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Skorokhod embeddings, minimality and non-centred target distributions

In this paper we consider the Skorokhod embedding problem for target distributions with non-zero mean. In the zero-mean case, uniform integrability provides a natural restriction on the class of embeddings, but this is no longer suitable when the target distribution is not centred. Instead we restrict our class of stopping times to those which are minimal, and we find conditions on the stopping times which are equivalent to minimality. We then apply these results, firstly to the problem of embedding non-centred target distributions in Brownian motion, and secondly to embedding general target laws in a diffusion. We construct an embedding (which reduces to the Azema-Yor embedding in the zero-target mean case) which maximises the law of \sup_{s \le T} B_s among the class of minimal embeddings of a general target distribution \mu in Brownian motion. We then construct a minimal embedding of \mu in a diffusion X which maximises the law of \sup_{s \le T} h(X_s) for a general function h.


Introduction
The Skorokhod embedding problem was first proposed, and then solved, by Skorokhod (1965), and may be described thus: Given a Brownian motion (B t ) t≥0 and a centred target law µ can we find a stopping time T such that B T has distribution µ?
Skorokhod gave an explicit construction of the stopping time T in terms of independent random variables, and in doing so showed that any zero-mean probability measure may be embedded in Brownian motion. Since the problem was posed many more solutions have been given, see for example Dubins (1968), Root (1969) and Chacon and Walsh (1976), and the comprehensive survey article of Ob lój (2004). With different solutions comes the question of optimal properties of the embeddings, and various optimal embeddings have been found -for example the embedding minimising the variance of T (Rost, 1976), the embedding minimising in convex order the law of the local time at zero (Vallois, 1992), or the embedding stochastically minimising the law of the maximum (Perkins, 1986).
An obvious extension of the problem is to consider more general classes of processes. Here the question of the existence of an embedding becomes more interesting. For a Markov process and an arbitrary target measure necessary and sufficient conditions are given by Rost (1971) and a construction in this general case is given by Bertoin and Le Jan (1992). In the case of diffusions on R simpler necessary and sufficient conditions are given in Grandits and Falkner (2000), Pedersen and Peskir (2001) and Cox and Hobson (2004), along with some constructions.
The work we present here was motivated by the following question: Given a diffusion (X t ) t≥0 and a target distribution µ X for which an embedding exists, which embedding maximises the law of sup s≤T X s (respectively sup s≤T |X s |)?
For Brownian motion, the question has been solved by Azéma and Yor (1979a) (respectively Jacka (1988)) under the condition that B t∧T is a UI-martingale.
There are several considerations that need to be made when moving from the Brownian case to the diffusion case. Firstly, the mean-zero assumption that is made by Azéma and Yor (1979a) and Jacka (1988) is no longer natural since we are no longer necessarily dealing with a martingale. The second aspect that needs to be considered is with what restriction should we replace the UI condition? That such a condition is desirable may be seen by considering a recurrent diffusion. Here the maximisation problem can easily seen to be degenerate by considering first running the diffusion until it hits a level x, allowing it to return to its starting point and then using the reader's favourite embedding. Clearly this dominates the unmodified version of the reader's favourite embedding.
In Pedersen and Peskir (2001) an integrability condition on the maximum (specifically that E(sup t≤T s(X t )) < ∞ where s is the scale function of X) was suggested to replace the UI condition in the Brownian case. In this work we propose using the following class of stopping times introduced by Monroe (1972) to provide us with a natural restriction on the set of admissible embeddings. Definition 1. A stopping time T for the process X is minimal if whenever S ≤ T is a stopping time such that X S and X T have the same distribution then S = T a.s..
The class of minimal stopping times provides us with a natural class of 'good' stopping times. In the Brownian case it includes as a subclass those embeddings for which E[sup s≤T B s ] is finite. Furthermore, there is a link to the uniformly integrable case as a consequence of the following result: Theorem 2. (Monroe, 1972, Theorem 3) Let T be an embedding in Brownian motion of a centred distribution µ. Then T is minimal if and only if the process B t∧T is uniformly integrable, or equivalently if E[B T |F S ] = B S for all stopping times S ≤ T .
