An FBSDE approach to the Skorokhod embedding problem for Gaussian processes with non-linear drift

We solve the Skorokhod embedding problem for a class of Gaussian processes including Brownian motion with non-linear drift. Our approach relies on solving an associated strongly coupled system of Forward Backward Stochastic Differential Equation (FBSDE), and investigating the regularity of the obtained solution. For this purpose we extend the existence, uniqueness and regularity theory of so called decoupling fields for Markovian FBSDE to a setting in which the coefficients are only locally Lipschitz continuous.


Introduction
The Skorokhod embedding problem (SEP) stimulates research in probability theory now for over 50 years. The classical goal of the SEP consists in finding, for a given Brownian motion W and a probability measure ν, a stopping time τ such that W τ possesses the law ν. It was first formulated and solved by Skorokhod [Sko61,Sko65] in 1961. Since then there appeared many different constructions for the stopping time τ and generalizations of the original problem in the literature. Just to name some of the most famous solutions to the SEP we refer to Root [Roo69], Rost [Ros71] and Azéma-Yor [AY79]. A comprehensive survey can be found in [Obł04].
Recently, the Skorokhod embedding raised additional interest because of its new applications in financial mathematics, as for instance to obtain model-independent bounds on lookback options [Hob98] or on options on variance [CL10,CW13,OdR13]. An introduction to this close connection of the Skorokhod embedding problem and robust financial mathematics can be found in [Hob11].
In this paper we construct a solution to the Skorokhod embedding problem for Gaussian process G of the form where G 0 ∈ R is a constant and α, β : [0, ∞) → R are suitable functions. Especially, this class of processes includes Brownian motions with non-linear drift. The SEP for Brownian motion with linear drift was first solved in the technical report [Hal68] and 30 years later again in [GF00] and [Pes00]. Techniques developed in these works can be extended to time-homogeneous diffusions, as done in [PP01], and can be seen as generalization of the Azéma-Yor solution. However, to the best of our knowledge there exists no solution so far for the case of a Brownian motion with non-linear drift.
The spirit of our approach is related to the one by Bass [Bas83], who employed martingale representation to find an alternative solution of the SEP for the Brownian motion. This approach was further developed for the Brownian motion with linear drift in [AHI08] and for time-homogeneous diffusion in [AHS15]. It rests upon the observation that the SEP may be viewed as the weak version of a stochastic control problem: the goal is to steer G in such a way that it takes the distribution of a prescribed law, which in case of zero drift is closely related to the martingale representation of a random variable with this law. We therefore propose in this paper to formulate and solve the SEP for G in terms of a fully coupled Forward Backward Stochastic Differential Equation (FBSDE).
In general terms, the dynamics of a system of FBSDE is expressed by the equations with coefficient functions µ, σ of the forward part, terminal condition ξ and driver f of the backward component. In recent decades the theory of FBSDE with its close connection to quasi-linear partial differential equations and their viscosity solutions has been propagated extensively, in particular in its numerous areas of applications as stochastic control and mathematical finance (see [EPQ97] or [PW99]). There are mainly three methods to show the existence of a solution for a system of FBSDE: the contraction method [Ant93,PT99], the four step scheme [MPY94] and the method of continuation [HP95,Yon97,MY99]. As a unified approach, [MWZZ15] (see also [Del02]) designed the theory of decoupling fields for FBSDE, which was refined and extended to a multidimensional setting in [FI13,Fro15]. It can be seen as an extension of the contraction method. In our approach of the SEP via FBSDE, we shall focus on

An FBSDE approach to the Skorokhod embedding problem
We consider a filtered probability space (Ω, F, (F t ) t∈[0,∞) , P) large enough to carry a one-dimensional Brownian motion W and with F := σ ( ∞ t=0 F t ). The filtration (F t ) is assumed to be generated by the Brownian motion and to be augmented by P-null sets. where G 0 ∈ R is a constant and α, β : [0, ∞) → R are deterministic measurable processes such that t 0 |α s | ds + t 0 β 2 s ds < ∞ for all t ≥ 0, find an integrable (F t )-stopping time τ together with a starting point c ∈ R such that c + G τ has the law ν.
In order to have a truly stochastic problem β should not vanish and ν should not be a Dirac measure. In fact, we will assume that β is bounded away from zero later on.
Our method of solving this problem is based on the observation that it may be viewed as the weak version of a stochastic control problem: We want to steer G in such a way that it takes the distribution of a prescribed law. The spirit of our approach is related to an approach to the original Skorokhod embedding problem by Bass [Bas83] that was later extended to the Brownian motion with linear drift in [AHI08]. The procedure of both papers can be briefly summarized and divided into the following four steps.
