When focusing on the exponential utility, we consider the σ-martingale mea-sures with finite entropy. The set of these meamea-sures is given by
Mef(S) =
Q ∈ Pe | S ∈ Mσ(Q), and E dQ
dP logdQ dP
< +∞
. (2.16)
Very frequently, throughout this section and Chapter 5, we will work with densities instead of probabilities. For this, we will use the following set
Zloce (S) := {Z ∈ Mloc(P ) |Z > 0, Z log(Z) is locally integrable, ZS ∈ Mσ(P )}.
(2.17) The following function will be used from time to time,
f1(x) :=
((x + 1) log(x + 1) − x, if x > −1;
+∞, otherwise. (2.18)
Throughout this section and Chapter 5, the main assumption on S is Z
{|x|>1}
|x|eλTxF (dx) < +∞, P ⊗ A − a.s., for all λ ∈ IRd. (2.19)
The next proposition will provide a necessary and sufficient characterization on σ-martingale density, which is expressed by Jacod parameters.
Proposition 2.2: Let Z = E (N ) be a positive local martingale and (β, f, g, N0) be the Jacod components of N . Then Z is a σ-martingale density for (S, P ) if and only if the following hold:
(i) We have Z
|x(1 + f (x)) − h(x)|F (dx) < +∞, P ⊗ A − a.e.
(ii) and
b · A + cβ · A + (x − h(x) + xf (x)) ? ν = 0. (2.20) Furthermore, if Z is a σ-martingale density for (S, P ), then the following holds:
Z
x(1 + f (x))F (dx)∆A = 0, P − a.s. (2.21) Proof. Thanks to Ito’s formula, we deduce that ZS is a σ-martingale if and only if S + [S, N ] is a σ-martingale. The last statement is equivalent to say that there exists a bounded and predictable positive process φ such that
φ · (S + [S, N ]) is a local martingale. (2.22)
Due to Theorem 2.2 (precisely the representation of ∆N given by (2.9)) and the representation of S given by (2.4), we derive that
S +[S, N ] = S0+Sc+h(x)?(µ−ν)+cβ ·A+b·A+[x−h(x)+x(f (x)+g(x))]?µ
Therefore, (2.22) holds if and only if the following two conditions hold:
φ[x − h(x) + x(f (x) + g(x))] ? µ is locally integrable (2.23)
φb · A + φcβ · A + φ(x − h(x) + xf (x)) ? ν = 0. (2.24) Since φ is predictable, positive and bounded, it is easy to deduce that (2.24) is equivalent to (ii). While, (2.23) is equivalent to
Z T 0
φt Z
[x − h(x) + xf (x)]Ft(dx)dAt< +∞, P − a.s.
which holds if and only if (i) is true.
Furthermore, by taking jumps on both sides of (2.24) and using ∆Ab = R xF (dx)∆A, ∆Ac = 0 (see the properties of predictable characteristics of S in Section 2.A for details), we have (2.21) immediately. This ends the proof of this proposition.
The following definitions on entropy-Hellinger process can be found in [16] and [17], to which we refer the readers for more details about the entropy-Hellinger process of a probability measure (which is also called Leibler-Kullback pro-cess).
Definition: (i) Let N ∈ M0, loc(P ) such that 1 + ∆N ≥ 0. Then, if the non-decreasing adapted process
Vt(E)(N ) := 1
2hNcit+ X
0<s≤t
h
(1 + ∆Ns) log(1 + ∆Ns) − ∆Nsi
(2.25)
is locally integrable (i.e. VE(N ) ∈ A+loc(P )), then its compensator (with respect to the probability P ) is called the entropy-Hellinger process of N , and is denoted by hE(N, P ).
(ii) Let Q ∈ Pa with density Z = E (N ). Then, we define the entropy-Hellinger process of Q with respect to P by
hEt (Q, P ) := hEt (Z, P ) := hEt (N, P ), 0 ≤ t ≤ T.
The expression of entropy-Hellinger process for a positive σ-martingale and its jump will be shown explicitly in the next lemma.
Lemma 2.4: Consider a positive σ-martingale density Z = E (N ) satisfying N = λ · Sc+ W ? (µ − ν),
Then, from (2.26), we derive
1 + ∆Nt= eλTt∆St
γt I{∆St6=0}+ 1
γtI{∆St=0}. After simplification, this leads to
P [(1 + ∆N ) log (1 + ∆N ) − ∆N ] =h
Then, by plugging this equation in (2.30) and compensating, we obtain
Hence, after simplification, (2.27) follows.