Our first result extends this theorem to non-centred target distributions.
Theorem 3. Let T be an embedding in Brownian motion of a non-centred distribution µ with mean m < 0. Then T is minimal if and only if the process B − t∧T is uniformly integrable, It is clear that the notion of a minimality fits well with the problem of embedding in any process, such as a diffusion, and not just Brownian motion. Our approach to embeddings in diffusions can be traced back to Azéma and Yor (1979b) and will be to map the diffusion into natural scale (so that, up to a time change, it resembles Brownian motion) and use techniques developed for embedding Brownian motion. Using this method on a transient diffusion one finds that the state space and target distribution for the Brownian motion is restricted to a half-line (or sometimes a finite interval). We will show minimality to be equivalent to stopping the Brownian motion before it leaves this interval, so that a minimal stopping time is necessarily before the first explosion time of X.
When we map from the problem of embedding µ X in X to the Brownian motion the target law µ we obtain for B is the image of µ X under the scale function. The key point is that there is no reason why this target law should have mean zero. Thus, unlike most of the other studies of Skorokhod embeddings in Brownian motion we are interested in non-centred target distributions, and non-UI stopping times. Instead, as described in Theorem 3, the class of 'good' stopping times satisfy slightly different integrability conditions.
Having characterised the class of minimal embeddings, we then turn to the problem of constructing optimal embeddings in Brownian motion for non-centred target distributions. In particular, amongst the class of minimal embeddings we find the stopping time which maximises the law of the maximum of the stopped process. In the centred case this embedding reduces to the classical Azema-Yor embedding. In fact most of the paper will concentrate on embedding non-centred target distributions in B, and we will only return to the diffusion case in a short final section.
The paper will proceed as follows. In Section 2 we prove some results concerning minimality of stopping times for non-centred target distributions, giving equivalent conditions to minimality in terms of the process. Next, in Section 3 we construct an extension of the Azema-Yor embedding for non-centred target distributions and show both that it is minimal, and that it retains the optimality properties of the original Azema-Yor embedding. In Section 4 we use these stopping times to construct an embedding maximising the distribution of sup s≤T h(B s ) for a general function h. Finally in Section 5 we apply these results to the problem of embedding optimally in diffusions.

Minimal Embeddings for Non-centred Distributions
In this section we examine the properties of minimal stopping times. In particular, for embeddings in Brownian motion we aim to find equivalent conditions to minimality when the target distribution is not centred.
We begin by noting the following result from Monroe (1972) which justifies the existence of minimal stopping times: Proposition 4 ( Monroe (1972), Proposition 2). For any stopping time T there exists a minimal stopping time S ≤ T such that B S ∼ B T .
For β ∈ R define H β = inf{t > 0 : B t = β}, the first hitting time of the level β. The main result of this section is the following: Theorem 5. Let T be a stopping time of Brownian motion which embeds an integrable distribution µ where m = R x µ(dx) < 0. Then the following conditions are equivalent: (i) T is minimal for µ; (ii) for all stopping times R ≤ S ≤ T , (1) (iii) for all stopping times S ≤ T , (v) as γ → ∞ γP(T > H −γ ) → 0; (vi) the family {B − S } taken over stopping times S ≤ T is uniformly integrable; In the case where supp(µ) ⊆ [α, ∞) for some α < 0 then the above conditions are also equivalent to the condition: (viii) Remark 6. (i) Of course the Theorem may be restated in the case where m > 0 by considering the process −B t . We will use this observation extensively in Section 3.
(ii) Equation (2) is suggestive of the fact that when T is minimal, (B (u/1−u)∧T ) 0≤u≤1 is a supermartingale. To check this we need to show also that EB − t∧T < ∞ for all t. We show this more generally, for a stopping time S ≤ T . Using (2), It follows from comparing parts (ii) and (iii) of Theorem 5 that if S ≤ T is a stopping time and T is minimal, then S is minimal provided EB S ≤ 0 and E|B S | < ∞. The first condition is a trivial consequence of (1) on taking R = 0, the second condition then follows from Remark 6(ii) on noting that, since EB S ≤ 0, we have EB + S ≤ EB − S < ∞. Consequently we have the following corollary of Theorem 5: Corollary 7. If T is minimal and S ≤ T for a stopping time S then S is minimal for L(B S ).
