Geometry of distributionconstrained optimal stopping problems
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Abstract
We adapt ideas and concepts developed in optimal transport (and its martingale variant) to give a geometric description of optimal stopping times \(\tau \) of Brownian motion subject to the constraint that the distribution of \(\tau \) is a given probability \(\mu \). The methods work for a large class of cost processes. (At a minimum we need the cost process to be measurable and \((\mathcal {F}^0_{t})_{t \ge 0}\)adapted. Continuity assumptions can be used to guarantee existence of solutions.) We find that for many of the cost processes one can come up with, the solution is given by the first hitting time of a barrier in a suitable phase space. As a byproduct we recover classical solutions of the inverse first passage time problem/Shiryaev’s problem.
Keywords
Distributionconstrained optimal stopping Optimal transport Inverse first passage problem Shiryaev’s problemMathematics Subject Classification
Primary 60G42 60G44 Secondary 91G201 Appetizer
To whet the reader’s appetite and to give some idea of the kind of problems that can be solved with the methods presented in this paper we would like to start with two corollaries to our main results. In Sect. 3 we will present these main results and in Sect. 4 we will use them to prove Corollary 1.1 from them.
Both Corollaries 1.1 and 1.2 assert that the solutions of certain optimal stopping problems can be described by a barrier in an appropriate phase space.
In this section, let \((B_{t})_{t \ge 0}\) be a Brownian motion started^{1} in 0 on some filtered probability space Open image in new window satisfying the usual conditions and let \(\mu \) be a measure on \( (0,\infty ) \). First we consider optimal stopping problems of the following form.
Problem
Corollary 1.1
\(\tau \) has the following uniqueness properties: On the one hand it is the a.s. unique stopping time which has distribution \(\mu \) and which is of the form (1.1) (we will later say that such a stopping time is the hitting time of a downwards barrier).
 Let \(p \ge 0\), assume \(\mu \) has finite moment of order \(\frac{1}{2} + p + \varepsilon \) for some \(\varepsilon > 0\) and let \(A: \mathbb {R}_{+}\rightarrow \mathbb {R}\) be strictly increasing and \(A(t) \le K (1 + t^p)\) for some constant K.^{2} Then we may choose$$\begin{aligned} \psi (B_t,t) = B_t A(t) \text {.}\end{aligned}$$
 Let \(p \ge 2\), assume \(\mu \) has finite moment of order \(\frac{p}{2} + \varepsilon \) for some \(\varepsilon > 0\) and let \(\phi : \mathbb {R}\rightarrow \mathbb {R}\) satisfy \( \phi ''' > 0\) as well as \(\left \phi (y)\right \le K (1 + y^p)\) for some constant K. Then we may choose$$\begin{aligned} \psi (B_t,t) = \phi (B_t) \text {.}\end{aligned}$$
To give an example of a slightly more complicated functional amenable to analysis with our tools consider
Problem
Corollary 1.2
2 Background: martingale optimal transport and Shiryaev’s problem
In this article we consider distributionconstrained stopping problems from a mass transport perspective. Specifically we find that problems of the form exemplified in \((\textsc {OptStop}^{\psi (B_t,t)})\) and \((\textsc {OptStop}^{B^*_{t}})\) are amenable to techniques originally developed for the martingale version of the classical mass transport problem. This martingale optimal transport problem arises naturally in robust finance; papers to investigate such problems include [8, 12, 16, 18, 20, 25, 31]. In mathematical finance, transport techniques complement the Skorokhod embedding approach (see [24, 32] for an overview) to modelindependent/robust finance.
A fundamental idea in optimal transport is that the optimality of a transport plan is reflected by the geometry of its support set which can be characterized using the notion of ccyclical monotonicity. The relevance of this concept for the theory of optimal transport has been fully recognized by Gangbo and McCann [19], based on earlier work of Knott and Smith [28] and Rüschendorf [36, 37] among others. Inspired by these ideas, the literature on martingale optimal transport has developed a ‘monotonicity principle’ which allows to characterize martingale transport plans through geometric properties of their support sets, cf. [6, 7, 9, 10, 22, 39].
The main contribution of this article is to establish a monotonicity principle which is applicable to distributionconstrained optimal stopping problems. This transport approach turns out to be remarkably powerful, in particular we will find that questions as raised in Problems \((\textsc {OptStop}^{\psi (B_t,t)})\) and \((\textsc {OptStop}^{B^*_{t}})\) can be addressed using a relatively intuitive set of arguments.
The distributionconstrained optimal stopping problem \((\textsc {OptStop})\) (and specifically \((\textsc {OptStop}^{B^*_{t}})\)) arises naturally in financial and actuarial mathematics. We refer the reader to [23] which describes various examples (unitlinked life insurances, stochastic modelling for health insurances, the liquidation of an investment portfolio, the valuation of swing options).
Bayraktar and Miller [5] consider the same optimization problem that we treat here. However their setup and methods are rather distinct from the ones used here: they assume that the target distribution is given by finitely many atoms and that the target functional depends solely on the terminal value of Brownian motion. Following the measure valued martingale approach of Cox and Källblad [5, 14] address the constrained optimal stopping problem using a Bellman perspective.
The problem to construct a stopping time \(\tau \) of Brownian motion such that the law of \(\tau \) matches a given distribution on the real line was proposed by Shiryaev in his Banach Center lectures in the 1970’s, it has since been called Shiryaev’s problem or inverse first passage problem. Dudley and Gutmann [17] provide an abstract measuretheoretic construction. An early barriertype solution to the inverse first passage problem was given by Anulova [3]. She constructs a symmetric twosided barrier (corresponding to the case \(a=0\) in the sixth picture of Fig. 1). Anulova discretises the measure \(\mu \) and concludes through approximation arguments. The solution to the inverse first passage problem given in Corollary 1.1 was derived by Chen et al. [13] based on a variational inequality which describes the corresponding barrier. Notably, this is predated by a (formal) PDE description of such barriers given by Avellaneda and Zhu [4] in the context of credit risk modeling. Ekström and Janson [13] relate this solution to an optimal stopping problem and provide an integral equation for the barrier. Analytic solutions to the inverse first passage problem are known only in a few cases ([1, 2, 11, 29, 33, 38]). An interesting connection between the inverse first passage problem and Skorokhod’s problem is provided by Jaimungal et al. [26].
