Tesi di Laurea Magistrale
An approach to detect
fair prices of exotic options:
the optimal transport under
marginal martingale constraints
Candidato: Relatore:
Sara Golfieri
Dott. Dario Trevisan
Controrelatore:Prof. Marco Romito
Abstract ii
1 Introduction 1
1.1 Financial Setting . . . 1
1.1.1 Samuelson-Black-Scholes model . . . 2
1.2 The problem of pricing exotic options . . . 4
2 Existence of a martingale transport plan 9 2.1 Properties of ΠM(µ, ν) . . . 9
2.2 Convex Order . . . 12
2.3 Strassen’s Theorem . . . 16
2.4 Potential functions . . . 19
2.5 An SDE proof of Strassen’s Theorem . . . 24
3 The Dual Problem 27 3.1 No-duality gap theorem . . . 28
3.2 Technical results . . . 33
4 The left-curtain transport plan 39 4.1 Extended Convex Order . . . 39
4.2 Maximal and Minimal Elements: the Shadow Embedding . . . 41
4.3 Associativity of Shadows . . . 45
4.4 Definition of the Left-Curtain Transport Plan . . . 53
4.5 Uniqueness of the Monotone Martingale Transport . . . 55
4.6 Structure of the Monotone Martingale Coupling . . . 59
4.7 Optimality Properties of the Monotone Martingale Transport . . . . 62
In this thesis we discuss results obtained from the martingale transportation problem as a tool to solve financial problems.
We know that there exist some financial models that allow us to price vanilla options: for instance, the Black-Scholes’ model give formulas to calculate the value of call and put options. When we consider options as simple as vanilla options, the model give us a unique value for these prices. In the last years several models which allow to capture the risk of exotic options have emerged. These models depend on various parameters which can be calibrated more or less accurately to market price of liquid options (such as call/put options). This calibration procedure does not uniquely set the dynamics of forward prices which are only required to be (local) martingales according to the no-arbitrage framework. This could lead to a wide range of various prices of a given exotic option when evaluated using different models calibrated on the same market data.
What we do in this thesis is to review some results from the mathematical litera-ture that determinate lower and upper bounds for prices of exotic options produced by models calibrated on the same market data, and therefore with similar marginals. More precisely, we fix an exotic option depending only on the value of a single asset S at discrete times t1 < t2 and we denote by Φ(S1, S2) its payoff, where Φ
is supposed to be some measurable function. In the no-arbitrage framework, the standard approach is to postulate a model, that is, a probability measure Q on R2
under which the process (Si)2i=1
Si: R2 → R, Si(s1, s2) = si, i = 1, 2,
is required to be a discrete martingale on its own filtration. By S0 = s0 we denote
the current spot price.
The fair value of Φ is then given as the expectation of the payoff EQ[Φ].
Additionally we impose that our model is calibrated to a continuum of call op-tions, which is equivalent to prescribe probability measures µ1, µ2 on the real line
such that the one dimensional marginals of Q satisfy Qi= LawSi = µi i = 1, 2.
Let ΠM(µ1, µ2) be the set of all martingale measures Q on R2 having marginals
Q1= µ1, Q2= µ2.
We concentrate on the lower bound of the price and consider the problem P = inf{EQ[Φ(s1, s2)] : Q ∈ ΠM(µ1, µ2)}. (0.0.1)
Instead, the upper bound can be found analyzing the same problem but with −Φ instead of Φ.
We show that, in the same way of the classical optimal transport, the set ΠM(µ1, µ2)
is a (weakly) compact set, and then, if we require that the "cost" function Φ(s1, s2)
is lower semi-continuous, this problem has a minimum, provided the set ΠM(µ1, µ2)
is nonempty. Then we review a necessary and sufficient condition (due to Strassen) for this set to be nonempty: the measure µ1 and µ2 must be in convex order. We
then consider the dual problem of (0.0.1), which has an interesting financial interpre-tation in terms of hedging strategies. Since the dual problem does not exhibit nice compactness properties and hence the dual extremizer is not necessarily attained, we review conditions on Φ(s1, s2) for the supremum to be attained, following the
approach in [4]. Finally we construct an optimizer for the problem (0.0.1) for some types of Φ, called the left-curtain transport plan, introducing the shadow embedding of a measure, following the approach in [5].
Introduction
The aim of this chapter is to motivate the problem of this thesis is to price complex options. To this aim we first give some basic notions in financial mathematics.
1.1
Financial Setting
In finance ([1]), an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price (market price) of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. For example, an option that gives the right to buy (sell) a product S at a specific time T and price K is called call option (put option). Options are classified into a number of styles, we are interested in two of them:
• Vanilla options
– American option - an option that may be exercised on any trading day on or before expiration.
– European option - an option that may only be exercised on expiry. • Exotic options - any of a broad category of options that may include complex
financial structures.
Given an option, we have to face up to these problems: the pricing that is, to give it the fair price, and the hedging that is an investment position intended to offset potential losses or gains that may be incurred by an individual or a companion investment.
Since we do not know the evolution of the price of an option, we build models which allow us to have a realistic prediction of this price. We consider models in which the time is discrete, that is the set of times is T = {t0, t1, . . . , tN}, we take also
a probability space (Ω, F, P) in which there is a filtration (Fi)Ni=0. In these models
we have that F0 = {∅, Ω} and FN = F. In the market we have a risk-free asset
(bond) S0
n = (1 + r)n and d risky assets, (Sni)Nn=0 with i ∈ {0, . . . , d} represented
by stochastic adapted processes evaluated at t0, . . . , tN. We call portfolio a couple
(H0
n, Hn)Nn=1, where (Hn0)n≥1 is a predictable stochastic process that takes values
in R and (Hn)n≥1 is a predictable stochastic process that takes values in Rd. The
portfolio value at time tn is Vn= Hn0Sn0+ Hn· Sn.
Definition 1.1.1. We say that a portfolio is self-financing if the following holds Hn0Sn0+ Hn· Sn= Hn+10 Sn0+ Hn+1· Sn ∀ n = 0, . . . , N.
An arbitrage is a self-financing portfolio with V0 = 0, VN ≥ 0and P{VN > 0} > 0.
Heuristically the arbitrage is the possibility of having a strictly positive profit without investing any money. We require that in the market there is no arbitrage, and we call this hypothesis N. A. hypothesis. We are now ready to state the first fundamental theorem of asset pricing (see [1] for a proof):
Theorem 1.1.2 (No Arbitrage Theorem). The following are equivalent • N. A. hypothesis holds;
• there exists a probability Q ∼ P such that every (Si
n)Nn=0is a martingale process.
Furthermore, we can choose Q with dQ dP ∈ L
∞.
The second theorem deals with uniqueness of Q. A market is said to be complete if there exists a price for every asset.
Theorem 1.1.3. If N. A. hypothesis holds, the following are equivalent: • the market is complete;
• there exists only one martingale probability equivalent to P. 1.1.1 Samuelson-Black-Scholes model
As an example we present one of the models that one can construct in case of continuous times. We suppose that the process Stsatisfies the following S.D.E.:
dSt= St(µ dt + σ dWt) t ∈ [0, T ] S0 t = ert
where S0
t represents the bond, σ and µ are constant and (Wt)0≤t≤T is the Wiener
process (also called standard Brownian motion). Hence, if we call ˜St = e−rtSt, we
have that the following holds by Itô’s formula d ˜St= ˜St((µ − r) dt + σ dWt) = ˜Stσ dWt+ µ − r σ dt . If we set W∗ t = Wt+ Rt 0 µ − r σ ds, we obtain that d ˜St= ˜Stσ dWt∗.
Using Girsanov’s Theorem (see [6]) we have the following theorem Theorem 1.1.4. Let P∗ be the probability defined by
dP∗ dP = exp − µ − r σ WT − 1 2 µ − r σ 2 T ! .
The process ( ˜St)0≤t≤T is a martingale under the probability P∗.
In the S.B.S. model the following theorem holds.
Theorem 1.1.5. If we call X the asset, that is a random variable which is FT -measurable, and X satisfies: X = f(St), where f is a measurable function, then the
value of self-financed portfolio is Vt= F (t, St), where
F (t, x) = e−r(T −t) Z R f (xe(r−σ22 )(T −t)+σ √ T −ty)e− y2 2 √ 2π dy.
Furthermore, if F ∈ C1 in the first variable and F ∈ C2 in the second variable then
Ht=
∂F ∂x(t, St). We call delta hedging the value of ∂F
∂x(t, St).
Since X = (St− K)+ or X = (K − St)+ satisfy the hypothesis of this model,
we can obtain the no-arbitrage price of a call/put option, that are the celebrated Black-Scholes’ formula.
Theorem 1.1.6 (Black-Scholes’ formulas). The value of call and put options (with strike price K) at time t, represented by Ct= C(t, St) and Pt= P (t, St) is given by
the following formulas
P (t, x) = Ke−r(T −t)Φ(−d2) − xΦ(−d1),
where
d1,2 =
log(Kx) + (r ± σ22)(T − t) σ√T − t
and Φ is the cumulative distribution function of the variable N(0, 1) that is Φ(x) = Z x −∞ ey22 √ 2πdy.
Moreover, in case of the call option, the delta hedging is given by Φ(d1), otherwise,
for the put option, the value of the delta hedging is given by −Φ(−d1).
1.2
The problem of pricing exotic options
We are now ready to explain what the main interest of this thesis is. Since the introduction of the Black-Scholes paradigm, several alternative models which allow to capture the risk (i.e. pricing and hedging) of exotic options have emerged. These models depend on various parameters which can be calibrated more or less accurately to market price of vanilla options (such as call/put options). This procedure does not uniquely set the dynamics of forward prices which are only required to be (local) martingales according to the no-arbitrage framework. This could lead to a wide range of various prices of a given exotic option when evaluated using different models calibrated on the same market data. We investigate lower and upper bounds for exotic options produced by models calibrated on the same market data, and therefore with same marginals.