For ease of exposition we divide the proof of Theorem 5 into a series of smaller results. The first is a key result which shows that the strongest of the stopping time conditions is sufficient for minimality. Throughout this section it is to be understood that µ is a distribution with negative mean and T a stopping time embedding µ.
Then T is minimal.
where we use (4) to deduce (5). However since we have equality in the first and last expressions, we must also have equality throughout and so Since this holds for all a ∈ R we must have and R = T . Hence T is minimal.
Lemma 9. If T is a stopping time such that B T ∼ µ and γP(T > H −γ ) → 0 as γ → ∞ then E|B S | < ∞ and EB S ≤ 0 for all stopping times S ≤ T .
Proof. We show that, for S ≤ T , EB − S < ∞ and EB + S ≤ EB − S from which the result follows. Suppose γ > 0. Since B t∧H−γ is a supermartingale, We may rewrite the term on the left of the equation as By monotone convergence the term on the left increases to EB + S , while by monotone convergence and the hypothesis of the Lemma the right hand side converges to EB − S . Consequently The previous two lemmas are crucial in determining sufficient conditions for minimality. Now we consider necessary conditions.
It is easy to see that for 0 it follows from the dominated convergence theorem that f is continuous. As a corollary if f (γ 0 ) > 0 for some γ 0 < 0, then there exists γ 1 ∈ (γ 0 , 0) such that f (γ 1 ) = 0. Given this γ 1 , and conditional on T > H γ1 , let then S embeds µ and S ≤ T but S = T , contradicting the minimality of T . Hence T ′′ is minimal. But then by Theorem 2, W t∧T ′′ is uniformly integrable and so, for γ < γ 1 by Monotone convergence and the minimality of T x respectively. Further, We now turn to the proof of the main result: Proof of Theorem 5. We begin by showing the equivalence of conditions By Lemma 9 E|B R | < ∞ and by dominated convergence the right hand side converges to For the term on the left we consider Again by Lemma 9 and dominated convergence the first term on the right converges to E(B S ; A) while the other term converges to 0 by (v). Hence on letting γ → ∞ in (6) we have and we have shown (ii).
We have already shown that minimality is equivalent to these conditions: We have shown equivalence between (i) -(vii). We are left with showing that if µ has support bounded below then (viii) is also equivalent. So assume that the target distribution µ has support contained in [α, ∞) for some α < 0 and that T is an embedding of µ. Then B t∧Hα is a continuous supermartingale, bounded below and therefore if S ≤ T ≤ H α , The reverse implication follows from considering the stopping time which is only possible if P(A) = 0.
We close this section with a discussion of the case where the mean of the target distribution is well defined, but infinite. In this case the notion of minimality still makes perfect sense, but many of the conditions outlined in Theorem 5 can be shown to be no longer equivalent to minimality.
Such a φ is an increasing function with φ(x) < x. Define J = inf{B u ; u ≤ H 1 }. By construction φ(J) ∼ µ and we can construct an embedding of µ by setting T = inf{u > H 1 : B u = φ(J)}. Further, at the stopping time T the Brownian motion is at a current minimum. Hence T is minimal. Now consider the alternative conditions in Theorem 5. In reverse order, (vii) and (vi) both clearly fail, whereas (v) may or may not hold depending on the target law µ, and (iv) holds trivially. (However (iv) holds for all embeddings of µ so that it is not sufficient for minimality.) The condition in (iii) also holds, since on stopping B is at a minimum value. However the choice R = 0 and S = H 1 shows that (ii) fails and is not necessary for minimality. Hence, in the case where m = R xµ(dx) is well defined but equal to −∞, the only condition from the list in Theorem 5 which could possibly be necessary and sufficient for minimality is (iii).
