The q -Optimal Absolutely Continuous Mar- Mar-tingale Measure

In Section 4.6, we see that under condition C_q⁻ an equivalent martingale measure does not exists if C_q⁺ is not satisfied. In this section, we modify and relaxC_q again. This is done in a slightly different way as in Section 4.2 leading to theq-optimal absolutely continuous martingale measure, qAMM.

4.3.1 Relaxing Condition C_q

To guarantee that our new candidate is absolutely continuous, we need to guarantee that the new Girsanov parameterh−1, in ¯E(f, h−1),driving the jumps is not smaller than−1 :

Assumption 4.3.1. (C_q^A, C_q^E)

C^A_q :There exists a ξ_q ∈R^N such that

egq(x) :=|(q−1)ξ_q^′(e^x−1) + 1|^q−11 1_{(q−1)ξ′

q(e^x−1)+1≥0} (4.24) satisfies

σσ^′ξ_q+ Z

R^N\{0}

((e^x−1)eg_q(x)−x1_kxk≤1)ν(dx) =β (4.25) and

Z

RN\{0}|eg^q

q(x)−1−q(eg

q(x)−1)|ν(dx)<∞. (4.26) C^A_q^p: C_q^Ap holds if C_q^A is satisfied but (4.26) is replaced by

Z

R^N\{0}

|(q−1)ξ_q^′(e^x−1) + 1|^q−1q −1−q(eg_q(x)−1)

ν(dx)<∞. (4.27) C^E_q : C_q^E is verified if eg^q_q⁻¹(x) = (q−1)ξ_q^′(e^x−1) + 1>0, ν-a.s.

Note, since there is a sphere around zero such that (q−1)ξ_q^′(e^x−1)+1>0 for values x in this sphere, the above integral is always well defined for x around zero. Our conditions are connected as follows:

Lemma 4.3.1. We have C_q^Ap ⇒C_q^A and

C_q^A∧C_q^E ⇔C_q^Ap∧C_q^E ⇔C_q⁻∧C_q⁺⇔Cq.

Note, the aboveC_q⁺/C_q^E is only seen in combination withC_q⁻orC_q^Aresp.

C_q^Ap. If C_q⁺ does not hold, (q−1)ξ^′_q(e^x−1) + 1 and (q−1)θ_q^′(e^x−1) + 1 might be different. IfC_q⁻ andC_q^Ahold,C_q⁺and C_q^E are equivalent. We just mentionC_q⁺ in the sequel.

Theorem 4.3.2. ¹ Suppose that C_q^A holds. Then Zˇ^(q)= ¯E(ξ_q^′σ,eg_q−1)

is the density process of theq-optimal absolutely continuous martingale mea-sure, qAMM. If in addition C_q⁰ holds, then the qAMM is equal to the qSMM and equal to the qEMM under C_q⁺.

1At this point I would like to thank Walter Schachermayer for the discussion about different types of utility problems. In particular, the reconsideration of the effect of an incorporation of consumption processes initiated the idea to extend the verification pro-cedure for signed measures to the absolutely continuous case.

The proof of Theorem 4.3.2 will be again based on a verification pro-cedure by a superhedging argument and duality. Superhedging means that we also allow for consumption, see Definition 1.2.7 and the discussion at the beginning of the chapter. If the utility function is decreasing it is better to consume. The proof will be given in Section 4.3.2.

We can extend Proposition 4.2.3 and see that C_q^A is a quite weak as-sumption. Intuitively, C_q^A means that for the existence of the qAMM we have to guarantee integrability of the density of the qAMM on the positive axis only. Otherwise the density is zero, but roughly on this set the signed measure might be negative. Integrability has to be guaranteed in the signed case. Hence,C_q⁻ has to include integrability also on the set ofx, whereC_q⁺ fails:

Proposition 4.3.3. Suppose N = 1, P is not a martingale measure, and (σ >0∨ν((−∞,0))6= 06=ν((0,∞))). Then,

C_q ⇒ C_q⁻ ⇔ C_q^Ap ⇔ Z

x≥1

e^pxν(dx)<∞

⇒ C_q^A ⇔ Z

x≥1

e^pxν(dx)<∞

or Z

R\{0}

((e^x−1)−x1_|x|≤1)ν(dx) + (b+1

2σ²)>0

!

where p=q/(q−1) is the conjugate exponent toq.

