From minimal embeddings to minimal diffusions

There is a natural connection between the class of diffusions, and a certain class of solutions to the Skorokhod Embedding Problem (SEP). We show that the important concept of minimality in the SEP leads to the new and useful concept of a minimal diffusion. Minimality is closely related to the martingale property. A diffusion is minimal if it minimises the expected local time at every point among all diffusions with a given distribution at an exponential time. Our approach makes explicit the connection between the boundary behaviour, the martingale property and the local time characteristics of time-homogeneous diffusions.


Introduction
This article shows that there is a one-to-one correspondence between a (generalised) diffusion in natural scale and a class of solutions to the Skorokhod Embedding Problem (SEP) for a Brownian motion. The main contribution of this article is to establish the concept of a minimal diffusion, which is motivated by the correspondence.
The link between embedding and diffusion involves an additive functional which can be used to define both a diffusion (via a time change) and a stopping time (the first time that the additive functional exceeds an independent exponentially distributed random variable). The fundamental connection was first observed by Cox et. al. in [9]. In that article, the authors proved existence of a (martingale) diffusion with a given law at an exponentially distributed time by exploiting the link between martingale diffusions and minimal embeddings. It was then demonstrated in [14] that a diffusion's speed measure can be represented in terms of its exponential time law without recourse to embedding theory.
The general theme of this paper is that we can exploit the correspondence between diffusions and the SEP to introduce notions from the theory of SEP into the diffusion setting. This leads us to introduce the novel concept of a minimal diffusion, which we argue is the canonical diffusion in the family of diffusions with given law at a random time, and corresponds to the minimal embedding.
In the literature on the SEP, the concept of a minimal stopping time is crucial. The problem of constructing stopping times so that a stopped process has a given law degenerates unless there is a notion of what constitutes a 'good' solution. An early definition of 'good' was 'finite expectation'. The concept of a 'minimal solution' was first introduced by Monroe, [17]. Loosely speaking, a stopping time is minimal if there is no almost surely smaller stopping time for the process which results in the same distribution. The property of minimality connects much of the modern literature on Skorokhod embeddings and their applications in mathematical finance. The main contribution of this article is to show that the concept of minimality extends naturally as a property of diffusions. In leading up to this result, we provide a novel characterisation of minimal stopping times in terms of the local times, which is important in transferring the concept of minimality from solutions of the SEP to diffusions.
From the point of view of classical diffusion theory, the article provides insight into the additive functional that defines a diffusion and introduces the intuitive concept of 'minimality' to diffusion theory. We argue that the property of minimality is in many ways more fundamental than the martingale property. The concept of minimality disentangles distributional properties of the process from its boundary behaviour. There may not exist a martingale diffusion with a given exponential time law, but there is always a minimal diffusion.
Our study of this problem was motivated in part by the observation that the construction of time-homogenous diffusions via time-change was closely related to the solution to the SEP given by Bertoin & Le Jan [4]. Indeed, one of the proofs in [9] relied on explicitly constructing a process which enabled the use of the results of [4]. The Bertoin Le-Jan (BLJ) embedding stops a process at the first time an additive functional grows larger than the accumulated local time at the starting point. However many of the same quantities appear in the construction of both the BLJ embedding and the time-homogenous diffusion. By considering the BLJ embedding, we are able to explain why this is the case, and thereby provide new insight into the BLJ construction, and its connection to time-homogenous diffusions.