3 Statement of main results
Assumption 1
Throughout we will assume that Open image in new window is a filtered probability space and that \( (B_{t})_{t \ge 0} \) is an adapted process which has continuous paths on Open image in new window , such that B can be regarded as a measurable map from \(\Omega \) to \(C(\mathbb {R}_{+})\), the space of continuous functions from \(\mathbb {R}_{+}\) to \(\mathbb {R}\). The cost function \(c\) will always be a measurable map \(C(\mathbb {R}_{+})\times \mathbb {R}_{+}\rightarrow \mathbb {R}\). \(\mu \) will denote a probability measure on \(\mathbb {R}_{+}\).
Then the problem we consider can be stated as follows.
Problem
Here we formulate our main optimization problem in terms of minimization, following the usual convention in the optimal transport literature (which is also used in the closely related paper [6]). Clearly, a sign change transforms this into a maximization problem and in our applications we will in fact turn to this latter version when resulting formulations appear more natural. We trust that this will not cause confusion.
Throughout we will also make the following assumptions without further mention:
Assumption 2
 1.
\( c\) is measurable, \((\mathcal {F}^0_{t})_{t \ge 0}\)adapted, where \((\mathcal {F}^0_{t})_{t \ge 0}\) is the filtration on \(C(\mathbb {R}_{+})\) generated by the canonical process \(\left( \omega \mapsto \omega (t)\right) _{t \in \mathbb {R}_{+}{}} \).
 2.
There is a Open image in new window measurable random variable U which is uniformly distributed on [0, 1] and independent of the process \( (B_{t})_{t \ge 0} \).
 3.
There is a probability measure \(\lambda \) s.t. \( (B_{t})_{t \ge 0} \) is a Brownian motion with initial law \(\lambda \), i.e. \(B_0 \sim \lambda \).
 4.
The problem is wellposed in the sense that \( \mathbb E[c(B,\tau )] \) is defined and \( > \infty \) for all stopping times \( \tau \sim \mu \) and that \( \mathbb E[c(B,\tau )] < \infty \) for at least one such stopping time.
 5.
\( {\int }t^{p_0} \,d\mu (t) < \infty \), where \(p_0 \ge 0\) is some constant that we fix here and that can be chosen when applying the results from this section.
A note on language: The adjective “adapted” is usually applied to processes whose time argument is written in subscript form. For any filtered measurable space \(\tilde{\Omega }\) and any function \(f : \tilde{\Omega } \times \mathbb {R}_{+}\rightarrow \mathbb {R}\) (or possibly \(f : \tilde{\Omega } \times \mathbb {R}_{+}\rightarrow [\infty ,\infty ] \)) we will interchangeably think of f simply as a function or as the process \( Y_t(\omega ) := f(\omega ,t) \). And so f being adapted means the same thing as \((Y_t)_{t \in \mathbb {R}_{+}}\) being adapted. Similarly for a subset \(\Gamma \) of \(\tilde{\Omega } \times \mathbb {R}_{+}\) we may also think of \(\Gamma \) as its indicator function or as the process \(Y_t(\omega ) := 1_{\Gamma }(\omega ,t)\) and will also say that the set \(\Gamma \) is adapted.
With that in mind, Assumption 2.1 should seem like an obvious thing to ask for from the cost function. Also, knowing about the existence of optional projections, it should be clear no later than Lemma 5.3 that Assumption 2.1 does not pose a real restriction on the class of problems we are treating.
The role of Assumption 2.2 should become clearer soon. We would like to note at this point though that often enough our results put together will imply that the solution of Problem \((\textsc {OptStop})\) for a space Open image in new window which satisfies Assumption 2.2 is essentially the same as the solution of the Problem for a space which may not satisfy said assumption, and we will find that we can describe this solution in detail. This can be seen executed in the proofs of the corollaries stated in the Appetizer.
The methods in this paper work not just for Brownian motion but for a class of processes which is conceptually bigger, but then turns out to not include much beyond Brownian motion—namely for any spacehomogeneous but possibly timeinhomogeneous Markov process with continuous paths which has the strong Markov property. (Here spacehomogeneous means that starting the process at location x and then moving its paths to start at location y results in a version of the process started at y.) If the reader so wishes, she may think of B as a process from this slightly larger class of processes. Care was taken not to reference any properties of Brownian motion beyond those stated here. In particular our results apply to multidimensional Brownian motion.
Assumption 2.4 is mostly just there to ensure that we are actually talking about an optimization problem in a meaningful sense. For the problems presented in the Appetizer, the moment conditions on \(\mu \) which are given in the statement of Corollary 1.1 and Corollary 1.2 ensure that Assumption 2.4 is satisfied (as we will see in the proofs of these corollaries).
The constant \(p_0\) in Assumption 2.5 will (implicitly) appear in the statement of Theorem 3.6, one of the main results. Its role is to ensure that \(\mathbb E[\varphi (B,\tau )]\) will be finite for some (class of) function(s) \(\varphi \) and any solution \(\tau \) of \((\textsc {OptStop})\). (The choice \(\varphi (B,\tau ) = \tau ^{p_0}\) is somewhat arbitrary here.)
The main results are Theorems 3.1 and 3.6.
We give two versions of Theorem 3.1. Version A is easier to state and may feel more natural, but we will need Version B (which is more general and has essentially the same proof as Version A) in the proof of the corollaries in the Appetizer.
Theorem 3.1
Version A. Assume that the cost function \(c\) is bounded from below and lower semicontinuous when we equip \(C(\mathbb {R}_{+})\) with the topology of uniform convergence on compacts. Then the Problem \((\textsc {OptStop})\) has a solution.
Version B. Assume that the cost function \(c\) is lower semicontinuous when we equip \(C(\mathbb {R}_{+})\times \mathbb {R}_{+}\) with the product topology of two Polish topologies which generate the right sigmaalgebras on \(C(\mathbb {R}_{+})\) and \(\mathbb {R}_{+}\) respectively and assume that the set \( \left\{ c_(B,\tau ) : \tau \sim \mu , \tau \text { is a stopping time} \right\} \) is uniformly integrable, where \(c_ := c\vee 0 \) denotes the negative part of \(c\). Then the Problem \((\textsc {OptStop})\) has a solution.
To state Theorem 3.6 we need a few more definitions.