3 An extension of the Azéma-Yor embedding to the non-centred case Let µ be a target distribution on R, and let B t be a Brownian motion with B 0 = 0. Define M T = sup s≤T B s and J T = inf s≤T B s . In this section we are interested in finding an embedding to solve the following problem: Given a Brownian motion (B t ) t≥0 and an integrable (but possibly not centred) target distribution µ with mean m, find a minimal stopping time T such that T embeds µ and is maximised over all minimal stopping times T embedding µ, uniformly in x.
We call an embedding with this property the max-max embedding, and denote it by T max . Without some condition on the class of admissible stopping times the problem is clearly degenerate -any stopping time may be improved upon by waiting for the first return of the process to 0 after hitting level x and then using the original embedding. For this improved embedding P(M T ≥ x) = 1. Further, since no almost surely finite stopping time can satisfy P(M T ≥ x) = 1 for all x > 0, there can be no solution to the problem above in the class of all embeddings. As a consequence some restriction on the class of admissible stopping times is necessary for us to have a well defined problem.
Various conditions have been proposed in the literature to restrict the class of stopping times. In the case where m = 0, the condition on T that B t∧T is a UI martingale was suggested by Dubins and Gilat (1978), and in this case the maximal embedding is the Azema-Yor embedding. When m = 0 Monroe (1972) tells us that minimality and uniform integrability are equivalent conditions, so the Azema-Yor stopping time is the max-max embedding. For the case where m > 0, Pedersen and Peskir (2001) showed that EM T < ∞ is another suitable condition, with the optimal embedding being based on that of Azema and Yor. We argue that the class of minimal embeddings is the appropriate class for the problem under consideration since minimality is a natural and meaningful condition, which makes sense for all m (and which, for m > 0, includes as a subclass those embeddings with E(M T ) < ∞.) We now describe the construction of the candidate max-max stopping time. This construction is an extension of the classical Azéma and Yor (1979a) stopping time. However we derive the stopping rule in a slightly non-standard way via a Chacon and Walsh (1976) PSfrag replacements style argument. This interpretation of the Azéma-Yor construction (in the centred case) is due to Meilijson (1983). In the following we treat the cases m > 0, m = 0 and m < 0 in one go, since the basic idea is identical. Define the convex function: We note that c is related to the potential of µ, that it is Lebesgue-almost everywhere differentiable with left-derivative c ′ − (x) = 1 − 2µ([x, ∞)) and c(x) − |x| → |m| ∓ m as x → ±∞. For θ ∈ [−1, 1] define u(θ) := inf{y ∈ R : c(y) + θ(x − y) ≤ c(x), ∀x ∈ R}, The intuition behind these definitions is as follows. Let θ denote a gradient and consider the unique tangent to c with gradient θ. Then z + (θ) is the x-coordinate of the point where this tangent crosses the line y = x. Similarly, b(x) is the x-coordinate of the point for which the tangent to c at that point passes through (x, x). The mathematical definitions above, and the fact that u is left-continuous, ensure that b is well defined. Note that when m = 0, a few lines of calculus are sufficient to check that b −1 is precisely the barycentre function which arises in the classical Azema-Yor construction.
Theorem 12. Let T be a stopping time of (B t ) t≥0 which embeds µ and is minimal. Then for x ≥ 0 Define the stopping time T max via Then T max embeds µ, is minimal, and attains equality in (8) for all x ≥ 0.
Remark 13. (i) By the comments before the theorem relating b to the barycentre function, when m = 0 the above theorem is a restatement of the classical Azema-Yor result, which by Theorem 2 can be stated in terms of minimal, rather than uniformly integrable embeddings. For m > 0, b(x) = −∞ for x < m, and consequently T max ≥ H m . Moreover, for x > m, b −1 is a shifted version of the barycentre function. Consequently, when m > 0 the embedding T max may be thought of as 'wait until the process hits m then use the Azema-Yor embedding,' and the conclusion of Theorem 12 is similar to Proposition 3.1 of Pedersen and Peskir (2001), except that T max is shown to be optimal amongst the larger class of minimal stopping times rather than the class of embeddings for which M T ∈ L 1 . However, the truly original part of the theorem is in the case m < 0. In that case the embedding 'wait until the process hits m and then use the Azema-Yor embedding' does not achieve equality in (8).