The problem of determining the interval of consistent prices of a given exotic option can be cast as a infinite-dimensional linear programming problem.
Setting We fix an exotic option depending only on the value of a single asset S at discrete times t1 < · · · < tn and we denote by Φ(S1, . . . , Sn) its payoff, where
Φ : Rn→ R is supposed to be some measurable function. In the no-arbitrage
frame-work, the standard approach is to postulate a model, that is, a probability measure Q on Rn under which the process (Si)ni=1
Si: Rn→ R, Si(s1, . . . , sn) = si, i = 1, . . . , n,
is required to be a discrete martingale on its own filtration. As an example, we have the S.B.S. model above.
The fair value of Φ at time t0is then EQ[Φ].Since we are interested in exotic options,
Φi,K(Si) = (Si− K)+, K ∈ R at each date ti and price
C(ti, K) = EQ[Φi,K] =
Z
R+
(s − K)+dLawSi(s).
We will see inRemark 2.4.3that this is equivalent to prescribing probability measures µ1, . . . , µn on the real line such that the one dimensional marginals of Q satisfy
Qi= LawSi = µi ∀ i = 1, . . . , n.
Primal formulation We call ΠM(µ1, . . . , µn) the set of all martingale measures
Q on Rn having marginals Q1 = µ1, . . . , Qn = µn. That is Q ∈ ΠM(µ1, . . . , µn) if
and only if EQ[Si|S1, . . . , Sn] = Si−1 for i = 2, . . . , n and EQ[Φi,K] = C(ti, K) for
all K ∈ R and i = 1, . . . , n. We concentrate on the lower bound and consider the problem
P = inf{EQ[Ψ] : Q ∈ ΠM(µ1, . . . , µn)}.
As we have already said, since the problem is finding a minimum of a linear function on a compact convex set, it can be viewed as a primal infinite-dimensional linear programming problem and then it has a natural dual formulation:
Dual formulation The dual formulation corresponds to the construction of a semi-static subhedging strategy consisting of the sum of a semi-static vanilla portfolio and a delta strategy. More precisely, we are interested in payoffs of the form
Ψ(ui),(∆j)(s1, . . . , sn) = n X i=1 ui(si) + n−1 X j=1 ∆j(s1, . . . , sj)(sj+1− sj), s1, . . . , sn∈ R,
where the functions ui: R → R are µi-integrable for i = 1, . . . , n and the functions
∆j: Rj → R are assumed to be bounded measurable for j = 1, . . . , n − 1. If these
functions lead to a strategy which is subhedging in the sense Φ ≥ Ψ(ui),(∆j),
we have, for every pricing measure Q ∈ ΠM(µ1, . . . , µn)the obvious inequality
EQ[Φ] ≥ EQ[Ψ(ui),(∆j)] = EQ " n X i=1 ui(Si) # = n X i=1 Eµi[ui]. We have used that
EQ n−1 X j=1 ∆j(S1, . . . , Sj)(Sj+1− Sj) = 0,
indeed we have n−1 X j=1 EQ[∆j(S1, . . . , Sn)(Sj+1− Sj)] = = n−1 X j=1 EQ[EQ[∆j(S1, . . . , Sn)(Sj+1− Sj)|Sj] = = n−1 X j=0 EQ[∆j(S1, . . . , Sj)EQ[Sj+1− Sj|Sj]] = 0
thanks to the martingale property. This leads to consider the dual problem
D = sup ( n X i=1 Eµi[ui] : ∃∆1, . . . , ∆n−1 : Ψ(ui),(∆j)≤ Φ ) .
In our thesis we will consider only the case n = 2, which corresponds to the simplest but exotic option.
To study the problem, we follow the tradition customary of in the optimal transport literature [7].
We first rename c = Φ. We will always ask for the price to satisfy the sufficient integrability condition, that is, there exists a constant k such that
c(x, y) ≥ −k(1 + |x| + |y|).
If µ and ν are in M1, where M1 is the set of finite measures having finite first
moments, we define the set of transport plans as Π(µ, ν) := {π ∈ M1(R × R) such
that π(A × R) = µ(A) and π(R × B) = ν(B) ∀ A ∈ B(R), B ∈ B(R)}. We also define ΠM(µ, ν) as the set of all transport plans π such that the disintegration in
probability measures (πx)x∈R with respect to µ satisfies
Z
y dπx(y) = x
for µ-almost every x. Then π ∈ Π(µ, ν) is a martingale measure if and only if Z
R2
ρ(x)(y − x) dπ(x, y) = 0 (1.2.1) for all bounded measurable functions ρ : R → R. Indeed, if (1.2.1) holds for every bounded function ρ, then, taking ρ = 1 we have that π ∈ ΠM(µ, ν). Instead, if π is
in ΠM(µ, ν), taking a Borel measurable set A ⊆ R we obviously have that
Z
R2
1A(x)(y − x) dπ(x, y) = 0.
Using standard approximation techniques one obtains that this (1.2.1) holds for every bounded function.
Note that to verify this condition is enough to test (1.2.1) for all functions of the form ρ = 1(−∞,x], x ∈ R or for all continuous bounded functions. Then our primal
problem becomes
P = inf{Eπ[c(x, y)] : π ∈ ΠM(µ, ν)}. (1.2.2)
The dual problem, instead, becomes
D = sup{Eµ[ϕ] + Eν[ψ] : ∃∆ : ϕ(x) + ψ(y) + ∆(x)(y − x) ≤ Φ(x, y)}, (1.2.3)
where ϕ, ψ ∈ S = ( u : R → R : u(x) = a + bx + m X i=1 ci(x − ki)+, a, b, ci, ki∈ R ) and ∆ ∈ Cb(R).
While c(x, y) = (x − y)2 is arguably the most important cost function in the
theory of classical optimal transport, it plays a different role in the martingale setup. Indeed, assume that LawX = µ and LawY = ν have second moments, then
Z
R2
(y − x)2dπ(x, y) =Eπ[(Y − X)2] = Eν[Y2] + Eµ[X2] − 2Eπ[XY ] =
=Eν[Y2] + Eµ[X2] − 2Eπ[E[XY |X]] = Eν[Y2] − Eµ[X2].
That is, the cost depends only on the marginal distribution, and then every martin-gale plan realizes the minimum.
We conclude this chapter giving a brief description of what are we going to do in this thesis.
In the second chapter we will give conditions for the existence of the minimum of the problem (1.2.2). We first show that the set ΠM(µ, ν) is a weakly compact set
and then we will give a sufficient and necessary condition for the set to be nonempty, following the approach in [5]
In the third chapter we will study the dual problem (1.2.3), in particular we will analyze under which hypothesis there is no duality gap (i.e. P = D), using the approach given in [4]. This is a model-free version of the replication theorem of mathematical finance. We will then state some results, using the duality, that will be used in the last chapter.
In the fourth chapter we will construct optimal plans for the problem (1.2.2) and we will give conditions for the uniqueness. To this aim we will introduce the fundamental tool of the shadow embedding with its properties, following the approach in [5].
Existence of a martingale
transport plan
In this chapter we will give a necessary and sufficient condition for the existence of an optimal martingale transport plan between two measures. We first prove, in the first chapter, in the same way of the classical optimal transport problem, the value of the minimization problem (1.2.2) is attained, provided that the set ΠM(µ, ν) is
nonempty. Then the only thing that we have to show to have an optimal martingale transport plan is that ΠM(µ, ν) 6= ∅. We will show that the nonemptiness of the set
ΠM(µ, ν)is guaranteed by convex order.
2.1
Properties of Π
M(µ, ν)
We give some basic properties of the set ΠM(µ, ν) that will allow us to deduce the
existence of the minimum in the problem (1.2.2).
We consider the set M1(the set of finite measures on R with finite first moments),
with the usual topology, that is, a sequence (νn)n converges weakly in M1 to an
element ν ∈ M1 if:
• (νn) converges weakly that is
Z R f (x) dνn(x) n→∞ −→ Z R f (x) dν(x),
for every f continuous and bounded function.
• Z R |x| dνnn→∞−→ Z R |x| dν. 9
Theorem 2.1.1 (lower semicontinuity of the function C). If the cost function c is positive and lower continuous (it can also attain +∞ value), then C : π 7→ C(π) = R
R2c dπ is lower continuous with respect to the weak topology.
Proof. ∀ k ∈ N, let us define ck(x, y) := inf
x0,y0∈R{c(x
0, y0) ∧ k + k|x − x0| + k|y − y0|}.
This function is continuous, bounded by k and (ck)k∈N is an increasing sequence
converging to c. Now consider {πi}i∈N which weakly converges to π in P(R × R).
Then, for every k ∈ N, we have lim inf
i→∞ C(πi) ≥ lim infi→∞
Z R×R ck(x, y)dπi(x, y) = Z R×R ck(x, y)dπ(x, y).
Now, sending k → ∞, by Beppo Levi’s theorem we obtain lim inf
i→∞ C(πi) ≥ supk∈N
Z
R×R
ck(x, y)dπ(x, y) = C(π)
that is exactly the lower semicontinuity.
We recall that a topological space (X, τ) is said to be Polish if there exists a distance d on X inducing τ such that (X, d) is complete. We have the following standard results
Lemma 2.1.2 (Ulam). For every µ ∈ M+(R) and for every ε > 0 there exists
K ⊂ R compact such that µ(R \ K) < ε.