Remark 3.2
We will find it convenient to talk about processes that don’t start at time 0 but instead at some time \( t > 0 \). Similarly we will consider stopping times taking values in \([t,\infty )\). These will be defined on the space \(C([t, \infty ))\) equipped with the filtration \((\mathcal {F}_{t}^{s})_{s \ge t}\), again generated by the canonical process \(\left( \omega \mapsto \omega (s)\right) _{s \ge t} \). We refer to the distribution of Brownian motion started at time t and location x by \(\mathbb {W}^{t}_{x}\). This is a measure on \(C([t, \infty ))\). For a probability measure \(\kappa \) on \(\mathbb {R}\) we write \(\mathbb {W}^{t}_{\kappa }\) for the distribution of Brownian motion started at time t with initial law \(\kappa \).
Definition 3.3
Definition 3.4
As hinted at earlier, the definition of StopGo pairs depends on the parameter \(p_0\) from Assumption 2.5. A larger \(p_0\) means that we are asking for more in Assumption 2.5 and implies that we get a larger set \(\mathsf {SG}\), as we are quantifying over fewer stopping times \(\sigma \) in the definition of \(\mathsf {SG}\). This in turn implies that the conclusion of Theorem 3.6 below will be stronger.
Definition 3.5
Theorem 3.6
The following lemma should give a first hint about how the Monotonicity can be applied.
Lemma 3.7
When applying this Lemma to show that some optimal stopping problem has a barriertype solution as symbolized for example by the pictures in Fig. 1 the process \(Y_t(B)\) is of course what we are labelling the vertical axes in the pictures with. So for the first picture \(Y_t(\omega ) = \omega (t)\), for the second one \(Y_t(\omega ) = \omega (t)  \sup _{s \le t}\omega (s)\), for the third \(Y_t(\omega ) = (\omega (t)  \sup _{s \le t}\omega (s))\) (the sign is flipped relative to the labelling in the picture because in this picture the barrier is drawn “up” instead of “down”), etc.
Notice that, contrary to customs, when we draw the barriers Open image in new window in the pictures in Fig. 1 the first coordinate is the vertical axis and the second coordinate is the horizontal axis. This is because, to make crossreferencing and comparison with [6] easier, we follow their convention of always having time as the second coordinate but still in the pictures it seems more natural to put the independent variable on the horizontal axis.
Note that a priori Open image in new window and \(\hat{\tau }\) need not be stopping times or even measurable, as we don’t know much about the sets Open image in new window and Open image in new window .
Using the properties of a concrete process \((Y_{t})_{t \ge 0}\) we will in the proofs of Corollaries 1.1 and 1.2 be able to show that Open image in new window a.s. (this should not be surprising as for each time t the barriers Open image in new window and Open image in new window differ by at most a single point) and therefore that the optimizer \(\tau \) is the hitting time of a barrier.
Proof of Lemma 3.7
Let \(\tilde{\omega } \in \Omega \) s.t. \(\left( B(\tilde{\omega }),\tau (\tilde{\omega })\right) \in \Gamma \). By assumption this holds for \(\mathbb {P}\)a.a. \(\tilde{\omega }\). Then Open image in new window and therefore Open image in new window .
Next we show that \(\hat{\tau }(\tilde{\omega }) \ge \tau (\tilde{\omega })\). Assume that \(\left( Y_t(B(\tilde{\omega })),t\right) \in \hat{\mathcal {R}}\). We want to show that \(t \ge \tau (\tilde{\omega })\). By the definition of \(\hat{\mathcal {R}}\) we find that there is \(\eta \in C(\mathbb {R}_{+})\) with \((\eta ,t) \in \Gamma \) and \(Y_t(B(\tilde{\omega })) < Y_t(\eta )\), so by (3.5) we know \(\left( (B(\tilde{\omega }),t),(\eta ,t)\right) \in \mathsf {SG}\). Assuming, if possible, \(t < \tau (\tilde{\omega })\) we get according to Definition 3.5 that \((B(\tilde{\omega }),t) \in \Gamma ^<\). Therefore we have that \(\left( (B(\tilde{\omega }),t),(\eta ,t)\right) \in \mathsf {SG}\cap \left( \Gamma ^<\times \Gamma \right) \), but this is a contradiction to \(\mathsf {SG}\cap \left( \Gamma ^<\times \Gamma \right) = \emptyset \), so we must have \(t \ge \tau (\tilde{\omega })\). \(\square \)
Remark 3.8
4 Digesting the appetizer
We will now demonstrate how to use the Monotonicity Principle of Theorem 3.6 to derive Corollary 1.1. The proof of Corollary 1.2 is very similar but relies on understanding a technical detail which does not add much to the story at this point, so we leave it for the end of the paper.
Both of the sets Open image in new window and Open image in new window in Lemma 3.7 have the property that (writing \(\mathcal {R}\) for the set in question) \((y,t) \in \mathcal {R}\) and \(y' \le y\) implies \((y',t) \in \mathcal {R}\). We call such sets (downwards) barriers. More specifically, for technical reasons in what follows it is slightly more convenient to talk about subsets of \([\infty ,\infty ]\times \mathbb {R}_{+}\) instead of subsets of \( \mathbb {R}\times \mathbb {R}_{+}\), giving the following definition.
Definition 4.1
Clearly, in Lemma 3.7, instead of talking about Open image in new window , we could have talked about Open image in new window without anything really changing, and likewise for \(\hat{\mathcal {R}}\).
The reader will easily verify the following lemma.
Lemma 4.2
What we will show now, on the way to proving Corollary 1.1 is that the first hitting time after 0 of any downwards barrier by Brownian motion is a.s. equal to the first hitting time after 0 of the closure of that barrier. This serves to both resolve the question whether the times in Lemma 3.7 are stopping times and to show that Open image in new window a.s.
Let us assume for the rest of this section that B is actually a Brownian motion started in 0.
Lemma 4.3
Proof
As \(\left( \overline{\tau }_{1/n}\right) _n\) is a decreasing sequence bounded below by \(\overline{\tau }\) we get that convergence holds almost surely. \(\square \)
The following is a particular case of [21, Corollary 2.3] (which in turn relies on arguments given in [30, 35]). Note that this lemma is purely a statement about barriertype stopping times and is not directly connected to the optimization problem under consideration.
Lemma 4.4
Proof
Is to be found in [21, Corollary 2.3]. \(\square \)
We now have the necessary prerequisites to use our main results in showing that the first optimization problem in the Appetizer admits a (unique) barriertype solution.
