A stability result on optimal Skorokhod embedding

0.19 MB·

by  Gaoyue Guo

A stability result on optimal Skorokhod embedding
∗
Gaoyue Guo
†
7 January 31, 2017
1
0
2

n
a
Abstract
J

7
This is a continuation of Guo, Tan & Touzi [10]. Motivated by the model-
2
independent pricing of derivatives calibrated to the real market, we consider an

] optimization problem similar to the optimal Skorokhod embedding problem, where
R
the embedded Brownian motion needs only to reproduce a finite number of prices
P
. of Vanilla options. As same as in [10], we derive in this paper the corresponding
h
dualitiesandthegeometriccharacterization ofoptimizers. Thenweshowastability
t
a
result,i.e. whenmoreandmoreVanillaoptionsaregiven,theoptimizationproblem
m
converges toanoptimalSkorokhodembeddingproblem,whichconstitutes thebasis
[
ofthenumericalcomputation inpractice. Inaddition,bymeansofdifferentmetrics

1
on the space of probability measures, a convergence rate analysis is provided under
v
4 suitable conditions.
0
Keywords. Skorokhod embedding,Duality,Monotonicityprinciple,L´evy-Prokhorov
2
8 metric, Wasserstein metric.
0
.
1
0 1 Introduction
7
1
: TheSkorokhodembeddingproblem(SEP)consists in representingacentered prob-
v
ability onthereallineasthedistributionofaBrownianmotion stoppedatachosen
i
X
stopping time, see e.g. the survey paper [17] of Ob l´oj for a comprehensive account
r of thefield. Motivated by thestudyof the model-independentpricingof derivatives
a
consistent with the market prices of Vanilla options, the associated optimization
problem over all embedding stopping times has received the substantial attention
from the mathematical finance community. According to the no-arbitrage frame-
work, the underlying asset is required to be a martingale, and additionally, the
market calibration allows to recover the marginal laws of the underlying at cer-
tain maturities, see e.g. Breeden & Litzenberger [2]. Therefore, based on the fact
that every continuous martingale is a time-changed Brownian motion by Dambis-
Dubins-Schwarz theorem, Hobson studied the robust hedging of lookback options
in his seminal paper [13] by means of the SEP. The main idea of his pioneering
work is to exploit some solution to the SEP satisfying some optimality criteria,
∗Gaoyue Guo is grateful for the helpful discussions he has had with Henry-Labord`ere, Ru¨schendorf,
Tan and Touzi, and especially the remarks given by Touzi.
†GaoyueGuo thankfullyacknowledgesthe financialsupportoftheERC321111Rofirmandthe ANR
Isotace. CMAP, Ecole Polytechnique, France. [email protected]
1
which yields the robust hedging strategy and allows to solve together the model-
independent pricing and robust hedging problems. Here after, various extensions
were achieved in the literature, such as Cox & Hobson [3], Hobson & Klimmek [15],
Cox, Hobson & Ob l´oj [4], Cox & Ob l´oj [5], Davis, Ob l´oj & Raval[7] and Ob l´oj &
Spoida [18], etc. A thorough literature is provided in Hobson’s survey paper [14].
This heuristic idea is generalized by Beiglb¨ock, Cox & Huesmann [1] and Guo,
Tan & Touzi [10, 11], where the optimal SEP is defined by a unifying formulation
that recovers the previous known results. Namely, their main results are twofold.
First, an expected duality is established, i.e. identity between theoptimal SEPand
thecorrespondingrobusthedgingproblem. Second, theyderiverespectively achar-
acterization of the optimizers by a geometric pathwise property, due to which, all
the known optimal embeddings can be interpreted through one unifying principle.
From thefinancialviewpoint, Vanilla options areassumedto beideally liquid in
the above literature, i.e. the prices of call options are known for all strikes at some
maturity, or equivalently, theunderlyinghas the uniquedistribution determined by
the market prices at this maturity. However, only a finite number of call options
are traded in practice, which leads to another optimization problem that we shall
consider in this paper.