By taking the jumps in both sides of (2.27), we get
∆hE(Z, P ) = 1 from the fact that Z is a σ-martingale density (see (2.21)). This completes the proof.
Theorem 2.3: Suppose that Zloce (S) 6= ∅ and that (2.19) holds. If eZ ∈ Zloce (S) is the MEH σ-martingale density, then, there exists eH ∈ L(S) such that
log( eZ) = eH · S + hE( eZ, P ). (2.32)
Proof. Notice that the assumptions of Theorem 3.3 in [17] are fulfilled. Hence, a direct application of this theorem implies
Z = E ( ee N ), N := ee β · Sc+ fW ? (µ − ν),
Wft(x) := (eγt)−1
eβetTx− 1
, eγt:= 1 − at+R eβetTxν({t}, dx).
Thus,
log( eZ) = eN −12h eN i +P[log(1 + ∆N ) − ∆ ee N ]
= eβ · Sc+ fW ? (µ − ν) −12βeTc eβ · A +P[log(eβT xe
eγ ) −eβT xe
eγ + 1]I{∆S6=0}
+P[log(1
eγ) − 1
eγ + 1]I{∆S=0}
= eβ · Sc+ fW ? (µ − ν) −12βeTc eβ · A +P[−eγ log(eγ)+eγ−1
eγ ] +eγ eβTx−eβT xe +1
γe ? µ.
Remark that 1
eγ(eγ eβTx − eβeTx+ 1) ? µ = eβT(x − h(x)) ? µ + 1
eγ(eγ eβTh(x) − eβeTx+ 1) ? (µ − ν)+
+eγ−1(eγ eβTh(x) − eβeTx+ 1) ? ν,
since the functional eγ−1(eγ eβTh(x) − eβeTx+ 1) is (µ − ν)-integrable which is due to the (µ − ν)−integrability ofγe−1(eβeTx− 1) = W (x) and the boundedness of h(x). Therefore, we get
log( eZ) = eβ · Sc+ eβTh(x) ? (µ − ν) + eβT(x − h(x)) ? µ+
+eγ−1(eγ eβTh(x) − eβeTx+ 1) ? ν − 12βeTc eβ · A +P
eγ−1(−eγ log(eγ) +eγ − 1).
Equivalently, we deduce that
log( eZ) = eβ ·S +1
2βeTc eβ ·A+βeTxeβeTx− eβeTx+ 1
eγ ?ν +X
eγ−1(−eγ log(eγ)+eγ −1), (2.33) due to
β · S = ee β · Sc+ eβTb · A + eβTh(x) ? (µ − ν) + eβT(x − h(x)) ? µ.
Therefore, a direct application of Lemma 2.4 for λ = eβ, (2.32) follows
imme-diately. This ends the proof of the theorem.
Next, we will give the variation of this entropy-Hellinger concept towards the change of probability measures.
Definition: (i) Let Q be a probability measure and Y be a Q-local martingale such that 1 + ∆Y ≥ 0. Then, if the RCLL nondecreasing process
VE(Y ) = 1
2hYci +X
[(1 + ∆Y ) log(1 + ∆Y ) − ∆Y ] , (2.34)
is Q-locally integrable (i.e. VE(Y ) ∈ A+loc(Q)), then its Q-compensator is called the entropy-Hellinger process of Y (or equivalently of E (Y )) with respect to Q, and is denoted by hE(Y, Q) (respectively hE(E (Y ), Q)).
(ii) Let N ∈ M0, loc(P ) such that 1 + ∆N > 0 and Y is a semimartingale such that Y E (N ) is a P -local martingale and 1 + ∆Y ≥ 0. Then, if the process
1
2hYci +X
(1 + ∆N )h
(1 + ∆Y ) log(1 + ∆Y ) − ∆Y + 1i
, (2.35)
is P -locally integrable, then its P -compensator is called the entropy-Hellinger process of E (Y ) with respect to E (N ), and is denoted by hE(E (Y ), E (N )).
Remark that the first definition above in (i) is a natural extension in proba-bility as well as in mathematical finance areas, due to the popular and useful technique of change of probability measures. The second definition in (ii), which we will use throughout the thesis, extends (i) to the case when the uniform integrability of the nonnegative local martingale E (N ) may not hold.