Proof of Corollary 1.1
The strategy is as follows: We choose a cost function and leverage Theorem 3.1 to show that an optimizer exists, the Monotonicity Principle in the form of Theorem 3.6 and Lemma 3.7 will—with some help from Lemma 4.3—show that any optimizer must be the hitting time of a barrier. Lemma 4.4 shows that any two barriertype solutions must be equal.
We now provide the details. Start with a cost function \(c(\omega ,t) := \omega (t) A(t)\) for a strictly monotone function \(A: \mathbb {R}_{+}\rightarrow \mathbb {R}\) which satisfies \(A(t) \le K(1+t^p)\) and assume that \(\mu \) has moment of order \(\frac{1}{2} + p + \varepsilon \) for some \(\varepsilon > 0\). To prove that a barriertype solution exists when \(\mu \) has first moment, choose a bounded strictly increasing A and \(p=0\), \(\varepsilon =\frac{1}{2}\) in this step. (These assumptions guarantee in particular that the optimization problems considered below have a finite value.) Clearly the problem (OptStop) for c corresponds to \((\textsc {OptStop}^{\psi (B_t,t)})\) for \(\psi (B_t,t) = B_t A(t)\) (i.e. \(\psi \) takes the role of \(c\) such that the minimal/maximal values agree up to a change of sign). We will deal with the case where \(\psi (B_t,t) = \phi (B_t)\) at the end of this proof.
We now check that the conditions in Version B of Theorem 3.1 are satisfied. We also need to check that Assumption 2 holds. Here we need the assumption that \(\mu \) has moment of order \(\frac{1}{2} + p + \varepsilon \), as well as the Hölder and BurkholderDavisGundy inequalities. The latter specialized to Brownian motion state that for all \(q > 0\) there are positive constants \(K_0\) and \(K_1\) such that for any stopping time \(\tau \) we have \( K_0 \, \mathbb E\left[ \tau ^{q/2}\right] \le \mathbb E\left[ (B^*_\tau )^q\right] \le K_1 \, \mathbb E\left[ \tau ^{q/2}\right] \) (where \(B^*_t = \sup _{s \le t} B_s \)). With these in hand a straightforward calculation allows us to bound \(B_\tau A(\tau )\) in the \(L^{1+\delta }\)norm for some \(\delta > 0\), independently of the stopping time \(\tau \sim \mu \).
This shows both that the uniform integrability condition in Version B of Theorem 3.1 is satisfied and that Assumption 2.4 is satisfied.
On \(C(\mathbb {R}_{+})\) we may choose the (Polish) topology of uniform convergence on compacts. For the topology on \(\mathbb {R}_{+}\) we start with the usual topology and turn A into a continuous function (if it wasn’t), by making use of the fact that any measurable function from a Polish space to a second countable space may be turned into a continuous function by passing to a larger Polish topology (with the same Borel sets) on the domain. (This can be found for example in [27, Theorem 13.11].)
In the statement of Corollary 1.1 we did not require that the probability space Open image in new window satisfy Assumption 2.2. To remedy this we can enlarge the probability space by setting \(\tilde{\Omega } := \Omega \times [0,1]\), Open image in new window and \(\tilde{\mathbb {P}} := \mathbb {P}\otimes \mathcal {L}\), where \(\mathcal {L}\) is Lebesgue measure on [0, 1]. On this space we consider the Brownian motion \(\tilde{B}_t (\omega , x) := B_t(\omega )\). Theorem 3.1 now gives us an optimal stopping time \(\tilde{\tau }\) on the enlarged probability space. If we can show that this stopping time is in fact the hitting time of a barrier, then it follows that \(\tilde{\tau } = \tau \circ ((\omega ,x) \mapsto \omega )\) for a stopping time \(\tau \) which is defined as the hitting time of the Brownian motion B of the same barrier. As there are more stopping times on Open image in new window than on Open image in new window in the sense that any stopping time \(\tau '\) on Open image in new window induces a stopping time \(\tilde{\tau }' := \tau ' \circ ((\omega ,x) \mapsto \omega )\) on Open image in new window we conclude that \(\tau \) must also be optimal among the stopping times on Open image in new window . With this out of the way, let us refer to our Brownian motion by B, to the optimal stopping time by \(\tau \) and to our filtered probability space by Open image in new window irrespective of whether this is the original process and space we started with, or an enlarged one.
Choosing \( p_0 := \frac{1}{2} + p + \varepsilon \) in Assumption 2.5 we apply Theorem 3.6 to obtain a set \(\Gamma \) on which \((B,\tau )\) is concentrated under \(\mathbb {P}\) and for which (3.4) holds. As \(\mu \) is concentrated on \((0,\infty )\), we may assume that \(\Gamma \cap (C(\mathbb {R}_{+})\times \{ 0 \}) = \emptyset \). Next we want to show that Lemma 3.7 applies with \(Y_t(\omega ) = \omega (t)\).
\(\Gamma \cap (C(\mathbb {R}_{+})\times \{ 0 \}) = \emptyset \) implies Open image in new window and therefore Open image in new window , and likewise for \(\hat{\mathcal {R}}\) and \(\hat{\tau }\). As Open image in new window it follows from Lemma 4.3 that Open image in new window a.s. and that \(\tau \) is of the form claimed in (1.1) with \(\beta (t) := \sup \{y\in \mathbb {R}: (y,t) \in \overline{\mathcal {R}}\}\). The uniqueness claims follow from Lemma 4.4 and what we have already proven.
We now treat the case where \(\psi (B_t,t) = \phi (B_t)\) with \(\phi ''' > 0\), \(\left \phi (y)\right \le K (1 + y^p)\) and \(\mu \) has finite moment of order \( \frac{p}{2} + \varepsilon \) for some \(\varepsilon > 0\). Most of the proof remains unchanged. Setting \(c(\omega ,t) = \phi (\omega (t))\) we may again use the BurkholderDavisGundy inequalities to show that \(c(B_\tau ,\tau )\) is bounded in \(L^{1+\delta }\)norm, independently of the stopping time \(\tau \sim \mu \), thereby showing both that Assumption 2.4 is satisfied and that the uniformintegrability condition in Version B of Theorem 3.1 is satisfied.
