A REGULARIZED KELLERER THEOREM IN ARBITRARY DIMENSION GUDMUND PAMMER ETH Z urich Z urich Switzerland

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A REGULARIZED KELLERER THEOREM IN ARBITRARY DIMENSION

GUDMUND PAMMER

ETH Z¨urich, Z¨urich, Switzerland

BENJAMIN A. ROBINSON

Universit¨at Wien, Vienna, Austria

WALTER SCHACHERMAYER

Universit¨at Wien, Vienna, Austria

Abstract. We present a multidimensional extension of Kellerer’s theorem on the existence of

mimicking Markov martingales for peacocks, a term derived from the French for stochastic pro-

cesses increasing in convex order. For a continuous-time peacock in arbitrary dimension, after

Gaussian regularization, we show that there exists a strongly Markovian mimicking martingale

Itˆo diﬀusion. A novel compactness result for martingale diﬀusions is a key tool in our proof.

Moreover, we provide counterexamples to show, in dimension d≥2, that uniqueness may not

hold, and that some regularization is necessary to guarantee existence of a mimicking Markov

martingale.

1. Introduction

Given a ﬁnite set of probability measures on Rdthat are increasing in convex order, Strassen

[42] showed in 1965 that there exists a Markov martingale whose marginals coincide with the

given probability measures. We call this latter property mimicking. For a family of measures

indexed by continuous time that are increasing in convex order, also called a peacock (Processus

Croissant pour l’Ordre Convexe), Kellerer [28] proved in 1972 that there exists a mimicking

strong Markov martingale in dimension one. The questions of continuity and uniqueness for

Kellerer’s mimicking martingale remained open until the work of Lowther [32,33,34,35,36]

completely clariﬁed the situation. Lowther showed that, in dimension one, there exists a unique

strong Markov mimicking martingale and, moreover this process has continuous paths when the

peacock is weakly continuous and the marginals have convex support. It is also known that

the strong Markov property is required to obtain uniqueness; Beiglb¨ock, Lowther, Pammer and

Schachermayer [5] construct a one-dimensional continuous Markov martingale whose marginals

coincide with those of Brownian motion but which does not have the strong Markov property.

E-mail addresses:gudmund.pammer@math.ethz.ch, ben.robinson@univie.ac.at,

walter.schachermayer@univie.ac.at.

Date: March 25, 2024.

2020 Mathematics Subject Classiﬁcation. 60G44, 60J60, 60H10, 60J25 (Primary) 60F99 (Secondary).

Key words and phrases. Kellerer’s theorem, Strassen’s theorem, mimicking martingales, peacocks, diﬀusion

processes, martingale optimal transport.

This research was funded in part by the Austrian Science Fund (FWF) [10.55776/Y782], [10.55776/P34743],

[10.55776/P35519], [10.55776/P35197]. For open access purposes, the author has applied a CC BY public

arXiv:2210.13847v3 [math.PR] 22 Mar 2024

A REGULARIZED KELLERER THEOREM IN ARBITRARY DIMENSION 2

While the problem of ﬁnding mimicking Markov martingales is thus very well understood

for one-dimensional peacocks, the higher dimensional case has remained wide open, although

50 years have passed since the publication of Kellerer’s result. In this paper, to the best of

our knowledge, we provide the ﬁrst known multidimensional extension of Kellerer’s theorem.

Given a peacock on Rd, we show that, after some Gaussian regularization, there exists a strongly

Markovian martingale Itˆo diﬀusion that mimics the regularized peacock.

To prove our result, we construct a martingale Itˆo diﬀusion that mimics the regularized

peacock on the dyadics, and then pass to a limit in ﬁnite dimensional distributions. In order

to take such a limit, we prove a compactness result for martingale Itˆo diﬀusions.

Additionally, we show that uniqueness does not necessarily hold in higher dimensions. We

consider an example of a martingale Itˆo diﬀusion studied by Robinson [38] and Cox–Robinson

[9], and we show that this martingale mimics the marginals of a two-dimensional Brownian

motion, while itself not being a Brownian motion. We also show, by means of counterexamples,

that the Gaussian regularization is necessary to guarantee the existence of a mimicking Markov

martingale.