Generalised Diffusions
Let m be a non-negative, non-zero Borel measure on an interval I ⊆ R, with left endpoint a and right endpoint b (either or both of which may be infinite). Let x 0 ∈ (a, b) and let B = (B t ) t≥0 be a Brownian motion started at B 0 = x 0 supported on a filtration F B = (F B u ) u≥0 with local time process {L x u ; u ≥ 0, x ∈ R}. Define Γ to be the continuous, increasing, additive functional and define its right-continuous inverse by If X t = B At then X = (X t ) t≥0 is a one-dimensional generalised diffusion in natural scale with X 0 = x 0 and speed measure m. Moreover, X t ∈ I almost surely for all t ≥ 0. In contrast to diffusions, which are continuous by definition, generalised diffusions may be discontinuous if the speed measure places no mass on an interval. For instance, if the speed measure is purely atomic, then the process is a birth-death process in the sense of Feller [11]. See also Kotani and Watanabe [16]. In the sequel, we will use diffusion to denote the class of generalised diffusions, rather than continuous diffusions. Let H x = inf{u : X u = x}. Then for λ > 0 (see e.g. [22]), x ≥ y, (2.1) where ϕ λ and φ λ are respectively a strictly increasing and a strictly decreasing solution to the differential equation The two solutions are linearly independent with Wronskian W λ = ϕ λ φ λ − φ λ ϕ λ , which is a positive constant.

The Skorokhod Embedding Problem
We recall some important notions relating to the Skorokhod Embedding Problem (SEP). The SEP can be stated as follows: given a Brownian motion (B t ) t≥0 (or, more generally, some stochastic process) and a measure µ on R, a solution to the SEP is a stopping time τ such that B τ ∼ µ. We refer to Obłój [18] for a comprehensive survey of the history of the Skorokhod Embedding Problem.
Many solutions to the problem are known, and it is common to require some additional assumption on the process: for example that the stopped process (B t∧τ ) t≥0 is uniformly integrable. In the case where B 0 = x 0 , this requires some additional regularity on µ -specifically that µ is integrable, and x 0 =x µ = y µ(dy). We recall a more general notion due to Monroe [17]: Definition 2.1. A stopping time τ is minimal if, whenever σ is another stopping time with B σ ∼ B τ then σ ≤ τ P-a.s. implies σ = τ P-a.s..
That is, a stopping time τ is minimal if there is no strictly smaller stopping time which embeds the same distribution. It was additionally shown by Monroe that if the necessary condition described above was true (µ integrable with mean x 0 ) then minimality of the embedding τ is equivalent to uniform integrability of the stopped process.
In the case where the means do not agree, minimality of stopping times was investigated in Cox and Hobson [8] and Cox [7]. Most natural solutions to the Skorokhod Embedding problem can be shown to be minimal. In some cases, constructions can be extended to non-minimal stopping times -see for example Remark 4.5 in [6].
To motivate some of our later results, we give the following alternative characterisation of minimality for stopping times of Brownian motion. We note that this condition was first introduced by Bertoin and Le Jan [4], as a property of the BLJ embedding. We write L a t for the local time at the level a. When a = 0, we will often simply write L t . Lemma 2.2. Let µ be an integrable measure, and suppose τ embeds µ in a Brownian motion (B t ) t≥0 with B 0 = x 0 . Let a ∈ R be fixed. Then τ is minimal if and only if τ minimises E[L a σ ] over all stopping times σ embedding µ. Proof. Without loss of generality, we may assume a = 0. Let τ be an embedding of µ. Note that L t − |B t | is a local martingale. Let τ N be a localising sequence of stopping times such that τ N ↑ τ and {L t∧τ N − |B t∧τ N |} is a family of martingales. We have This is true for any embedding τ , however in the case where τ is minimal, we observe that {(B τ N ) − } N ∈N is a UI family, by Theorem 5 of [8], and so we have the equality: For the converse, again using Theorem 3 of [8], we observe that it is sufficient to Note that for any integrable measure µ a minimal embedding exists, and therefore any embedding τ which minimises E[L τ ] over the class of embeddings must have

Diffusions with a given law at an exponential time
Let us begin by recalling the construction of a diffusion's speed measure in terms of its exponential time law in [14].
