Optimal transport and Skorokhod embedding

1.1 A motivating example: Root’s construction

To illustrate our approach we introduce Root’s construction [48], which will serve as inspiration in the rest of the paper. Root’s construction is one of the earliest solutions to (SEP), and it is prototypical for many further solutions to (SEP) in that it has a simple geometric description and possesses a certain optimality property in the class of all solutions.

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1.2 Optimal Skorokhod embedding problem

The Root stopping time solves (OptSEP) in the case where \(\gamma (f,s)= s^2\). Other examples where the solution is known include functions depending on the running maximum \(\gamma ((f,s)):= {\bar{f}}(s):= \max _{t\le s} f(t)\) or functions of the local time at 0.

The solutions to (SEP) have their origins in many different branches of probability theory, and in many cases, the original derivation of the embedding occurred separately from the proof of the corresponding optimality properties. Moreover, the optimality of a given construction is often not immediate; for example, the optimality property of the Root embedding was first conjectured by Kiefer [36] and subsequently established by Rost [50].

In contrast to existing work, we will start with the optimization problem (OptSEP) and we seek a systematic method to determine the minimizer for a given function \(\gamma \). To develop a general theory for this optimization problem we interpret stopping times in terms of a transport plan from the Wiener space \(({C_0(\mathbb {R}_+)},{\mathbb {W}})\) to the target measure \(\mu \), i.e. we want to think of a stopping time \(\tau \) as transporting the mass of a trajectory \((B_t(\omega ))_{t \in \mathbb {R}_+}\) to the point \(B_{\tau (\omega )}(\omega )\in \mathbb {R}.\) Note that this is not a coupling between \({\mathbb {W}}\) and \(\mu \) in the usual sense and one cannot directly apply optimal transport theory. Nevertheless the transport perspective provides a powerful intuition that guides us to develop an analogous theory, which in particular accounts for the adaptedness properties of stopping times. To this end, it is necessary to combine ideas and results from optimal transport with concepts and techniques from stochastic analysis.

As in optimal transport, it is crucial to consider (OptSEP) in a suitably relaxed form, i.e. in (OptSEP) we will optimize over randomized stopping times (see Definition 3.7 below). These can be viewed as usual stopping times on a possibly enlarged probability space but in our context it is more natural to interpret them as stopping times of ‘Kantorovich-type’ (in the sense of optimal transport), i.e. stopping times which terminate a given path not at a single deterministic time instance but according to a distribution.

This relaxation will allow us to transfer many of the convenient properties of classical transport theory to our probabilistic setup. Exactly as in classical transport theory, (OptSEP) can be viewed as a linear optimization problem. The set of couplings in mass transport is compact and similarly the set of all randomized stopping times solving (SEP) on Wiener space is compact in a natural sense. Under the standing assumption that B is defined on a sufficiently rich stochastic basis, these considerations allow us to prove:

Here we can talk about the continuity properties of \(\gamma \) since S possesses a natural Polish topology (cf. (3.1)).

In the language of linear optimization, Theorem 1.1 is a primal problem. It is therefore natural to expect that there exists a corresponding dual problem, and our second main result concerns this duality:

We will prove this result in Sect. 4, and variants of this result will prove to be important in establishing later results. Theorem 1.2 has close analogues in the literature. In particular, using Hobson’s time change argument [30, 31], Theorem 1.2 is comparable to the work of Dolinsky and Soner [20, 21]. Similar duality results in a discrete time framework are established by Bouchard and Nutz [9] among others.

1.3 Geometric characterization of optimizers: monotonicity principle

A fundamental idea in optimal transport is that the optimality of a transport plan is reflected by the geometry of its support set. Often this is key to understanding the transport problem. On the level of support sets, the relevant notion is c-cyclical monotonicity. The relevance of this concept for the theory of optimal transport has been fully recognized by Gangbo and McCann [24], based on earlier work of Knott and Smith [37] and Rüschendorf [51, 52] among others.

Inspired by these results, we establish a monotonicity principle which links the optimality of a stopping time \(\tau \) with ‘geometric’ properties of \(\tau \). Combined with Theorem 1.1, this principle will turn out to be surprisingly powerful. For the first time, all the known solutions to (SEP) with optimality properties can be established through one unifying principle. Moreover, the monotonicity principle allows us to treat the optimization problem (OptSEP) in a systematic manner, generating further embeddings as a by-product.

Our third main result states:

If (1.4) holds, we will loosely say that \(\Gamma \) supports \(\tau \). The significance of Theorem 1.3 is that it links the optimality of the stopping time \(\tau \) with a particular property of the set \(\Gamma \), i.e. \(\gamma \)-monotonicity. In applications, the latter turns out to be much more tangible. We emphasize that we do not require continuity assumptions on \(\gamma \) in this result. This will be important when we apply our results.

Here \((\mathcal {F}_t^B)_{t \ge 0}\) denotes the natural filtration generated by the Brownian motion B. A consequence of considering only \((\mathcal {F}_t^B)_{t \ge 0}\)-stopping times is that the set \(\mathsf {SG}\) does not depend on the particular choice of the underlying stochastic basis.

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We note that a swapping of paths (as illustrated in Fig. 2) was used by Hobson [31, p. 34] to provide a heuristic derivation of the optimality properties of the Root embedding. Indeed Hobson’s approach was the starting point of the present paper.