The relationship between the two definitions is obvious. Indeed, let (Tn)n≥1 be a sequence of of stopping times that increases stationarily to T such that E(N )Tn is a true martingale. Then, by putting Qn:= ETn(N ) · P , we obtain
hEt∧Tn(E (Y ), E (N )) = hEt (E (YTn), Qn), 0 ≤ t ≤ T.
What we actually need in current thesis is the MEH local martingale den-sity under change of probability measure. In the remaining part of this section, we focus on describing the MEH σ-martingale density when we change prob-ability. This case can be derived easily from the more general case where one works with respect to a positive local martingale density, Z, that may not be uniformly integrable. First, we generalize the characterization of the MEH σ-martingale density for the case when S may not be bounded nor quasi-left continuous. For the case of bounded and quasi-left continuous S, a more elaborate result is given in [16].
In what follows, we denote by Z a positive local martingale given by
Z := E (N ), N := β ·Sc+W ?(µ−ν)+g ?µ+N , Wt(x) := ft(x)+ fbt 1 − at
I{at<1}, (2.36) where β, f, g, N are the Jacod components of N. Here, we define:
Zloce (S, Z) := {Z | Z > 0, ZZ ∈ Zloce (S)}, (2.37)
where Zloce (S) is given by (2.17).
Theorem 2.4: Consider Z defined in (2.36) and suppose that
Zloce (S, Z) 6= ∅, and Z
{|x|>1}
eλTx(1 + f (x))F (dx) < +∞, ∀ λ ∈ IRd.
Then, the minimization problem
min
Z∈Zloce (S,Z)
hE(Z, Z), (2.38)
admits a solution eZ = E ( eN ) given by
N = ee β · Sc,Z+ fW ? (µ − νZ), fWt(x) = eβeTtx− 1
1 − aZt +R eβeTtyνZ({t}, dy),
where eβ is the root of
0 = bZ+ cλ + Z
(eλTxx − h(x))FZ(dx). (2.39)
Here Sc,Z, bZ, aZ, νZ and FZ are given by
Sc,Z := Sc−cβ·A, bZ := b+cβ−
Z
f (x)h(x)F (dx), aZt := νZ({t}, IRd\{0})
and νZ(dt, dx) := FtZ(dx)dAt, FtZ(dx) := (1 + ft(x))Ft(dx).
Proof. Consider a sequence of stopping times, (Tn)n≥1, stationarily increasing to T (i.e., P (Tn = T ) → 1 as n → ∞) such that ZTn is a true martingale, for a fixed but arbitrary n we denote Q := ZTn · P . Remark that all the equations in the theorem are robust with stopping. Then due to Lemma 2.5, it is enough to prove that the theorem is valid on [0, Tn]. Then, we obtain that νQ(dt, dx) = (1 + ft(x))I{t≤Tn}ν(dt, dx), Zloce (S, Q) 6= ∅ and that R
{|x|>1}eλTxFQ(dx) < +∞ for λ ∈ IRd.
Therefore, the assumptions of Theorem 3.3 in [17] are fulfilled. Hence a direct application of this theorem for STn and under the measure Q = ZTn· P , we deduce that the problem defined in (2.38) admits a solution eZQ= E
NeQ , where eNQ is given, on [0, Tn], by
NeQ = eβ · Sc,Q+ fW ? (µ − νQ), fWt(x) = eβeTtx− 1
1 − aQt +R eβetTyνQ({t}, dy). Herein Sc,Q is the continuous local martingale part of S under Q and νQ is the Q-compensator measure of µ, and aQt = νQ {t}, IRd\ {0}. Moreover, βe is given by the equation
0 = bQ+ cλ + Z
eλTxx − h(x)
FQ(dx)
=h
bZ+ cλ +R (eλTxx − h(x))FZ(dx)i I[0,Tn].
(2.40)
It is then clear that eNQcoincides with eN of the theorem and that the equation (2.40) is exactly the equation (2.39) of the theorem. This ends the proof of theorem.
Theorem 2.5: Let Z be a positive local martingale and eZ ∈ Zloce (S, Z). If the assumptions of Theorem 2.4 are fulfilled, and eZ is the MEH local martingale density with respect to Z, then
log( eZ) = eβ · S + hE( eZ, Z). (2.41)
Proof. The proof of this theorem follows the same arguments as in the proofs of Theorems 2.3 and 2.4.