1. Construct a function g : R → R such that g(W 1 ) has the given law ν. 2. Use the martingale representation property of the Brownian motion for α ≡ 0 and β ≡ 1 or BSDE techniques for α ≡ κ = 0 and β ≡ 1 to solve Y t = g(W 1 ) − κ 0 Z s dW s into a Brownian motion B. This also provides a random timeτ := 1 0 Z 2 s ds fulfilling Bτ + κτ + Y 0 = g(W 1 ), which is why Bτ + κτ + Y 0 has the law ν. 4. Show thatτ is a stopping time with respect to the filtration generated by B through an explicit characterization using the unique solution of an ordinary differential equation. With this description transform the embedding with respect to B into one with respect to the original Brownian motion W to obtain the stopping time τ as the analogue toτ .
The first step of the algorithm just sketched is fairly easy. Let F : R → [0, 1] such that F (x) := ν((−∞, x]) is the cumulative distribution function associated with ν and define F −1 : (0, 1) → R via F −1 (y) := inf{x ∈ R : F (x) ≥ y}. Denoting by Φ the distribution function of the standard normal distribution, we define g : R → R by g(x) := F −1 (Φ(x)). It is straightforward to prove that g has the following properties.
An FBSDE approach to the Skorokhod embedding problem Since we want to require as little regularity as possible for the processes involved, we use the concept of weak differentiability. We recall that a measurable f : for any smooth test function ϕ : R n → R with compact support, for almost all ω ∈ Ω. Now we define a measurable functionδ : ). Conversely, for every weakly differentiable functionδ : [0, ∞) → R we can set G 0 :=δ(0) and α s :=δ (s).
Note that H is weakly differentiable, monotonically increasing and starts at 0. If we assume that β is bounded away from 0, H becomes strictly increasing and invertible such that the inverse function H −1 is monotonically increasing and Lipschitz continuous. In this case we can define δ :=δ • H −1 . Notice, if β ≡ 1, then H = Id and thus δ =δ.
For the second step we assume that β is bounded away from 0 and observe that the random time change, which turns the martingale · 0 Z s dW s into a Gaussian process of the form · 0 β s dB s simultaneously turns the scale process . 0 Z 2 s ds into · 0 β 2 s ds = H. This means we have to modify the classical martingale representation of g(W 1 ) to which amounts to finding a solution (Y, Z) to the equation For δ(t) ≡ 0 this would be just the usual martingale representation with respect to the Brownian motion. Also for a linear drift δ(t) = κt and β ≡ 1 equation (2.3) can be rewritten as In the entire paper we assume that β is bounded away from 0, i.e. inf s∈[0,∞) |β s | > 0.
Observe that the condition r <τ is equivalent to r 0 β 2 s ds < 1 0 Z 2 s ds. Since Y σr is a continuous martingale with quadratic variation H(r) = r 0 β 2 s ds, we can define a Brownian motion B by As an immediate consequence of the previous lemma we observe the following fact:

Decoupling fields for fully coupled FBSDEs
The theory of FBSDEs, closely connected to the theory of quasi-linear partial differential equations and their viscosity solutions, receives its general interest from numerous areas of application among which stochastic control and mathematical finance are the most vivid ones in recent decades (see [EPQ97] or [PW99]). Owing to their general significance, we treat the theory of FBSDEs and their decoupling fields in a more general framework than might be needed to obtain a solution to our equation (2.3).
Although in Section 3.2 we will focus on the Markovian case, which means that all involved coefficients are purely deterministic, let us dwell in a more general setting first.

General decoupling fields
For a fixed finite time horizon T > 0, we consider a complete filtered probability space (Ω, F, (F t ) t∈[0,T ] , P), where F 0 contains all null sets, (W t ) t∈[0,T ] is a d-dimensional Brownian motion independent of F 0 , and F t := σ(F 0 , (W s ) s∈[0,t] ) with F := F T . The dynamics of an FBSDE is classically given by An FBSDE approach to the Skorokhod embedding problem for s, t ∈ [0, T ] and X 0 ∈ R n , where (ξ, (µ, σ, f )) are measurable functions such that for d, n, m ∈ N. Throughout the whole section µ, σ and f are assumed to be progressively measurable with respect to (F t ) t∈[0,T ] .
A decoupling field comes with an even richer structure than just a classical solution.
T ] with t 1 ≤ t 2 and any F t1 -measurable X t1 : Ω → R n there exist progressively measurable processes (X, Y, Z) on [t 1 , t 2 ] such that for all s ∈ [t 1 , t 2 ]. In particular, we want all integrals to be well-defined.
Some remarks about this definition are in place.
• The first equation in (3.1) is called the forward equation, the second the backward equation and the third will be referred to as the decoupling condition.
• Note that, if t 2 = T , we get Y T = ξ(X T ) a.s. as a consequence of the decoupling condition together with u(T, ·) = ξ. At the same time Y T = ξ(X T ) together with decoupling condition implies u(T, ·) = ξ a.e. • If t 2 = T we can say that a triplet (X, Y, Z) solves the FBSDE, meaning that it satisfies the forward and the backward equation, together with Y T = ξ(X T ). This relationship Y T = ξ(X T ) is referred to as the terminal condition.
In contrast to classical solutions of FBSDEs, decoupling fields on different intervals can be pasted together.