It remains to show that \(\omega (t) < \eta (t)\) implies \(((\omega ,t),(\eta ,t)) \in \mathsf {SG}\). \(\phi ''' > 0\) implies that the map \(y \mapsto \phi (\eta (t) + y)  \phi (\omega (t) + y)\) is strictly convex. By the strict Jensen inequality \(\mathbb E[ \phi (\eta (t) + \tilde{B}_\sigma )  \phi (\omega (t) + \tilde{B}_\sigma ) ] > \phi (\eta (t))  \phi (\omega (t))\) for any stopping time \(\sigma \) on \(C([t, \infty ))\) which is almost surely finite, satisfies optional stopping and is not almost surely equal to t. As we may choose \(p_0 := \frac{p}{2} + \varepsilon \), which is greater than 1, we may assume that the \(\sigma \) in the definition of \(\mathsf {SG}\) has finite first moment, which is enough to guarantee that it satisfies optional stopping. Rearranging the last inequality gives (3.2). \(\square \)
5 Existence of an optimizer
The proof of existence of solutions to the Problem (OptStop) crucially depends on thinking of stopping times as the joint distribution of the process to be stopped and the stopping time. We introduce some concepts to make this precise and give a proof of Theorem 3.1 at the end of this section.
Lemma 5.1
If \(G \in C_b\left( C([t, \infty ))\right) \), then Open image in new window .
Proof
Obvious. \(\square \)
Here we use \(C_b(X)\) to denote the set of continuous bounded functions from a topological space X to \(\mathbb {R}\). The last sentence of the lemma is of course true for any topology on \(C([t, \infty ))\) for which the map \(\omega \mapsto \omega \odot \theta \) is continuous for all \(\theta \), but we will only need it for the topology of uniform convergence on compacts.^{3}
Given spaces X and Y we will denote the projection from \(X \times Y\) to X by \(\mathsf {proj}_{X}\) (and similarly for Y). For a measurable map \(F: X \rightarrow Y\) between measure spaces and a measure \(\nu \) on X we denote the pushforward of \(\nu \) under F by \(F_*(\nu ) := D \mapsto \nu (F^{1}\left[ D\right] )\).
Definition 5.2
In this definition the topology on \(C([t, \infty ))\) is that of uniform convergence on compacts and the topology on \([t, \infty )\) is the usual topology.
In any of these, if we drop the superscript t then we will mean time \(t = 0\), while, if we drop the subscript \(\kappa \), then we mean that the initial distribution \(\kappa = \delta _0\), i.e. the Brownian motion to be stopped is started deterministically in 0.
To explain the qualifier finite it may help to imagine that for a nonfinite randomized stopping time of mass \(\alpha < 1\), the mass \(1\alpha \) which is missing is placed along \(C([t,\infty ))\times \{\infty \}\).
The following Lemma 5.3 from [6] shows that the problem (OptStop) is equivalent to the following optimization problem \((\textsc {OptStop'})\) in the sense that a solution of one can be translated into a solution of the other and vice versa. This of course also implies that the values of the two problems are equal, thereby showing that the concrete space Open image in new window has no bearing on this value, as long as Assumptions 1 and 2 are satisfied.
Problem
Lemma 5.3
\(\xi \) is a finite randomized stopping time iff \(\tau \) is a.s. finite.
Proof of Theorem 3.1
Now for the details. On each of the spaces \(C(\mathbb {R}_{+})\) and \(\mathbb {R}_{+}\) we are dealing with two topologies, one coming from the Definition 5.2 of randomized stopping times (to wit, the topology of uniform convergence on compacts on the space \(C(\mathbb {R}_{+})\) and the usual topology on \(\mathbb {R}_{+}\)) and one coming from the assumptions in the statement of this theorem. We can equip each of these spaces with the smallest topology which contains the two topologies in question. These are again Polish topologies and they still generate the standard sigmaalgebras on the respective spaces. For the remainder of this proof all topological notions are to be understood relative to these topologies. So the topology on \(C(\mathbb {R}_{+})\times \mathbb {R}_{+}\) is the product topology of these two topologies, and the weak topology on the space of measures on \(C(\mathbb {R}_{+})\times \mathbb {R}_{+}\) is to be understood relative to this product topology. The cost function \(c\) of course remains lower semicontinuous and by Lemma 5.1 the functions Open image in new window appearing in Definition 5.2 are continuous.
6 Geometry of the optimizer
This section is devoted to the proof of Theorem 3.6. The proof closely mimicks that of Theorem 1.3/Theorem 5.7 in [6]. For the benefit of those readers already familiar with said paper we will first describe the changes required to the proofs there to make them work in our situation and then—for the sake of a more selfcontained presentation—indulge in reiterating the main arguments and only citing results from [6] that we can use verbatim.
Sketch of differences in the proof of Theorem 3.6 relative to [6, Theorem 5.7]
Again the strategy is to show that for a larger set Open image in new window we can find a set Open image in new window such that Open image in new window . The definition of Open image in new window must of course be adapted analoguously to the changes required to the definition of \(\mathsf {SG}\).
Apart from that the only real changes are to [6, Theorem 5.8]. Whereas previously it was essential that the randomized stopping time \(\xi ^{r(\omega ,s)}\) is also a valid randomized stopping time of the Markov process in question when started at a different time but the same location \(\omega (s)\), we now need that \(\xi ^{r(\omega ,s)}\) will also be a randomized stopping time of our Markov process when started at the same time s but in a different place. Of course, when we are talking about Brownian motion both are true, but this difference is the reason why in the case of the Skorokhod embedding the right class of processes to generalize the argument to is that of Feller processes while in our setup we don’t need our processes to be timehomogeneous but we do need them to be spacehomogeneous. That we are able to plant this “bush” \(\xi ^{r(\omega ,s)}\) in another location is what guarantees that the measure \(\xi _1^\pi \) defined in the proof of Theorem 5.8 of [6] is again a randomized stopping time.
Whereas in the Skorokhod case the task is to show that the new better randomized stopping time \(\xi ^\pi \) embeds the same distribution as \(\xi \) we now have to show that the randomized stopping time we construct has the same distribution as \(\xi \). The argument works along the same lines though—instead of using that \(\left( (\omega ,s),(\eta ,t)\right) \in \widehat{\mathsf {SG}}^\xi \) implies \(\omega (s)=\eta (t)\) we now use that \(\left( (\omega ,s),(\eta ,t)\right) \in \widehat{\mathsf {SG}}^\xi \) implies \(s=t\). \(\square \)
We now present the argument in more detail.