Theorem 2.1.3 (Prokhorov). Let Z be a Polish space, then F ⊂ M(Z) is sequen-tially relatively compact with respect to the weak topology if and only if F is equi-tight, i.e. for every ε > 0 there exists K ⊂ Z compact such that µ(Z \ K) < ε for every µ ∈ F .
Proof. We prove only the implication necessary for our aims: if F is equi-tight then it is sequentially relatively compact with respect to the weak topology. If Z is compact then by Riesz Theorem M(Z) = C(Z)∗ = (C
b(Z))∗ and, since C(Z) is separable,
by Banach-Alaoglu theorem, this is sequentially compact with respect to the weak* topology. Let us analyze the general case. Take (µn)n∈N ⊂ F, then there exist
K1 ⊂ · · · ⊂ Kk ⊂ Kk+1 ⊂ · · · ⊂ Z with ωk = supn∈Nµn(Z \ Kk) → 0 as k → ∞.
By a diagonal argument, thanks to the compact case, there exists a sequence nlsuch
that, for every k ∈ N, there exists νk∈ M+(Z) supported on Kk with µnlxKk * νk as l → ∞. Since νk ≤ νk+1 by construction, we can define ν := supk∈Nνk and we
obtain Z Z ϕd(µnl− ν) ≤ Z Z ϕd(µnl) − Z Z ϕdµnlxKk + + Z Z ϕdµnlxKk− Z Z ϕdνk + Z Z ϕdνk− Z Z ϕdν . Taking l → ∞, we have lim sup l→∞ Z Z ϕd(µnl− ν) ≤ sup k∈N |ϕ|ωk+ Z Z |ϕ|d(ν − νk)
which leads to the conclusion as k → ∞.
Proposition 2.1.4. The set Π(µ, ν) is compact with respect to the weak topology. Proof. Thanks to the Prokhorov theorem,we need to prove that Π(µ, ν) ⊂ M(R×R) is equi-tight. Since R is a Polish space, ∃ K, ˜K ⊂ R compact, such that µ(R\K) < ε 2 and ν(R \ ˜K) < ε
2. Thus we have that
π(R × R \ K × ˜K) ≤ π((R \ K) × R) + π(R × (R \ ˜K)) = µ(R \ K) + ν(R \ ˜K) < ε and consequently Π(µ, ν) is relatively compact with respect to the weak topology. Furthermore Π(µ, ν) is closed, indeed, if we take πn→ π we have that
R Rf (x) dµ(x) = R Rf (x) dπn(x, y) n→∞ −→ R Rf (x) dπ(x, y), R Rg(y) dν(y) = R Rg(y) dπn(x, y) n→∞ −→ R Rg(y) dπ(x, y),
for every f, g ∈ Cb(R). Then π is in Π(µ, ν), and hence we obtain the compactness.
Theorem 2.1.5. If c : R × R → [0, ∞] is a lower semicontinuous cost, then the minimum in the primal problem is obtained.
Proof. It is a direct corollary ofTheorem 2.1.1 andProposition 2.1.4, thanks to the existence of a minimum, for a lower semicontinuous function, on a compact set. Theorem 2.1.6. The set ΠM(µ, ν) is compact in the weak topology.
Proof. First notice that ΠM(µ, ν) ⊆ Π(µ, ν). We have already proved that Π(µ, ν)
is compact in the weak topology, so it is enough to prove that ΠM(µ, ν) is weakly
We can rewrite ΠM(µ, ν)in the following way ΠM(µ, ν) = \ π ∈ Π(µ, ν) : Z R2 f (x)(y − x) dπ(x, y) = 0
where the intersection is on the continuous and bounded functions f : R → R. Since RR2f (x)(y − x) dπ(x, y) are continuous functions with respect to π, the sets {π ∈ Π(µ, ν) : R
R2f (x)(y − x) dπ(x, y) = 0}are closed because they are zero locus of
continuous functions. The theorem follow from the fact that intersection of weakly closed sets is a weakly closed set.
Remark 2.1.7. If π1 ∈ ΠM(µ1, ν1) and π2 ∈ ΠM(µ2, ν2) then π1 + π2 ∈ ΠM(µ1 +
µ2, ν1+ ν2).
2.2
Convex Order
We now introduce the concept of convex order between two measures. It will be the necessary and sufficient condition for the set ΠM(µ, ν) to be nonempty.
Definition 2.2.1. Two measures µ, ν are in convex order if:
(1) they have finite mass and finite first moments, i.e. they are in M. (2) RRϕ dµ ≤R
Rϕ dν for every convex function ϕ : R → R.
In that case, we write µ C ν.
If µ C ν, then one can apply (2) to all affine functions. Using ϕ(x) ≡ 1 and
ϕ(x) ≡ −1we have µ(R) = Z R dµ ≤ Z R dν = ν(R), µ(R) = Z R dµ ≥ Z R dν = ν(R). Using ϕ(x) = x and ϕ(x) = −x we have
Z R x dµ ≤ Z R x dν Z R x dµ ≥ Z R x dν.
Then µ and ν have the same mass and the same barycenter.
We note that ΠM(µ, ν) 6= ∅ then µ C ν: if π ∈ ΠM(µ, ν) and ϕ is a convex
function we have Z R×R ϕ(y) dν(y) = Z R×R ϕ(y) dπ(x, y) = = Z R Z R ϕ(y) dπx(y) dµ(x) ≥ Z R ϕ(x) dµ(x)
where, in the last inequality, we have used that RRy dπx(y) = x and Jensen’s
in-equality.
Remark 2.2.2. If µ and ν have the same mass and the same barycenter, then R
Rf dµ =
R
Rf dν for every f : R → R affine function. Indeed if f(x) = ax + b
for some a, b ∈ R, we have Z R f dµ =a Z R x dµ + b Z R dµ = aµ(R)B(µ) + bµ(R) = =aν(R)B(ν) + bν(R) = a Z R x dν + b Z R dν = Z R f dν.
It is sufficient to test hypothesis (2) inDefinition 2.2.1against suitable subclasses of convex functions. For instance, if µ and ν have the same finite mass and the same first moments then they are in convex order if and only if
Z R (x − k)+dµ(x) ≤ Z R (x − k)+dν(x) ∀ k ∈ R.
In the same way, it is sufficient to check (2) for positive convex functions with finite asymptotic slope in −∞ and +∞.
Lemma 2.2.3. if µ is an atom of mass α > 0 at the point x, µ = αδx, then αδxC ν ⇔ ν has mass α and barycenter x.
Proof. We have already proved that, if µ C νthen the two measures have the same
mass and the same barycenter. Conversely, thanks to what we said above, if µ and ν have the same mass and the same barycenter, then we have that
Z R f dµ = Z R f dν
for every f : R → R affine function. Now, we take ϕ: R → R convex, we can write ϕas the supremum of affine functions ϕ = supi{fi}.
αfi(x) = Z R fidµ = Z R fidν ≤ Z R sup i fidν = Z R ϕ dν αfi(x) ≤ Z R ϕ dν
Now, letting i → ∞ we have that RRϕ dµ = αϕ(x) ≤R
Rϕ dν, then µ C ν.
Lemma 2.2.4. If µi C νi for every i = 1, . . . , n then Pi=1n µi≤Pni=1νi.
Lemma 2.2.5. If two measures µ and µ0 have the same mass and the same barycen-ter, µ is concentrated on [a, b] and µ0 is concentrated on R \ (a, b) then µ
C µ0.
Proof. Given ϕ convex function, we can build ψ linear function such that ψ(a) = ϕ(a) and ψ(b) = ϕ(b). We know that, thanks to the convexity of ϕ, ϕ ≤ ψ on [a, b] and ϕ ≥ ψ on R \ (a, b). Then, R
Rϕ dµ ≤
R
Rψ dµ because µ is concentrated on [a, b],
R Rψ dµ 0 ≤R Rϕ dµ. Then Z R ϕ dµ ≤ Z R ψ dµ = Z R ψ dµ0 ≤ Z R ϕ dµ0 Note that we have usedRemark 2.2.2in the equality above.
Lemma 2.2.6. If two measures µ and µ0 have the same mass and the same barycen-ter, µ − (µ ∧ µ0)is concentrated on [a, b] and µ0− (µ ∧ µ0)is concentrated on R \ (a, b)
then µ C µ0.
Proof. As in the proof ofLemma 2.2.5we take ϕ convex and we build ψ. Then Z R ϕ dµ − (µ ∧ µ0) ≤ Z R ψ dµ − (µ ∧ µ0) = = Z R ψ dµ0− (µ ∧ µ0) ≤Z R ϕ dµ0− (µ ∧ µ0) .
Using the linearity of µ 7→ RRf dµwe obtain Z R ϕ dµ − Z R ϕ d(µ ∧ µ0) ≤ Z R ϕ dµ0− Z R ϕ d(µ ∧ µ0). Then Z R ϕ dµ ≤ Z R ϕ dµ0. It follows that µ C µ0.
We introduce two technical Propositions that will be used in the next section. Proposition 2.2.7(Approximation of a measure in the convex order). Assume that γ ∈ M. There exists a sequence (γ(n))n of finitely supported measures such that
γ(n)
C γ(n+1), the sequence (γ(n))n converges weakly to γ in M and γ(n)≤ γ holds
for every n.
We give a proof of this Proposition using a probabilistic approach.
Proof of Proposition 2.2.7. We consider γ ∈ M and without loss of generality we
the space of probability (R, B, γ), more precisely X = id : (R, B, γ) → (R, B, γ). We now construct the filtration Fn using the partition Jk,N defined as follows:
Jk,N = 2kN −1 [ i=−2kN i 2k, i + 1 2k ∪ (N, +∞] ∪ (−∞, −N ].