Before we give an example and state a proof, we present an interesting corollary of Theorem 4.2.2 and 4.3.2:

Corollary 4.3.4. Suppose N = 1, and (σ > 0 ∨ ν((−∞,0)) 6= 0 6= ν((0,∞))). If, for some q > 1, C_q⁻ holds and the qSMM is not equivalent, then the qEMM does not exist.

Proof. By Proposition 4.3.3, C_q⁻ and C_q^A hold. We suppose that the qEMM exists. Then the qAMM coincides with the qEMM and consequently egq(x) > 0 ν-almost surely. This implies that the qAMM and the qSMM both coincide with the qEMM and therefore the qSMM is equivalent.

We extend Example 4.2.4:

Example 4.3.5 (Example 4.2.4 (continued)). SupposeN = 1 and (σ >0∨ν((−∞,0))6= 06=ν((0,∞))).

(i) Set:

Z

R\{0}

((e^x−1)−x1_|x|≤1)ν(dx) + (b+1

2σ²)<0.

If ν(dx) behaves (up to a slowly varying function) as e^−γ+xdx forx → ∞, thenC_q^A holds for p:=q/(q−1) < γ₊ and fails for p > γ₊.

(ii) We next assume that Z

R\{0}

((e^x−1)−x1_|x|≤1)ν(dx) + (b+1

2σ²)>0,

which is equivalent to the existence of the minimal entropy martingale mea-sure, if R

x≥1e^θexν(dx) =∞ see Fujiwara and Miyahara [54] and Theo-rem 4.4.1 below. Then,

• C_q fails for all q >1.

• C_q⁻ holds for q such thatp < γ₊ and fails forq such that p > γ₊.

• C_q^Aholds for all q >1.

In fact, in this situation ξ_q from condition C_q^A is negative. Therefore, the qEMM does not exist, and the qAMM and the qSMM are different in the range ofq, whereC_q⁻ is satisfied.

We need some preparation to prove Proposition 4.3.3. For N = 1, we modify the function Φ a bit: with

eq^ξ

q(x) := 1_{(q−1)ξ(e^x_−1)+1≥0}(e^x−1)|(q−1)ξ(e^x−1) + 1|^q−11 , we define

Φ(q, ξ) = −β+σ²ξ+ Z

R\{0}

(eq^ξ_q(x)−x1_|x|≤1)ν(dx) which are, for fixedq >1, again given on the domain

Dom^(a)_q =

ξ : Z

x≥1

(e^x−1)((q−1)ξ(e^x−1) + 1)

1 q−1

+ ν(dx)<∞

. With this modified definition at hand, we are able to extend Lemma 3.2.6.1 and Lemma 4.2.5. Ifσ = 0, we need to assume that positive and negative jumps appear to guarantee strict monotonicity. The only case, we exclude in an arbitrage-free market is the pathological case that we have no diffusion component, only jumps in one direction, and a drift term with the reverse sign.

Lemma 4.3.6. (i) Let N = 1. Suppose σ 6= 0 or ν((−∞,0)) 6= 0 6= ν((0,∞)). Then the mapping Φ(q,·) is continuous and strictly increasing on Dom^(a)q .

(ii) Let N = 1. Then we have for all q > 1 that 0 ∈ Dom^(a)q ∩Dom_e and Φ(q,0) = Φ(q,0) = Φ_e(0).

(iii) LetN = 1 and q= _p−^p₁. We have (−∞,0]⊂Dom^(a). Suppose further Z

x≥1

e^pxν(dx)<∞. Then,Dom^(a)q =R.