We want to remark that, if u is a decoupling field andũ is a modification of u, i.e. for each s ∈ [t, T ] the functions u(s, ω, ·) andũ(s, ω, ·) coincide for almost all ω ∈ Ω, thenũ is also a decoupling field to the same problem. Hence, u could also be referred to as a class of modifications and a progressively measurable representative exists if the decoupling field is Lipschitz continuous in x (Lemma 2.1.3 in [Fro15]).
For the following we need to fix briefly further notation.
Let I ⊆ [0, T ] be an interval and u : I × Ω × R n → R m a map such that u(s, ·) is measurable for every s ∈ I. We define where inf ∅ := ∞. We also set L u,x := ∞ if u(s, ·) is not measurable for every s ∈ I. One can show that L u,x < ∞ is equivalent to u having a modification which is truly Lipschitz continuous in x ∈ R n .
We denote by L σ,z the Lipschitz constant of σ w.r.t. the dependence on the last component z and w.r.t. the Frobenius norms on R m×d and R n×d . We set L σ,z = ∞ if σ is not Lipschitz continuous in z.
Lσ,z we mean 1 Lσ,z if L σ,z > 0 and ∞ otherwise. For an integrable real valued random variable F the expression E t [F ] refers to E[F |F t ], while Et ,∞ [F ] refers to ess sup E[F |F t ], which might be ∞, but is always well defined as the infimum of all constants c ∈ [−∞, ∞] such that E[F |F t ] ≤ c a.s. Additionally, we write F ∞ for the essential supremum of |F |.
In practice it is important to have explicit knowledge about the regularity of (X, Y, Z).
For instance, it is important to know in which spaces the processes live, and how they react to changes in the initial value.
1. We say u to be weakly regular if L u,x < L −1 σ,z and sup s∈[t,T ] u(s, ·, 0) ∞ < ∞. 2. A weakly regular decoupling field u is called strongly regular if for all fixed t 1 , t 2 ∈ [t, T ], t 1 ≤ t 2 , the processes (X, Y, Z) arising in (3.1) are a.e unique and for each constant initial value X t1 = x ∈ R n . In addition they are required to be measurable as functions of (x, s, ω) and even weakly differentiable w.r.t. x ∈ R n such that for every s ∈ [t 1 , t 2 ] the mappings X s and Y s are measurable functions of (x, ω) and even weakly differentiable w.r.t. x such that ess sup x∈R n sup 3. We say that a decoupling field on [t, T ] is strongly regular on a subinterval [t 1 , t 2 ] ⊆ [t, T ] if u restricted to [t 1 , t 2 ] is a strongly regular decoupling field for (u(t 2 , ·), (µ, σ, f )).
A brief discussion of existence and uniqueness of classical solutions can be found in Remark 2.2.4 in [Fro15]. For later reference we give the following remarks (cf. Remarks 2.2.2 and 2.2.3 in [Fro15]).
Remark 3.5. It can be observed from the proof that the supremum of all h = T − t, with t satisfying the properties required in Theorem 3.4 can be bounded away from 0 by a bound, which only depends on the Lipschitz constant of (µ, σ, f ) with respect to the last 3 components, T , L σ,z , L ξ and L ξ · L σ,z < 1, and which is monotonically decreasing in these values.
Furthermore, we notice from the proof that our decoupling field u on More precisely, C depends only on T , L, L ξ,x , L ξ,x L σ,z and is monotonically increasing in these values.
This local theory for decoupling fields can be systematically extended to global results based on fairly simple "small interval induction" arguments (Lemma 2.5.1 and 2.5.2 in [Fro15]).

Markovian decoupling fields
A system of FBSDEs given by (ξ, (µ, σ, f )) is said to be Markovian if these four coefficient functions are deterministic, that is, if they depend only on (t, x, y, z). In the Markovian situation we can somewhat relax the Lipschitz continuity assumption and still obtain local existence together with uniqueness. What makes the Markovian case so special is the property which comes from the fact that u will also be deterministic. This property allows us to bound Z by a constant if we assume that σ is bounded.
Lemma 3.7 ([Fro15], Lemma 2.5.13, 2.5.14 and 2.5.15). Let (ξ, (µ, σ, f )) be deterministic functions and satisfy (SLC). Suppose that there exist a weakly regular decoupling field u on an interval [t, T ] 1. The decoupling field u is deterministic in the sense that it has a modification which is a function of (r, If u is a strongly regular and deterministic decoupling field, then u is continuous in the sense that it has a modification which is a continuous function In the Markovian case this boundedness of Z motivates the following definition, which will allow us to develop a theory for non-Lipschitz problems via truncation. Definition 3.8. Let t ∈ [0, T ] and let (ξ, (µ, σ, f )) be deterministic functions. We call a function u : u is a decoupling field in the sense of Definition 3.1 and additionally Z ∞ < ∞.