Toillustratetheidea,letusstartbyasimplecasewhereallcalloptionsareofthe
same maturity. Let K = (K ) be a collection of strikes and C = (C )
i 1≤i≤n i 1≤i≤n
be the set of corresponding prices of call options. Then, roughly speaking, the
embedding problem aims to find a stopping time τ on some Brownian motion B =
(B ) such that the stopped process B := B is uniformly integrable
t t≥0 τ∧· τ∧t t≥0
(UI) and
(cid:0) (cid:1)
E (B K )+ = C , for all i= 1, ,n.
τ i i
− ···
The stopping time(cid:2)τ is called (cid:3)a (K,C) embedding. In particular, if the price
−
vector C is given by some centered probability µ on R, i.e.
C = (x K )+dµ(x), for all i= 1, ,n,
i i
− ···
Z
then every µ embedding is clearly a (K,C) embedding. Following the spirit of
− −
theoptimalSEP,foranygiven K andC,wemayconsideranoptimization problem
among all (K,C) embeddings, that may write formally as
−
sup E[Φ(B,τ)],
τ: (K,C)−embedding
where Φ denotes a measurable reward function to be specified later.
Let theno-arbitrage condition hold, namely, there exists a centered distribution
µ that determines the priceof each call option. Since the distribution µ is generally
unknowntopractitioners,weask: withtheincreasinginformationKn = (Kn)
i 1≤i≤n
and Cn = (Cn = (x Kn)+dµ) given by the market, could we recover
i − i 1≤i≤n
asymptoticallyµ? Moreimportantly,doestheoptimizationproblemaboveconverge
R
to the optimal SEP with target distribution µ:
sup E[Φ(B,τ)] sup E[Φ(B,τ)] as n .
−→ −→ ∞
τ: (Kn,Cn)−embedding τ: µ−embedding
Ifthisconvergence holds,wewillseelaterthatthecomputationofP(µ)reducestoa
finite-dimensional optimization problem, and moreover, it is of interest to estimate
the convergence rate.
2
The paper is organized as follows. In Section 2, we first recall the formulation
of the optimal SEP and then provide the formulation of our optimal embedding
problem as well as its dual problems. We establish the two dualities and show
further the expected convergence when given more and more market information.
Next, the monotonicity principle is derived in Section 3. In Section 4, by means of
different metrics on the space of probability measures, we provide an estimation of
the convergence rate. We finally give the related numerical computation in Section
??.
Notations.
(i) Let Ω be the space of all continuous functions ω = (ω ) on R such that
t t≥0 +
ω = 0, B = (B ) the canonical process, i.e. B (ω) := ω and F = ( ) the
0 t t≥0 t t t t≥0
F
canonical filtration generated by B, i.e. := σ(B ,s t). Denote by P the
t s 0
F ≤
Wiener measure and by Fa = ( a) the augmented filtration under P .
Ft t≥0 0
(ii) For every integer k 1, define
≥
Rk resp. Rk := (x , ,x ) Rk : x > > x (resp. x > > x )
< > 1 ··· k ∈ 1 ··· k 1 ··· k
(cid:16) (cid:17) n o
and the corresponding closure Rm (resp. Rm).
≤ ≥
(iii)Letm 1beafixedintegerandsetΘm := Rm Rm. Definetheproductcanon-
≥ ≤ ∩ +
ical space by Ω := Ω Θm with its elements denoted by ω¯ := (ω,θ := (θ , ,θ )).
1 m
× ···
Denote further by (B,T := (T , ,T )) the canonical element on Ω, i.e. B(ω¯) :=
1 m
···
ω and T(ω¯) := θ for all ω¯ = (ω,θ) Ω. The corresponding canonical filtration is
∈
denoted by F= ( ) , where
t t≥0
F
:= σ(B , for s t) σ( T s , for s t and i= 1, ,m).
t s i
F ≤ ∨ { ≤ } ≤ ···
In particular, all random variables T , ,T are F stopping times.
1 m
··· −
(iv) We endow Ω with the uniform convergence topology, and Ω with the product
topology,thenΩandΩarebothPolishspaces(metrizable,separableandcomplete).
2 An optimal embedding problem
In this section, we first give the problem formulation as well as its dual problems,
see Guo, Tan & Touzi [10] for more details. We emphasize that the problem is
formulated in a weak setting, i.e. the stopping times are identified by means of
probability measures on the enlarged space Ω.