We set Fn= σ(Jn,n). Then we define the discrete random variables
Xn= E[X|Fn],
they are constant on each piece of Jn,n, equals to the mass of the piece concentrated
on the barycenter, then their laws are sums of atomic masses. Xn are martingales,
indeed
E[Xm|Fn] = E[E[X|Fm]|Fn] = E[X|Fn] = Xn, ∀ m ≥ n.
We also note that X is integrable since γ ∈ M. Hence we can apply the theorem of convergence for martingales, and we obtain that Xn → X a.s., using that the
convergence a.s. implies the convergence in law, we have that γn → γ, and this
complete the proof.
Definition 2.2.8. For a random variable Xµ with law µ we write briefly Xµ ∼ µ. We pick Xµ∼ µand Yρ∼ ρindependent variables then µ∗ρ is the law of the variable
Xµ+ Yρ.
Proposition 2.2.9(Approximation of measure in convex order with regularization). Let µ and ν be in M such that µ C ν. Let ρ ∈ M, ρ(R) = 1. Then µ ∗ ρ C ν ∗ ρ.
Proof. Without loss of generality we assume that µ(R) = ν(R) = 1. We need to show that for every ϕ convex function we have
E[ϕ(Xµ+ Yρ)] ≤ E[ϕ(Xν+ Yρ)]
Indeed
E[ϕ(Xµ+ Yρ)] =E[E[ϕ(Xµ+ Yρ)|Yρ]] = E[E[ϕ(Xµ+ y)]|Yρ] =
=E Z R ϕ(x + y) dµ(x) ≤ E Z R ϕ(x + y) dν(x) =
=E[E[ϕ(Xν+ y)]|Yρ] = E[ϕ(Xν + Yρ)].
2.3
Strassen’s Theorem
Theorem 2.3.1 (Strassen’s Theorem). Let µ, ν ∈ M. The condition µ C ν is
necessary and sufficient for the existence of a martingale transport plan in ΠM(µ, ν).
We have already shown that this condition is necessary, we will give two proofs of the fact that this is also sufficient.
The first will be done describing a procedure which allows to obtain a martingale transport plan. This proof will be done in several steps: we will first give the result in the case where µ is concentrated in finitely many points, then with an approximation argument we will show that the theorem is true for every µ ∈ M.
The second proof, instead, follows from an argument due to Bruno Dupire that uses SDE’s (but we do not provide all details here).
Proposition 2.3.2. Assume that µ = Pni=1δi, where δi is an atomic measure. If ν
satisfies µ C ν, then ΠM(µ, ν) is nonempty.
Before doing the proof of the theorem, we want to show how to perform the inductive step and, to this aim we need to understand how to couple a single atom, say δ := δ1, with a properly chosen portion ν0 of ν so that the other atoms satisfy
(Pn
i=2δi) C ν −ν
0. Assume that δ has mass α and barycenter x. UsingLemma 2.2.3
we can pick some ν0 so that it has mass α and barycenter x, furthermore, since ν0 is
a portion of ν we have that ν0≤ ν. As δ is a part of µ and µ
C ν, we can introduce
the measure ˜µ := µ − δ which has mass ˜µ(R) = µ(R) − α = ν(R) − α (µ and ν have the same mass because they are in convex order). We write t := ν(R) − α. Obviously we have that µ = ˜µ + δ C ν. We are looking for the measure ν0 among
the measures {νs: s ∈ [0, t]}obtained as the restriction of ν between two quantiles s
and s0 = s + α. More precisely, we consider ν
s= G#L[s,s+α] where G: [0, t + α] → R
is the quantile function of ν. The barycenter B(s, ν) of νs depends continuously on
the parameter s ∈ [0, t] and we have that
B(0, ν) ≤ x B(t, ν) ≥ x.
Indeed, using (δ + ˜µ) C ν applied to the functions u 7→ (u − G(α))− and u 7→
(u − G(t))+ we obtain Z R (u − G(t)) dδ(u) ≤ Z R (u − G(t))+dδ(u) ≤ ≤ Z R (u − G(t))+dν(u) = Z R (u − G(t)) dνt(u)
But we have also RR(u − G(t)) dδ(u) = αx − G(t)α and R
νt(R)B(t, ν) − G(t)νt(R), then, thanks to the inequality above
αx − G(t)α ≤ νt(R)B(t, ν) − G(t)νt(R)
Since νt(R) = α we have x ≤ B(t, ν). Analogously, in the second case:
Z R −(u − G(α)) dδ(u) ≤ Z R (u − G(t))−dδ(u) ≤ ≤ Z R (u − G(t))−dν(u) = Z R −(u − G(t)) dνt(u)
Choosing t = 0, we obtain −xα + G(α)α ≤ −ν0(R)B(0, ν) + G(α)ν0(R) and then
x ≥ B(0, ν). Using the intermediate value theorem, the continuity of s 7→ B(s, ν) implies that ∃ s ∈ [0, t] such that νshas barycenter x. By construction we know that
νs(R) = α, and so, thanks to Lemma 2.2.3 δ C νs. Furthermore we note that if
B(s, ν) = B(s0, ν), the measures νs and νs0 are equal (it follows from the invariance of the Lebesgue measure under translations) so that there exists a unique measure with barycenter x and we denote it by ν0.
This proof leads to the following lemma:
Lemma 2.3.3. Let µ be of the form µ = ˜µ + δ, where δ is an atom and assume that µ C ν. Then there exists a unique splitting of the measure ν into two positive
measures ν0 and ˜ν = ν − ν0 in such a way that:
(1) δ C ν0,
(2) ˜ν(I) = 0 where I = conv(spt(ν◦ 0)) is the interior of the smallest interval con-taining the support of ν0.
Moreover, the measures ˜µ and ˜ν satisfies ˜µ C ν˜.
Proof. We have already constructed ν0 in the argument above. We note that I =
(G(s), G(s + α)), it remains to show (2) and the fact that ˜µ is smaller in convex order than ˜ν. ˜ν(I) = ν(I) − ν0(I) = 0 because G is the quantile function of ν and
ν0 is the restriction of ν between the quantiles s and s + α. We have to prove that ˜
µ C ν˜. To this aim, we take a nonnegative, convex function ϕ which satisfies
lim sup |x|→+∞ ϕ(x) x < +∞ (2.3.1)
and we will show that RRϕ d˜µ ≤ R
Rϕ d˜ν. We introduce a new function ψ which
equals ϕ on R \ I and it is linear on I. Thanks to (2.3.1) we can choose ψ to be convex and ψ ≥ ϕ. The functions ψ and ϕ coincide on the boundary of I, then
Z R ϕ d˜µ ≤ Z R ψ d˜µ = Z R ψ dµ − Z R ψ dδ.
But as ψ is linear on I, byRemark 2.2.2 and the fact that ν0 is concentrated on I,
one has RRψ dδ =R
Rψ dν
0 and because µ
C ν one has also RRψ dµ ≤
R Rψ dν. It follows that Z R ϕ d˜µ ≤ Z R ψ dν − Z R ψ dν0 = Z R ψ d˜ν = Z R ϕ d˜ν.
The last equality follows from the fact that ˜ν is concentrated on R \ I and ψ = ϕ on this set.
Thanks to the discussion above we know how to perform the inductive step, indeed we write µ = δ + Pn
i=2δi and we can find ν
0 (that we have called ν s) such
that δ C νsand Pni=2δi C ν−νs. We are now ready to prove theProposition 2.3.2.
Proof of Proposition 2.3.2. We use the induction on the number of atoms. If n = 1 we have µ = δ. If δ C ν, fromLemma 2.2.3we know that ν has the same mass and
the same barycenter of δ, then δ ⊗ ν is in ΠM(µ, ν). To perform the inductive step
on µ = Pn
i=2δi, we applyLemma 2.3.3 to the measures δ = δ1 and ˜µ = Pni=2δi to
obtain a splitting ν = ˆν1+ ˆνthat satisfies δ1C νˆ1 and ˜µ C ν˜. As we noticed in the
basic step we have that ΠM(δ1, ˆν1) consists of a single element π1:= δ1⊗ ˆν1. In the
next step, we repeat the procedure with ˜µ and ˜ν in the place of µ and ν and continue until the n − th step where δn can be martingale-transported to the remaining part
of ν. More precisely, at the (n − 1) − th step we have that δn−1 C νn−1ˆ and
δn C (ν −Pn−1i=1 νˆi) and this is the only condition that we need to
martingale-transport δnto ν − Pn−1i=1 νˆi. Hence, we have obtained recursively a sequence ( ˆνi)ni=1
such that δi C νˆi for every i = 1, . . . , n and ˆν1+ · · · + ˆνn= ν. We have constructed
nmartingale transport plans π1, . . . , πnwhere πi is the unique element of ΠM(δi, ˆνi).
Thus π1+ · · · + πn is an element of ΠM(µ, ν).
Using Proposition 2.2.7 we can extendProposition 2.3.2to the general case of a non-atomic mass.