By the monotonicity principle, Theorem 1.3, an optimal stopping time is supported by a set \(\Gamma \) such that \(\Gamma ^<\times \Gamma \) contains no stop-go pair \(\big ((f,s),(g,t)\big )\). Intuitively, such a pair gives rise to a possible modification, improving the given stopping rule: as \(f(s) = g(t)\), we can imagine stopping the path (f, s) at time s, and allowing (g, t) to go on by transferring all paths which extend (f, s), the ‘remaining lifetime’, onto (g, t), which is now going (see Fig. 2). By (1.6) this guarantees an improved value of \(P_\gamma \), contradicting the optimality of our stopping rule. Observe that the condition \(f(s)=g(t)\) is what guarantees that a modified stopping rule still embeds the measure \(\mu \). In Sect. 2 below we will briefly indicate how the monotonicity principle can be used to derive existing solutions to the Skorokhod embedding problem as well as a whole family of novel solutions to the Skorokhod embedding problem; many further examples will be provided in Sect. 6.

Importantly, the transport-based approach readily admits a number of strong generalizations and extensions. With only minor changes the existence result, Theorem 1.1, the duality result, Theorem 1.2, and the monotonicity principle, Theorem 1.3 below, extend to general starting distributions and Brownian motion in \(\mathbb {R}^d\), and more generally to sufficiently regular Markov processes; see Sects. 5 and 7. This is notable since previous constructions usually exploit rather specific properties of Brownian motion.

The monotonicity principle, Theorem 1.3, represents the culmination of the three main results, and the proof of this result will be the most complex part of this paper, requiring substantial preparation in order to combine the relevant concepts from stochastic analysis and optimal transport. The preparation and proof of this result will therefore comprise the majority of the paper. In fact the proof will automatically imply a stronger version (Theorem 5.7) of Theorem 1.3. For our applications, it will also be helpful to introduce a version of this result which incorporates a secondary optimization, Theorem 5.16.

The ‘classical’ optimal transport version of Theorem 1.3 can be established through fairly direct arguments, at least in a reasonably regular setting, cf. [3, Thms. 3.2,3.3] and [57, p. 88f]. However, these approaches do not extend easily to our setup: stopping times are of course not couplings in the usual sense and there is no reason for particular combinatorial manipulations to carry over in a direct fashion. Another substantial difference is that the procedure of transferring paths described below Definition 1.5 necessarily refers to a continuum of paths while the classical notion of cyclical monotonicity is concerned with rearrangements along finite cycles. The argument given subsequently is more in the spirit of [6, 8] and requires a fusion of ideas from optimal transport and stochastic analysis.

1.4 New horizons

1.5 Background

Since the first solution to (SEP) by Skorokhod [54] the embedding problem has received frequent attention in the literature, with new solutions appearing regularly, and exploiting a number of different mathematical tools. Many of these solutions also prove to be, by design or accident, solutions of (OptSEP) for a particular choice of \(\gamma \), e.g. [4, 33, 45, 48, 50, 56]. The survey [43] is a comprehensive account of all the solutions to (SEP) up to 2004 and references many articles which use or develop solutions to the Skorokhod embedding problem. More recently, novel twists on the classical Skorokhod embedding problem have been investigated by: Last et. al. [38], who consider the closely related problem of finding unbiased shifts of Brownian motion (and where there are also natural connections to optimal transport); Hirsch et. al. [29], who have used solutions to the Skorokhod embedding problem to construct Peacocks; and Gassiat et. al. [25], who have exploited particular properties of Root’s solution to construct efficient numerical schemes for SDEs.

The Skorokhod embedding problem has also recently received substantial attention from the mathematical finance community. This goes back to an idea of Hobson [30]: through the Dambis-Dubins-Schwarz Theorem, the optimization problems (OptSEP) are related to the pricing of financial derivatives, and in particular to the problem of model-risk. We refer the reader to the survey article [31] for further details.

Recently there has been much interest in optimal transport problems where the transport plan must satisfy additional martingale constraints. Such problems arise naturally in the financial context, but are also of independent mathematical interest, for example—mirroring classical optimal transport—they have important consequences for the study of martingale inequalities (see e.g. [9, 28, 44]). The first papers to study such problems include [7, 19, 23, 32], and this field is commonly referred to as martingale optimal transport. The Skorokhod embedding problem has been considered in this context by Galichon et. al.  in [23]; through a stochastic control problem they recover the Azéma–Yor solution of the Skorokhod embedding problem. Notably, their approach is very different from the one pursued in the present paper.

1.6 Outline of the article

In Sect. 2 we establish the Root and the Rost embeddings as a consequence of Theorems 1.1 and 1.3, as well as constructing a family of new embeddings. The results presented in this section are intended as a motivation for the rest of the paper. In the derivation of these embeddings we highlight the interplay between arguments of a probabilistic nature, and arguments relating to the pathwise space S introduced in (1.2). A major benefit of working in these two separate domains is that it is typically relatively easy to prove pointwise statements in the setup of the space S; on the other hand, the associated probabilistic arguments are usually straightforward. However neither set of arguments naturally transfers to the other setup.

The link between these distinct domains is provided by Theorems 1.1 and 1.2, and in particular the monotonicity principle Theorem 1.3 which we establish in Sects. 3–5. In Sect. 3, we introduce a framework that allows us to view classical probabilistic concepts on the pathwise space S and establish a number of auxiliary results that will be needed later on. In Sect. 4 we prove our first two main results. As in the transport case, Theorem 1.1 will be a simple consequence of lower semi-continuity plus compactness of the set of solutions to the Skorokhod problem. To establish Theorem 1.2, we use classical duality results from optimal transport. In Sect. 5 we prove Theorem 1.3 based on a combination of arguments from optimal transport with Choquet’s capacitability theorem and ingredients from stochastic analysis.