(iv) Suppose conditionC_q^A is satisfied and there is anˆǫ >0such that σσ^′ ≥ IN×Nˆǫ then kξqk ≤R, where R:= ˆǫ⁻¹kR

((e^x−1)−x1_kxk≤1)ν(dx)−βk. Proof of Lemma 4.3.6. (i)-(iv) are in analogy to Lemma 4.2.5:

(i) We can split up the integral in two parts (−) and (+) on the sets: {x : (q−1)ξ(e^x−1)+1} ≤0 and{x: (q−1)ξ(e^x−1)+1}>0.The latter integral is strictly increasing in ξ. We thus show that either (+) is different from zero or σ is positive and (−) is increasing to obtain that Φ(q,·) is strictly increasing. There is only one critical point if (q −1)ξ(e^x −1) + 1 = 0.

In this case if e^x−1 > 0, then ξ < 0 and vice versa. So if e^x −1 > 0, (e^x−1)|(e^x−1)((q−1)ξ(e^x−1) + 1|^q−11 increases ifξincreases. Ife^x−1<0, (e^x−1)|(e^x−1)((q−1)ξ(e^x−1) + 1|^q−11 decreases ifξ decreases. If σ = 0, there are ν-a.s. positive and negative jumps. Hence, for ξ ≥ 0 we have ν({x : (q−1)ξ(e^x −1) + 1 > 0}) ≥ ν((0,∞)) > 0 and for ξ ≤ 0 we get ν({x : (q −1)ξ(e^x −1) + 1 > 0}) ≥ ν((−∞,0)) > 0, thus (+) does not vanish, is strictly increasing, and (−) is increasing, so together Φ(q,·) is strictly increasing. If σ 6= 0 and (q−1)ξ(e^x −1) + 1 = 0, σ²ξ is strictly increasing. Hence, Φ(q,·) is strictly increasing for q > 1. The remaining assertions again follow directly from the monotone convergence theorem.

(ii) The assertion is trivial.

(iii) For ξ ≤ 0, eq^ξ_q(x) is bounded from above, thus (−∞,0] ⊂ Dom^(a) . Since _q−¹₁ =p−1, there is a constantK_p >0 such that forx >1 andξ∈R

(e^x−1)|(q−1)ξ(e^x−1) + 1|^q−11 ≤ K_p

p−1|ξ|(e^x−1)^p+K_p(e^x−1), which isν-integrable over (1,∞) as R

x≥1e^pxν(dx)<∞. (iv) By a direct calculation, as in proof of Lemma 3.2.5.1,

ξ^′Φ(q, ξ) (4.28)

= Z

R^N\{0}

ξ^′(e^x−1)|(q−1)ξ^′(e^x−1) + 1|^q−11 1_{(q−1)ξ^′_(e^x_−1)+1≥0}

−ξx1_kxk≤1

ν(dx)−ξ^′β+ξ^′σσ^′ξ (4.29)

≥ ξ^′( Z

R^N\{0}

((e^x−1)−x1_kxk≤1)ν(dx)−β) +ξ^′σσ^′ξ

≥ −kξkk Z

R^N\{0}

((e^x−1)−x1_kxk≤1)ν(dx)−βk+kσξk², (4.30)

as (q−1)ξ^′(e^x−1) + 1≥1 forξ^′(e^x−1)>0,

0≤1_{(q−1)ξ^′_(e^x_−1)+1≥0}|(q−1)ξ^′(e^x−1) + 1|^q−11 ≤1 forξ^′(e^x−1)<0 and 1_{(q−1)ξ′(e^x−1)+1≥0} = 1,and

ξ^′(e^x−1)≤0 =ξ^′(e^x−1)1_{(q−1)ξ^′_(e^x_−1)+1≥0}|(q−1)ξ^′(e^x−1) + 1|^q−11 forξ^′(e^x−1)<0 and 1_{(q−1)ξ′(e^x−1)+1≥0} = 0.

Proof of Proposition 4.3.3. We start considering the first line of assertions of Proposition 4.3.3. By Proposition 4.2.3, it remains to show that C_q^Ap ⇔ (4.11).

⇒: SupposeC_q^Ap holds for some fixed q. Then, by definition R

x>1

|(q−1)ξq(e^x−1) + 1|^q−1q −1

−q(((q−1)ξ_q(e^x−1) + 1)

1 q−1

+ −1)

ν(dx)<∞.