The decomposition of the speed measure in Theorem 3.1 is essentially related to the λ-potential of a diffusion. Recall that for a diffusion X, the λ-potential (also known as the resolvent density) of X is defined as u ] is the expected local time of X at y up until the exponentially distributed time T λ .
where W λ ≤ 6. The speed measure of the consistent diffusion charges only the points 0, 1/2 and 1 and is given by Consistent diffusions are birth-death processes in the sense of Feller [11]. dΓt dt > 0 whenever B t ∈ {0, 1, 2} and the process is 'sticky' there. The process Γ t is constant whenever B t / ∈ {0, 1, 2}, so that A t skips over these time intervals and X t = B At spends no time away from these points. Note that if W λ = 6 then m({0}) = m({1}) = ∞ so that Γ t = ∞ for t greater than the first hitting time of 0 or 1 implying that the endpoints are absorbing. If W λ < 6, the endpoints are reflecting but sticky.
Let H x = inf{u ≥ 0 : X u = x} and let y ∈ (a, b).
Proof. We prove the second statement, the first follows similarly.
It follows by Theorem 51.2 in [21] (which holds in the generalised diffusion case) that ∞ xm(dx) = ∞. Now suppose instead that b < ∞.
Observe that C µ (x) is a convex function on I, so it has left and right derivatives, while its second derivative can be interpreted as a measure; moreover, C µ (b) = 0 = (C µ ) (b+) and (C µ ) (dx) = µ(dx).
From the convexity and other properties of C µ , we note that ( and = ∞ for u < b. Using Fubini/integration by parts, and the fact that (C µ ) (x) is increasing and negative, Letting first v → b, the right hand side is equal (in the limit) to − (C µ ) (u−) C µ (u) , but we observed above that this is unbounded as u b, and since the whole expression is increasing in u, it must be infinite, as required.
As Lemma 3.4 demonstrates, the behaviour of X at the boundaries is determined by the value of W λ . When I is unbounded, the boundary behaviour determines whether or not X is a martingale diffusion. Suppose that a is finite, b = ∞. Then Kotani [15] (see also Delbaen and Shirakawa ([10])) show that X t∧Ha∧H b is a martingale if and only if ∞− xm(dx) = ∞ (i.e. ∞ is not an entrance boundary). By Lemma 3.4 this is equivalent to the conditions x 0 ≤x µ and 1/W λ = C µ (x 0 ) being satisfied. An analogous observation holds when b is finite and a is infinite. Theorem 3.1 and Lemma 3.4 provide a natural way of determining boundary properties by inspection of the decomposition of the speed measure in terms of P(X T λ ∈ dx). Furthermore, the decomposition gives us a canonical way of constructing strict local martingales with a given law at a random time. For instance if b is infinite, we can generate strict local martingale diffusions by choosing a measure µ and setting W λ < 1/C µ (x 0 ).
In the example above, the strict local martingale property of the process follows from the fact that C µ is 'shifted up' in the decomposition of the speed measure (3.1). In general, we expect reflection at the left boundary X = a, when we have had to 'shift the function P µ up', and we expect the process stopped at t = H a ∧ H b to be a strict local martingale when we 'shift C µ up'. From minimal embeddings to minimal diffusions Example 3.6. Let us reconsider the diffusion in Example 3.5. Let us construct a diffusion Y with the same law as the inverse Bessel process of dimension three at an exponential time, such that Y t∧H0 is a martingale (and therefore, such that Y 0 =x µ ).
As we are shifting the starting point, we must adjust the speed measure betweenx µ and the starting point of the diffusion in Example 3.5. By Lemma 3.4 we know that the speed measure of the martingale diffusion must satisfy

The BLJ embedding for Brownian motion
The BLJ construction is remarkably general and can be used to embed distributions in general Hunt processes. Our interest, however, lies in the specific case of embedding a law µ in Brownian Motion. In this setting, the rich structure of the embedding translates into a one-to-one correspondence to the family of diffusions with a given law at an exponential time.