For η > 0, let γηdenote the centered Gaussian law on Rdwith covariance ηid, and let ∗

denote the convolution operator between measures. Our main result is the following.

Theorem 1.1 (existence of mimicking martingales).Let (µt)t∈[0,1] be a weakly continuous

square-integrable peacock on Rd. Fix δ, ε > 0and, for each t∈[0,1], deﬁne the regularized

measure µr

t:=µt∗γε(t+δ). Then there exists a strongly Markovian martingale Itˆo diﬀusion

(Mt)t∈[0,1] mimicking the regularized peacock (µr

t)t∈[0,1].

More precisely, there exists a measurable function (t, x)7→ σt(x)on [0,1] ×Rd, taking values

in the set of positive deﬁnite matrices such that, on any probability space (Ω,F,P)supporting

a standard Rd-valued Brownian motion (Bt)t∈[0,1] and an independent random variable ξ∼µ0,

the martingale Msatisﬁes

(1.1) dMt=σt(Mt)dBt, M0=ξ.

The map (t, x)7→ σt(x)2:=σt(x)σt(x)⊤is locally Lipschitz continuous in the variable x, uni-

formly in t∈[0,1] and, for every x∈Rd, there exist constants cx, Cx>0such that, for

t∈[0,1], we have the bounds

cxid ≤σt(x)2≤Cxid.

Moreover, the martingale Mis a Feller process.

Note that, in particular, the mimicking martingale that we construct in Theorem 1.1 is

continuous and strongly Markovian. A key ingredient in the construction of this mimicking

martingale is the following result that allows us to pass to limits of martingale Itˆo diﬀusions,

the details of which are presented in Section 5.

Theorem 1.2 (compactness of martingale Itˆo diﬀusions).A set of martingale Itˆo diﬀusions

satisfying Assumptions 5.1 (A1)–(A5) is precompact in the set of martingale Itˆo diﬀusions with

respect to convergence in ﬁnite dimensional distributions.

Our next main result is that, in dimension d≥2, mimicking martingales of the form (1.1)

may not be unique.

Theorem 1.3 (non-uniqueness of mimicking martingales).Let (Bt)t∈[0,1] be a standard Brow-

nian motion on R2with initial law Law(B0) = η, where ηis rotationally invariant with ﬁnite

second moment. Deﬁne a peacock µby µt= Law(Bt), for t∈[0,1].

Then there exists a continuous strongly Markovian martingale diﬀusion (Mt)t∈[0,1] of the form

(1.1), that is not a Brownian motion, such that Law(Mt) = µt, for all t∈[0,1].

We further construct a series of counterexamples in dimension d= 4 which show that, without

regularization, Theorem 1.1 does not hold in full generality, even without imposing continuity

of the mimicking martingale, let alone the Itˆo diﬀusion property.

A REGULARIZED KELLERER THEOREM IN ARBITRARY DIMENSION 3

Theorem 1.4 (necessity of regularization).There exists a weakly continuous square-integrable

peacock (µt)t≥0on R4such that, for the peacock (µt∗γt)t≥0, there exists no mimicking Markov

martingale.

While previous authors have considered the problem of ﬁnding mimicking martingales in

general dimensions, to the best of our knowledge the present work is the ﬁrst to provide a

multidimensional extension of Kellerer’s theorem. Prior to Kellerer’s work, Doob [15] proved

the existence of mimicking martingales taking values in an abstract compact space in continuous

time, but notably did not consider the Markov property. More recently, Hirsch and Roynette

[22] proved existence for continuous-time peacocks on Rd,d≥1, with right-continuous paths,

again without the Markov property.

Juillet [25] considered generalizing Kellerer’s theorem in two diﬀerent directions, ﬁrst showing

that when a peacock on Ris indexed by a two-parameter family with some partial order,

mimicking martingales may not exist at all. Moreover, [25] provides an example of a peacock

on R2for which there exists no mimicking martingale that additionally satisﬁes the so-called

Lipschitz Markov property, deﬁned in [25, Deﬁnition 6]. The Lipschitz Markov property implies

the Feller property and, for c`adl`ag processes, the strong Markov property; see [35, Lemma 4.2].