In Sect. 6 we use our results to establish all known solutions to (OptSEP) as well as further embeddings. We also give an example in which (OptSEP) admits only optimizers depending on additional randomization. For readers who are mainly interested in these applications, it should be possible to read this section immediately after Sect. 2.

In Sect. 7 we describe a number of extensions of our previous results. In particular we consider general starting distributions and show that our main results extend to continuous Feller processes under certain assumptions which we are able to verify for a large class of processes. As a special case of the results in this section, we also show that, as usual, the moment condition on \(\mu \) can be dropped when the second condition in (SEP) is recast in terms of uniform integrability resp. minimality (cf. (2.1)).

1.7 Frequently used notation

2.1 The Root embedding

We recall the definition of the Root embedding, \(\tau _{\text {Root}}\), from (1.1), and we wish to recover Root’s result [48] from an optimization problem. Remember that, according to Root’s terminology, a (closed) set \({\mathcal {R}} \subseteq \mathbb {R}_+ \times \mathbb {R}\) is a barrier if \((s,x) \in {\mathcal {R}}\) implies \((t,x) \in {\mathcal {R}}\) whenever \(t>s\). Then Root’s construction of a solution to the Skorokhod embedding problem can be summarized as follows:

Proof

Step 1. We first pick—by Theorem 1.1—a stopping time \({\hat{\tau }}\) which attains \(P_\gamma .\) By Theorem 1.3 there exists a set \(\Gamma \subseteq S\) such that \(\left( \left( B_{s}\right) _{s \le {\hat{\tau }}}, {\hat{\tau }}\right) \in \Gamma \) almost surely, and such that \((\Gamma ^{<} \times \Gamma ) \cap \mathsf {SG}= \emptyset \).

Step 2. Next, consider paths \((f,s),(g,t)\in S\) such that \(f(s)=g(t)\). We consider when \(\big ((f,s),(g,t)\big )\in \mathsf {SG},\) i.e. under which conditions (f, s) should be stopped and Brownian motion should continue to go after (g, t). In the present case (1.6) amounts to

$$\begin{aligned} \mathbb {E}[h(s+\sigma )] + h(t) > h(s) + \mathbb {E}[h(t+\sigma )]. \end{aligned}$$

(2.2)

Thus, by strict convexity of \(h,((f,s),(g,t)) \in \mathsf {SG}\) iff \(t <s\). We define two barriers by

$$\begin{aligned} {\mathcal {R}}_\textsc {cl}:=\{(s,x):\exists (g,t)\in \Gamma , g(t)=x, t\le s\},\\ {\mathcal {R}}_\textsc {op}:=\{(s,x):\exists (g,t)\in \Gamma , g(t)=x, t<s\}. \end{aligned}$$

Fix \((g,t) \in \Gamma \). Then we have \((t,g(t)) \in {\mathcal {R}}_{\textsc {cl}}\). Suppose for contradiction that \(\inf \{s \in [0,t]: (s,g(s)) \in {\mathcal {R}}_{\textsc {op}}\} < t\). Then there exists \(s<t\) such that \((f,s) := (g_{\upharpoonright [0,s]},s) \in \Gamma ^{<}\) and \((s, f(s)) \in {\mathcal {R}}_{\textsc {op}}\). By definition of \({\mathcal {R}}_{\textsc {op}}\), it follows that there exists another path \((k,u) \in \Gamma \) such that \(k(u) = f(s)\) and \(u < s\). But then \(( (f,s), (k,u)) \in \mathsf {SG}\cap (\Gamma ^<\times \Gamma )\) which cannot be the case. Hence,

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Step 3. Now consider \(\omega \in \Omega \) such that \((g,t) = ((B_{s}(\omega ))_{s \le {\hat{\tau }}(\omega )}, {\hat{\tau }}(\omega )) \in \Gamma \). Then it follows immediately that:

$$\begin{aligned} \tau _{\textsc {cl}}(\omega ):= & {} \inf \{s : (s,B_s(\omega )) \in {\mathcal {R}}_{\textsc {cl}}\} \nonumber \\\le & {} {\hat{\tau }}(\omega ) \le \inf \{s : (s,B_s(\omega )) \in {\mathcal {R}}_{\textsc {op}}\} =: \tau _{\textsc {op}}(\omega ). \end{aligned}$$

(2.3)

We finally observe that \(\tau _\textsc {cl}= \tau _\textsc {op}\) a.s. by the strong Markov property, and the fact that one-dimensional Brownian motion immediately returns to its starting point. \(\square \)

A consequence of this proof is that (on a given stochastic basis) there exists exactly one solution of the Skorokhod embedding problem which minimizes \(\mathbb {E}[h(\tau )]\); this property was first established in [50], together with the optimality property of Root’s solution. To see this, assume that minimizers \(\tau _1\) and \(\tau _2\) are given. Then we can use an independent coin-flip to define a new minimizer \({\bar{\tau }}\) which is with probability 1 / 2 equal to \(\tau _1\) and with probability 1 / 2 equal to \(\tau _2\). By Theorem 2.1, \({\bar{\tau }}\) is of barrier type and hence \(\tau _1=\tau _2\).