Embeddings and diffusions
We will now show that each BLJ embedding τ κ of µ in B corresponds to a localmartingale diffusion X such that X T λ ∼ µ, for an independent exponentially distributed random variable T λ . The correspondence is a consequence of the role played by the functional Γ u = R L x u m(dx), where m is defined as in (3.1) for a fixed starting point x 0 and a target law µ on (a, b). We have already seen that B A T λ ∼ µ, where A t = Γ −1 t . We will now observe that for κ ≥ κ 0 , the BLJ embedding τ κ introduced above can be re-written as τ κ = inf{t ≥ 0 : Γ t λκ > L x0 t }. The remarkable role played by the additive functional Γ in both settings, and the resulting correspondence between embeddings and diffusions, is the subject of this section. In order to better understand the connection, we first make the following distinction: let τ * κ = inf{t ≥ 0 : Γ t > T λ }, where T λ is an independent exponential random variable with mean 1/λ, and m(dx) is ECP 19 (2014), paper 34. given by (3.1), with κ = 2/W λ . It follows that B τ * κ ∼ µ for κ ≥ 2 max{C µ (x 0 ), P µ (x 0 )}, and (from Corollary 3.2) we have E x0 [L x0 T λ ] = κ.
The connection between the quantities that define BLJ embeddings and those that define diffusions follows from relating the λ-potential of diffusions, the potential U µ of µ and V x0 µ . A number of solutions to the Skorokhod Embedding Problem, most notably the solutions of Chacon-Walsh [5], Azéma-Yor ( [2], [3]) and Perkins [19] can be derived directly from quantities related to the potential U µ .
Hx 0 ] = |y − x 0 | + |x 0 − x| − |y − x| is simply the potential kernel for B x0 killed at x 0 . Integrating we find The result follows after some re-arrangement.
Recalling the definition of κ 0 in the BLJ embedding, we now observe that κ 0 = max{2C µ (x 0 ), 2P µ (x 0 )}, which is the smallest constant c such that c − V x0 µ (x) ≥ 0 for all x. The situation is illustrated in Figure 1 below.
x µ x 0 κ 0 x x Figure 1: then the potential must be shifted upwards to lie above |x 0 − x| everywhere. Thus, if x 0 <x µ , figure).
The following Lemma follows from the Corollary in [4], page 540.
In comparison, the Corollary on p. 540 in [4] µ (x). It is helpful to visualise this relationship pictorially, see Fig. 1. The expected local time at x of Brownian motion started at x 0 until the minimal BLJ stopping time is the distance at x between the minimally shifted potential function and the potential kernel. So far we have focused on the minimal embedding τ κ0 for µ. The picture for the suboptimal embeddings τ κ , κ > κ 0 is analogous. The picture simply reflects the additional expected local time accrued between the minimal and the non-minimal stopping time. The expected local time of a non-minimal embedding is the expression in (5.2) plus the . The graph of the expected local time (see Fig. 1) is given by shifting the potential picture κ 0 − V x0 µ (x) upwards by κ − κ 0 .
Remark 5.4. The assumption µ({x 0 }) = 0 made in Section 4 is necessary in the construction of the BLJ embedding in [4], but not in the construction of diffusions with a given exponential time law in [14]. To make the correspondence independent of this assumption on the target law, a modified version of the BLJ embedding with external randomisation can be constructed.
With the close connection indicated by Proposition 5.3, it is natural to consider if we can more explicitly explain the connection between τ * κ and τ κ . To do this, we consider the excursion theory that is behind the correspondence between diffusions and embeddings. We refer to Rogers [20] for background on excursion theory. Let Consider the Brownian motion B t , and Γ t = R L x t m(dx), which is increasing. For a test function g, define: φ(u) := n(df )g(f (Γ −1 u ) 1 {Γ ζ(f ) ≥u} , the expected value of g(B Γ −1 u ) averaged over the excursion measure, conditioned on the event that Γ ζ(f ) ≥ u on the excursion.