The key property of the class of c`adl`ag Lipschitz Markov processes is compactness with respect

to convergence in ﬁnite dimensional distributions, as shown in [35, Lemma 4.5]. On the other

hand, it is well known that the class of Markov martingales is not closed with respect to this

mode of convergence; see, e.g. [3, Example 1]. All proofs of Kellerer’s theorem that are known

to us make use of the compactness of Lipschitz Markov processes; see, e.g. [3,24,28,35]. In

light of the result of [25], the notion of Lipschitz Markovianity does not lend itself well to

the higher-dimensional problem. In its place, we consider a class of Feller processes that are

martingale Itˆo diﬀusions with particular properties. We show in Theorem 1.2 that this set of

processes is compact with respect to convergence in ﬁnite dimensional distributions.

We have seen that, in dimension one, uniqueness holds in the class of continuous strong

Markov mimicking martingales when the marginals of the peacock have convex support. The-

orem 1.3 shows that strong Markovianity is not suﬃcient to guarantee uniqueness in higher

dimensions, by exhibiting a continuous two-dimensional strong Markov martingale with Brow-

nian marginals that is not itself a Brownian motion. The question of the existence of martingales

distinct from Brownian motion that have Brownian marginals goes back to Hamza and Kle-

baner [21], who showed that such a fake Brownian motion with discontinuous paths exists in

one dimension. As already mentioned, the culmination of this one-dimensional investigation

was the construction [5] of a continuous Markovian fake Brownian motion. Of course Brow-

nian motion is the unique continuous strong Markov martingale with Brownian marginals in

one dimension. In two dimensions however, we show in Theorem 1.3 that there exists a fake

Brownian motion that is continuous and strongly Markovian.

We remark that the mimicking martingale of Theorem 1.1 is an Itˆo diﬀusion process with

Markovian diﬀusion coeﬃcient. Finding mimicking martingales of this form has also received

extensive interest since the work of Krylov [29] and Gy¨ongy [20]. In fact we twice apply a more

recent result of Brunick and Shreve [7] on mimicking Markovian diﬀusions in our construction

in Section 2.

For a more detailed review of the existing literature, we refer the reader to the surveys of

Hirsch, Roynette and Yor [24] and Beiglb¨ock, Pammer and Schachermayer [4], and the references

therein.

The structure of the present article is as follows. In Section 2, we construct a strongly

Markovian mimicking martingale Itˆo diﬀusion, thus proving Theorem 1.1. We then prove The-

orem 1.3 in Section 3, by providing a counterexample to uniqueness of mimicking martingales.

We present further examples in Section 4, which show that existence may fail without regu-

larization, thus proving Theorem 1.4. Finally, in Section 5, we prove the compactness result

Theorem 1.2 for martingale Itˆo diﬀusions, which is key to the proof of Theorem 1.1 in Section 2.

A REGULARIZED KELLERER THEOREM IN ARBITRARY DIMENSION 4

We introduce some notation and terminology that will be used throughout the paper. The

notation |· | represents the Euclidean norm on Rd, and BRdenotes the closed ball with radius

R > 0 centered at the origin. We denote by P2(Rd) the set of probability measures on Rd

with ﬁnite second moment. We denote by FXthe natural ﬁltration of a stochastic process X,

enlarged as necessary to satisfy the usual conditions. For measures µ, ν, we write µ⪯νto

denote that µis dominated by νin convex order; i.e. for any convex function f,Rfdµ≤Rfdν.

A family of measures (µt)t∈Iis called a peacock if it is increasing in the convex order.1We say

that a process (Xt)t∈Imimics (µt)t∈Iif Law(Xt) = µtfor all t∈I.

For matrices A, B ∈Rd×dthe notation A≤Bdenotes that the matrix B−Ais positive

semideﬁnite. When working with matrices, we will always use the Hilbert–Schmidt norm (also

known as the Frobenius norm): for A∈Rd×d, we write ∥A∥= Tr(AA⊤) = Pd

i,j=1 A2

ij =

i=1 λ2

i, where λiare the eigenvalues of A. We further denote the square matrix A:=AA⊤.