We can relate the right-hand-side of (10) to the slope of a line joining (x, x) with (λ, c(λ)). In taking the infimum over λ we get a tangent to c and a value for the slope in [−1, 1]. Thus the bound on the right-hand-side of (8) lies in [0, 1].
(iii) T max has the property that it maximises the law of M T over minimal stopping times which embed µ. If we want to minimise the law of the minimum, or equivalently we wish to maximise the law of −J T , then we can deduce the form of the optimal stopping time by reflecting the problem about 0, or in other words by considering −B. Let T min be the embedding which arises in this way, so that amongst the class of minimal stopping times which embed µ, the stopping time T min maximises simultaneously for all x ≥ 0.
We now turn to the proof of Theorem 12.
Proof. The following inequality for x > 0, λ < x may be verified on a case by case basis: In particular, on {M T < x}, (11) reduces to and on {M T ≥ x} we get Then taking expectations, If m ≤ 0 then by Theorem 5(vii) and the minimality of T we know E(B T ∧Hx ) = 0 and so Conversely if m > 0, by Theorem 5(iii) applied to −B, and so Since λ was arbitrary in either case, (8) must hold. It remains to show that T max attains equality in (8), embeds µ and is minimal. We begin by showing that it does attain equality in (8). Since (8) is attained by a value λ * with the property that a tangent of c at λ * intersects the line y = x at (x, x). By the definition of b we can choose λ * = b(x). In particular, since {M Tmax < x} ⊆ {B Tmax ≤ b(x)} and {M Tmax ≥ x} ⊆ {B Tmax ≥ b(x)}, the stopping time T max attains equality almost surely in (12) and (13). Assuming that T max is minimal, we are then done for m ≤ 0. If m > 0 we do not always have equality in (15). If x < m then E(B Tmax∧Hx ) = x, but then λ * = −∞ and so the term (14) is zero. As a result equality is again attained in (8). Otherwise, if x ≥ m then T max ≥ H m and the properties of the classical Azema-Yor embedding ensure that E(B Tmax∧Hx ) = m and there is equality both in (15) and (8). Fix a value of y which is less than the supremum of the support of µ. Let b −1 be the left-continuous inverse of b. Then, since we have equality in (8), we deduce: where c ′ − is the left-derivative of c. It is easy to see from the definition (7) that this last expression equals µ([y, ∞)). Hence T max embeds µ.
Now we consider minimality of T max . It is well known that in the centred case T max is uniformly integrable, and hence by Theorem 2, T max is minimal. Suppose m > 0. By an analogue of Theorem 5(v), in order prove minimality it is sufficient to show that lim x↑∞ xP(T max > H x ) → 0. But, by the arguments in the previous paragraph This last quantity is exactly the height above 0, when it crosses the y-axis, of the tangent to c which passes through (x, x). As x ↑ ∞ this height decreases to zero. Now suppose m < 0. By Theorem 5(v), in order to prove minimality it is sufficient to show that as x ↓ −∞, |x|P(T max > H x ) → 0. We have It is easy to see from the representation of b that b −1 (x) tends to zero as x → −∞.
4 An embedding to maximise the modulus Jacka (1988) shows how to embed a centred probability distribution in a Brownian motion so as to maximise P(sup t≤T |B t | ≥ y). Our goal in this section is to extend this result to allow for non-centered target distributions with mean m = 0. In fact we solve a slightly more general problem. Let h be a measurable function; we will construct a stopping time T mod which will maximise P(sup t≤T |h(B t )| ≥ y) simultaneously for all y where the maximum is taken over the class of all minimal stopping times which embed µ. The reason for our generalisation will become apparent in the application in the next section.
Without loss of generality we may assume that h is a non-negative function with h(0) = 0 and such that for x > 0 both h(x) and h(−x) are increasing. To see this, observe that for arbitrary h we can define the functioñ Thenh has the desired properties and since sup s≤T |h(B s )| = sup s≤Th (B s ) + |h(0)| the optimal embedding forh will be an optimal embedding for h.
