoto and M. Yor introduced the following transforms from In their paper these authors have shown that if the law of Z is equivalent to the Lebesgue measure, then the laws of the processes T Z B (µ) s , s ≤ t, and In what follows we study T Z B (µ) in its own filtration and we characterize the variables Z such that T Z B (µ) is a diffusion.As a consequence of this characterization, we will be able to give a new proof of Matsumoto-Yor's theorem 1.More precisely, in all this paper, we deal with the process (X t ) t≥0 defined as follows Theorem 3 Assume that the law of Z admits a bounded C 2 density with respect to the law Then (X t ) t≥0 is a diffusion in its own filtration if and only if there exists δ ≥ 0 such that: Moreover, in this case there exists a standard Brownian motion (β t ) t≥0 adapted to the natural filtration of (X t ) t≥0 such that s ds with infinitesimal generator Moreover, under the assumption (2.1) , for t > 0 and x, z > 0 where (X t ) t≥0 is the natural filtration of (X t ) t≥0 .
In the following remarks, we assume that (2.1) is satisfied for some δ ≥ 0.

The law (2.1) is called a generalized inverse Gaussian distribution. These laws have been widely discussed by Barndorff-Nielsen [1], Letac-Wesolowski
3. We get the theorem 1.1.by taking δ = 0, because in this case 4. The stochastic differential equation (2.2) solved by (X t ) t≥0 , enjoys the pathwise uniqueness property.Indeed, a simple computation shows that a scale function is given by where I µ is the classical modified Bessel function with index µ.Using Wronskian relation for Bessel functions, we get then the speed measure And finally, Feller's test for explosions (see for example [17] pp.386) gives where e is the lifetime of a solution.
5. If we apply Itô's formula conditionally to Z, we see that in the natural filtration of (X t ) t≥0 enlarged by +∞ 0 X 2 s ds the process (X t ) t≥0 is a semimartingale whose decomposition is 2) becomes dX t = −δX 2 t dt + X t dβ t and this equation is solved by We have then the following surprising identity in law for which it would be interesting to have a direct proof.
For µ = 1 2 , it seems difficult to solve explicitly the equation (2.2) , even in the case µ = n + 1 2 , with n ∈ N. Let us just mention that [11], [15], [16], and [20] the BES (µ, δ ↓) process, i.e. the diffusion process with the infinitesimal generator 1 2 From [6] (see also [3]), it implies then that there exists a Bessel process Q with index µ, starting at x 0 and independent of the first hitting T 0 of 0 for ρ such that

Proof of proposition 2
This proof is very simple and inspired from [6].As we have Let now y > 0 and denote Q y the law of the process (X t ) t≥0 conditioned with Y = y.From theorem 1.5. of [6], we see that the following absolute continuity relation takes place where (χ t ) t≥0 is the coordinate process, (Ξ t ) t≥0 its natural filtration, and P −µ the law of the process (x 0 exp (−µt + B t )) t≥0 .By integrating this absolute continuity relation with respect to the law of Y, i.e.
where ξ is the bounded density of Y with respect to 2 x 2 0 γµ , we deduce after some elementary computation that the law Q of our process (X t ) t≥0 satisfies the following equivalence relation Hence, by Girsanov theorem, if X is a diffusion in its own filtration, then the infinitesimal generator of this diffusion can be written where b : R * + → R is related to From (3.4) the homogeneity of b implies that there exist two functions f and g such that but ϕ is a solution of the following partial differential equation it immediately implies that there exists a constant C such that We note now that the constant C is negative, because ξ is bounded and we have the limit condition lim Denote the constant C by − δ 2 2 with δ ∈ R + and conclude that x 1+µ dx , x > 0 It proves the first part of our theorem.On the other hand, if (2.1) is satisfied the following equivalence relation takes place (it suffices to use the equivalence relation (3.3)) which implies, by Girsanov theorem where (β t ) t≥0 is a P standard Brownian motion adapted to the natural filtration of (X t ) t≥0 .
Let us consider a sequence (Z n ) n>0 of random variables independent of (B t ) t≥0 and such that We associate with this sequence (Z n ) n>0 the sequence of processes defined by We have 1 But, one can show that On the other hand, from theorem 3, , t ≥ 0 is a diffusion with infinitesimal generator Now, to conclude the proof of the first part of the corollary, we use proposition 2 which asserts that The computation of the conditional probabilities is easily deduced from (3.6) and proposition 6 of [9], by a change of probability, namely the absolute continuity (3.6).

Proof of proposition 6
A simple computation shows that for t ≥ 0 where D is the Malliavin's differential and (χ t ) t≥0 the coordinate process on the Wiener space.

2
+∞ 0 e 2χs−2µs ds} implies then that for all t ≥ 0 As noticed in the remarks of section 2, in the enlarged filtration X t ∨ σ +∞ 0 X 2 s ds t≥0 , our process (X t ) t≥0 is a semimartingale whose decomposition is By projecting this decomposition on (X t ) t≥0 , using the classical filtering formulas, we get

Opening
To conclude the paper, let us relate our result to another simple transformation of the Brownian motion (first considered by T. Jeulin and M. Yor and then generalized by P.A. Meyer, see [14]).
It is easily seen that for a standard Brownian motion (B t ) 0≤t<1 , the process is well defined and is a Brownian motion in its own filtration.Now, let us consider a random variable Z independent of (B s ) 0≤s<1 .Consider the process We have the following proposition (for further details on it, we refer to [2]).

Proposition 8
The process is well defined and is a Brownian motion in its own filtration which is independent of X 1 .Moreover, (X t ) 0≤t<1 is a (homogeneous) diffusion in its own filtration if and only if, there exist α, β ∈ R such that where C > 0 is a normalization constant.In this case dX t = α tanh (αX t + β) dt + dW t , 0 ≤ t < 1 where (W t ) 0≤t<1 is a standard Brownian motion adapted to the natural filtration of (X t ) 0≤t<1 .
Hence, a somewhat vague question arise: Are there other natural transformations of the Brownian motion for which the same phenonemon of loss of information takes place ?