The regularity properties for Markovian decoupling fields are analogously defined as for the standard decoupling fields (cf. Definition 3.3), which a slightly modification for strongly regular decoupling fields. Definition 3.9. Let u : [t, T ] × Ω × R n → R m be a Markovian decoupling field to (ξ, (µ, σ, f )). We call a weakly regular u strongly regular if for all fixed t 1 , t 2 ∈ [t, T ], t 1 ≤ t 2 , the processes (X, Y, Z) arising in the defining property of a Markovian decoupling field are a.e. unique for each constant initial value X t1 = x ∈ R n and satisfy (3.2). In addition they must be measurable as functions of (x, s, ω) and even weakly differentiable w.r.t. x ∈ R n such that for every s ∈ [t 1 , t 2 ] the mappings X s and Y s are measurable functions of (x, ω), and even weakly differentiable w.r.t. x such that (3.3)

holds.
Let us introduce the assumption on the coefficients for which an existence and uniqueness theory will be developed.
We begin by providing a local existence result.
Theorem 3.10. Let (ξ, (µ, σ, f )) satisfy (MLLC). Then there exists a time t ∈ [0, T ) such that (ξ, (µ, σ, f )) has a unique weakly regular Markovian decoupling field u on [t, T ]. This u is also strongly regular, deterministic, continuous and satisfies sup t1,t2,Xt 1 Z ∞ < ∞, where t 1 < t 2 are from [t, T ] and X t1 is an initial value (see the definition of a Markovian decoupling field for the meaning of these variables).
Proof. For any constant H > 0 let χ H : R m×d → R m×d be defined as It is easy to check that χ H is Lipschitz continuous with Lipschitz constant L χ H = 1 and bounded by H.
The boundedness of χ H together with its Lipschitz continuity makes (µ H , σ H , f H ) Lipschitz continuous with some Lipschitz constant L H . Furthermore, L σ H ,z ≤ L σ,z . Also (µ H , σ H , f H ) have linear growth in (y, z) as required by Lemma 3.7. According to Theorem 3.4 we know that the problem given by (ξ, (µ H , σ H , f H )) has a unique weakly regular decoupling field u on some small interval [t , T ] where t ∈ [0, T ). We also know that this u is strongly regular, u is deterministic (by Lemma 3.7), and continuous (by Lemma 3.7).
We will show that for sufficiently large H and t ∈ [t , T ) it will also be a Markovian decoupling field to the problem (ξ, (µ, σ, f )). By Remark 3.5 we obtain L u(t,·), For any t 1 ∈ [t , T ] and F t1 -measurable initial value X t1 consider the corresponding unique (X, Y, Z) on [t 1 , T ] satisfying the forward equation, the backward equation and the decoupling condition for µ H , σ H , f H and u. Using Lemma 3.7 we have Now we only need to choose H large enough such that becomes smaller than H 4 , and then in the second step choose t close enough to T such that L σ,z C H (T − t) 1 4 becomes smaller than 1 T ] the process Z a.e. does not leave the region in which the cutoff is "passive", i.e. the ball of radius H. Therefore, u restricted to the interval [t, T ] is a decoupling field to (ξ, (µ, σ, f )), not just to (ξ, (µ H , σ H , f H )). It is even a Markovian decoupling field due to the boundedness of Z. As a Markovian decoupling field it is weakly regular, because it is weakly regular as a decoupling field to For the uniqueness we assume than there is another weakly regular Markovian and an x ∈ R n as initial condition X t1 = x, and consider the corresponding processes (X,Ỹ ,Z) that satisfy the corresponding FBSDE on [t 1 , T ], together with the decoupling condition viaũ. At the same time consider (X, Y, Z) solving the same FBSDE on [t 1 , T ], but associated with the Markovian decoupling field u. SinceZ, Z are bounded, the two triplets (X,Ỹ ,Z) and (X, Y, Z) also solve the Lipschitz FBSDE given by (ξ, (µ H , σ H , f H )) on [t 1 , T ] for H large enough. The two conditionsỸ s =ũ(s,X s ) and Y s = u(s, X s ) imply by Remark 2.2.4 in [Fro15] that both triplets are progressively measurable processes Strong regularity of u as a Markovian decoupling field to (ξ, (µ, σ, f )) follows directly from the above argument about uniqueness of (X, Y, Z) for deterministic initial values and bounded Z, and the strong regularity of u as decoupling field to (ξ, (µ H , σ H , f H )).
Remark 3.11. We observe from the proof that the supremum of all h = T − t with t satisfying the hypotheses of Theorem 3.10 can be bounded away from 0 by a bound, which only depends on L ξ,x , L ξ,x · L σ,z , σ(·, ·, ·, 0) ∞ , T , L σ,z and the values ( to the last 3 components, where B H ⊂ R m×d denotes the ball of radius H with center 0. This bounded is monotonically decreasing in these values. The following natural concept introduces a type of Markovian decoupling fields for non-Lipschitz problems (non-Lipschitz in z), to which nevertheless Lipschitz results can be applied.