So suppose that h has the properties listed above. We want to find an embedding of µ in B which is minimal and which maximises the law of sup t≤T h(B t ). (Since h is non-negative we can drop the modulus signs.) Suppose also for definiteness that µ has a finite, positive mean m = R xµ(dx) > 0. In fact our construction will also be optimal when m = 0 (the and note Let µ + (respectively µ − ) be the measure obtained by conditioning a random variable with law µ to lie in the upper p th (respectively lower (1 − p) th ) quantile of its distribution. Recall that we have taken m > 0 and that Then using (18) in the definition (16) we have that In particular z + (θ 0 ) is the mean of µ + , since µ + ({u(θ 0 )}) = 1 − 1 p µ((u(θ 0 ), ∞)). When we repeat the calculation for z − (θ 0 ) using (19) we find that .
Note that −z − (θ 0 ) is strictly smaller than the mean of µ − . We now describe the candidate stopping time T mod ≡ T h mod . Note that this stopping time will depend implicitly on the function h via z ± (θ 0 ). Let T 0 := inf{t > 0 : B t ∈ (−z − (θ 0 ), z + (θ 0 ))}, and define Here we use θ T0 to denote the shift operator, and T µ+ max is the stopping time constructed in Section 3 for a zero-mean target distribution, so that T µ+ max is a standard Azema-Yor embedding of the centred target law µ + . (Recall that z + (θ 0 ) is the mean of the corresponding part of the target distribution.) Similarly T µ− min is the stopping time applied to −B started at −z − (θ 0 ) which maximises the law of the maximum of −B. In this case the mean of the target law µ − is larger than −z − (θ 0 ) so that in order to define T µ− min we need to use the full content of Section 3 for embeddings of non-centred distributions.
The following theorem asserts that this embedding is indeed an embedding of µ, that it is minimal, and that it has the claimed optimality property.
Theorem 14. Let µ be a target distribution such that m > 0. Then within the class of minimal embeddings of µ in B, the embedding T mod as defined above has the property that it maximises Proof. By construction T mod embeds µ. We need only show that it is optimal and minimal.
For x ≤ h(−z − (θ 0 )) ∧ h(z + (θ 0 )) we know the probability that {sup t≤T mod h(B t ) ≥ x} is one and so, for such x, T mod is clearly optimal. Indeed if h is discontinuous at −z − (θ 0 ) or z + (θ 0 ) slightly more can be said. Note first that if z + (θ 0 ) coincides with the supremum of the support of µ, then by Theorem 5(iii) and the minimality of T mod (see below), the stopped Brownian motion can never go above z + (θ 0 ). Hence we may assume that h is constant on the interval to the right of the supremum of its support. Define and take x ≤ L. Then either Similar considerations apply for B T0 = −z − (θ 0 ) except that then −J T mod > z − (θ 0 ) in all cases. We deduce that for x ≤ L P sup t≤T mod h(B t ) ≥ x = 1 and hence T mod is optimal for such x. So suppose that x > L. For any stopping time T embedding µ, the following holds: We will show that the embedding T mod attains the maximal values of both terms on the right hand side, and further that for T mod the two events on the right hand side are disjoint. Hence T mod is optimal. By the definition of θ 0 , x > (h(z + (θ 0 ))) ∨ (h(−z − (θ 0 ))). It follows that and by the definition of T mod and the properties of T max , we deduce where here T µ max is the embedding of Section 3 applied to µ. A similar calculation can be done for the minimum. In particular T mod inherits its optimality property from the optimality of its constituent parts T µ+ max and T µ− min Finally we note that T mod is indeed minimal. Consider the family of stopping times S ≤ T mod . On the set where B T0 = −z − (θ 0 ) we have that B + S is bounded, whereas on the set B T0 = −z − (θ 0 ) the minimality of T µ− min ensures that B + S is uniformly integrable. Hence, combining these two cases, B + S is a uniformly integrable family and the minimality of T mod follows from 5(vi) applied to a target distribution with positive mean.
Remark 15. If the restrictions of h to R + and R − are strictly increasing then T mod will be essentially the unique embedding which attains optimality in Theorem 14. If however h has intervals of constancy then other embeddings may also maximise the law of sup t≤T |h(B t )|.