As may be clear by now, what we will show is that if \(\xi \in \mathsf {RST}_{\lambda }(\mu )\) is a solution of \((\textsc {OptStop'})\), then there is a measurable, \((\mathcal {F}^0_{t})_{t \ge 0}\)adapted set \(\Gamma \subseteq C(\mathbb {R}_{+})\times \mathbb {R}_{+}\) such that \(\mathsf {SG}\cap \left( \Gamma ^< \times \Gamma \right) = \emptyset \). Using Lemma 5.3 this implies Theorem 3.6.
We need to make some preparations. To align the notation with [6] and to make some technical steps easier it is useful to have another characterization of measurable, \((\mathcal {F}^0_{t})_{t \ge 0}\)adapted processes and sets. To this end define
Definition 6.1
[6, Theorem 3.2], which is a direct consequence of [15, Theorem IV. 97], asserts that a process X is measurable, \((\mathcal {F}^0_{t})_{t \ge 0}\)adapted iff X factors as \(X=X'\circ r\) for a measurable function \(X' : S \rightarrow \mathbb {R}\). So a set \(D \subseteq C(\mathbb {R}_{+})\times \mathbb {R}_{+}\) is measurable, \((\mathcal {F}^0_{t})_{t \ge 0}\)adapted iff \(D = r^{1}\left[ D'\right] \) for some measurable \(D' \subseteq S\).
Given an optimal \(\xi \in \mathsf {RST}_{\lambda }(\mu )\) we may therefore rephrase our task as having to find a measurable set \(\Gamma \subseteq S\) such that \(r_*(\xi )\) is concentrated on \(\Gamma \) and that \(\mathsf {SG}' \cap \left( \Gamma ^< \times \Gamma \right) = \emptyset \), where Open image in new window .
Note that for \(\Gamma \subseteq S\) although \(\left( r^{1}\left[ \Gamma \right] \right) ^<\) is not equal to \( r^{1}\left[ \Gamma ^<\right] \) we still have \(\mathsf {SG}\cap \left( r^{1}\left[ \Gamma ^<\right] \times r^{1}\left[ \Gamma \right] \right) = \emptyset \) iff \(\mathsf {SG}\cap \left( (r^{1}\left[ \Gamma \right] )^< \times r^{1}\left[ \Gamma \right] \right) = \emptyset \).
One of the main ingredients of the proof of [6, Theorem 1.3] and of our Theorem 3.6 is a procedure whereby we accumulate many infinitesimal changes to a given randomized stopping time \(\xi \) to build a new stopping time \(\xi ^\pi \). The guiding intuition for the authors is to picture these changes as replacing certain “branches” of the stopping time \(\xi \) by different branches. Some of these branches will actually enter the statement of a somewhat stronger theorem (Theorem 6.8 below), so we begin by describing these. Our way to get a handle on “branches”—i.e. infinitesimal parts of a randomized stopping time—is to describe them through a disintegration (wrt \(\mathbb {W}^{0}_{\lambda }\)) of the randomized stopping time. We need the following statement from [6] which should also serve to provide more intuition on the nature of randomized stopping times.
Lemma 6.2
[6, Theorem 3.8] Let \(\xi \) be a measure on \(C(\mathbb {R}_{+})\times \mathbb {R}_{+}\). Then \( \xi \in \mathsf {RST}_{\lambda } \) iff there is a disintegration \((\xi _{\omega })_{\omega \in C(\mathbb {R}_{+})}\) of \(\xi \) wrt \(\mathbb {W}^{0}_{\lambda }\) such that \((\omega ,t) \mapsto \xi _\omega ([0,t])\) is measurable, \((\mathcal {F}^0_{t})_{t \ge 0}\)adapted and maps into [0, 1].
Using Lemma 6.2 above let us fix for the rest of this section both \(\xi \in \mathsf {RST}_{\lambda }(\mu )\) and a disintegration \(\left( \xi _{\omega }\right) _{\omega \in C(\mathbb {R}_{+})}\) with the properties above. Both Definition 6.3 below and Theorem 6.8 implicitly depend on this particular disintegration and we emphasize that whenever we write \(\xi _{\omega }\) in the following we are always referring to the same fixed disintegration with the properties given in Lemma 6.2. Note that the measurability properties of \(\left( \xi _{\omega }\right) _{\omega \in C(\mathbb {R}_{+})}\) imply that for any \( I \subseteq [0,s] \) we can determine \( \xi _\omega (I) \) from Open image in new window alone. For \((f,s) \in S\) we will again overload notation and use \(\xi _{(f,s)} \) to refer to the measure on [0, s] which is equal to Open image in new window for any \( \omega \in C(\mathbb {R}_{+})\) such that \(r(\omega ,s) = (f,s)\).
Definition 6.3
Here \(\delta _s\) is the Dirac measure concentrated at s. Really, the definition in the case where \( \xi _{(f,s)}([0,s]) = 1 \) is somewhat arbitrary—it’s more a convenience to avoid partially defined functions. What we will use is that Open image in new window .
Definition 6.4
 1.
\(\xi ^{(f,t)}\left( C(\mathbb {R}_{+})\times \mathbb {R}_{+}\right) < 1\) or \({\int }s^{p_0} \,d\xi ^{(f,t)}(\theta ,s) = \infty \)
 2.
the integral on the right hand side equals \(\infty \)
 3.
either of the integrals is not defined
Lemma 6.6 below says that the numbered cases above are exceptional in an appropriate sense and one may consider them a technical detail. Note that when we say \(\left( (f,t),(g,t)\right) \in \mathsf {SG}^\xi \) we are implicitly saying that \( \xi _{(f,t)}([0,t]) < 1 \).
Note that the sets \(\mathsf {SG}^\xi \) and \(\widehat{\mathsf {SG}}^\xi \) are measurable (in contrast to \(\mathsf {SG}\), which may be more complicated).
Definition 6.5
We call a measurable set \(F \subseteq S\) evanescent if \(r^{1}\left[ F\right] \) is evanescent, that is, if \(\mathbb {W}^{0}_{\lambda }\left( \mathsf {proj}_{C(\mathbb {R}_{+})}\left[ r^{1}\left[ F\right] \right] \right) = 0\).