• We call u controlled in z if there exists a constant C > 0 such that for all t 1 , t 2 ∈ [t, T ], t 1 ≤ t 2 , and all initial values X t1 , the corresponding processes (X, Y, Z) from the definition of a Markovian decoupling field satisfy |Z s (ω)| ≤ C, for almost all (s, ω) ∈ [t, T ] × Ω. If for a fixed triplet (t 1 , t 2 , X t1 ) there are different choices for (X, Y, Z), then all of them are supposed to satisfy the above control. • We say that a Markovian decoupling field on [t, T ] is controlled in z on a subin- Remark 3.13. Our Markovian decoupling field from Theorem 3.10 is obviously controlled in Z: consider (3.4) together with the choice of t ≤ t 1 made in the proof.
Remark 3.14. Let (ξ, (µ, σ, f )) satisfy (MLLC), and assume that we have a Markovian decoupling field u on some interval [t, T ], which is weakly regular and controlled in z. Then u is also a solution to a Lipschitz problem obtained through a cutoff as in Theorem 3.10. As such it is strongly regular (Theorem 3.6) and deterministic (Lemma 3.7). But Lemma 3.7 is also applicable, since due to the use of a cutoff we can assume the type of linear growth required there. Thus, u is also continuous.
and T − t is small enough as required in Theorem 3.10 resp. Remark 3.11, then u is controlled in z on [s, T ].
Proof. Clearly, u is not just controlled in z on [s, t], but also on [t, T ] (with a possibly different constant), according to Remark 3.13. Define C as the maximum of these two constants.
We only need to control Z by C for the case s ≤ t 1 ≤ t ≤ t 2 ≤ T , the other two cases being trivial. For this purpose consider the processes (X, Y, Z) on the interval [t 1 , t 2 ] corresponding to some initial value X t1 and fulfilling the forward equation, the backward equation and the decoupling condition. Since the restrictions of these processes to [t 1 , t] still fulfill these three properties we obtain |Z r (ω)| ≤ C for almost all r ∈ [t 1 , t], ω ∈ Ω.
At the same time, if we restrict (X, Y, Z) to [t, t 2 ], we observe that these restrictions satisfy the forward equation, the backward equation and the decoupling condition for the interval [t, t 2 ] with X t as initial value. Therefore, |Z r | ≤ C holds for a.s. for r ∈ [t, t 2 ].
The following important result allows us to connect the (MLLC)-case to (SLC). This shows S = [t, T ] by small interval induction (Lemma 2.5.1 and 2.5.2 in [Fro15]).
Note that Theorem 3.16 implies together with Remark 3.14 that a weakly regular Markovian decoupling field to an (MLLC) problem is deterministic and continuous.
2. Global regularity: If that there exists a weakly regular Markovian decoupling field u to this problem on some interval [t, T ], then u is strongly regular.
Proof. 1. We know that u (1) and u (2) are controlled in z. Choose a passive cutoff (see proof of Theorem 3.10) and apply 1. of Theorem 3.6.
2. u is controlled in z. Choose a passive cutoff (see proof of Theorem 3.10) and apply 2. of Theorem 3.6.
Then for any initial condition Proof. Existence follows from the fact that u is also strongly regular according to 2. of Theorem 3.17 and controlled in z according to Theorem 3.16.
Uniqueness follows from Corollary 3.6: Assume there are two solutions (X, Y, Z) and (X,Ỹ ,Z) to the FBSDE on [t, T ] both satisfying the aforementioned bound. But then they both solve an (SLC)-conform FBSDE obtained through a passive cutoff. So they must coincide according to Corollary 3.6. Unfortunately, the maximal interval might very well be open to the left. Therefore, we need to make our notions more precise in the following definitions.

Solution to the Skorokhod embedding problem
In this section we present a solution to the Skorokhod embedding problem as stated in (SEP) at the beginning of Section 2 based on solutions of the associated system of FBSDEs.

Weak solution
Let us therefore return to our FBSDE (2.3) that can be rewritten slightly more generally as With the general results of Section 3.2 at hand we are capable to solve this system of equations. In other words, we perform the second step of our algorithm to solve the SEP.
Lemma 4.1. Assume that δ and g are Lipschitz continuous. Then for the FBSDE (4.1) there exists a unique weakly regular Markovian decoupling field u on [0, T ]. This u is strongly regular, controlled in z, deterministic and continuous.
Proof. Using Theorem 3.21 we know that there exists a unique weakly regular Markovian decoupling field u on I M max . This u is strongly regular, controlled in z, deterministic and continuous. It remains to prove I M max = [0, T ]. Due to Lemma 3.22 it is sufficient to show the existence of a constant C ∈ [t, ∞] such that L u(t,·),x ≤ C < L −1 σ,z for all t ∈ I M max . In our case L −1 σ,z = ∞, so we have to prove that the weak partial derivatives of u with respect to x (1) and x (2) are both uniformly bounded.
Fix t ∈ I M max and consider the corresponding FBSDE on [t, T ]: First notice that the associated triplet (X, Y, Z) depends on the initial value x = (x (1) , x (2) ) ∈ R 2 , even in a weakly differentiable way with respect to the initial value x, according to the strong regularity of u. For more details about weak derivatives we refer to Chapter 2 of [Fro15], Section 2.1.2.