Let (Ω) be the space of all (Borel) probability measures on Ω, and define
P
:= P (Ω) : B is an F Brownian motion and B is UI under P .(2.1)
P ∈ P − Tm∧·
n o
Remark 2.1. (i) For any P , T , ,T can be considered as ordered random-
1 m
∈ P ···
ized stopping times under P, see e.g. Beiglbo¨ck, Cox & Huesmann [1].
(ii) Throughout the paper, m is a fixed integer which stands for the number of
maturities. In the following, we consider different embedding problems according to
the constraints on B , ,B .
T1 ··· Tm
3
2.1 Optimal embedding problems
Optimal Skorokhod embedding problem We begin by recalling the opti-
mal SEP. Let µ := (µ , ,µ ) be a vector of probability distributions on R and
1 m
···
denote, for any integrable function φ: R R,
→
µ (φ) := φ(x)dµ (x), for all i = 1, ,m.
i i
···
R
Z
The vector µ is called a peacock if
µ has a finite first moment, i.e. µ (x ) < + , for all i= 1, ,m;
i i
| | ∞ ···
the map i µ (φ) is non-decreasing for any convex function φ.
i
7→
A peacock µ is called centered if µ (x) = 0 for all i= 1, ,m. Denote by P(cid:22) the
i
···
collection of all centered peacocks. For every vector µ = (µ , ,µ ), define the
1 m
···
set of µ embeddings
−
P
(µ) := P : B µ , for all i = 1, ,m . (2.2)
P ∈P Ti ∼ i ···
n o
AsaconsequenceofKellerer’stheorem, (µ)isnonemptyifandonlyifµ P(cid:22). In
P ∈
the following, let us fix a non-anticipative function Φ :Ω R, i.e. Φ is measurable
→
and Φ(ω,θ) = Φ ω ,θ holds for all (ω,θ) Ω. We define the optimal SEP for
θm∧· ∈
every µ P(cid:22) by
∈ (cid:0) (cid:1)
P(µ) := sup EP[Φ(B,T)]. (2.3)
P∈P(µ)
It follows by Guo, Tan & Touzi [10] that, under some suitable metric on P(cid:22), the
map µ P(µ) is concave and upper-semicontinuous, which leads to the required
7→
dualities. As for the optimal embedding problem in the following, we will proceed
with an analogous analysis to derive the dualities.
Optimal embedding problem Next let us turn to define the optimization
problem described in Section 1, where the embedded Brownian motion is only
required to be consistent with a finite number of market prices of call options. For
the sake of clarity, we assume that the call options of different maturities have
the same set of strikes. Throughout the paper, K := (K ) Rn is reserved
i 1≤i≤n ∈ <
for the vector of strikes and C := (C ) for the price matrix of call
i,j 1≤i≤m,1≤j≤n
options, indexed by maturity and strike. Then a probability P is called a
∈ P
(K,C) embedding if
−
EP (B K )+ = C , for all i= 1, ,m and j = 1, ,n. (2.4)
Ti − j i,j ··· ···
Denote by(cid:2) (K,C) the(cid:3)collection of all (K,C) embeddings and by (K) Rmn
P − A ⊂ +
thesetofall matrices C such that (K,C)is nonempty. Similarly, for each matrix
P
C (K), we may define the optimal embedding problem by
∈ A
P(K,C) := sup EP[Φ(B,T)]. (2.5)
P∈P(K,C)
It follows by definition that (K) is convex, and we pursue to derive the corre-
A
sponding dualities by a similar argument. However, the set (K,C) is generally
P
4
not compact, see Example 2.2 below. Notice that the map C P(K,C) is also
7→
concave, then a classical result in convex analysis is applied to obtain the required
result.
Example 2.2. Take m = n = 1, K = 0 and C = 2. Let (µk) be a sequence of
k≥2
probability distributions defined by
1 2 1
µk(dx) := δ (dx)+ 1 δ (dx)+ δ (dx).
k {−k} − k {−k−k2} k {2k}
(cid:18) (cid:19)
It follows by a straightforward computation that
µk(x) = 0 and µk(x+) = 2.
