ISSN: 1083-589X ELECTRONIC COMMUNICATIONS in PROBABILITY Illustration of various methods for solving partly Skorokhod’s embedding problem ∗

We show that excursion theory and Az\'ema's exponential result allow to solve partly Skorokhod's embedding problem.


Some particular Brownian stopping times
Throughout the paper, (B t ) denotes one-dimensional standard Brownian motion, and (L t ) is its local time.In the sequel, we look at some variant of the Azéma-Yor algorithm [2] for solving Skorokhod's embedding problem with the help of stopping times depending only on Brownian motion and its supremum.

1.1
In this paper, we wish to identity the law of B θ F , for the stopping time: The function F : R + × R + −→ R + is a continuous function with the following property: denoting F (σ, x) ≡ F σ (x), we assume that F σ is strictly increasing from 0 to ∞, and we denote by F −1 σ (•) the inverse of F σ : Thus, we may rewrite: where: .
where e is a standard exponential variable, H(x) = x 0 dy h(y).
As an illustration, we note that for h 1 (l) = l, It would be interesting to know which class of distributions is obtained from the RHS of (1.1).In fact, if h H −1 (u) = ϕ(u) is Lipschitz, then H is the only solution of the ordinary differential equation H (t) = ϕ(H(t)); H(0) = 0. Thus, the family of laws obtained from (1.1) is quite rich; for example take for h a positive power of l.

1.2
The remainder of this paper consists in three sections: • in Section 2, we use Azéma exponential result to obtain (1.1); • in Section 3, we use an excursion argument for the same purpose; • in Section 4, we mention two points to be looked at carefully; • in Section 5, we sketch how the previous arguments allow to recover the Azéma-Yor algorithm for solving Skorokhod's embedding problem.
Then, under the hypothesis (CA): all martingales are continuous, and L avoids all (F t ) stopping times T , i.e.: P (L = T ) = 0, the variable A L ∞ is a standard exponential variable with mean 1.

2.2
We compute A L (h) Proof.We use the balayage formula (see, e.g.[3], Chapter VI) to assert that, for any bounded predictable process (K s ), one has: where g t = sup{s < t : B s = 0}.In fact, we shall use the following variant: Illustration of various methods for solving partly Skorokhod's embedding problem Thus, applying the optional stopping theorem, we get: which yields the desired result.

2.3
As a consequence of the definition of θ (h) , we get from Proposition 2.2, we deduce: which proves, together with (2.1) that (1.1) is satisfied.

3.1
Call (τ l , l ≥ 0) the inverse local time.The excursion theory argument runs in the following equalities between the random sets: From excursion theory, we now deduce: hence (1.1) holds.
Proof.By excursion theory, the process is an inhomogeneous Poisson process, whose intensity measure may be expressed simply in terms of the Itô measure n; precisely, we have: hence, denoting by ε the generic excursion, and V (ε) its life time: Illustration of various methods for solving partly Skorokhod's embedding problem hence the result.For the second equality, we have used: 4 Taking some care In our discussion, two points need to be looked at carefully.
(i) First, we want θ (h) < ∞ a.s.This may be ensured as follows: there is the representation: as a consequence of Dubins-Schwarz and the balayage formula, where (β(u), u ≥ 0) is a reflecting Brownian motion.Thus, (ii) Some care also has to be taken in the application of the optional stopping theorem in our proof of Proposition 2.2.But, in fact, replacing θ (h) by θ (h) ∧ n, and using dominated convergence, and monotone convergence, we justify the use of the optional stopping theorem.

A relation with the Azéma-Yor algorithm for solving Skorokhod's problem
We note that the arguments in Section 2 and Section 3 allow (almost) to recover the Azéma-Yor result for Skorokhod embedding.Azéma-Yor [2] have obtained an explicit solution to Skorokhod's embedding problem, as follows: given a probability µ(dx) on R, with first moment, and centered, if: ECP 18 (2013), paper 48.Page 2/5 ecp.ejpecp.org

thenB
Tµ ∼ µ, where S t = sup s≤t B s , and ψ µ(x) = 1 µ[x,∞) [x,∞) tdµ(t)is the Hardy-Littlewood function attached to µ.Indeed, similar calculation as above show that, ifG µ = sup{t ≤ T µ : S t − B t = 0},then the increasing process associated to G µ is: Σ(S t∧Tµ ), where Σ(x) = x 0 dy y−φ(y) , with φ the inverse of ψ µ .Thus, Σ(S Tµ ) (law) = e, a condition we assume in the paper.Indeed, that these two integrals are infinite simultaneously follows from the general fact that the bracket of a martingale at infinity is infinite if and only if its local time at infinity is infinite.Here, this martingale is M t = h(L t )B t , whose local time is H(L t ).