Let us look at the matrix d dx X. We observe that d dx (1) X (1) ist not 0. We will see later that it remains positive on the whole interval allowing us to apply the chain rule of Lemma A.7 in order to write d dx u(s, X s ) d dx X s . But let us first proceed by differentiating the backward equation in (4.1) with respect to x (2) : To be precise the above holds a.s. for every s ∈ [t, T ], for almost all x = (x (1) , x (2) ) ∈ R 2 . Now define a stopping time τ via s , X (2) Then the dynamics of d dx (2) X (2) s −1 can be expressed by for an arbitrary stopping timeτ < τ with values in [t, T ]. We also have d Applying Itô's formula and using the dynamics of d dx (2) Y and d dx (2) X (2) we easily obtain an equation describing the dynamics of V s∧τ :  Note that, since V and (−2ZV ) are bounded processes,Z1 [·≤τ ] is in BM O(P) according to Theorem A.5 with a BM O(P)-norm which does not depend onτ < τ , and so in particular E[ τ t |2Z rZr | 2 dr] < ∞. From (4.2) we can actually deduce that τ = T must hold almost surely. Indeed, (4.2) implies that This means that V s can be viewed as the conditional expectation of with respect to F s and some probability measure, which turnsW into a Brownian motion on [t, T ]. Note here that 2Z r V r is bounded on [t, T ] because ||Z|| ∞ < ∞. Hence, we conclude that V t and therefore d dx (2) u(t, x (1) , x (2) ) is bounded by δ ∞ for almost all x = (x (1) , x (2) ) ∈ R 2 . This value is independent of t.
Secondly, we have to bound d dx (1) u(t, x (1) , x (2) ). To this end we differentiate the equations in (4.1) with respect to x (1) : s , which allows us to deduce the dynamics of U from the dynamics of d dx (1) Y , d dx (1) X (2) and V using Itô formula: By the same argument as for the process V we deduce that U and therefore d dx (1) u(t, x (1) , x (2) ) is bounded by g ∞ = L g for almost all (x (1) , x (2) ), where L g is the Lipschitz constant of g, i.e. the infimum of all Lipschitz constants.
This shows that I M max = [0, T ]. Finally, Lemma 3.18 shows that there is a unique solution (X, Y, Z) to the FBSDE on [0, T ] for any initial value (X which is equivalent to the simpler condition Z ∞ < ∞ as we claim.
If Z ∞ < ∞, then according to the forward equation where L g and L δ are Lipschitz constants of g and δ, respectively.
In the next lemma we investigate the properties of the control process Z which was obtained in Lemma 4.1.
Lemma 4.2. Assume that δ and g are Lipschitz continuous. Let u be the unique weakly regular Markovian decoupling field associated to the problem (4.1) on [0, T ] constructed in Lemma 4.1. Then for any t ∈ [0, T ) and initial condition (X Furthermore, if the weak derivative d dx (1) u has a version which is continuous in the first two components (s, Proof. We already know that Z is bounded according to Lemma 4.1, but not in the form of the more explicit bound Z ∞ ≤ L g .
Notice that lim h↓0 On the other hand we can use the decoupling condition to write After applying conditional expectations to both sides of the above equation we investigate the two summands on the right hand side separately.
which clearly tends to 0 as h → 0.

Conclusion: We have shown
(1) u is continuous in the first two components on [0, T ) × R 2 and bounded by g ∞ otherwise.
In order to formulate the weak solution of the Skorokhod embedding problem in the next theorem, we use the notations of Section 2. As before we assume that β is bounded away from 0. Under this condition H −1 is well-defined and Lipschitz continuous. Therefore, δ =δ • H −1 is Lipschitz continuous ifδ is Lipschitz continuous, which is equivalent to α being bounded. Moreover,τ = H −1 1 0 Z 2 s ds is bounded by H −1 (L 2 g ) since Z is bounded by L g and H −1 is increasing.  [AHI08]. Therefore, we shall prove a sufficient criterion for this in terms of regularity properties of the Markovian decoupling field u.
Remark 4.5. The boundedness of the stopping time solving the SEP has not been investigated so frequently. However, very recently it gained attention in [AS11] and [AHS15]. Especially, its economic interest comes from its applications in the context of game theory (see [SS13]).

Strong solution
This subsection is devoted to the fourth step of our algorithm, i.e. to translate the results of the preceding section into a solution of the Skorokhod embedding problem in the strong sense.