Lemma 6.6

\(\left\{ (f,s) \in S : \xi ^{(f,s)}\left( C(\mathbb {R}_{+})\times \mathbb {R}_{+}\right) < 1 \right\} \)

\(\left\{ (f,s) \in S : {\int }F((f,s) \odot \theta ,u) \,d\xi ^{(f,s)}(\theta ,u) \not \in \mathbb {R}\right\} \)
Proof
See [6].\(\square \)
Lemma 6.7
Proof
Can be found in [6]. Note that they fix \(p_0 = 1\). \(\square \)
Theorem 6.8
Our argument follows [6, Theorem 5.7]. We also need the following two auxilliary propositions, which in turn require some definitions.
Definition 6.9
Proposition 6.10
Let \( \xi \) be a solution of \((\textsc {OptStop'})\). Then \( \left( r \times \mathsf {Id}\right) _*(\pi )(\mathsf {SG}^\xi ) = 0 \) for all \( \pi \in \mathsf {JOIN}_{\lambda }(r_*(\xi )) \).
Here we use \(\times \) to denote the Cartesian product map, i.e. for sets \(X_i,Y_i\) and functions \(F_i : X_i \rightarrow Y_i\) where \(i \in \{1,2\}\) the map \(F_1 \times F_2 : X_1 \times X_2 \rightarrow Y_1 \times Y_2\) is given by \((F_1 \times F_2)(x_1,x_2) = (F_1(x_1),F_2(x_2))\). Proposition 6.10 is an analogue of [6, Proposition 5.8] and it is where the material changes compared to [6] take place. We will give the proof at the end of this section.
Proposition 6.11
 1.
\( \left( r \times \mathsf {Id}\right) _*(\pi )(E) = 0 \) for all \(\pi \in \mathsf {JOIN}_{\lambda }(\upsilon )\)
 2.
\( E \subseteq (F \times Y) \cup (S \times N) \) for some evanescent set \(F \subseteq S\) and a measurable set \(N \subseteq Y\) which satisfies \(\upsilon (N) = 0\).
Proposition 6.11 is proved in [6] and we will not repeat the proof here.
Proof of Theorem 6.8
Setting \(F_2 := \left\{ (f,s) \in S : \xi _{(f,s)}([0,s]) = 1 \right\} \) and arguing on the disintegration \(\left( \xi _\omega \right) _{\omega \in C(\mathbb {R}_{+})}\) we see that \( r_*(\xi )(F_2^>) = 0 \), so \(r_*(\xi )(F^>) = 0\) for \(F := F_1 \cup F_2\).
This shows that \(S {\setminus } (N \cup F^>)\) has full \(r_*(\xi )\)measure. Let \(\Gamma \) be a Borel subset of that set which also has full \(r_*(\xi )\)measure.
Lemma 6.12
Proof
We use the criterion in Lemma 6.2. Let \( (\alpha _\omega )_{\omega \in C(\mathbb {R}_{+})} \) be a disintegration of \( \alpha \) wrt \( \mathbb {W}^{0}_{\lambda } \) for which \( (\omega ,t) \mapsto \alpha _\omega ([0,t]) \) is measurable, \((\mathcal {F}^0_{t})_{t \ge 0}\)adapted and maps into [0, 1]. Then \((\hat{\alpha }_\omega )_\omega \) defined by \( \hat{\alpha }_{\omega } := F \mapsto {\int }F(t) G(\omega ,t) \,d\alpha _{\omega }(t) \) is a disintegration of the measure in (6.5) for which \((\omega ,t) \mapsto \hat{\alpha }_\omega ([0,t]) \) is measurable, \((\mathcal {F}^0_{t})_{t \ge 0}\)adapted and maps into [0, 1]. \(\square \)
Lemma 6.13
Proof
Lemma 6.14
Remark 6.15
The intuition behind the Gardener’s Lemma is that we are replacing certain branches \( \beta ^{(\omega ,t)} \) of the randomized stopping time \( \xi \) by other branches \( \gamma ^{(\omega ,t)} \) to obtain a new stopping time \( \hat{\xi } \). This process happens along the measure \(\alpha \). Note that (6.6) implies that \({\int }1_{D}\left( (\omega ,t) \odot \tilde{\omega }\right) \,d\mathbb {W}^{t}_{0}(\tilde{\omega }) \,d\alpha (\omega ,t) \le \mathbb {W}^{0}_{\lambda }(D) \) for all Borel \(D \subseteq C(\mathbb {R}_{+})\). The authors like to think of \(\alpha \) as a stopping time and of the maps \((\omega ,t) \mapsto \beta ^{(\omega ,t)}\) and \((\omega ,t) \mapsto \gamma ^{(\omega ,t)}\) as adapted (in some sense that would need to be made precise). As these assumptions aren’t necessary for the proof of the Gardener’s Lemma, they were left out, but it might help the reader’s intuition to keep them in mind.
Proof of Lemma 6.14
We need to check that the \(\hat{\xi }\) we define is indeed a measure, that \((\mathsf {proj}_{C(\mathbb {R}_{+})})_*(\hat{\xi }) = \mathbb {W}^{0}_{\lambda }\) and that (5.1) holds for \(\hat{\xi }\).
Checking that \(\hat{\xi }\) is a measure is routine—we just note that (6.6) guarantees that \(\hat{\xi }(D) \ge 0 \) for all Borel D.
Proof of Proof of Proposition 6.10
We prove the contrapositive. Assuming that there exists a \( \pi ' \in \mathsf {JOIN}_{\lambda }(r_*(\xi )) \) with \( \left( r \times \mathsf {Id}\right) _*(\pi ')(\mathsf {SG}^\xi ) > 0 \), we construct a \( \xi ^\pi \in \mathsf {RST}_{\lambda }(\mu ) \) such that \( {\int }c\,d\xi ^\pi < {\int }c\,d\xi \).
If \(\pi ' \in \mathsf {JOIN}_{\lambda }(r_*(\xi ))\), then for any two measurable sets \(D_1,D_2 \subseteq S\), because Open image in new window and by making use of Lemma 6.12 we can deduce that Open image in new window . Using the monotone class theorem this extends to any measurable subset of \(S \times S\) in place of \(D_1 \times D_2\). So we can set Open image in new window and know that \((\mathsf {proj}_{C(\mathbb {R}_{+})\times \mathbb {R}_{+}})_*(\pi ) \in \mathsf {RST}_{\lambda }\) and that \(\pi \) is concentrated on \(\mathsf {SG}^\xi \).