First proof of sufficiency in Theorem 2.3.1. As shown in Proposition 2.2.7, take a sequence of finitely supported measures (µn)n≥1 such that µnC ν and µn* µ(in
this proof we don’t need that the measures are increasing in convex order). We have already solved the problem of transporting a discrete distribution. Pick martingale measures (πn)n≥1 which transport µn to ν for each n. To be able to pass to a limit,
we need to show that
Ω := ΠM(µ, ν) ∪ ∞ [ n=1 ΠM(νn, ν) = ∞ [ n=0 ΠM(µn, ν)
(we have written µ0 instead of µ to have a simpler definition of Ω) is weakly
in Ω and, of course π is a martingale transport plan. We first show that Ω is weakly sequentially relatively compact, then we show that π is a transport martingale plan. Since R is a Polish space, it is equivalent proving that Ω is equi-tight, i.e. for every ε > 0 there exist two compact sets K, ˜K ⊂ R such that π(R × R \ K × ˜K) < ε, for every µ ∈ Ω. Since R is a Polish space, we already know that if we take µ ∈ M, for every ε > 0 we can find a compact set A ⊂ R such that µ(R \ A) < ε. Hence, if we pick π ∈ Ω, there will be an n such that π ∈ ΠM(µn, ν), and ε > 0 we can find
Kn, ˜K ⊂ R compact sets, such that µn(R \ Kn) < ε/2and ν(R \ ˜K) < ε/2, then
π(R × R \ Kn× ˜K) < π((R \ Kn) × R) + π(R × (R \ ˜K)) < ε.
We show that π ∈ ΠM(µ, ν). We already know that
Z R ϕ(x) dµn= Z R×R ϕ(x) dπn(x, y)
Using the fact that πn* π and µn* µwe have
Z R ϕ(x) dµn→ Z R ϕ(x) dµ Z R×R ϕ(x) dπn(x, y) → Z R×R ϕ(x) dπ(x, y)
for every ϕ ∈ Cb. Moreover
Z R ϕ(y) dν = Z R×R ϕ(y) dπn(x, y) → Z R×R ϕ(y) dπ(x, y)
Then π is a martingale transport plan.
2.4
Potential functions
Given a measure, we now define the potential function of this measure. These tions allow to turn the relations between measures into relations between real func-tions.
Definition 2.4.1. For each µ ∈ M we define the potential function uµ: R → R by uµ(x) = Z ∞ −∞ |y − x| dµ(y) for x ∈ R. Set k = µ(R) and m = 1 k R Rx dµ.
Proposition 2.4.2. If µ is in M, then uµ has the following properties: (i) uµ is convex
(ii) limx→−∞uµ(x) − k|x − m| = 0 and limx→+∞uµ(x) − k|x − m| = 0.
Conversely, if f is a function satisfying these properties for some numbers m ∈ R and k ∈ [0, +∞), then there exists a unique measure µ ∈ M such that f = uµ. The
measure µ satisfies
µ=d 1 2f
00
.
Proof. We first show that the potential function satisfies (i) and (ii). We need to show that for every λ ∈ (0, 1) we have uµ(λx + (1 − λ)z) ≤ λuµ(x) + (1 − λ)uµ(z).
Z R |y − (λx + (1 − λ)z)| dµ(y) = Z R |λ(x − y) + (1 − λ)(y − z)| dµ(y) ≤ ≤ Z R
(λ|y − x| + (1 − λ)|y − z|) dµ(y) =
= λ Z R |y − x| dµ(y) + (1 − λ) Z R |y − z| dµ(y)
where we used the fact that x 7→ |x| is a convex and positive function. We show that limx→+∞uµ(x) − k|x − m| = 0 uµ(x) = Z {y≥x} (y − x) dµ(y) + Z {y<x} (y − x) dµ(y) = = Z {y≥x}
y dµ(y) − xµ({y ≥ x}) + xµ({x ≥ y}) − Z {y<x} y dµ(y) x→+∞ −→ xµ(R) − Z R y dµ(y) = k|x − m|,
we have used that R{y≥x}y dµ(y) and xµ({y ≥ x}) go to 0 when x → +∞.
Analo-gously limx→−∞uµ(x) − k|x − m| = 0 uµ(x) = Z {y≥x} (y − x) dµ(y) + Z {y<x} (y − x) dµ(y) = = Z {y≥x}
y dµ(y) − xµ({y ≥ x}) + xµ({x ≥ y}) − Z {y<x} y dµ(y) x→−∞ −→ Z R y dµ(y) − xµ(R) = −k|x − m|.
Now we show that 2 hµ, ϕi = hf00, ϕifor every ϕ ∈ C 0, this implies µ d = 12f00. f00, ϕ = f, ϕ00 = Z Z R ϕ00(x)|y − x| dµ(y)dx = = Z R Z {y≥x} ϕ00(y − x) dx dµ(y) + Z R Z {y<x} ϕ00(x − y) dx dµ(y) = = Z R Z y −∞ ϕ00(x)y dx dµ(y) − Z R Z y −∞ ϕ00(x)x dx dµ(y)+ + Z R Z +∞ y ϕ00(x)x dx dµ(y) − Z R Z +∞ y ϕ00(x)y dx dµ(y) = = Z R
yϕ0(y) dµ(y) − Z
R
yϕ0(y) dµ(y)− − Z R ϕ0(x)x y −∞ − Z y −∞ ϕ0(x) dx dµ(y)+ + Z R ϕ0(x)x +∞ y − Z +∞ y ϕ0(x) dx ! dµ(y) = = − Z R ϕ0(y)y dµ(y) + Z R ϕ(y) dµ(y) + Z R ϕ0(y)y dµ(y) + Z R ϕ(y) dµ(y) = =2 Z R
ϕ(y) dµ(y) = 2 hϕ, µi .
Conversely, let f be a function that satisfies (i) and (ii), we define µ:=d 12f00. Thanks to (i) we have that µ is a positive measure. We note that (ii) implies that f behave as k(x − m) for x → +∞ and as −k(x − m) for x → −∞, then there exist the limits of f0(x)for x → ±∞. Hence we can derivate (ii) and we find that
lim x→+∞f 0(x) − k = 0 lim x→−∞f 0(x) + k = 0 (2.4.1) which implies lim x→+∞f 0 (x) = lim x→−∞f 0 (x).
Then we have that
f0(x) = µ((−∞, x)) − µ((x, +∞)) = Z
R
[1(−∞,x)(y) − 1(x,+∞)] dµ(y).
Using again (2.4.1) we have µ((−∞, +∞)) = µ(R) = k. Moreover, (ii) implies that limx→+∞f (x) = limx→−∞f (x), and we can write
f (x) = 1 2 Z x −∞ f0(y) dy − Z +∞ x f0(y) dy = Z R |x − y| dµ(y).
If we show that the barycenter of µ is m we complete the proof. We use (ii) lim x→+∞ Z R |x − y| dµ(y) − k(x − m) = 0, Writing Z R |x − y| dµ(y) = Z {x≥y} (x − y) dµ(y) + Z {x<y} (x − y) dµ(y)x→+∞−→ Z R (x − y) dµ(y) We obtain 0 = lim x→+∞f (x) − k(x − m) = xk − Z R y dµ(y) + km − kx which implies m = 1 k Z R y dµ(y).
Then we have shown that f(x) = uµ(x).
We note that, if µ and ν have the same mass and the same barycenter, thanks to the elementary equality 2x+= x + |x|, we have that the knowledge of u
µ and uν is equivalent to that of vµ(x) = Z R (x − y)+dµ(y), vν(y) = Z R (x − y)+dν(y).
Remark 2.4.3. We can now prove that prescribing the payoff of call options is equiv-alent to prescribe the marginals of the plan. Indeed, if we set Φi,y = (Si − y)+ for
i = 1, 2 and y ∈ R the payoffs, and we call µ = LawS1 and ν = LawS2, we have C(t1, y) = Z R (x − y)+dµ(y), C(t2, y) = Z R (x − y)+dν(y).
Then, give the payoff of call options is equivalent to give the potential functions of the law of the options,i.e. uµ and uν. Thanks to Proposition 2.4.2, u00µ
d
= µ and u00ν = νd , and then this is equivalent to require that the first marginal of π is equal to µand the second marginal of π is equal to ν.
We now list some relevant properties of potential functions. Proposition 2.4.4. Let µ and ν be in M.
(a) If µ and ν have the same mass and the same barycenter, µ C ν is equivalent
to uµ≤ uν.
(c) A sequence of measures (µn)nin M with mass k and mean m converges weakly
in M to some µ if and only if (uµn)n converges pointwise to the potential function of some µ0 ∈ M. In that case µ = µ0.
Proof. (a) Since y 7→ |x − y| is a convex function, if µ C ν then uµ ≤ uν.
Conversely, we have uµ≤ uν. We note that if µ and ν have the same barycenter,
using |x| = 2x+− x, we have that
Z R |x − y| dµ(y) ≤ Z R |x − y| dν(y) is equivalent to Z R (x − y)+dµ(y) ≤ Z R (x − y)+dν(y) Then µ C ν.
(b) If µ ≤ ν then µ − ν is a positive measure, then it belongs to M. We have also that uν − uµ= Z R |x − y| dν − Z R |x − y| dµ = Z R |x − y| d(ν − µ) = u(ν−µ) and, since ν − µ ∈ M u(ν−µ) is a potential function, then it is convex.
Con-versely, uν − uµ= u(ν−µ) is a convex function, we define ν − µ as one-half the
second derivative of u(ν−ν) in the sense of distribution. Since the function is
convex we have that the measure ν − µ is positive and then ν ≥ µ. (c) We first suppose that µn* µ. We note that µ ∈ M indeed
Z R |x| dµ(x) ≤ sup n∈N Z R |x| dµn(x) < ∞.