In Sect. 6.3 we will prove generalizations of Theorem 2.1 which admit similar conclusions in \(\mathbb {R}^d\) and for general initial distributions.

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2.2 The Rost embedding

A set \({{\mathcal {R}}} \subseteq \mathbb {R}_+\times \mathbb {R}\) is an inverse barrier if \((s,x)\in {{\mathcal {R}}}\) and \(s > t\) implies that \((t,x)\in {{\mathcal {R}}}\). It has been shown by Rost [50] that under the condition \(\mu (\{0\})=0\) there exists an inverse barrier such that the corresponding hitting time (in the sense of (1.1)) solves the Skorokhod problem (see Fig. 3(A)). It is not hard to see that without this condition some additional randomization is required. We derive this using an argument almost identical to the one above.

As in the case of the Root embedding we obtain that the minimizer of \(\mathbb {E}[h( \tau )]\) is unique.

2.3 The Cave embedding

In this section we give an example of a new embedding that can be derived from Theorem 1.3. It can be seen as a unification of the Root and Rost embeddings. A set \({\mathcal {R}} \subseteq \mathbb {R}_+\times \mathbb {R}\) is a cave barrier if there exists \(t_0\in \mathbb {R}_+\), an inverse barrier \({\mathcal {R}}^0\subseteq [0,t_0]\times \mathbb {R}\) and a barrier \({\mathcal {R}}^1 \subseteq [t_0,\infty )\times \mathbb {R}\) such that \({\mathcal {R}}={\mathcal {R}}^0\cup {\mathcal {R}}^1.\) We will show that there exists a cave barrier such that the corresponding hitting time (in the sense of (1.1)) solves the Skorokhod problem (Fig. 3(B)). We derive this using an argument similar to the one above:

Since this construction does not already appear in the literature, we emphasize that the result remains true for integrable (centered) measures \(\mu \) (see Sect. 7).

2.4 Remarks

In Sect. 6.3 we will show that the arguments above can be adapted to prove the existence of Rost and Root embeddings in a more general setting. Specifically, in Sects. 6 and 7 we will show that this approach generalizes to a multi-dimensional setup and (sufficiently regular) Markov processes. In the case of the Root embedding it does not matter for the argument whether the starting distribution is a Dirac in 0 as in our setup or a more general distribution \(\lambda \). For the Rost embedding a general starting distribution is slightly more difficult. In the case where \(\lambda \) and \(\mu \) have common mass, then it may be the case that \({{\mathrm{proj}}}_{\mathbb {R}_+}({\mathcal {R}}_\textsc {cl}\cap (A \times \mathbb {R}_+)) = \{0\}\) for some set A—that is, all paths which stop at \(x \in A\) do so at time zero. In this case it is possible that \({\hat{\tau }} < \tau _\textsc {op}\) when the process starts in A, and in general, some proportion of the paths starting on A must be stopped instantly. As a result, in the case of general starting measures, independent randomization is necessary. In the Rost case, it is also straightforward to compute the independent randomization which preserves the embedding property.

Other recent approaches to the Root and Rost embeddings can be found in [13, 14, 25, 26]. These papers largely exploit PDE techniques, and as a consequence, are able to produce more explicit descriptions of the barriers, however the methods tend to be highly specific to the problem under consideration.

3.1 Spaces and filtrations

For our arguments it will be important to be precise about the relationship between the sets \({{C_0(\mathbb {R}_+)}} \times \mathbb {R}_+\) and S. We therefore discuss the underlying filtrations in some detail.

We consider two different filtrations on the Wiener space \({{C_0(\mathbb {R}_+)}}\), the canonical or natural filtration \(\mathcal {F}^0=(\mathcal {F}_t^0)_{t\in \mathbb {R}_+}\) as well as its usual augmentation \(\mathcal {F}^a=(\mathcal {F}^a_t)_{t\in \mathbb {R}_+}\). As Brownian motion is a continuous Feller process, all right-continuous \(\mathcal {F}^a\)-martingales are continuous ([47, Theorem VI. 15.4]) and hence all \(\mathcal {F}^a\)-stopping times are predictable and the \(\mathcal {F}^a\)-optional and \(\mathcal {F}^a\)-predictable \(\sigma \)-algebras coincide [46, Corollary IV 5.7]. By [16, Theorem IV. 97, Rem. IV. 98] we also have that the \(\mathcal {F}^0\)-predictable, \(\mathcal {F}^0\)-optional and \(\mathcal {F}^0\)-progressive \(\sigma \)-algebras coincide because \({{C_0(\mathbb {R}_+)}}\) is the set of continuous paths. Moreover, we will use the following result.

The following result is a particular case of [16, Theorem IV. 97] (in somewhat different notation).

The mapping r is not a closed mapping: it is easy to see that there exist closed sets in \({{C_0(\mathbb {R}_+)}} \times \mathbb {R}_+\) with a non-closed image under r. However this does not happen for closed optional sets: it is straightforward that an \(\mathcal {F}^0\)-optional set \(A\subseteq {{C_0(\mathbb {R}_+)}} \times \mathbb {R}_+\) is closed iff the corresponding set r(A) is closed in S.

It is trivially true that an S-continuous process is continuous in the usual pathwise sense. The converse is not generally true—consider the case where \(X_t(\omega )=\text{ sign }(\omega (1))(t-2)_+\). This is a continuous, optional process, however the corresponding function \(X^S\) is not a continuous mapping from S to \(\mathbb {R}\). Other examples arise from functions connected to the local time of Brownian motion, cf. Sect. 6.2.