If T λ is an independent exponential random variable with parameter λ then we compute, using the fact that the process B is recurrent, and the memoryless property of the exponential, that for some constant c. In the BLJ setting, consider the stopping time τ κ = inf{u ≥ 0 : Γ u λκ > L x0 t }. Observe that Γ L −1 s is an increasing process with independent, identically distributed increments, and hence Y s := s − λκΓ L −1 s is a Lévy process which goes below 0 on the excursion for which τ κ occurs. In particular, we have where L −1 τκ = inf{t ≥ 0 : Y t ≤ 0} and the first equality follows from the Maisonneuve exit system, see [4], p. 544.
In addition we observe that Y t is a Lévy process and so, using Lemma 1 in [4], we can rewrite this formula as )]). The behaviour of ψ is determined by properties of Y t . In particular, it is convex and ψ −1 (0) = inf{t : It follows that E[Y 1 ] ≤ 0, since otherwise with positive probability, Y t ∞ and hence L −1 τ = ∞ with positive probability, and hence the same is true for the BLJ stopping time.
Equating the terms in (5.4) and (5.3), we now have an identity involving only φ and we can choose κ so that ψ −1 (0) = λ. Note that the constant c appearing in (5.3) is equal to 1 λ . Moreover, by the embedding property of τ κ and the construction of Γ, both equations are equal to g(x)µ(dx).

Minimal diffusions
The purpose of this final section is to define a notion of minimality for diffusions which will be analogous to the notion of minimality for the BLJ Skorokhod embeddings.
Suppose we have fixed a starting point x 0 and a law µ and we are faced with a (Skorokhod embedding) problem of finding a stopping time τ , such that B τ ∼ µ, where B is a standard Brownian motion started at x 0 . Then (and especially when we are interested in questions of optimality) it is most natural to search for stopping times τ which are minimal.
In the family of BLJ embeddings, the embedding τ κ0 is minimal, and hence (by Lemma 2.2) so too is τ * κ0 , and the notion of minimal embedding carries over into a notion of minimality for diffusions in natural scale. Definition 6.1. We say that a time homogeneous diffusion X in natural scale and started at x 0 is a λ-minimal diffusion if, whenever Y is another time-homogeneous diffusion in natural scale, with X T λ ∼ Y T λ where T λ an independent exponential random variable with mean 1 . We say that X is a minimal diffusion in natural scale if X is λ-minimal for all λ > 0. By Proposition 5.3, the λ-minimal local-martingale diffusion corresponds to the minimal BLJ embedding τ κ0 . As an analogue of the notion of minimality for the BLJ solution to the Skorokhod Embedding Problem, minimality is a natural probabilistic property. A minimal diffusion has the lowest value of E x0 [L y T λ ] for all y > 0 among the diffusions with a given law µ at time T λ started at x 0 (this follows from Theorem 3.1). In terms of diffusion dynamics, the minimal diffusion moves around the state space slower than all other diffusions with the same exponential time law. Minimality is a necessary condition for X t∧Ha∧H b to be a martingale. For instance, if a is finite and b is infinite, then X t∧Ha∧H b is a martingale if and only if X is minimal and x 0 ≤x µ , i.e. b is not an entrance boundary. When I is a finite interval, minimality is a more fundamental property than the martingale property; every (stopped) diffusion on a finite interval is a martingale diffusion, but there is only one minimal diffusion for every exponential time law.
Note also that these definitions extend in an obvious way to diffusions which are not in natural scale: a diffusion which is not in natural scale is minimal if and only if it is minimal when it is mapped into natural scale. (See Example 6.3.) We collect these observations in the following result: φ(x) = cosh(2−x) cosh (2−x0) . Note that this diffusion is non-minimal. To construct the minimal diffusion with the same marginal law at an exponential time, let η = cosh(x 0 ) −1 and set ϕ η (x) = ϕ(x) − η = cosh(x)−1 cosh x0 , and similarly φ η (x) = φ(x) − η = cosh(2−x) cosh(2−x0) −