2. Construction of a mimicking martingale

Let d≥2 and let (µt)t∈[0,1] be a weakly continuous peacock in P2(Rd); i.e. µt0⪯µt1for

all t0≤t1, supt∈[0,1] R|x|2µt(dx) = R|x|2µ1(dx)<∞, and t7→ Rfdµtis continuous for any

bounded continuous function f. Fix δ, ε > 0 and deﬁne the regularized peacock µrby

(2.1) µr

t=µt∗γε(t+δ), t ∈[0,1].

Note that the process (µr

t)t∈[0,1] is a peacock satisfying µt⪯µr

t, for all t∈[0,1].

Remark 2.1. The relevant feature of the function

(2.2) φ(t) = ε(t+δ), t ∈[0,1],

is that ϕ(0) >0 and t7→ ϕ(t) is strictly increasing. It will become clear from the construction

below that we can replace (2.2) with any such function.

In this context, we also normalize the peacock (µt)t∈[0,1] by making a deterministic time

change so that R|x|2µt+h(dx)−R|x|2µt(dx) = h, for t∈[0,1), h > 0. For convenience, we still

take φas in (2.2) after the time change.

Moreover, in place of the Gaussian family (γε(t+δ))t∈[0,1], one may take another family of

centered probability measures (ηt)t∈[0,1] ⊆ P2(Rd) that is weakly continuous and strictly in-

creasing in convex order. Provided that these measures have smooth densities that are uni-

formly bounded from below on Rd, one could apply similar arguments to prove an analogue of

Theorem 1.1. However, the proof would become signiﬁcantly more involved.

Fix n∈Nand consider the dyadics Sn:={2−n,2·2−n,...,2n·2−n} ⊆ [0,1]. For k∈

{0,...,2n}, denote tn

k:=k2−n. We will construct a martingale Itˆo diﬀusion that mimics µron

the dyadics Sn. Theorem 5.4 will allow us to pass to a limit. We ﬁrst construct martingale

Itˆo diﬀusions on each dyadic interval, before concatenating these intervals. This step is rather

standard; cf. [23,24]. For our purposes it is convenient to use the concept of stretched Brownian

motion introduced in [1]. We now ﬁx k∈ {0,...,2n−1}and consider the interval [tn

k, tn

k+1).

Recall that Bdenotes a standard Brownian motion on Rd.

Lemma 2.2. Let (µt)t∈[0,1] be a weakly continuous square-integrable peacock. Then there exists

a strongly Markovian martingale diﬀusion (¯

Mn,k

t)t∈[tn

k,tn

k+1], with the representation

(2.3) d ¯

Mn,k

t= ¯σn,k

t(¯

Mn,k

t)dBt,on [tn

k, tn

k+1),

for some measurable function (t, x)7→ ¯σn,k

t(x), taking values in the set of positive semideﬁnite

matrices, such that Law( ¯

Mn,k

k) = µtn

kand Law( ¯

Mn,k

k+1 ) = µtn

k+1 .

1The terminology peacock was introduced by Hirsch, Profeta, Roynette and Yor [23] as a pun on the French

Processus Croissant pour l’Ordre Convexe (PCOC), meaning a process increasing in convex order.

A REGULARIZED KELLERER THEOREM IN ARBITRARY DIMENSION 5

Proof. Let ( ¯

Mn,k)t∈[tn

k,tn

k+1]be the stretched Brownian motion with Law( ¯

Mn,k

k) = µtn

kand

Law( ˜

Mn,k

k+1 ) = µtn

k+1 , as deﬁned in [1, Deﬁnition 1.6]. By deﬁnition, ¯

Mn,k is a martingale with the

representation d ¯

Mn,k

t=θn,k

tdBt, for some FB-predictable process θn,k taking values in the set

of positive semideﬁnite matrices. Moreover, by [1, Corollary 2.5], ¯

Mn,k is a strong Markov pro-

cess. Thus Lemma A.2 gives the existence of a measurable function ¯σn,k

·: [tn

k, tn

k+1)×Rd→Rd×d

such that (2.3) holds. □

We do not yet have a control on the matrix norm of (¯σn,k

t(¯

Mn,k))t∈[tn

k,tn

k+1]. In order to achieve

an upper bound on the diﬀusion matrix, we make a ﬁrst convolution with a Gaussian. This

will have an averaging eﬀect and allow us to control the diﬀusion from above almost surely.