Clearly, (3.3) defines an \(\mathcal {F}^0_t\)-measurable function which is a version of the classical conditional expectation; subsequently, it will be useful to have this function defined for all \(\omega \). In accordance with Definition 3.3 we write \(X^{M,S}\) for the function satisfying \(X^M = X^{M,S}\circ r\).

For \(X\in C_b({{C_0(\mathbb {R}_+)}})\), \(X^M\) is a martingale with continuous paths and hence satisfies the optional stopping theorem. Using the functional monotone class theorem, we see that the optional stopping theorem holds for \(X^M\) for all bounded measurable \(X: {C_0(\mathbb {R}_+)}\rightarrow \mathbb {R}\). Also one can prove that \(X^M\) has almost surely continuous paths, even if X itself was not continuous, but we will not use this fact.

3.2 Randomized stopping times

Working on the probability space \(({C_0(\mathbb {R}_+)}, {\mathbb {W}})\), a stopping time \(\tau \) is a mapping which assigns to each path \(\omega \) the time \(\tau (\omega ) \) at which the path is stopped. If the stopping time depends on external randomization, then we may consider a path \(\omega \) which is not stopped at a single point \(\tau (\omega )\), but rather that there is a sub-probability measure \(\tau _\omega \) on \(\mathbb {R}\) which represents the probability that the path \(\omega \) is stopped at a given time, conditional on observing the path \(\omega \). The aim of this section is to make this idea precise, and to establish connections with related properties in the literature. Specifically, the notion of a randomized stopping time has previously appeared in e.g. [5, 40, 49].

Subsequently we will identify randomized stopping times as a subset of the well studied \({\mathbf {P}}\)-measures: A finite measure \(\xi \) on \({C_0(\mathbb {R}_+)}\times \mathbb {R}_+\) is a \({\mathbf {P}}\) -measure (wrt \({\mathbb {W}}\)) if it does not charge any \({\mathbb {W}}\)-evanescent set. A basic result of Doléans [18] is the following

Below, it will sometimes be convenient to represent randomized stopping times on an extension of the space \(({C_0(\mathbb {R}_+)}, \mathcal {F}^0, (\mathcal {F}^0_t)_{t\ge 0},{\mathbb {W}})\): we will consider \(({{\overline{C}}_0(\mathbb {R}_+)},{\bar{\mathcal {F}}},({\bar{\mathcal {F}}}_t)_{t \ge 0},\overline{{\mathbb {W}}})\), where \({{\overline{C}}_0(\mathbb {R}_+)}= {C_0(\mathbb {R}_+)}\times [0,1], \overline{{\mathbb {W}}}(A_1\times A_2) = {\mathbb {W}}(A_1) \mathcal {L}(A_2)\) (where \(\mathcal {L}\) denotes Lebesgue measure), \({\bar{\mathcal {F}}}\) is the completion of \(\mathcal {F}^0 \otimes {\mathcal {B}}([0,1])\), and \({\bar{\mathcal {F}}}_t\) the usual augmentation of \((\mathcal {F}_t^0 \otimes {\mathcal {B}}([0,1]))_{t \ge 0}\). We will write \({\bar{B}}=({\bar{B}}_t)_{t\ge 0}\) for the process given by \({\bar{B}}_t(\omega ,u)=\omega _t.\) Observe that if \(Y_t(\omega ,u) = u\), then \(({\bar{B}}_t, Y_t)\) is (trivially) a continuous Feller process, and hence by the same arguments as above, the \({\bar{\mathcal {F}}}\)-predictable and \({\bar{\mathcal {F}}}\)-optional \(\sigma \)-algebras coincide.

Randomized stopping times play a key role in this paper; depending on the respective context, the following different characterizations will be useful:

An immediate consequence of Theorem 3.8 (3) is the following

The next lemma implies that optimizing over usual stopping times on a rich enough probability space in (OptSEP) is equivalent to optimizing over randomized stopping times on Wiener space.

3.3 Randomized stopping times solving the Skorokhod problem and compactness

For us it is crucial that randomized stopping times have the following property:

Our use of randomization to achieve compactness of a set of stopping times has similarities to the work of Baxter and Chacon [5]. However their setup is different, and their intended applications are not connected to Skorokhod embedding.

3.4 Joinings

4.1 The primal problem

By Lemma 3.11, Theorem 1.1 is a consequence of this result.

In Sect. 7 below we establish existence of a minimizing stopping time in the case where the measure \(\mu \) does not necessarily admit a finite second moment. However we will then replace Assumption (4.2) by the requirement that \(\gamma \) is bounded from below.

4.2 The dual problem

The following result implies Theorem 1.2.

We will establish Theorem 4.2 as a consequence of the following auxiliary duality result, where we write T for the projection map \({C_0(\mathbb {R}_+)}\times \mathbb {R}_+\times \mathbb {R}\rightarrow \mathbb {R}_+,T(\omega ,t,y)=t\).

Proposition 4.3 should be compared to the (formally) very similar classical duality theorem of optimal transport, see e.g. [58, Section 5] for a proof as well as for a discussion of its origin and related literature.

The strategy of the proof of Proposition 4.3 is to establish the duality relation \((\star )\) for \(\pi \), resp. \((\varphi , \psi ) \) taken from certain larger candidate sets, in which case the duality relation follows from Theorem 4.4. Then we introduce additional constraints via a variational approach to obtain an improved duality through the following min-max theorem.