We consider the function f : y 7→ |x − y|, |x − y| ≤ |x| + |y|, and we define fk := sup[inf(f, k), −k]. Since fk is a continuous and bounded function we
have lim nN Z R fk(y) dµn(y) = Z R fk(y) dµ(y). If k ≥ |x| then
|f (y) − fk(y)| = (|f (y)| − k)+≤ (|x| + |y| − k)+ ≤ |y|1{|y|≥k−|x|},
and, since µn has mean equals to m for every n ∈ N, we have
sup k Z R f (y) dµn(y) − Z R fk(y) dµn(y) ≤ sup k Z {|y|≥k−|x|} |y| dµn n→∞ −→ 0
. This implies Z
R
f (y) dµ(y) = lim
k→∞n→∞lim Z R fk(y) dµn(y) = = lim n→∞k→∞lim Z R
fk(y) dµn(y) = lim n→∞
Z
R
f (y) dµn(y).
Remembering that f(y) = |x − y| we have shown that uµn(x) → uµ(x) for every x ∈ R. Conversely, if uµn(x) → uµ(x), by equicontinuity
lim
n→∞uµn = uµ
uniformly on compact sets of R, and hence in the sense of distributions. Con-sequently, since (µn) (resp. µ) is the second derivative, in the sense of
distri-butions, of the function uµn (resp. uµ), lim
n→∞µn= µ
in the sense of distributions. Using the fact that µnand µ have finite mass we
have the statement.
2.5
An SDE proof of Strassen’s Theorem
We give a sketch of the proof of theTheorem 2.3.1, using Proposition 2.2.9.
Sketch of a second proof of sufficiency in Theorem 2.3.1. We have µ C ν. We first
suppose that the two measures have density µ = ρ0 > 0 and ν = ρ1 > 0. Set
ρt= tρ1+ (1 − t)ρ0. We want to show that we can find a martingale Xtwith law ρt.
To this aim we look for the solution of an SDE of the type dXt= σ(t, Xt)dBt.
We need to establish who σ(t, Xt) is. We know that, for every f ∈ C02
df (Xt) = f0(Xt)σ(t, Xt)dBt+
1 2f
00(X
s)σ2(s, Xs)ds
from Ito’s formula. Hence, f (Xt) − f (X0) = Z t 0 f0(Xs)dXs+ 1 2 Z t 0 f00(Xs)σ2(s, Xs) ds.
Now, if we suppose that Xthas law ρt we obtain ∂t Z R f ρ = 1 2 Z R f00(σ2ρ)
Integrating by parts two times, remembering that f ∈ C0 we have
∂t Z R f ρt= 1 2 Z R f (σ2ρt)00 But, ρt= tρ1+ (1 − t)ρ0, ∂t Z R f (tρ1+ (1 − t)ρ0) = 1 2 Z R f (σ2ρ)00 Z R f ρ1− Z R f ρ0= 1 2 Z R f00(σ2ρ)00
Since this equality holds for every f ∈ C0 we need to have that ρ1− ρ0 = (σ2ρt)00,
then σt= s 2R R (ρ1− ρ0) ρt .
We need to show that Rz −∞ Ry −∞(ρ1(x) − ρ0(x)) dx dy ≥ 0. By Proposition 2.4.2 we have that ρ0 d = u000 ρ1 d = u001
where u0 and u1 are the potential functions of ρ0 and ρ1 respectively. Then we
choose σt= s 2(u1− u0) ρt .
Since ρ0 C ρ1 thanks toProposition 2.4.4 we have that
u1− u0≥ 0.
Note that we have that ρt > 0 because ρ1 and ρ0 are greater than 0. In order to
complete the proof we should verify that
• σ > 0 (it follows from the fact that ρt> 0).
• dXt = σ(t, Xt)dBt is well posed, e.g. we should give the following conditions
on σ: |σt(x) − σt(y)| ≤ k|x − y|;
• The law of Xt is ρt.
We skip those technical conditions and we conclude with the sufficiency in Propo-sition 2.2.9 in the general case. We have µ C ν, then thanks to Proposition 2.2.9
µ ∗ ρε C ν ∗ ρε where ρε are smooth densities (e.g. Gaussians with ε variations).
Namely ρε
0 := µ ∗ ρε and ρ1 := ν ∗ ρε, we apply the discussion above to ρ0 and ρ1
and we obtain a martingale (Xε
t)t∈(0,1). If we take (X0ε, X1ε), we have that its law is
a martingale transport plan, indeed the marginal of Xε
0 is ρε0 and the marginal of X1ε
is ρε
The Dual Problem
We now construct a semi-static subhedging portfolio consisting of the sum of static vanilla options ϕ(S1), ψ(S2) and an investment in the risky asset according to the
self-financing trading strategy ∆(S1) . Hence, the portfolio is represented as follows
ϕ(x) + ψ(y) + ∆(x)(y − x)
where ϕ: R → R is in L1(µ), ψ : R → R is in L1(ν) and ∆: R → R is a continuous
and bounded function. These functions lead to a subhedging strategy if the following holds
c(x, y) ≥ ϕ(x) + ψ(y) + ∆(x)(y − x).
Then, if we take a pricing measure π ∈ ΠM(µ, ν), we obtain the obvious inequality
Eπ[c] ≥ Eπ[ϕ(x) + ψ(y) + ∆(x)(y − x)] = Eπ[ϕ(x) + ψ(y)] = Eµ[ϕ] + Eν[ψ], (3.0.1)
where we used that π ∈ ΠM(µ, ν) and then π has µ and ν as marginals and
R
Rf (x)(y − x) dπ = 0 for every function f ∈ Cb. This leads us to consider the
dual problem
D = sup {Eµ[ϕ] + Eν[ψ] : ∃∆ s.t. ϕ(x) + ψ(y) + ∆(x)(y − x) ≤ c(x, y)}
which, by (3.0.1) satisfies
P ≥ D.
We note the financial interpretation of the inequality above: suppose that some-body offers the option c at a price q < D. Then, since D is a supremum, there exist ϕ, ψ and ∆ such that ϕ(x)+ψ(y)+∆(x)(y −x) ≤ c(x, y) and Eµ[ϕ] + Eν[ψ] > q. We
want to show that, buying c and going short in Ψ(x, y) = ϕ(x) + ψ(y) + ∆(x)(y − x) we can have arbitrage. Indeed, the price of Ψ is Eπ[Ψ] = Eµ[ϕ] + Eν[ψ] which is
strictly larger than q, thanks to the fact that q < D. According with the notation of Chapter 1 we have a self-financing portfolio, with
V0 = Eπ[Ψ] − q
VT = c − q + Eπ[Ψ] − Ψ
where T represents the time at which we go short in Ψ. If we show that V0≥ 0 and
VT > V0 and P[VT > V0] > 0we have arbitrage. We rewrite VT in the following way
c − Ψ + Eπ[Ψ] − q = c − Ψ + V0,
and we note that c ≥ Ψ, then VT ≥ V0. If we show that the expectation of c − Ψ is
strictly positive we obtain that VT > V0 with positive probability.
Eπ[c − Ψ + Eπ[Ψ] − q] = Eπ[c] − Eπ[Ψ] + Eπ[Ψ] − q = Eπ[c] − q,
but Eπ[c] ≥ P ≥ D > pand then we have shown that VT > 0.
The main result of this chapter is showing that the extremal martingale prices of a financial derivative correspond to the minimal or maximal initial capital necessary for sub-/super-replication replication respectively. This is exactly the replication theorem of mathematical finance, with the novelty that this is a model-free version.
3.1
No-duality gap theorem
Theorem 3.1.1. Assume that µ and ν are probability measure on R such that µ C ν
(and then ΠM(µ, ν) 6= ∅). Let c: R2→ (−∞, ∞]be a lower semi-continuous function
such that
c(x, y) ≥ −K(1 + |x| + |y|)
on R2 for some constant K. Then there is no duality gap, i.e. P = D.
The dual supremum is in general not attained.
We note that an upper bound for the price of the option c can be given by means of semi-static superhedging. Applying the following to the function −c we obtain the following:
Corollary 3.1.2. Assume that µ and ν are probability measure on R such that µ C ν (and then ΠM(µ, ν) 6= ∅). Let c: R2 → (−∞, ∞]be an upper semi-continuous
function such that
on R2 for some constant K. Then there is no duality gap
P = sup {Eπ[c] : π ∈ ΠM(µ, ν)} =
= inf {Eµ[ϕ] + Eν[ψ] : ∃∆ ϕ + ψ + ∆ ≥ c} = D.
The supremum is attained i.e. there exists a maximizing martingale measure. We will prove this theorem using the Monge-Kantorovich duality theory. To this aim we state the following results.
Lemma 3.1.3. We can restrict the set of the dual maximizers to the following set S = ( u : R → R : u(x) = a + bx + m X i=1 ci(x − ki)+, a, b, ci, ki ∈ R )
We will give the proof of this Lemma after the proof of the main theorem. Theorem 3.1.4 (Monge-Kantorovich duality theorem). Let c: R2 → (−∞, ∞]be a lower semi-continuous function satisfying
c(x, y) ≥ −K(1 + |x| + |y|)
on R2 for some constant K and let µ and ν be probability measures on R having
finite first moments. Then PM K(c) = inf Z R2 c dπ = = sup Z R u1dµ + Z R u2dν : u1⊕ u2 ≤ c, ui∈ S = DM K(c),
where we have set u1⊕ u2(x, y) = u1(x) + u2(y).
Theorem 3.1.5. Let M, T be convex subsets of vector spaces V1 and V2, where V1
is locally convex and let f : M × T → R.If • M is compact,
• f(·, y) is lower semicontinuous and quasi-convex on M for every y ∈ T , • f(x, ·) is upper semicontinuous and quasiconcave on T for every x ∈ M, then
sup
y∈T
inf
x∈Mf (x, y) = infx∈Msupy∈Tf (x, y).
For a proof see Theorem 1.9 in [7].