Our main goal is to show that if g and δ are sufficiently smooth, thenτ and B constructed so far have the property thatτ is indeed a stopping time with respect to the filtration F B s s∈[0,∞) generated by the Brownian motion B, and thus a functional of the trajectories of B. The same functional applied to the trajectories of the original Brownian motion W will then provide the strong solution. For this purpose, we assume that g and δ are three times weakly differentiable with bounded derivatives. We also require that g is non-decreasing and not constant. Our arguments shall be based on a deep analysis of regularity properties of the associated decoupling field u. In the whole subsection we denoted by u the unique weakly regular Markovian decoupling field to the problem (4.1) as constructed in Lemma 4.1, assume for convenience T = 1 and use the notation as in Section 2. Proof. We consider the system (4.1) for t = 0 and x (1) = x (2) = 0. According to Lemma 4.2 we can assume Z = d dx (1) u ·, X (1) · , X (2) · and, thereby, we have r 2 dr for all s ∈ [0, T ]. Hence, we can assume that X (1) starts in 0, and is Lipschitz continuous and strictly increasing in s due to positivity of d dx (1) u 2 on [0, 1) × R 2 . Therefore, for every ω ∈ Ω the mapping H −1 X (2) It is straightforward to see that this inverse is given by the process σ from the proof of Lemma 2.2. Let us calculate the weak derivative of σ: Firstly, note H −1 (x) = (H (H −1 (x))) −1 and also H −1 (X (2) σr (ω)) = r or equivalently X (2) σr (ω) = H(r). So, we obtain σr , X (2) σr (4.5) on {σ r < 1}. Observe at this point that {σ r < 1} = r < H −1 X (2) 1 = {r <τ }. If we define σ r := 1 for r >τ , then σ is still continuous and we So, if we define Σ r := W σr , we have the dynamics where r ∈ [0,τ ). Note that this dynamical system is locally Lipschitz continuous in (σ, Σ). Moreover, for any K 1 , K 2 > 0 and K 3 ∈ (0, 1) define a bounded random variable τ K1,K2,K3 via Note that σ and Σ both remain bounded on [0, τ K1,K2,K3 ]. Therefore, on [0, τ K1,K2,K3 ] the pair (σ, Σ) coincides with the unique solution (σ K1,K2,K3 , Σ K1,K2,K3 ) to a Lipschitz problem, which is automatically progressively measurable w.r.t. the filtration (F B · ). Note An FBSDE approach to the Skorokhod embedding problem In order to deduce sufficient conditions for Theorem 4.6 to hold, we need to investigate higher order derivatives of u. For this purpose we consider the following system: where u is the unique weakly regular Markovian decoupling field to the problem (4.1).
In particular, u is twice weakly differentiable w.r.t. x with uniformly bounded derivatives.
Proof. The proof is in parts akin to the proof of Lemma 4.1 and we will seek to keep these parts short.
Let u (i) , i = 0, 1, 2, be the unique weakly regular Markovian decoupling field on I M max . We can assume u (i) to be continuous functions on I M max × R 2 (cf. Theorem 3.21). Let t ∈ I M max . For an arbitrary initial condition x ∈ R 2 we consider the corresponding processes X (1) , X (2) , Y (0) , Y (1) , Y (2) , Z (0) , Z (1) and Z (2) on [t, T ]. Note that X (1) , X (2) , Y (0) , Z (0) solve the FBSDE (4.1), which implies that they coincide with the processes X (1) , X (2) , Y, Z from (4.1) if we assume according to Lemma 3.18. This condition is fulfilled due to strong regularity and the fact that we work with Markovian decoupling fields.
is the maximal interval for the problem given by (4.6). We now claim that Y (1) and Y (2) are bounded processes: Using the backward equation we have and, therefore, for s ∈ [t, T ], which using Gronwall's lemma implies This in turn automatically implies boundedness of Y (1) according to its dynamics. Furthermore, Y (1) , Z (1) and Y (2) , Z (2) satisfy the BSDE which is also fulfilled by the processes U,Ž and V,Z from the proof of Lemma 4.1 (see (4.3) and (4.4)) and so in particular Using the boundedness of Z (0) , Z (2) and V this implies using Lemma A.4 that Y (2) − V is 0 almost everywhere. Therefore, after settingW s : SinceW is a Brownian motion under some probability measure equivalent to P we also have Z (2) −Z = 0 a.e.
Similarly, one shows that Y (1) and U as well as Z (1) andŽ coincide so Similarly, we get u (2) = d dx (2) u. Further, note that u (1) = d dx (1) u is continuous. This makes Lemma 4.2 applicable, so (4.7) Thereby Y (1) and Y (2) satisfy the following dynamics: which implies using the chain rule of Lemma A.7:  Using the chain rule of Lemma A.7 and the decoupling condition, we have Let us set We can apply the Itô formula to deduce dynamics of Y (12) and Y (11) from dynamics of (2) s −1 , so we can write using (4.12) Using the definitions of Y (12) , Y (22) and Z (12) we can simplify this to for almost all (s, ω) ∈ [t, T ] × Ω.
Furthermore, in this case the processes can be bounded uniformly, i.e. independently of (t, x).
Proof. The first part of the proof works analogously to the proof of Lemma 4.2. So we keep our arguments short. For i = 0, 1, s+h (W s+h − W s )|F s ] for small h > 0. As in the proof of Lemma 4.2, we use Itô's formula applied to (4.6) to obtain and also On the other hand we can use the decoupling condition to rewrite Let us deal separately with the two summands. For the first one recall that X (1) s and X (2) s are F s -measurable, X (1) s+h = X (1) s + (W s+h − W s ), W s+h − W s is independent of F s , and u is deterministic, i.e. is assumed to be a function of s, x (1) , x (2) ∈ [0, T ] × R 2 . A combination of these properties leads to For the second summand recall that u (i) is also Lipschitz continuous in the last component with some Lipschitz constant L and X which tends to 0 as h → 0.