Next consider the constraints

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Using the min–max theorem (Theorem 4.5) with the function \(F(\pi ,\alpha )= \int c +\alpha (T-V) \, d\pi \), the set of \(\pi \) satisfying (\(p[t_0]\)), and \(\alpha \ge 0\) we thus obtain

$$\begin{aligned} \inf _{\pi \text { sat. } (p[t_0, V])}{\int } c\, d\pi&= \inf _{\pi \text { sat. } (p[t_0])} \left( {\int } c\, d\pi + \sup _{\alpha \ge 0} \alpha \int T-V \, d\pi \right) \nonumber \\&=\sup _{\alpha \ge 0} \, \inf _{\pi \text { sat. } (p[t_0])}{\int } c +\alpha (T-V) \, d\pi \end{aligned}$$

(4.7)

$$\begin{aligned}&=\sup _{\alpha \ge 0} \, \sup _{(\varphi ,\psi ) \text { sat. } (d[c +\alpha (T-V),t_0]) } {\mathbb {W}}(\varphi ) + \mu (\psi )\nonumber \\&=\sup _{(\varphi ,\psi )\text { sat. } (d[c ,t_0, V])} {\mathbb {W}}(\varphi ) + \mu (\psi ), \end{aligned}$$

(4.8)

where we applied Claim 1 to the function \({\tilde{c}}=c +\alpha (T-V)\) to establish the equality between (4.7) and (4.8). Hence we obtain:

In the next step we will drop \(t_0\) and consider the constraints

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Recalling (3.5), \(\pi \in \mathsf {JOIN}^{1,V}(\mu )\) if and only if

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here, k enforces the condition that \(\mathsf {proj}_{ {{C_0(\mathbb {R}_+)}}\times \mathbb {R}_+}(\pi _{\upharpoonright {{C_0(\mathbb {R}_+)}}\times \mathbb {R}_+ \times D}) \in \mathsf {RST}\) for all Borel sets D. We will apply the min-max theorem to \( F(\pi , h )= \int c+ h\, d\pi , \) where \(\pi \) satisfies (p[V]) and

$$\begin{aligned} h(\omega ,s,y)= {\sum }_{i=1}^{n} f_i(s) (g_i-\mathbb {E}[g_i|\mathcal {F}_{t_i}^0])(\omega )k_i(y), \end{aligned}$$

(4.9)

\(n\in \mathbb {N},f_i\in C_b(\mathbb {R}_+), {{\mathrm{supp}}}\,f_i\subseteq [0,t_i],t_i\ge 0\), \(g_i\in C_b({C_0(\mathbb {R}_+)}),k_i\in C_b(\mathbb {R})\).

It follows that \((\varphi ,\psi )\) satisfy (\(d^M[c,V]\)). Thus (4.10) yields the non-trivial part of \((\star )\) for the constraints \((p^M[V])\), (\(d^M[c,V]\)) in the case of continuous bounded c. As above, the extension to lsc c is straightforward. \(\square \)

4.3 General starting distribution

5.1 Proof of Proposition 5.8

If we are able to construct such a pair \(\xi _0^\pi , \xi _1^\pi \), then \(\xi ^\pi :=(\xi _0^\pi +\xi _1^\pi )/2 \in \mathsf {RST}(\mu )\) is strictly better than \(\xi \) and therefore yields the desired contradiction.

5.2 Proof of Proposition 5.9

Only the implication (1) \(\Rightarrow \) (2) of Proposition 5.9 is non-trivial. The proof is based on Choquet’s capacitability theorem and the following auxiliary duality result which is closely related to Proposition 4.3. We fix \(t_0\in \mathbb {R}_+\) and set \(S_{t_0}:=\{(f,s)\in S: s\le t_0\}\).

We now state several consequences of Proposition 5.11 in which we switch the roles of \(\inf \) and \(\sup \) to provide a more natural formulation.

5.3 A secondary minimization result

We need an extended version of the stop-go pairs introduced in Definition 5.3.

Then we have the following generalization of Theorem 5.7.

The proof given for Theorem 5.7 also applies in the present situation. Hence, the result follows immediately from the following straightforward variant of Proposition 5.8.

6.1 Recovering classical embeddings

In this section we derive a number of classical embeddings as well as establish new embeddings. Figure 4 shows graphical representations of some of these constructions. We highlight the common feature of all these pictures: when plotted in an appropriate phase space, the stopping time is the hitting time of a barrier type set. Identifying the appropriate phase space, and determining the exact structure of the barrier will be the key step in deriving the solutions to (SEP) in this section.

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For subsequent use, it will be helpful to write, for \((f,s) \in S,\bar{f} = \sup _{r \le s} f(r),\underline{f} = \inf _{r \le s} f(r)\) and \(|f|^* = \sup _{r \le s} |f(r)|\).

Proof

Fix a bounded and strictly increasing continuous function \(\varphi :\mathbb {R}_+\rightarrow \mathbb {R}\) and consider the continuous functions \(\gamma ((f,s)) = -\bar{f}\) and \({\tilde{\gamma }}((f,s)) = \varphi (\bar{f})(f(s))^2\). Then \((\hbox {OptSEP}_2)\) is well posed and by Theorem 6.1 there exists a minimizer \(\tau _{AY}\). By Theorem 6.4, pick a \({\tilde{\gamma }}|\gamma \)-monotone set \(\Gamma \subseteq S\) supporting \(\tau _{AY}\). We claim that

$$\begin{aligned} {\mathsf {SG}_2}\supseteq \{((f,s),(g,t))\in S\times S: g(t)=f(s), \bar{g} <\bar{f} \}. \end{aligned}$$

(6.6)

This is represented graphically in Fig. 5.