Note that p=q/(q−1). Hence, for large x the term e^px dominates in the above integrand and we conclude that (4.11) holds.

⇐: Suppose that condition (4.11) holds for some fixed q= _p−1^p >1. Recall that Dom^(a)q = R thanks to Lemma 4.3.6, (iii). Now, in view of Lemma 4.3.6, (i), it suffices to show that

ξlim↓−∞Φ(q, ξ) =−∞ and lim

ξ↑∞Φ(q, ξ) =∞. (4.31) By assumption σ 6= 0 or ν((−∞,0)) 6= 0 6= ν((0,∞)). We first treat the case σ 6= 0. Dividing (4.30) by ξ implies (4.31) immediately. If σ = 0, then ν((−∞,0)) 6= 0 6= ν((0,∞)). As shown in the last proof, for every given x, (e^x −1)|(q−1)ξ^′(e^x −1) + 1|^q−11 1_{(q−1)ξ′(e^x−1)+1} is monotone in ξ, on the setν((−∞,0)) 6= 0, it converges to−∞ forξ → −∞, and on the set ν((0,∞)) 6= 0, it converges to +∞ forξ →+∞. Hence, the monotone convergence theorem yields (4.31) in the latter case. Obviously, condition (4.11) ensures that (4.27) holds.

It remains to prove the second equivalence. (4.6) holds if and only if, Φ(0, q) < 0. If (4.6) does not hold, then a root ξ_q exists in (−∞,0] ⊂ Dom^(a) as Φq is strictly increasing and continuous and lim_ξ↓−∞Φ(q, ξ) =

−∞. Hence eq^ξ_q(x) is bounded from above andC_q^Afollows (see also Lemma 4.3.6, (iii)). If (4.6) is satisfied, we have R

x≥1e^pxν(dx) < ∞. In view of (4.31), we just need to prove sufficient integrability. The last step of “⇐” follows from Lemma 4.3.6, (iii), implying Dom^(a) =R.

⇒: SupposeC_q^Aholds for some fixed q. As before, (4.6) holds if and only if

ξ_q > 0.If ξ_q >0, then C_q^A and C_q^Ap are equivalent as ((q−1)ξ_q^′(e^x−1) + 1)

1 q−1

+ =|(q−1)ξ_q^′(e^x−1)+1|^q−11 for largex. Hence, (4.11) holds. Otherwise, (4.6) does not hold, i.e.R

R\{0}((e^x−1)−x1_|x|≤1)ν(dx) + (b+¹₂σ²)>0.

We continue Example 4.2.6 focusing on the case q= 2:

Example 4.3.7. Setb6= 0. We need to solve:

σ²ξ_q+γ(e^a−1)((e^a−1)ξ_q+ 1)1_{(e^a_{−1)ξq+1≥0}} =−µ

We start with Case 2: setσ = 0 and q = 2,i.e. the variance case, then we have one unique solution of (4.16):

ξ_q=

−b γ(e^a−1) −1

e^a−1 and eg₂(a) =ξq(e^a−1) + 1 = −b γ(e^a−1),

if and only if ξq(e^a−1) + 1 = _γ(e^−ba−1) > 0 which holds true if and only if sgn(a) =−sgn(b).

• C_q⁻ always holds and a unique solution θ2 exists. In the case b 6= 0, C_q⁺ and C_q^A hold if and only if sgn(a) = −sgn(b). Ifsgn(a) = sgn(b), we have arbitrage although the qSMM still exists by C_q⁻. C_q^A does not hold. Proposition 4.3.3 cannot be applied as we have arbitrage, in particular σ = 0 and there do not exist positive and negative jumps.

• Ifb= 0, we have arbitrage, butC_q⁻andC_q^Astill hold, C_q⁺fails. ξ_q=θ_q. We have Me = ∅. Note that this case is not covered in Proposition 4.3.3.