Proof of Theorem 3.1.1. We will restrict ourselves to dual candidates Ψ = ϕ+ψ +∆ satisfying ϕ, ψ ∈ S and ∆ ∈ Cb. If the assertion holds true for a function c and
if ϕ, ψ ∈ S then the assertion carries over to c0 = c + ϕ ⊕ ψ. Then, without loss
of generality we may assume that c ≥ 0. Moreover we also assume, for now, that c ∈ Cb(R2), we will get rid of this extra condition later.
We apply Theorem 3.1.5 to the compact convex set M = Π(µ, ν) (we have shown the compactness of this set in Theorem 2.1.6), the convex set T = Cb(R) and the
function
f (π, ∆) = Z
R2
c(x, y) − ∆(x)(y − x) dπ(x, y).
We need to show that f satisfies the hypothesis of Theorem 3.1.5. The convexity of f (·, y) is an easy consequence of the linearity of f : π 7→ f(π, ∆). We only have to show that f(π, ·) is concave on Cb(R). If we call F (∆) = f (π, ∆), we need to show
F (λ∆1+ (1 − λ)∆2) ≥ λF (∆1) + (1 − λ)F (∆2): F (λ∆1+ (1 − λ)∆2) = Z R2 [c(x, y) − (λ∆1+ (1 − λ)∆2)(x)(y − x)] dπ(x, y) = =λ Z R2 [c(x, y) − ∆1(x)(y − x)] dπ(x, y)+ +(1 − λ) Z R2 [c(x, y) − ∆2(x)(y − x)] dπ(x, y) = =λF (∆1) + (1 − λ)F (∆2).
We can apply Theorem 3.1.5.
D ≥ sup ϕ,ψ∈S,∆∈Cb(R),ϕ+ψ+∆≤c Z R ϕ dµ + Z R ψ dν = (3.1.1) = sup ∆∈Cb(R) sup ϕ,ψ∈S,ϕ(x)+ψ(y)≤c(x,y)−∆(x)(y−x) Z R ϕ dµ + Z R ψ dν = (3.1.2) = sup ∆∈Cb(R) inf π∈Π(µ,ν) Z R2 c(x, y) − ∆(x)(y − x) dπ(x, y) = (3.1.3) = inf π∈Π(µ,ν) ∆∈Csupb(R) Z R2 c(x, y) − ∆(x)(y − x) dπ(x, y) = (3.1.4) = inf π∈ΠM(µ,ν) Z R2 c(x, y) dπ(x, y) = P. (3.1.5) Here Theorem 3.1.4 is applied to c(x, y) − ∆(x)(y − x) to establish the equality between (3.0.3) and (3.0.4). We only need to show that c(x, y)−∆(x)(y −x) satisfies the hypothesis of the Theorem. Since ∆ is a bounded function, there exist a constant M such that |∆| ≤ M, taking K0= K +M, where K is the constant in the statement of the Theorem, we obtain c(x, y) − ∆(x)(y − x) ≥ −(K + M)(1 + |x| + |y|). The
equality between (3.0.4) and (3.0.5) is guaranteed by Theorem 3.1.5. We have to justify the equality between (3.0.5) and (3.0.6): indeed, if π is in ΠM(µ, ν)for every
bounded function ∆ we have that RR2∆(x)(y−x) = 0, instead, if π is not in ΠM(µ, ν)
there is a function ∆ such that B =
Z
R2
∆(x)(y − x) 6= 0.
By appropriately scaling ∆ the value of B can be made arbitrarily large.
Now we assume that c: R2 → (0, ∞] is merely lower semi-continuous and pick a
sequence of bounded continuous functions c1 ≤ c2 ≤ . . . such that c = supk≥0ck. In
what follows we will write P (c), D(c), P (ck), D(ck)to emphasize the dependence on
the cost function. For each k pick πk∈ ΠM(µ, ν)such that
P (ck) ≥ Z R2 c dπk− 1 k.
Passing to a subsequence if necessary, since ΠM(µ, ν) is a weakly compact set, we
may assume that (πk) converges weakly to some π ∈ ΠM(µ, ν). Then
P (c) ≤ Z R2 c dπ = lim m→∞ Z R2 ckdπ = lim m→∞ lim k→∞cmdπk ≤ ≤ lim m→∞ lim k→∞ Z R2 ckdπk = lim k→∞P (ck).
Since P (ck) ≤ P (c), letting k go to ∞ we obtain that limk→∞P (ck) ≤ P (c),
com-bining with the result above we have limk→∞P (ck) = P (c). Then D(c) ≥ D(ck) =
P (ck), where we have used the equality proved for a bounded continuous function.
Now, letting k go to ∞ we have that
D(c) = P (c).
Proof of Lemma 3.1.3. As in the proof ofTheorem 3.1.1, it is sufficient to prove the duality equation in the case c ≥ 0.
Given a bounded continuous function f and ε > 0, there is some u ∈ S such that f ≥ u, RRf − u dµ < εand R
Rf − u dν < ε. Therefore it suffices to prove
PM K(c) = sup Z R u1dµ + Z R u2dν : u1⊕ u2 ≤ c, ui ∈ Cb(R) . (3.1.6) We will first show under the additional assumption that c ∈ Cc(R2). By Theorem
that PM K(c) − Z R u1dµ + Z R u2dν ≤ η
and u1⊕ u2 ≤ c. Note that the latter inequality implies that u1 and u2are uniformly
bounded since c is bounded from above. We now show how we can replace u1 by a
function in Cb. We consider H(x, y) = c(x, y) − u2(y) and define
˜
u1(x) := inf
y∈RH(x, y) ∀ x ∈ R.
We want to show that ˜u1 is uniformly continuous. Since c ∈ Cc(R2) it is uniformly
continuous, hence, for every ε > 0 there exists δ > 0 such that whenever x, x0 ∈ R,
|x − x0| < δ, then |H(x, y) − H(x0, y)| = |c(x, y) − c(x0, y)| < ε. Thus we obtain | ˜u1(x) − ˜u2(x0)| = inf
y∈RH(x, y) − infy∈RH(x 0 , y) ≤ ε whenever |x − x0| < δ. By definition ˜u
1 is bounded from below, indeed, since there
exist a constant K2 such that |u2| ≤ K2 we have that
˜ u1= inf
y∈R[c(x, y) − u2(y)] ≥ K − K2,
furthermore ˜u1 ≥ u1, indeed rewriting the inequality as follows ˜u1− u1 ≥ 0we have
inf
y∈R[c(x, y) − u2(y)] − u1(x) = infy∈R[c(x, y) − u2(y) − u1(x)] ≥ 0.
Finally ˜u1⊕ u2 ≤ c, it is sufficient showing that ˜u1(x) ≤ c(x, y) − u2(y),by definition
˜
u1 = infy∈R[c(x, y) − u2(y)], then the inequality holds. Replacing u2 in the same
fashion, we obtain (3.1.6) in the case c ∈ Cc(R2). Using the same argument as in
the proof of Theorem 3.1.1, we obtain the Theorem in the case of a general, lower semi-continuous function c: R2 → [0, ∞].
As we have already said in the chapter 1, the existence of a primal optimizer π is a consequence of the compactness of the set of all martingale transport plans. The dual set of sub-hedges does not exhibit nice compactness properties and the supremum is not necessarily attained. We comment on the consequences of attainment of the dual problem. Assume that there exists a dual maximizer, that is there exist ϕ ∈ L1(µ)
and ψ ∈ L1(ν)and a continuous bounded function ∆ such that the subhedge satisfies
Ψ := ϕ + ψ + ∆ ≤ c
and
Eµ[ϕ] + Eν[ψ] = D.
Let π be a primal optimizer, i.e. a martingale measure satisfying Eπ[c] = P. Then
we have
0 ≤ Eπ[c − Ψ] = P − D = 0.
The financial interpretation is that under the market model π, the payoff c is perfectly replicated through the semi-static hedge corresponding to ϕ, ψ and ∆.
3.2
Technical results
We now give some technical results that use tools from the section above (theory of duality) and that will be fundamental in the next chapter.
Definition 3.2.1. Let α a measure on R × R with finite first moment in the second variable. We say that α0, a measure on the same space, is a competitor of α if α0 has
the same marginals as α and for (projx
#)-a.e. x ∈ R Z R y dαx(y) = Z R y dα0x(y),
where (αx)x∈R and (α0x)x∈R are disintegrations of the measures with respect to
proj#xα.
The following lemma can be viewed as a substitute for the characterization of optimality through the notion of c-cyclical monotonicity in the classical setup (see chapter one).
Lemma 3.2.2 (Variational Lemma). Assume that µ, ν are probability measures in convex order and that c : R2 → R is a Borel measurable cost function satisfying the
sufficient integrability condition. Assume that π ∈ ΠM(µ, ν)is an optimal martingale
transport plan which leads to finite costs. Then there exists a Borel set Γ with π(Γ) = 1 such that the following holds: If α is a measure on R × R with |spt(α)| < ∞ and spt(α) ⊆ Γ, then we have R c dα ≤ R c dα0 for every competitor α0 of α.
For the proof of Lemma 3.2.2, we need the following result:
Lemma 3.2.3. Let (Z, ζ) be a Polish probability space and M ⊆ Zn. Then either of the following holds true:
(1) There exist subsets (Mi)i of Zn such that ζ(projiMi) = 0 for i = 1, . . . , n and M ⊆ n [ i=1 Mi.
(2) There exists a measure γ on Zn such that γ(M) > 0 and proji
#γ ≤ ζ for
i = 1, . . . , n.
Lemma 3.2.3is a corollary of the following more general result.
Definition 3.2.4. Let X1, . . . , Xnbe Polish spaces with finite Borel measures µ1, . . . , µn.