Moreover, µk converges weakly to the measure µ which puts the unit mass on 1.
−
Take an arbitrary sequence of measures (P ) with P (µk) (0,2), then
k k≥2 k
∈ P ⊂ P
it admits a convergent subsequence denoted again by (P ) , see e.g. Theorem 2.8
k k≥2
or Lemma 4.3 in Guo, Tan & Touzi [10]. Moreover, any accumulation point P
satisfies EP[B ]= 1 and EP[B+]= 0, which implies further P / (0,2).
T − T ∈ P
2.2 Dual problems and dualities
In this section, we introduce the corresponding dual problems. Recall that P
0
is the Wiener measure on Ω under which the canonical process B is a standard
Brownian motion, F = ( ) is its natural filtration and Fa = ( a) is the
Ft t≥0 Ft t≥0
augmented filtration by P . Denote by a the collection of all increasing families
0
T
of Fa stopping times τ = (τ , ,τ ) such that the process B is uniformly
− 1 ··· m τm∧·
integrable. Define also by Λ the space of continuous functions λ : R R with
→
linear growth and by Λm its m product. For µ = (µ , ,µ ) P(cid:22), λ =
1 m
− ··· ∈
(λ , ,λ ) Λm, x = (x , ,x ) Rm and ω,θ = (θ , ,θ ) Ω, we
1 m 1 m 1 m
··· ∈ ··· ∈ ··· ∈
denote
(cid:0) (cid:1)
m m
µ(λ) := µ (λ ), λ(x) := λ (x ) and ω := (ω , ,ω ).
i i i i θ θ1 ··· θm
i=1 i=1
X X
Then the first dual problem of (2.3) is given by
D (µ) := inf supEP[Φ(B,T) λ(B )]+µ(λ) (2.6)
0 T
λ∈Λm(P∈P − )
= inf sup EP0[Φ(B,τ) λ(B )]+µ(λ) . (2.7)
τ
λ∈Λm(cid:26)τ∈Ta − (cid:27)
As for the second dual problem, we return to the enlarged space Ω. An F adapted
−
continuous process S = (S ) is called a strong supermartingale if
t t≥0
P−
EP S S , P a.s.
τ2 Fτ1 ≤ τ1 −
forallF stoppingtimesτ (cid:2) τ (cid:12)and(cid:3)allP . Denoteby thesetofall strong
− 1 ≤ 2(cid:12) ∈ P S P−
supermartingales starting at zero and put
:= (λ,S) Λm :λ(ω )+S (ω) Φ(ω,θ), for all (ω,θ) Ω .
D ∈ ×S θ θm ≥ ∈
Then the se(cid:8)cond dual problem is given by (cid:9)
D(µ) := inf µ(λ). (2.8)
(λ,S)∈D
5
Remark 2.3. (i) By penalizing the marginal constraints, we obtain the first dual
problem D (µ), where a multi-period optimal stopping problem appears for every
0
fixed λ Λm.
∈
(ii)The second dual problem D(µ) is slightly different to that in [10], where we wrote
S as an stochastic integral. As our main concern is to study the model-independent
pricing, we show later that the formulation D(µ) is enough to deduce the required
results.
Following the idea above, we may define similarly the dual problems of (2.5). For
C = (C ) , α = (α ) Rmn and x = (x , ,x ) Rn,
i,j 1≤i≤m,1≤j≤n i,j 1≤i≤m,1≤j≤n 1 n
∈ ··· ∈
set
m,n m,n
α C := α C , α x := α x and x+ := (x+, ,x+).
· i,j · i,j · i,j · j 1 ··· n
i=1,j=1 i=1,j=1
X X
Hence, the dual problems of (2.5) are defined respectively by
D (K,C) := inf supEP Φ(B,T) α (B K)+ +α C (2.9)
0 T
α∈Rmn − · − ·
P∈P
n o
(cid:2) (cid:3)
= inf sup EP0 Φ(B,τ) α (B K)+ +α C .(2.10)
τ
α∈Rmn τ∈Ta − · − ·
n o
(cid:2) (cid:3)
and
D(K,C) := inf α C. (2.11)
(α,S)∈D(K) ·
where
(K) := (α,S) Rmn :α (ω K)++S (ω) Φ(ω,θ), for all (ω,θ) Ω .