Embeddings in diffusions
Our original motivation in considering the embeddings of the previous sections was their use in the investigation of the following question: Given a regular (time-homogeneous) diffusion (X t ) t≥0 and a target distribution µ X , find (if possible) a minimal stopping time which embeds µ X and which maximises the law of sup t≤T X t (alternatively sup t≤T |X t |) among all such stopping times.
Note that in the martingale (or Brownian) case it is natural to consider centred target laws, at least in the first instance. However in the non-martingale case this restriction is no longer natural, and as we shall see below is completely unrelated to whether it is possible to embed the target law in the diffusion X. It was this observation which led us to consider the problem of embedding non-centred distributions in B.
The key idea (see Azéma and Yor (1979b)) is that we can relate the problem of embedding in a diffusion to the case where we are dealing with Brownian motion via the scale function. There exists a continuous, increasing function s such that s(X t ) is a local martingale, and hence a time-change of Brownian motion. Then the requirement X T X ∼ µ X translates to finding an embedding of a related law in a Brownian motion B, and the criterion of maximising sup t≤T X t also has an equivalent statement in terms of B.
We first recall the properties of the scale function (see e.g. Rogers and Williams (2000, V.45)). If (X t ) t≥0 is a regular (time-homogeneous) diffusion on an interval I ⊆ R with absorbing or inaccessible endpoints and vanishing at zero, then there exists a continuous, strictly increasing scale function s : I → R such that Y t = s(X t ) is a diffusion in natural scale on s(I). We may also choose s such that s(0) = 0. In particular Y t is (up to exit from the interior of s(I)) a time change of a Brownian motion with strictly positive speed measure. For definiteness we write Y t = B τt .
We suppose also that our target distribution µ X is concentrated on the interior of I. Then we may define a measure µ = µ Y on s(I) • by: µ(A) = µ Y (A) = µ X (s −1 (A)), A ∈ s(I) • , Borel The original problem of embedding µ X in X is equivalent to the problem of embedding µ Y in Y before Y exits the interval s(I) • . Since Y is a time change of a Brownian motion, we need only consider the problem of embedding µ Y in a Brownian motion before exit from s(I) • . If T is an embedding of µ in B then T X ≡ τ −1 (T ) is simultaneously an embedding of µ Y in Y and µ X in X.
The first question is when does any embedding exist? If we define m = I s(x)µ X (dx), then the following lemma (see Pedersen and Peskir (2001) and Cox and Hobson (2004)) gives us necessary and sufficient conditions for an embedding to exist.
Lemma 16. There are three different cases: (i) s(I) • = R, in which case X is recurrent and we can embed any distribution µ X on I • in X.
In each case it is clear that: T X is minimal for X ⇐⇒ T X is minimal for Y ⇐⇒ T is minimal for B, where T = τ (T X ). Further, since µ = µ Y is concentrated on s(I) • , if the stopping time T is minimal then T will occur before the Brownian motion leaves s(I) • (this is a consequence of Theorem 5 in case (ii) and Theorem 2 in case (iii)), and then T X will be less than the first explosion time of X.
It is now possible to apply the results of previous sections to deduce a series of corollaries about embeddings of µ X in X. Suppose that µ X can be embedded in X or equivalently that µ Y can be embedded in B, before the Brownian motion leaves s(I) • . Suppose further that in the recurrent case where s(I) • = R the law µ = µ Y is integrable. Let T max and T h mod be the optimal embeddings of µ in B as defined in Sections 3 and 4. (Observe that from now on we make the dependence of T h mod on h explicit in the notation.) Then we can define T X max and T X,h mod by Corollary 17. T X max is optimal in the class of minimal embeddings of µ X in X in the sense that it maximises P max t≤T X t ≥ y uniformly in y ≥ 0.
Corollary 18. T X,h mod is optimal in the class of minimal embeddings of µ X in X in the sense that it maximises P max t≤T (h • s)(X t ) ≥ y uniformly in y ≥ 0.
Corollary 19. T X,|s −1 | mod is optimal in the class of minimal embeddings of µ X in X in the sense that it maximises P max t≤T |X t | ≥ y uniformly in y ≥ 0.