We continue with Case 4: setσ 6= 0 andb=−¹₂σ²+µ, and q= 2. Hence, ξ_q= −(µ+γ(e^a−1))

σ²+γ(e^a−1)² if

eg₂(a) = (e^a−1)ξ_q+ 1 = −µ(e^a−1) +σ²

σ²+γ(e^a−1)² >0⇔σ² > µ(e^a−1).

Thus, ifσ²≤µ(e^a−1),then ξ_q=−µ/σ² and−(e^a−1)µ/σ²+ 1≤0:

• C_q^A always holds and a unique solution ξ₂ exists. We do not have arbitrage in all cases as σ² >0.

• C_q⁺ holds if and only if σ²> µ(e^a−1). θ_q=ξ_q.

• If σ² ≤ µ(e^a−1), then θ_q 6= ξ_q. −ξ_q = µ/σ², which is equal to the optimal exponential portfolio in a Brownian setting.

4.3.2 Verification of the q-Optimal Absolutely Continuous Solution

This section is devoted to the proof of Theorem 4.3.2. As a first step we derive a verification theorem for the q-optimal absolutely continuous mar-tingale measure. Again this theorem does not rely on the L´evy setting, but holds for general semimartingale models.

Theorem 4.3.8. Suppose Z¯ ∈ D^aq, q = _p−^p₁ and, for some x < p, the˜ contingent claim

X^(p,˜x)( ¯Z) :=p− sgn( ¯Z_T)|Z¯_T|^p−11





p−x˜ E(|Z¯|

p p−1

T )



 (4.32)

is superreplicable with initial wealth x˜ and superhedging strategy (ϑ, C) in A^p×C^p or A^(p,c) in the sense of Definition 1.2.7, i.e. X^(p,˜x)( ¯Z) is super-replicable with respect to Wi(˜x) for i=C or i=AT.

Then Z¯ is the density process of the q-optimal absolutely continuous mar-tingale measure.

Before we proceed with a proof we like to mention a property similarly applied in the proof of Lemma 1.2.8, Lemma 3.3.2, and Remark 4.2.8:

Remark 4.3.9. Suppose q = _p−1^p , ϑ ∈ A^p, and x ∈ R, then we know from Remark 4.2.8 that

E[Z_T(x+ Z _T

0

ϑ^′_udS_u)] =E(Z₀x) =x.

AsZ ∈ Da^q is non-negative and so E(Z_TC_T)≥0,we have E[ZT(x+

Z T 0

ϑ^′_udSu−CT)] =x−E(ZTCT)≤x.

The last assertion is obvious if (ϑ, C)∈ A^(p,c).

Proof of Theorem 4.3.8. Letq =p/(p−1). We consider the following max-imization problems with utility functionu_p(x) =−|1−^x_p|^p:

Max₁: X⁽¹⁾ := arg max{E(u_p(X))|X ∈L^p s.t. E( ¯Z_TX)≤x˜}

Max₂: X⁽²⁾ := arg max{E(u_p(X))|X ∈L^p s.t. ∀Z ∈ Da^q: E(Z_TX)≤x˜} Max₃: X⁽³⁾ := arg max{E(u_p(X))|X ∈ G_C^p(˜x)}

Max4: X⁽⁴⁾ := arg max{E(up(X))|X ∈ G_AT^p (˜x)}

where G_C^p(˜x) =

X∈L^p(FT) : ∃ϑ∈ A^p, C ∈C^p s.t. X= ˜x+ Z T

0

ϑ^′_udS_u−C_T

and GAT^p (˜x) =

X∈L^p(FT) : ∃(ϑ, C) ∈ A^(p,c) s.t. X= ˜x+ Z _T

0

ϑ^′_udS_u−C_T

. From Lemma 1.2.8, we know that Max₂ dominates Max₄,which further dominates Max3,i.e.V(˜x)_C,dyn≤ V(˜x)_AT,dyn ≤ V(˜x)a,stat.If the superhedg-ing strategy is inA^p×C^p, thanks to Remark 4.3.9 we derive that

E(u_p(X⁽¹⁾)) ≥ E(u_p(X⁽²⁾))≥E(u_p(X⁽⁴⁾))

≥E(u_p(X⁽³⁾))≥E(u_p(X^(p,˜x)( ¯Z))).