Now we will argue that the integrand in the right hand side of (6.8) is negative \(\bar{\pi }\)almost everywhere. This will conclude the proof.
7 Variations on the theme
We proceed to prove Corollary 1.2. This is closely modelled on the treatment of the AzemaYor embedding in [6, Theorem 6.5]. As is the case there we run into a technical obstacle, though one which can be overcome by combining the ideas we have already seen in slightly new ways.
 (4’)
\( \mathbb E[c(B,\tau )] \in \mathbb {R}^n \) for all stopping times \( \tau \sim \mu \).
To get an existence result we may assume that \(c=(c_1,c_2)\) is componentwise lower semicontinuous and that both \(c_1\) and \(c_2\) are bounded below (in either of the ways described in the two versions of Theorem 3.1). Note that—because we are talking about the lexicographic order—\(\xi \in \mathsf {RST}_{\lambda }(\mu )\) is a solution of \((\textsc {OptStop'})\) for \(c\) iff \(\xi \) is a solution of \((\textsc {OptStop'})\) for \(c_1\) and among all such solutions \(\xi '\), \(\xi \) minimizes \({\int }c_2 \,d\xi '\). By Theorem 3.1 in the form that we have already proved the set of solutions of \((\textsc {OptStop'})\) for \(c_1\) is nonempty. It is also a closed subset of a compact set and therefore itself compact. This allows us to reiterate the argument that we used in the proof of Theorem 3.1 to find inside this set a minimizer of \(\xi ' \mapsto {\int }c_2 \,d\xi '\). This minimizer is the solution of \((\textsc {OptStop'})\) for \(c\).
With this in hand we may pick up our
Proof of Corollary 1.2
The same arguments as in the proof of Corollary 1.1 apply, so we may assume that our probability space satisfies Assumption 2.2. We start with a cost function \(c(\omega ,t) := (c_1(\omega ,t),c_2(\omega ,t)) := (\omega ^*(t), (\omega ^*(t)  \omega (t))^3)\). \(c_1(B,\tau ) _{L^{3}} \le \left B\right ^*_{\tau } _{L^{3}} \le K_1 \tau _{L^{3/2}}^{1/2}\), by the BurkholderDavisGundy inequalities, so \((c_1)_\) satisfies the uniform integrability condition and \(\mathbb E[c(B,\tau )]\) is finite for all stopping times \(\tau \sim \mu \). \(c_2 \ge 0\) and by the BurkholderDavisGundy inequalities \(\mathbb E[c_2(B,\tau )] \le \mathbb E[(B^*(\tau ))^3] \le K_1 \mathbb E[\tau ^{3/2}] = K_1 {\int }t^{3/2} \,d\mu (t)\) for some constant \(K_1\). The last term is finite by assumption.
For the right hand side of (7.4) we get the same expression with \(\omega \) replaced by \(\eta \). This concludes the proof that \(\omega (t)  \omega ^*(t) < \eta (t)  \eta ^*(t)\) implies \(((\omega ,t),(\eta ,t)) \in \mathsf {SG}\) and Lemma 3.7 gives us barriers Open image in new window , Open image in new window such that for their hitting times Open image in new window , Open image in new window by \(B_t  B^*_{t}\) we have Open image in new window a.s.
Again we want to show that Open image in new window a.s. and that they are actually stopping times. Again we do so by showing that they are both a.s. equal to the hitting time of the closure of the respective barrier. If Open image in new window then this works in exactly the same way as in Lemma 4.3. (This time we define \(\overline{\tau }_\varepsilon := \inf \{ t > 0 : (B^\varepsilon _t(\omega )  (B^\varepsilon )^*_{t}(\omega ), t) \in \overline{\mathcal {R}}\}\) where \(B^\varepsilon _t(\omega ) := B_t(\omega ) + A(t) \varepsilon \).) If Open image in new window then \((B^\varepsilon _t(\omega )  (B^\varepsilon )^*_{t}(\omega ), t) \in \overline{\mathcal {R}}\) and \(t>0\) need not imply \( B_t(\omega )  B^*_{t} (\omega ) < B^\varepsilon _t(\omega )  (B^\varepsilon )^*_{t}(\omega )\), which is essential for the topological argument showing that the hitting time of \(\mathcal {R}\) is less than or equal \(\overline{\tau }_\varepsilon \). But if Open image in new window , then Open image in new window and Open image in new window are both almost surely \(\le T\) where \(T := \inf \{ t > 0 : (0,t) \in \overline{\hat{\mathcal {R}}}\}\), so in the step where we show that the hitting time of \(\mathcal {R}\) is less than \(\overline{\tau }_\varepsilon \) we can argue under the assumption that \(\overline{\tau }_\varepsilon (\omega ) < T\). In this case we do have that \((B^\varepsilon _t(\omega )  (B^\varepsilon )^*_{t}(\omega ), t) \in \overline{\mathcal {R}}\) and \(t>0\) implies \( B_t(\omega )  B^*_{t} (\omega ) < B^\varepsilon _t(\omega )  (B^\varepsilon )^*_{t}(\omega )\). \(\square \)
Remark 7.1
We hope that the proofs of Corollaries 1.1 and 1.2 have given the reader some idea of how to apply the main results of this paper to arrive at barriertype solutions of constrained optimal stopping problems, as depicted in Fig. 1.
We would like to conclude by giving a couple of pointers to the interested reader who may want to work through the proofs corresponding to the remaining pictures in Fig. 1.
For the problem of minimizing \(\mathbb E[B^*_{\tau }]\), it may actually happen that the times Open image in new window from Lemma 3.7 do not coincide. Specifically one has to expect this to happen on a nonnegligible set when Open image in new window contains parts of the time axis which \(\hat{\mathcal {R}}\) does not contain. Under these circumstances an optimizer may turn out to be a true randomized stopping time, with a proportion of a path hitting the time axis at a certain point needing to be stopped while the rest continues. In this situation the picture alone does not completely describe the optimal stopping time.
For the problems involving absolute values one needs to make a minor modification in the proof of Proposition 6.10. Specifically one can allow “mirroring” the paths which are “transplanted” using the Gardener’s Lemma. This leads to a slightly different definition of StopGo pairs, which is perhaps most easily described by saying that in Fig. 2 the green paths which are stoppen by \(\sigma \) may be flipped upsidedown on either side.
Footnotes
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF).
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