D ∈ ×S · θ − θm ≥ ∈
(cid:8) (cid:9)
Assumption 2.4. Φ : Ω R is bounded from above and the map ω¯ Φ(ω¯) is
→ 7→
upper-semicontinuous.
Theorem 2.5. Let Assumption 2.4 hold.
∗
(i) There exists a P (µ) such that
∈ P
∗
EP [Φ(B,T)] = P(µ) = D (µ) = D(µ).
0
(ii) Assume further C (K), then one has
∈ C
P(K,C) = D (K,C) = D(K,C).
0
Proof. The proof of D (K,C) = D(K,C) is as same as that of D (µ) = D(µ) by
0 0
by Guo, Tan & Touzi [10], then It remains to prove P(K,C) = D (K,C).
0
One has by Corollary 4.2 of Davis & Hobson [6] that (K) (K) (K),
C ⊂ A ⊂ C
where (K) denotes the closure of (K) and (K) Rmn consists of all matrices
C C C ⊂ +
C = (C ) such that for all i= 1, ,m and j = 1, ,n
i,j 1≤i≤m,1≤j≤n
··· ···
(C ) Rm and (C ) Rn,
i,j 1≤i≤m ∈ < i,j 1≤j≤n ∈ >
C C C C
C > ( K )+ and 1 > i,j − i,j+1 > i,j−1− i,j.
i,j j
− K K K K
j+1 j j j−1
− −
6
For the sake of simplicity, we set P(C) P(K,C) throughout the proof. It
≡
follows by definition that the map C P(C) is concave on (K). Notice that
(K) Rmn is convex and open, then7→C P(C) is continuouCs on (K). Hence,
C ⊂ 7→ C
we may follow the reasoning in the proof of Theorem 3.10 in Guo, Tan & Touzi [12]
and extend the map P from (K) to Rmn by
C
P(C), if C (K),
P(C) := ∈ C
( , otherwise.
−∞
e
The concavity and upper semicontinuity of the map C P(C) follow immediately
7→
from the definition. Then, one obtains by Fenchel-Moreau theorem
e
P(C) = P∗∗(C),
where P∗∗ denotes the biconjugae of P. In particular, for C (K) one has
e e
∈ C
P(C) = P(C) = P∗∗(C)
e e
= inf α C P∗(α) = inf α C inf α C′ P(C′)
α∈Rmn · e − e α∈Rmn · −C′∈Rmn · −
n o
inf (cid:8)α C +esup(cid:9) sup EP Φ(B,T) α (B(cid:8) K)+e (cid:9)
T
≥ α∈Rmn · C∈C(K) P∈P(K,C) − · −
n n oo
(cid:2) (cid:3)
= inf α C + sup EP Φ(B,T) α (B K)+
T
α∈Rmn · − · −
n P∈∪C∈C(K)P(K,C)n oo
(cid:2) (cid:3)
= inf α C + sup EP Φ(B,T) α (B K)+ ,
T
α∈Rmn · − · −
n P∈∪C∈C(K)P(K,C)n oo
(cid:2) (cid:3)
where the last inequality follows from the fact that (K,C) is included
∪C∈C(K)P
in the closure of (K,C) under the weak convergence. Hence
C∈C(K)
∪ P
P(C) = P(C) = P∗∗(C) D (C) P(C),
0
≥ ≥
which yields the required duality.
e e
Remark 2.6. Notice that the set (K,C) is generally not compact with respect to
P
the weak convergence, due to which the existence of optimizers of P(K,C) can not
be ensured.