If the superhedging strategy is not inA^p×C^p,but in A^(p,c),we still have E(u_p(X⁽¹⁾))≥E(u_p(X⁽²⁾))≥E(u_p(X⁽⁴⁾))≥E(u_p(X^(p,˜x)( ¯Z))).

In both cases inequalities turn into equalities and the remaining assertions follow exactly as in the proof of Theorem 4.2.7.

We next suppose thatC_q^Aholds for some q >1. We can therefore define Z¯= ¯E(ξ_q^′σ,eg_q−1).

Z¯ is q-integrable due to (4.26), i.e. E|Z¯|^q < ∞. Moreover, applying the Dol´eans-Dade formula and integration by parts, one easily verifies that ¯ZS is a local martingale thanks to (4.25).

In order to prove Theorem 4.2.2, it remains to show that X^(p,˜x)( ¯Z) is superreplicable within the class A^(p,c) or A^p ×C^p for q = p/(p −1) and some ˜x < p. We will again derive a superreplicating strategy constructively.

This closely follows the approach in Section 4.2, but we need to balance out terms that cannot be included into the strategy ϑ. Roughly speaking, we plug them into a new consumption component. The strategy will be inA^(p,c) under C_q^A. This is perfectly fine for deriving the qAMM. For our portfolio problem, we also like to know when the strategy is inA^p×C^p. A sufficient condition is given byC_q^Ap.

As before at first, we have to find an appropriate wealth process that reaches the claimX^(p,˜x)( ¯Z).We fix some ˜x < p and get:

Lemma 4.3.10. We have X^(p,˜x)( ¯Z) = ¯Y_T, where Y¯_t := p−(p−x) ¯˜ Et((q−1)ξ_q^′σ,eg^q−1

q −1) (4.33)

×e^{(q−1)ξ′qR0tS−1u−dAu}e^t

R

RN\{0}((q−1)ξ_q^′(e^x−1)+1)⁻ν(dx)

=: p−(p−x) ¯˜ M_tA¯_t=:p−(p−x) ¯˜ M_tA¯⁽¹⁾_t G_t.

Proof. We easily see thatX^(p,˜x)( ¯Z) = ˜Y_T,where

by applying Lemma 4.2.1 twice. With eg

q(x) = ((q−1)ξ_q^′(e^x−1) + 1)^1/(q₊ ⁻¹⁾

Hence, applying the Dol´eans-Dade formula to ¯M and integration by parts, we get

Thus, is positive andC^(p,C) is therefore increasing. 1_τ≤t is furthermore an RCLL process. Thanks to Lemma 4.3.10: P-supermartingale for every Z ∈ D^qa.

Proof. Set without loss of generality ˜x = 0. We fix Z ∈ Da^q and define dQ = Z_TdP. Then, R_t

0(ϑ^(p,C)u )^′dS_u is a local martingale under Q. We denote by τ_n a localizing sequence of stopping times. Then, the stopped processes ¯Y_t∧τn are supermartingales under Q. Due to Bayes formula, for s≤t, we thus have

E(Zt∧τnY¯_t∧τn|Fs) =Zs∧τnE^Q( ¯Y_t∧τn|Fs)≤Zs∧τnY¯_s∧τn →ZsY¯_s, n→ ∞.

Finally, E(Z_t∧τnY¯_t∧τn|Fs)→E(Z_tY¯_t|Fs) by dominated convergence, since

|Z_t∧τnY¯_t∧τn| ≤ sup

0≤u≤T|Z_u| sup

0≤u≤T|Y¯_u| and

E( sup

0≤u≤T|Z_u| sup

0≤u≤T|Y¯_u|)≤E( sup

0≤u≤T|Z_u|^q)^1/qE( sup

0≤u≤T|Y¯_u|^p)^1/p. The first factor is finite, since Z is a q-integrable martingale. For the sec-ond one, we observe that ¯A_t is deterministic and bounded and ¯M_t is a p-integrable martingale due to (4.26), (following the same argument as in Lemma 4.2.11 or also see the proof of Lemma 1.2.8). Therefore, the second factor is finite as well.

