In this section, we take a look at an application of Theorem 3.2.2, which will give us a condition for the existence of an equivalent martingale measure. We prove the theorems for Banach lattices. The application of these theorems to Orlicz hearts will be a special case. The proofs are adapted from Stricker [150].
Let H be a predictable, simple process, i.e. H =Pn−1
i=1 λi1(ti,ti+1], where 0 ≤ t0 < t1 <
· · · < tn ≤ 1 and λi = (λ1i, . . . , λdi) ∈ Rd is a random Ft-measurable vector. We denote
by (H · X)t the stochastic integral of the predictable process H = (H1, . . . , Hd) ∈ Rd with
respect to the semimartingale X = (X1, . . . , Xd) ∈ Rd. We suppose that (H · X)0 = 0. If
H is locally bounded, then it is well known that (H · X)t= d X j=i Z t 0 Hsj dXsj.
Let X be a Rd-valued, c`adl`ag, adapted semimartingale with Xt ∈ E for all t ∈ [0, 1].
In this section, we consider the following special case of the convex set K of the previous section. Let
K = {(H · X)1 : H is predictable, simple and bounded}.
By considering this cone K, we may add another equivalent condition to those in Theorem 3.2.2.
Theorem 3.2.3. The following are equivalent. (v) E+∩ (K − L+∞(P )) = {0}.
(vi) There exists a probability measure Q equivalent to P and with density dQdP ∈ E∗ such
that X is a Q-martingale.
Proof. (v) ⇒ (vi): Assume that condition (v) holds, then condition (i) of Theorem 3.2.2 also holds. Thus, by Theorem 3.2.2, there exists a random variable z∗ ∈ (E∗)+ such that sup
ξ∈K
z∗(ξ) < ∞. Hence, z∗(k) ≤ supξ∈Kz∗(ξ) for all k ∈ K. Since K is a cone, we get for all n ∈ N that
z∗(k) ≤ 1 nξ∈Ksup
z∗(ξ).
Thus, by the Archimedean property, we have that z∗(k) = 0 for all k ∈ K. In other words, z∗((H · X)1) = 0 for all predictable, simple and bounded processes H. Let dQ = z∗dP .
Then dQdP ∈ E∗ and X is a martingale under Q. Thus, we have shown that there exists a probability measure Q equivalent to P and with Radon-Nikod´ym derivative dQdP ∈ E∗ such
that X is a Q-martingale.
(vi) ⇒ (v): If Q is equivalent to P , the Radon-Nikod´ym derivative dQdP = z∗ belongs to E∗ and z∗ > 0. If X is a martingale under Q, then z∗((H · X)1) = 0 for all previsible,
simple, bounded H. Thus z∗(k) = 0 for all k ∈ K, i.e. sup
k∈K
z∗(k) = 0 < ∞.
Hence, condition (i) of Theorem 3.2.2 is verified with z∗ = dQdP and (v) holds.
If X is continuous we can weaken condition (v) of Theorem 3.2.3 by replacing K − L+∞(P )
with K, as is shown in the next theorem.
Theorem 3.2.4. Let X be a continuous, adapted process with values in Rd. Let Xt ∈ E
for all t ∈ [0, 1]. Then there exists a probability Q equivalent to P and with Radon-Nikod´ym derivative dQdP ∈ E∗ such that X is a Q-martingale if and only if E+∩ K = {0}.
Proof. ⇒: Assume that there exists a probability Q equivalent to P and with Radon- Nikod´ym derivative dQdP ∈ E∗ such that X is a Q-martingale. Then, by Theorem 3.2.3, we
have that E+∩(K − L+∞(P )) = {0}. As K ⊆ K − L+∞(P ), we must have that E+∩K = {0}.
⇐: Conversely, suppose that E+∩ K = {0}. We show, using a contradiction argument,
that1A∈ K − L/ +∞(P ) is satisfied for all A ∈ F with P (A) > 0. Assume there exists A ∈ F
with P (A) > 0 such that 1A ∈ K − L+∞(P ). Then we can find a sequence of positive,
bounded, random variables (Bn), a sequence of previsible, simple, bounded processes (Hn)
and a set A ∈ F with P (A) > 0, such that (Hn· X)1− Bn converges to 1A.
We will construct two sequences (Un) and (Bn0) such that
- Bn0 ∈ L+ ∞(P ),
- Un is a previsible, simple, bounded process,
- ((Un· X)1) is bounded in E and
- ((Un· X)1− Bn0) converges to1A.
To construct these sequences, we introduce the stopping time Tn=
(
1 if (Hn· X)t< 1 for all t ∈ [0, 1]
inf{t > 0 : (Hn· X)t≥ 1} otherwise.
Note that (Hn· X)−1 ≥ (Hn· X)−Tn≥ 0. In fact (Hn· X)−1 tends to 0 in E. We also have that
0 ≤ (Hn· X)+Tn ≤ 1. Thus, the sequence (Hn· X)Tn = (Hn· X)
+
Tn− (Hn· X)
−
Tn is bounded
in E.
However, the process 1[0,Tn]Hn is not simple.
It is well known that there exists a decreasing sequence of stopping times (Tm)m∈N with
lim m→∞Tm= Tn. Let Tm0 = ( Tm if |(Hn· X)Tn− (Hn· X)Tm| ≤ 1 1 otherwise. Then, we see that
lim
m→∞k(Hn· X)Tn− (Hn· X)T
0
mk = 0. (3.2)
We can thus choose m such that k(Hn· X)Tn− (Hn· X)Tm0 k ≤
1 n.
Let Un = 1[0,T0
m]Hn and B
0
n = 1{Tn<1}∩Ac +1{Tn=1}Bn. Then Un is a simple and
bounded process and Bn0 ∈ L+
∞(P ). But (Hn· X)Tn− B 0 n= (Hn· X)1− Bn on {Tn= 1} 0 on Ac∩ {Tn< 1} 1 on A ∩ {Tn< 1}.
Therefore, (Hn· X)Tn− B
0
nconverges to 1Ain E. From (3.2), we see that (Hn· X)T0 m− B
0 n
also converges to1A.
Since ((Un· X)1) is bounded, we can extract a sub-sequence which converges weakly
to a random variable Y . As K is convex and closed in the strong topology, Y ∈ K. But (Hn· X)1− Bnconverges to1Ain E, with the result that the weak convergence of (Hn· X)1
leads to the weak convergence of Bn to a random variable B ≥ 0. Thus, Y = B +1A6= 0.
This shows that Y ∈ E+, contradicting our assumption that E+∩ K = {0}.
Hence,1A∈ K − L/ +∞(P ). Hence, there exists a probability Q equivalent to P and with
Radon-Nikod´ym derivative dQdP ∈ E∗ such that X is a Q-martingale.
Let (Φ, Ψ) be complementary finite Young functions. As HΦ(P ) is a Banach lattice
with order continuous norm, e = 1 is a quasi-interior point of HΦ(P ) and we have that
L∞(P ) ⊆ HΦ(P ) ⊆ L1(P ), we can specialise Theorem 3.2.2 and Theorem 3.2.3 to Orlicz
hearts by letting E = HΦ(P ).
In incomplete markets, there does not exist a unique equivalent martingale measure that can be used to price contingent claims. One method of choosing an equivalent pricing measure, is to incorporate a preference structure into the pricing techniques. This is done via utility funtions, which will be introduced in the next section.