2.3 Convergence of optimal embedding problems
Let us study here the asymptotic behavior of the upper bound with respect to the
market information. Assume that the market is consistent with a centered peacock
µ= (µ , ,µ ), then we ask: when more and more call options are traded, does
1 m
···
the upper bound converge to P(µ)? Namely, let Kn = (Kn) Rn be the
i 1≤i≤n ∈ <
vector of strikes that are available in the market and Cn = (Cn ) be
i,j 1≤i≤m,1≤j≤n
the corresponding price matrix given by Cn = µ (x Kn)+ for all i= 1, ,m
i,j i − j ···
and j =1, ,n. Define the bound and the mesh(cid:16)of K by (cid:17)
···
Kn := (Kn)− (Kn)+ and ∆Kn := max Kn Kn .
| | 1 ∧ n 1<i≤n i − i−1
(cid:0) (cid:1)
Loosely speaking, to capture asymptotically µ by (Kn,Cn), a necessary condition
for the sequence (Kn) is the following:
n≥1
7
Assumption 2.7. The sequence (Kn) satisfies
n≥1
lim Kn = + and lim ∆Kn = 0.
n→∞| | ∞ n→∞
Motivated by financial applications, the sequence (Kn) is assumed to be in-
n≥1
creasing, i.e. Kn Kn+1 for all n 1 if Kn and Kn+1 are viewed as sets. It
⊂ ≥
follows by definition that the map n P(Kn,Cn) is non-increasing and
7→
P(Kn,Cn) P(µ), for all n 1.
≥ ≥
Theorem2.8. LetAssumption2.4hold. Thenforanyincreasingsequence(Kn)
n≥1
satisfying Assumption 2.7, one has
lim P(Kn,Cn) = P(µ),
n→∞
where Cn is defined by µ as above.
Proof. Notice that the sequence P(Kn,Cn) is non-increasing and thus the limit
exists. Let (P ) be a sequence such that P (Kn,Cn) and
n n≥1 n
∈ P
lim P(Kn,Cn) = lim EPn[Φ(B,T)].
n→∞ n→∞
Then repeating the reasoning of Lemma 4.3 in Guo, Tan & Touzi [10], we deduce
that the sequence (P ) is tight and any accumulation point P of (P ) be-
n n≥1 n n≥1
longs to . Without loss of generality, denote again by (P ) the convergent
n n≥1
P
subsequence with limit P, then it follows by Lemma 2.10 that
P
B µ , for all i= 1, ,m,
Ti ∼ i ···
which implies that P (µ). The proof is fulfilled by Fatou’s lemma:
∈ P
P(µ) lim P(Kn,Cn) = lim EPn[Φ(B,T)] EP[Φ(B,T)] P(µ).
≤ n→∞ n→∞ ≤ ≤
Remark 2.9. Combining Theorems 2.8 and 2.5, the λ appearing in D (µ) and
0
D(µ) can be restricted to take values in the set of Lipschitz functions.
Lemma 2.10. Let µ be a probability measure on R and set c(K) := µ((x K)+)
−
for every K R. Let (µn) be a weakly convergent sequence of probability mea-
n≥1
∈
sures such that
µn (x Kn)+ = µ (x Kn)+ , for all i = 1, ,n,
− i − i ···
where (Kn =(cid:0)(Kn) )(cid:1) is a(cid:0)n increasing(cid:1)sequence satisfying Assumption 2.7.
i 1≤i≤n n≥1
Then
lim µn = µ.
n→∞
8
Proof. Set cn(K) := µn((x K)+) (resp. c(K) := µ((x K)+)) for all K R.
− − ∈
Since (x a)+ (x b)+ a b , the function K cn(K) (resp. K c(K)) is
| − − − | ≤ | − | 7→ 7→
1 Lipschitz. Moreover, for every K Kn, one has cn(K) = c(K) for n large
n≥1
− ∈ ∪
enough, which implies that cn converges uniformly to c as Kn is dense on R.
n≥1
∪
Let f : R R be twice differentiable with compact support, then it follows by
→
Carr-Madan’s formula that
f(x) = f′′(K)(x K)+dK.
−
R
Z
We obtain in view of Fubini’s Theorem
µn(f) = f′′(K)cn(K)dK,
R
Z
which implies that µn(f) µ(f) as n . As any continuous function with
→ → ∞
compact support can be uniformly approximated by a twice differentiable function
with compact support, one has lim µn = µ.
n→∞
2.4 Additional market information: Power option
As described in Remark 2.6, the set of embeddings (K,C) is generally not com-
P
pact. For technical reasons, we consider in the following two subsets and
V
P
(K,C):
V
P
:= P :EP[B p] = V and (K,C) := (K,C),
PV ∈ P | Tm| PV PV ∩ P
n o
where p > 1 and V are fixed throughout the paper. The restriction of the em-
beddings comes from a new information: the Power option of the last maturity is
observed in the market. Consequently, this implies the unknown peacock µ must
satisfy
µ (x p) = V.
m
| |
Put similarly
PV(K,C) := sup EP[Φ(B,T)].