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Indeed, pick \(((f,s),(g,t))\in S\times S\) with \(f(s)=g(t)\) and \( \bar{g}< \bar{f}\) and a stopping time \(\sigma \) with positive and finite expectation. Then (6.1) amounts to

$$\begin{aligned} \mathbb {E}\big [{\bar{f}} \vee (f(s)+{\bar{B}}_\sigma )\big ] + \bar{g} \le \bar{f} + \mathbb {E}\big [{\bar{g}} \vee (g(t) + {\bar{B}}_\sigma )\big ] \end{aligned}$$

with a strict inequality unless \({\bar{g}} \ge g(t) + {\bar{B}}_\sigma \) a.s. However in that case (6.2) is trivially satisfied and (6.3) amounts to

$$\begin{aligned} \mathbb {E}\big [\varphi ({\bar{f}}) (f(s)+B_\sigma )^2 \big ] + \varphi ({\bar{g}} ) g(t)^2 >\varphi ({\bar{f}} )f(s)^2 + \mathbb {E}\big [\varphi ({\bar{g}}) (g(t)+B_\sigma )^2 \big ] \end{aligned}$$

which holds since \(g(t)=f(s)\). Summing up, \(((f,s),(g,t))\in \mathsf {SG}\subseteq {\mathsf {SG}_2}\) in the former case and \(((f,s),(g,t))\in {\mathsf {SG}_2}\) in the latter case, proving (6.6).

In complete analogy with the derivation of the Root embedding (Theorem 2.1) we define

$$\begin{aligned} {\mathcal {R}}_\textsc {cl}&:= \left\{ (m,x): \exists (g,t) \in \Gamma , \bar{g} \le m, g(t) = x\right\} ,\\ {\mathcal {R}}_\textsc {op}&:= \left\{ (m,x): \exists (g,t) \in \Gamma , \bar{g} < m, g(t) = x \right\} , \end{aligned}$$

and write \(\tau _\textsc {cl}, \tau _\textsc {op}\) for the first times the process \((\bar{B}_t(\omega ),{B}_t(\omega ))\) hits the sets \({\mathcal {R}}_\textsc {cl}\) and \({\mathcal {R}}_\textsc {op}\) respectively. Then we claim \(\tau _\textsc {cl}\le \tau _{AY} \le \tau _\textsc {op}\) a.s. Note that \(\tau _\textsc {cl}\le \tau _{AY}\) holds by definition of \(\tau _\textsc {cl}.\) To show \(\tau _{AY} \le \tau _\textsc {op}\), consider \(\omega \) satisfying \(((B_s(\omega ))_{s\le \tau _{AY}(\omega )},\tau _{AY}(\omega ))\in \Gamma \) and assume for contradiction that \(\tau _\textsc {op}(\omega )<\tau _{AY}(\omega ).\) Then there exists \(s\in [\tau _\textsc {op}(\omega ),\tau _{AY}(\omega ))\) such that \(f:=(B_r(\omega ))_{r\le s}\) satisfies \(({\bar{f}}, f(s))\in {\mathcal {R}}_\textsc {op}\). Since \(s< \tau _{AY}(\omega )\) we have \((f,s)\in \Gamma ^<\). By definition of \({\mathcal {R}}_\textsc {op}\), there exists \((g,t)\in \Gamma \) such that \(f(s)= g(t)\) and \({\bar{g}} < {\bar{f}}\), yielding a contradiction.

Finally, we define

$$\begin{aligned} \psi _0(m) = \sup \{x: (m,x) \in {\mathcal {R}}_\textsc {cl}\}. \end{aligned}$$

It follows from the definition of \({\mathcal {R}}_\textsc {cl}\) that \(\psi _0(m)\) is increasing, and we define the right-continuous function \(\psi _+(m) = \psi _0(m+)\), and the left-continuous function \(\psi _-(m) = \psi _0(m-)\). It follows from the definitions of \(\tau _{\textsc {op}}\) and \(\tau _{\textsc {cl}}\) that:

$$\begin{aligned} \tau _+ := & {} \inf \{t \ge 0: B_t \le \psi _+(\bar{B}_t)\} \le \tau _{\textsc {cl}} \le \tau _{\textsc {op}} \\\le & {} \inf \{t \ge 0: B_t < \psi _-(\bar{B}_t)\} =: \tau _-. \end{aligned}$$

It is then easily checked that \(\tau _- = \tau _+\) a.s., and the result follows on taking \(\psi = \psi _+\). \(\square \)

6.2 The Vallois-embedding and optimizing functions of local time

We say that a \(\mathcal {G}\)-adapted process \(\mathfrak {L}^x\) is a local time in x if it is a (right-continuous, increasing) compensator of \(|B-x|\) and we suppress x in the case of local time at 0. This determines \(\mathfrak {L}^x\) up to indistinguishability (and clearly the choice of \(\mathfrak {L}^x\) is irrelevant for (6.7)).

For us it is convenient to allow local time to assume the value \(+\infty \) on an evanescent set. Using this convention, Theorem 4.1 implies that there exists a Borel function \(L^x:S\rightarrow [0,\infty ]\) such that \(L^x\circ r \) is a (right-continuous, increasing) \(\mathcal {F}^0\)-predictable local time on Wiener space. We will call such a process \(L^x\) a raw local time in x. We note that the value \(+\infty \) cannot be avoided here, see [42].