Namely, we take a centered Gaussian random variable Γn,k with covariance matrix (ε[tn

k+δ])id,

independent of FBand F¯

Mn,k , and deﬁne

(2.4) ˜

Mn,k

t:=¯

Mn,k

t+ Γn,k, t ∈[tn

k, tn

k+1].

Then, for the initial law in this interval, we have Law( ˜

Mn,k

k) = µtn

k∗γε(tn

k+δ)=µr

k, and for

the terminal law, we have the ordering Law( ˜

Mn,k

k+1 ) = µtn

k+1 ∗γε(tn

k+δ)⪯µtn

k+1 ∗γε(tn

k+1+δ)=

µr

k+1 , where we recall the deﬁnition of µrfrom (2.1). Later we will make a second Gaussian

convolution, which will allow us to also bound the squared diﬀusion matrix from below. We

now prove that the square of the diﬀusion matrix obtained after the ﬁrst convolution is locally

bounded and locally Lipschitz.

Lemma 2.3. For n∈N,k∈ {0,...,2n}and ¯σn,k as in (2.3), deﬁne the matrix-valued function

(t, x)7→ ˜σn,k

t(x)as the unique positive semideﬁnite square root of

(2.5) ˜σn,k

t(x)2:=R¯σn,k

t(y)2gn,k(x−y) ¯mn,k

t(dy)

Rgn,k(x−y) ¯mn,k

t(dy), t ∈[tn

k, tn

k+1), x ∈Rd,

where gn,k is the density of a Gaussian with mean zero and covariance matrix (ε[tn

k+δ])id, and

¯mn,k

t= Law( ¯

Mn,k

t).

Then for every compact set K⊆Rd, there exist constants CK, LK, independent of t,kand

n, such that

∥˜σn,k

t(x)2∥ ≤ CK,(t, x)∈[tn

k, tn

k+1]×K,

and x7→ ˜σn,k

t(x)2is Lipschitz on Kwith Lipschitz constant LK, for all t∈[tn

k, tn

k+1].

Proof. Choose R > 0 such that R|x|d(µ1∗γε(1+δ))(x)≤R/2. Applying Doob’s maximal

inequality and the convex ordering of the marginals gives the bound ¯mn,k

t(BR)≥1

2, for all

t∈[tn

k, tn

k+1] and all k∈ {1,...,2n−1}. Then, for an arbitrary compact set K⊆Rd, we can

bound the normalising constant in the denominator of (2.5) by

ZRd

gn,k(x−y) ¯mn,k

t(dy)≥ZBR

gn,k(x−y) ¯mn,k

t(dy)

≥¯

2(2πε[tn

k+δ])−d

2,for x∈K,

(2.6)

where ¯

CK:= inf{exp{−ε−1δ−1|x−y|2}:x∈K, y ∈BR}>0, independent of tand k. We

bound the numerator of (2.5) in the Hilbert–Schmidt norm by



Z¯σn,k

t(y)2gn,k(x−y) ¯mn,k

t(dy)



≤(2πε[tn

k+δ])−d

2E[∥¯σn,k

t(¯

Mn,k

t)∥2].

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AREGULARIZEDKELLERERTHEOREMINARBITRARYDIMENSIONGUDMUNDPAMMERETHZ¨urich,Z¨urich,SwitzerlandBENJAMINA.ROBINSONUniversit¨atWien,Vienna,AustriaWALTERSCHACHERMAYERUniversit¨atWien,Vienna,AustriaAbstract.WepresentamultidimensionalextensionofKellerer’stheoremontheexistenceofmimickingMarkovmartingalesforpea...

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