P∈PV(K,C)
and
DV(K,C) := inf sup EP Φ(B,T) α (B K)+ +α C
0 α∈Rmn − · T − ·
nP∈PV o
(cid:2) (cid:3)
= inf sup EP0 Φ(B,τ) α (B K)+ +α C ,
τ
α∈Rmn τ∈Ta − · − ·
n V o
(cid:2) (cid:3)
where a denotes the subsetof a consisting of elements τ such that EP0[B p]=
TV T | τm|
V. Then using exactly the same arguments in Guo, Tan & Touzi [10], one has the
following theorem.
Theorem 2.11. Let Assumption 2.4 hold. Then for any C (K), there exists
∗ ∈ A
a P (K,C) such that
V
∈ P
∗
EP [Φ(B,T)] = PV(K,C) = DV(K,C).
0
9
m
3 Monotonicity principle: = 1
Similar to the monotonicity principle introduced in Beiglb¨ock, Cox & Huesmann
[1], we give on one-marginal case another principle that links the optimality of an
embedding and the geometry of its support set.
For every ω¯ = (ω,θ),ω¯′ = (ω′,θ′) Ω, we define the concatenation ω¯ ω¯′ Ω
∈ ⊗ ∈
by
ω¯ ω¯′ := (ω ω′,θ+θ′),
θ
⊗ ⊗
where ω ω′ Ω is defined by
θ
⊗ ∈
ω ω′ := ω 1 (t)+ ω +ω′ 1 (t), for all t R .
⊗θ t t [0,θ) θ t−θ [θ,+∞) ∈ +
(cid:0) (cid:1) (cid:0)< (cid:1)
Let Γ Ω be a subset, we define Γ by
⊆
Γ< := ω¯ = (ω,θ) Ω :ω¯ = ω¯′ for some ω¯′ Γ with θ′ > θ .
∈ θ∧· ∈
Definition 3.1.(cid:8)A pair (ω¯,ω¯′) Ω Ω is said to be a stop-go pair (cid:9)relative if
∈ ×
ω = ω′ and
θ θ′
ξ(ω¯)+ξ(ω¯′ ω¯′′) > ξ(ω¯ ω¯′′)+ξ(ω¯′) for all ω¯′′ Ω+,
⊗ ⊗ ∈
+
where Ω := ω¯ = (ω,θ) Ω : θ > 0 . Denote by SG the set of all stop-go pairs.
∈
By exactly th(cid:8)e same arguments in G(cid:9)uo, Tan & Touzi [11], we have the following
theorem.
Theorem 3.2. Suppose that the optimal embedding problem P(K,C) admits an
optimizer P∗ (K,C), and the duality P(K,C) = D(K,C) holds true. Then
∈ P ∗
there exists a Borel subset Γ Ω such that
⊆
∗ ∗ ∗< ∗
P Γ = 1 and SG Γ Γ = . (3.12)
∩ × ∅
(cid:2) (cid:3) (cid:0) (cid:1)
4 Analysis of convergence rate
As we have shown in Section 2.3 the convergence of P(Kn,Cn) to P(µ) for any
increasing sequence (Kn) satisfying Assumption 2.7, we continue to estimate
n≥1
the convergence rate in this section. First, notice by definition that
P(µ) PV(Kn,Cn) P(Kn,Cn), for all n 1,
≤ ≤ ≥
which implies by Theorem 2.8 that
lim PV(Kn,Cn) = P(µ).
n→∞
Throughout this section we focus on the asymptotic behavior PV(Kn,Cn) by as-
suming that V = µ (x p) < + , where p > 1 is a fixed number and q > 1 is
m
| | ∞
defined by 1/p+1/q = 1.
10