Our next goal is to verify that (6.7) admits an optimizer.

Note added in revision. Guo et al. [27] were able to relax the continuity assumption in our existence and duality results Theorems 1.1 and 1.2. Based on the work of Jacod and Memin [34] they establish these results under the assumption that \(t\mapsto \gamma \circ r(\omega , t)\) is lsc for every \(\omega \in {C_0(\mathbb {R}_+)}\). In particular their results would imply a more general version of Proposition 6.13.

We are now able to show:

6.3 Root and Rost embeddings in higher dimensions

In the case \(d=2\) it follows from Falkner’s results [22] that the Skorokhod problem admits a solution (i.e. \(\mathsf {RST}(\mu )\ne \emptyset \)) if (6.10) is satisfied for \(u(x,y)=-\ln |x-y|\) and then (6.11) applies.

In either case, assuming that we do have a solution satisfying (6.11), then the existence result as well as the monotonicity principle carry over to the present setup (with identical proofs) and we are able to state the following:

The proof of this result is much the same as that of Theorem 2.1, except we no longer show that \(\tau _\textsc {cl}= \tau _\textsc {op}\). In higher dimensions with general initial laws, it is easy to construct examples where there are common atoms of \(\lambda \) and \(\mu \), but where the size of the atom in \(\lambda \) is strictly larger than the atom of \(\mu \). By the transience of the process, it is clear that the optimal (indeed, only) behaviour is to stop mass starting at such a point immediately with a probability strictly between 0 and 1, however the stopping times \(\tau _\textsc {cl}\) and \(\tau _\textsc {op}\) will always stop either all the mass, or none of this mass respectively. For this reason, we do not say anything about the behaviour of \({\hat{\tau }}\) when \({\hat{\tau }} = 0\). Trivially, the above result tells us that the solution of the optimal embedding problem is given by a barrier if there exists a set D such that \(\lambda (D) = 1 = \mu (D^c)\).

We now consider the generalization of the Rost embedding. Recall that \((\min (\lambda , \mu ))(A) := \inf _{B \subseteq A} \left( \lambda (B)+ \mu (A{\setminus } B)\right) \) defines a measure.

6.4 An optimal Skorokhod embedding problem which admits only randomized solutions

By analogy with optimal transport, we might interpret a ‘natural stopping time’ (i.e. a stopping time wrt to the Brownian filtration) which solves (OptSEP) as a Monge-type solution whereas stopping times which depend on additional randomization are of Kantorovich-type. With the exception of the Rost solution, all optimal stopping times encountered in the previous section are natural stopping times, and in the Rost case external randomization is only needed at time 0. One might ask whether the optimal Skorokhod embedding problem always admits a solution \(\tau \) which is natural on \(\{\tau >0\}\). We sketch an example, showing that this is not the case:

In optimal transport it is a difficult and interesting problem to understand under which conditions transport problems admit solutions of Monge-type. An interesting subject for future research would be to understand when Monge-type solutions exist for the optimal Skorokhod embedding problem.

7.1 Sketch of proofs

As in Sect. 3 we consider the canonical setup \((C(\mathbb {R}_+,\mathbb {R}^d),\mathcal {F}^0,\mathbb {Q})\) (where \(\mathbb {Q}\) denotes the law of the Feller process) and we write Y for the canonical process. It follows from continuity of Y (resp. Z) and the Feller property that the \(\mathcal {F}^a\)-optional and the \(\mathcal {F}^a\)-predictable \(\sigma \)-algebra on the canonical space agree; similarly Proposition 3.5 on the definition of S-continuous martingales extends to the present context. We define \(\mathsf {RST},\mathsf {JOIN}\) and related notions as before with \(\mathbb {Q}\) replacing \({\mathbb {W}}\). We say that \(\xi \in \mathsf {RST}\) is a minimal embedding of \(\mu \) if the corresponding stopping time \(\rho \) (cf. (3.6)) on the enlarged probability space \((C(\mathbb {R}_+,\mathbb {R}^d)\times [0,1],{\bar{\mathbb {Q}}})\) constitutes a minimal embedding. (Representing randomized stopping times as in Theorem 3.8 (1), the stopping time \(\xi \) constitutes a minimal embedding iff there is no randomized stopping time \(\xi '\ne \xi \) embedding the same measure which satisfies \(A^{\xi '}\ge A^\xi \).) For \(\mu \in {\mathcal {P}}(\mathbb {R}^d)\) we define \(\mathsf {RST}(\mu )\) to be the set of all minimal randomized stopping times embedding the measure \(\mu \).

Recalling the argument from Theorem 3.14, we see that the existence of a function \(\zeta : S \rightarrow \mathbb {R}\) such that \(\zeta \circ r\) increases to \(\infty \) and \(\sup _{\xi \in \mathsf {RST}(\mu )} \xi (\zeta \circ r)<\infty \) implies that \(\mathsf {RST}(\mu )\) is compact. (Vice versa, if \(\mathsf {RST}(\mu )\) is compact then such a function exists and can be chosen so that \(\zeta \circ r\) is deterministic). Hence, by (7.3) resp. (7.1), \(\mathsf {RST}(\mu )\) is compact.

7.2 Examples

We now provide a list of Examples in which Assumption 7.1 is satisfied and Theorems 7.2, 7.3, and 7.4 apply.