Therefore, we can conclude is continuous in the first two components on [t, T ) × R 2 , for i = 0, 1, 2.
For the subsequent results we employ the following notation: • For a real number H > 0 let χ H : R → R be defined via χ H (x) := (−H) ∨ (x ∧ H) for x ∈ R. In particular, χ H is bounded, Lipschitz continuous and coincides with the identity function on the interval [−H, H].
• For real numbers y (ij) and y (i) we denote by y (ij)∧H and y (i)∧H the values χ H (y (ij) ) and χ H (y (i) ) for i, j = 1, 2.
To prove sufficiently regularity properties of the decoupling field u, we need to consider for H > 0 the following even higher dimensional system of equations: This already implies that d dx (i) Y (jk) , i, j, k = 1, 2, is uniformly bounded according to Lemma A.4. The lemma is applicable since is either 0 or has the structure g (3) (X (1) has a non-degenerate normal distribution w.r.t. P. Therefore its distribution is equivalent to the Lebesgue measure. But since Q ∼ P the distribution of X (1) T w.r.t. Q must also be equivalent to the Lebesgue measure. This In Theorem 4.10 we implicitly solved the Skorokhod embedding problem. To obtain a strong solution is now an immediate consequence.
Corollary 4.11. Provided G 0 , α and β as in (2.1) together with a probability measure ν such that the corresponding g and δ fulfil the requirements of Theorem 4.10. Then, there exists a bounded stopping time τ and a constant c ∈ R such that c + G τ has the law ν.

A BMO-Processes and their properties
Let (Ω, F T , (F t ) t∈[0,T ] , P) be a complete filtered probability space such that the filtration satisfies the usual hypotheses. Moreover, we assume that there exists a ddimensional Brownian motion W on [0, T ] independent of F 0 and that F t = σ(F 0 , F W t ), where F W is the natural filtration generated by W and F 0 contains all null sets.
For a probability measure Q and any q > 0 and m ∈ N we define H q (R m , Q) as the space of all progressively measurable processes (Z t ) t∈[0,T ] with values in R m such that Z q H q := E Q T 0 |Z s | 2 ds q/2 < ∞. By vector-valued we mean that Z assumes values in some normed vector space.
The smallest constant C such that the above bound holds is denoted by Also, if a progressively measurable process Z is only defined on a subinterval of [0, T ], the statement Z ∈ BM O(Q) means that its natural extension to [0, T ], obtained by setting it equal to 0 everywhere outside its initial domain, is in BM O(Q).
In the following we provide auxiliary results concerning BM0-processes. Lemma A.2. For a probability measure Q ∼ P let Z ∈ BM O(Q) be R m -valued. Then Z ∈ H 2n (R m , Q) for all n ∈ N and Z H 2n (R m ,Q) ≤ 2n √ n! Z BM O(Q) .
Proof. Let us define A t := t 0 |Z s | 2 ds, t ∈ [0, T ], which is progressively measurable, non-decreasing, starts at 0 and satisfies E Q [A T − A t |F t ] ≤ Z 2 BM O(Q) for all t ∈ [0, T ].
to see that the series converges absolutely and is monotonically increasing in K.
Lemma A.4. For some N ∈ N let Y be an R 1×N -valued progressively measurable bounded process on [0, T ], the dynamical behavior of which is described by • Y T is R 1×N -valued, F T -measurable and bounded, • Z is some R d×N -valued progressively measurable process s.t.
T 0 |Z| 2 r dr < ∞ a.s., which can also be interpreted as a vector (Z i ) i=1,...,d of R 1×N -valued progressively measurable processes Z i , i = 1, . . . , d, • α is an R 1×N -valued BM O(P)-process, • δ is some non-negative progressively measurable process with T 0 δ s ds < ∞ a.s., • I N ∈ R N ×N is the identity matrix, which using Cauchy-Schwarz inequality can be further controlled by Due to Lemma A.3 the first of the two factors above can be controlled by a finite constant, which depends only on p, β BM O(Q) , γ ∞ and T and is monotonically increasing in these values. Notice, that β BM O(Q) can be controlled by β BM O(P) and µ BM O(P) by Theorem A.1.6 in [Fro15] (or see [Kaz94], Theorem 2.4. and Theorem 3.6.). The second factor can be estimated using Doob's martingale inequality: Using Cauchy-Schwarz inequality and Doob's martingale inequality again, the above value can be controlled by This value is bounded by a finite constant, which depends only on p, T and γ ∞ and is monotonically increasing in these values: For instance use Theorem 2.1 in [Kaz94] by applying it to finitely many sufficiently small subintervals of [t, T ] such that 2p γ ∞ multiplied by the square root of the size of every subinterval is smaller 1/5. Also, use the triangle inequality and the tower property after splitting up the stochastic integral. One implication of the above control for sup s∈[t,T ] |Γ s | is that the stochastic integral in (A.2) represents a uniformly integrable martingale with respect to Q since