B ⊆ X1×· · ·×Xnis called L-shaped null set if there exist null sets N1⊆ X1, . . . Nn⊆
Xn such that B ⊆ Sni=1(proji)−1(Ni).
Proposition 3.2.5. Let X1, . . . , Xn, with n ≥ 2 Polish spaces with finite Borel
measures µ1, . . . , µn. Set for every Borel set B ⊆ X1× · · · × Xn
P (B) := sup{π(B) : π ∈ Π(µ1, . . . , µn)} L(B) := inf ( X µi(Bi) : Bi⊆ Xi, B ⊆ n [ i=1 (proji)−1(Bi) ) . Then P (B) ≥ 1
nL(B). In particular B satisfies one of the following:
• B is L-shaped null,
• there exists π ∈ Π(µ1, . . . , µn) such that π(B) > 0.
The main ingredient in the proof of the Proposition above is the following duality theorem due to Kellerer, for a proof see Lemma 1.8(a), Corollary 2.18 in [3].
Theorem 3.2.6 (Kellerer’s Theorem). Let X1, . . . , Xn, with n ≥ 2 Polish spaces
with finite Borel measures µ1, . . . , µn, and assume that c : X1 × · · · × Xn → R is
Borel and measurable and that c := supxc and c := infxc are finite. Set
I(c) := Z X c dπ π ∈ Π(µ1, . . . , µn) S(c) := XZ Xi ϕidµi : c(x1, . . . , xn) ≥ X ϕi(xi), c n− (c − c) ≤ ϕi ≤ c n . Then I(c) = S(c).
Proof of Proposition 3.2.5. Note that −I(−1B) = P (B)and
−S(−1B) = inf ( n X i=1 Z Xi χidµi : 1B(x1, . . . , xn) ≤ n X i=1 χi(xi), 0 ≤ χi≤ 1 ) . (3.2.1)
Note that the condition 0 ≤ χi≤ 1follows from the definition of S(c) with c = −1B,
inTheorem 3.2.6. Indeed, in this case, c = 0 and c = −1 and then c
n−(c−c) ≥ ϕi ≥ c n
becomes o ≤ χi ≤ 1. By Kellerer’s Theorem −S(−1B) = −I(−1B). Thus, it remains
to show that −S(−1B) ≥ n1L(B). Fix functions χi as in Equation 3.2.1. Then, for
(x1, . . . , xn) ∈ B we have that 1 = 1B(x1, . . . , xn) ≤ Pni=1χi(xi) and hence there
exists some i such that χi(xi) ≥ n1. Thus B ⊆ S n
i=1(proji)−1({χi ≥ 1n}). It follows
that −S(−1B) ≥ inf ( XZ Xi χidµi : B ⊆ n [ i=1 (proji)−1 χi ≥ 1 n , 0 ≤ χi≤ 1 ) ≥ ≥ inf ( X1 nµi χi ≥ 1 n : B ⊆ n [ i=1 (proji)−1 χi ≥ 1 n ) ≥ 1 nL(B).
>From this we deduce that either L(B) = 0 or there exists π ∈ Π(µ1, . . . , µn) such
that π(B) > 0.
The last assertion of the Proposition follows from the following Lemma due to Richárd Balka and Márton Elekes.
Lemma 3.2.7. If L(B) = 0 for a Borel set B ⊆ X1 × · · · × Xn. Then B is an L-shaped null set.
Proof. Fix ε > 0 and B(k) 1 , . . . , B
(k)
n Borel sets with µi(B (k)
i ) ≤ 2εk such that, for every k
B ⊆ (proj1)−1(B1(k)) ∪ · · · ∪ (projn)−1(B(k)n ). Let Bi:=S∞k=1Bi(k) for i = 2, . . . , n such that
B ⊆ (proj1)−1(B1(k)) ∪ (proj2)−1(B2) · · · ∪ (projn)−1(Bn)
for each k ∈ N. Thus, with B1 =T∞k=1B1(k),
B ⊆ (proj1)−1(B1) ∪ (proj2)−1(B2) · · · ∪ (projn)−1(Bn)
Hence we can assume from now on that µ1(B1) = 0and µi(Bi)is arbitrarily small for
i = 2, . . . , n. Iterating this argument in the obvious way we get the statement. Note that if n = 2 we have L(B) = P (B) for every Borel set B ⊆ X1× X2.
Proof of Lemma 3.2.3. Using Proposition 3.2.5 with X1 = · · · = Xn= Z and µ1 =
· · · = µn= ζ we obtain the statement.
Proof of Lemma 3.2.2. Fix a number n ∈ N. We want to construct a set Γn for
which the optimality property holds for every α satisfying |spt(α)| ≤ n. This set will satisfy π(Γn) = 1. Then Γ = ∩nΓn is the set required in the lemma, indeed
π(Γc) = π((∩nΓn)c) = π(∪Γcn) ≤
X
π(Γcn) = 0
In the last equality we have used that n ∈ N and countable union of negligible sets is negligible. Furthermore the set Γ has the optimality property because it is the intersection of sets for which the property holds. For a fixed n ∈ N we define the set M as follows M := (1) α is a measure on R × R (xi, yi)ni=1 : ∃α s.t. (2) spt(α) ⊆ {(xi, yi) : i = 1, . . . , n}
(3) ∃ a competitor α0satisf ying R
Rc dα 0 <R Rc dα
We now apply Lemma 3.2.3 to the space (Z, ζ) = (R2, π) and the set M. If we are
in case (1), let N = Sn
i=1proji(Mi) so that π(N) = 0 and M ⊆ (N × Zn−1) ∪ · · · ∪
(Zn−1× N ) = Zn \ (Z \ N )n because we already knew that M ⊆ (proj1(M 1) ×
Zn−1) ∪ · · · ∪ (Zn−1 × projn(M
n)). Hence, if we define Γn := z \ N = R2 \ N
thanks to condition (3), we obtain a set which does not support any nonoptimal α with |spt(α)| ≤ n. Moreover, since the set N is π−negligible, his complementary set has π−measure equal to 1. We want to show that the case (2) cannot occur. Striving for a contradiction, we assume that there exists a measure γ such that γ(M ) > 0and proj#i γ ≤ πfor i = 1, . . . , n. Restricting γ to M, we can assume that γ(R × R \ M ) = 0. Rescaling γ, if necessary, we can also assume that proj#i γ ≤
1 nπ.
We consider the measure ω = Pn
i=1proj#i γ on R. ω = P n
i=1proj#i γ ≤ nn1π = π
and it has positive mass. In particular µω = projx#ω ≤ proj#xπ = µ. We want find a
competitor ω0 such that ω0 leads to smaller costs than ω, that is,
Z R×R c(x, y) dω0 < Z R×R c(x, y) dω.
If such a measure exists then the measure π0 := π − ω + ω0 is a martingale optimal
transport plan which leads to smaller costs than π, contradicting the optimality of π. We first show that π0 is a transport plan. It has µ and ν as marginals because of the definition of competitor: we know that ω and ω0have the same marginals, hence,
if we perform π −ω +ω0 then only the marginals of π remain. π0 is a martingale plan
Z R y d(π − ω + ω)x(y) = Z R y dπx(y) − Z R y dωx(y) + Z R y dωx0(y) = = Z R y dπx(y) = x
using again the definition of competitor. Furthermore π0 leads smaller costs than π Z R×R c(x, y) dπ0 = Z R×R c(x, y) dπ − Z R×R c(x, y) dω0+ Z R×R c(x, y) dω < < Z R×R c(x, y) dπ.
It remains to show how to construct such a ω0.
For each p = ((x1, y1), . . . , (xn, yn)) ∈ (R × R)n, let αp the measure that is uniformly
distributed on the set {(x1, y1), . . . , (xn, yn)}. Then
ω = Z
p∈(R×R)n
αpdγ(p),
indeed, ω is the sum of the projection of γ and αp only give the right weight to γ
on the set considered. For each p = ((x1, y1), . . . , (xn, yn)) ∈ (R × R)n, let α0p an
optimizer of the following problem M inimize
Z
(x,y)∈R×R
c(x, y) dβ(x, y) β competitor of αp.
We underline that such an α0 exists and can be taken to depend measurably on p.
Indeed, since the problem above is a problem of optimal transport for finite space we can use, for instance, the simplex algorithm to find α0
p.As γ is concentrated on
M, for γ−almost all points p,thanks to (3) in the definition of M and thanks to the fact that α0 is an optimizer, the measure α0
p satisfies Z (x,y)∈R×R c(x, y) dα0p(x, y) < Z (x,y)∈R×R c(x, y) dαp(x, y).
Then, we define ω0 as follows
ω0 = Z
p∈(R×R)n
α0pdγ(p)
if we show that ω0 is a competitor of ω and leads to smaller costs than ω we have
found what we were looking for. For the first condition, using the fact that α0 p is a competitor of αp we have Z R y dωx(y) = Z R Z R y d(αp)x(y) dγx(p) = Z R Z R y d(α0p)x(y) dγx(p) = Z R y dωx0(y) Z R×R ϕ(x) dω(x, y) = Z p∈(R×R)n Z R×R ϕ(x) dαp(x, y) dγ(p) = = Z p∈(R×R)n Z R×R ϕ(x) dα0p(x, y) dγ(p) = Z R×R ϕ(x) dω0(x, y)
for every ϕ ∈ Cb and analogously for the other marginal. Now we show that ω0 leads
to smaller costs than ω. Z R×R c dω0 = Z p∈(R×R)n Z (x,y)∈R×R c(x, y) dαp0(x, y) dγ(p) < < Z p∈(R×R)n Z (x,y)∈R×R c(x, y) dαp(x, y) dγ(p) = Z R×R c dω.