Thus, we have proved the following lemma establishing Theorem 4.3.2:

Lemma 4.3.12. Suppose C_q^A holds. Define ϑ^(p,C)_t = −p−x˜

p−1E¯t−((q−1)ξ_q^′σ,((q−1)ξ_q^′(e^·−1))₊−1)

×e^{t(q−1)ξ′q}

−β+R

RN\{0}(e^x−1−x1_kxk≤1)ν(dx)

S⁻_t−¹ξq

and

C_t^(p,C) = 1_τ≤t(p−x) ¯˜ M_τ−A¯_τ((q−1)ξ_q^′(e^{∆ ˇXτ} −1) + 1)₋ (4.34) where τ = inf{t : ((q −1)ξ^′_q(e^{∆ ˇXt} −1) + 1)₋ > 0}. Then for x < p˜ and Z¯ = ¯E(ξ^′_qσ,eg_q−1) the contingent claim

X^(p,˜x)( ¯Z) :=p−sgn( ¯Z_T)|Z¯_T|^p−11 p−x˜ E(|Z¯T|^p−1p )

!

, q= p p−1

is superreplicable with initial wealth x˜and the predictable strategy ϑ^(p,C) and increasing RCLL process C^(p,C) with (ϑ^(p,C), C^(p,C))∈ A^(p,c).

Remark 4.3.13. (i) For t > τ, ϑ^(p,C)_t = 0 and C_t^(p,C) changes at most once at τ. Once the investment in the stocks has overshot p, we consume the rest and stop all investments into the stocks. Until the end of the period, expected utility then keeps constant atp.

(ii) If ˜x≥p,we just consume ˜x−p,the investment into the stocks is zero. As the scaling component of the dual problemYp(˜x) is zero, the dual measure can be chosen arbitrarily. Hence the optimization problem does not help to find the qAMM, if ˜x≥p.

Finally, we like to prove that the superhedging strategy is also inA^p×C^p. As

Y¯_t = p−(p−x) ¯˜ M_tA¯⁽¹⁾_t G_t=p−(p−x) ¯˜ M_tA¯_t

and ¯M_t is p-integrable, it remains to show that ϑ ∈ A^p, since then C_T is automatically inL^p:

Lemma 4.3.14. Suppose C_q^Ap holds. Then ϑ^(p,C) constructed in Lemma 4.3.12 belongs to the classA^p.

Proof. Recall that by the definition of eg_q and the proof of Lemma 4.2.11:

E|E¯t((q−1)ξ_q^′σ,(((q−1)ξ_q^′(e^·−1) + 1)₊)−1)|^p thanks to Taylor’s formula for small jumps and due to (4.3) for large jumps.

Hence, ¯Et((q−1)ξ^′_qσ,(q−1)ξ_q^′(e^·−1)) and ¯Et((q−1)ξ^′_qσ,(((q−1)ξ^′_q(e^·−1) + 1)+)−1) arep-integrable martingales underC_q^Ap thanks to (4.27) (also see Lemma 4.2.1).

By Doob’s inequality, we immediately observe thatϑ^(p,C)satisfies (4.19).

To prove (4.18), we first notice that, for some constant K_q>0, E

applying the Dol´eans-Dade formula for the last identity. By the Burkholder-Davis-Gundy inequality and since ¯E((q − 1)ξ^′_qσ,(q − 1)ξ^′_q(e^· − 1)) is p-integrable, the term on the right hand side is finite.

At the end of this section, it should be noted that the above (su-per)replicating strategies can exist, although there exists no equivalent mar-tingale measure. That means a (super)replicating strategy exists and we have arbitrage, see Example 4.2.6/4.3.7 and 4.4.7 and Section 4.5.

4.4 Convergence to the Minimal Entropy

Im Dokument Portfolio Optimization and Optimal Martingale Measures in Markets with Jumps (Seite 140-153)