Convergence of the Terminal Values and the Value Functions

This section is devoted to the convergence of the terminal values in a contin-uous semimartingale setting. We slightly extend the results inKohlmann and Niethammer[76] to the p-th problem.

2.2.1 Some Facts about Martingale Measures

By Assumption 2.1.1, S is continuous and therefore satisfies the Struc-ture Condition and admits the decomposition S = S₀+M +R_·

0dhMiuˆλ_u, whereM is a continuous local martingale and ˆλis a predictableR^N−valued process, as defined in Schweizer [118]. If the Dol´eans-Dade exponential Zˆ =E(−R ˆλ^′dM) is a martingale, the minimal martingale measure is given by

dQˆ = ˆZ_TdP.

Further, the minimal entropy martingale measure can be described by a backward stochastic differential equation (BSDE hereafter). This works

as follows. From Theorem 1 in Schweizer [118], we know that every equivalent martingale measure can be represented as ^dQ_dP = Z_Q, Z_Q = ET(M^Q), M^Q ∈ M_loc. Further, using the notation EtT(M^Q) = ^ET(MQ)

E^t(M^Q), Mania et al. [90] prove the following characterization of the minimal entropy martingale measure (Theorem 3.1.):

Theorem 2.2.1. Let all local (F, P)-martingales be continuous and P_f,e(P)6=∅.Then the value process V˜_t, given by

V˜_t= ess inf

Q∈P_f,e(P)E_Q(logEtT(M^Q)|Ft),

is a special semimartingale withV˜_t=m_t+a_t+ ˜V₀, where m∈M²_loc,(M²_(loc) denotes the space of all (local) martingales M˜ with ksup_tM˜_t²kL¹ <∞) and ais a locally bounded variation predictable process. Therefore the Galtchouk-Kunita-Watanabe (G-K-W) decomposition exists: m_t = R_t

0φ^′_sdM_s + ˜m_t, hm, M˜ i= 0. Furthermore, V˜_t is the solution of the following BSDE:

Yˇ_t= ˇY₀−ess inf

Q∈Pf,e(P)

1

2hM^Qit+hM^Q,Lˇit

+ ˇL_t, Yˇ_T = 0 (2.5) Moreover, Q_min is the minimal entropy martingale measure if and only if

dQmin

dP =ET(M^Qmin), M_t^Qmin =− Z t

0

λˆ^′_sdMs−m˜t. (2.6) Suppose, in addition, the minimal martingale measure exists, i.e. Zˆ is a martingale, and satisfies the Log-Reverse H¨older inequality, for a definition see e.g. Grandits and Rheinl¨ander [60]. Then, V˜ uniquely solves the above BSDE (2.5) and is bounded.

A similar characterization is proven for theq-optimal martingale measure inMania et al. [89]:

Theorem 2.2.2. If M^qe 6= ∅ and all P−local martingales are continuous, then the following assertions are equivalent:

1. The martingale measure Q_q isq-optimal 2. Q_q is a martingale measure satisfying

dQq =ET(M^Qq)dP, (2.7)

where

M_t^Qq =− Z t

0

ˆλ^′_sdM_s− 1 q−1

Z t 0

1

V˜s(q)dm˜_s(q). (2.8)

V˜_t(q) = V˜₀(q) + m_t(q) + a_t(q) is almost surely equal to ess inf_Q∈M^q_eE((EtT(M^Q))^q|Ft), it uniquely solves the following BSDE:

Yˇ_t = Yˇ₀−ess inf

M^Q∈M^qe

1

2q(q−1) Z t

0

Yˇ_sdhM^Qis+qhM^Q,Lˇit

+ ˇL_t, t < T, Yˇ_T = 1.

˜

m(q) denotes the orthogonal part of the G-K-W-decomposition of m(q):

mt(q) = Z t

0

φ^′_s(q)dMs+ ˜mt(q) (2.9) IfE(−R_t

0 ˆλ^′_sdM_s)is a martingale, i.e. the minimal martingale measure exists and in addition it satisfies the Reverse H¨older inequality, then the value process V˜(q) above is the unique solution of the above BSDE and there exist positive constants k andK such that almost surely for all t∈[0, T] :

k≤V˜_t(q)≤K

Simple consequences of Theorem 2.2.1 and Theorem 2.2.2 are Corollary 3.4 in Mania et al. [90] and Corollary 3 in Mania et al. [89] (also see Santacroce [114]). They state that the minimal entropy martingale measure, the minimal martingale measure, and the q-optimal martingale measures q > 1 coincide almost surely, if ˆK_T is deterministic. Under the weaker Assumption 2.1.1,Santacroce[114] establishes that:

E

m(q) q−1 −m

T

→0, q ↓1 (2.10)

Furthermore,

Esup

t≤T |Z_t^(q)−Z_t^min| →0, q ↓1, (2.11) in particular, Z_T^{(q) L}→¹ Z_T^min, q ↓ 1, where (Z_t^(q))_t and (Z_t^(min))_t are density processes of the q-optimal martingale measures and the minimal entropy martingale measure, respectively. The last assertion, using a duality ap-proach, is also proven inGrandits and Rheinl¨ander [60]. Assumptions are more or less the same, they obtain convergence in entropy.

2.2.2 Replacing Technical Assumptions

We already know that the dual solution of the optimization problem with utility function−|1−^x_p|^pis the_p−^p₁−optimal martingale measure timesYp(˜x) and the dual of the exponential problem is the minimal entropy martingale measure times Yexp(˜x). So the above considerations already show that the

dual measures converge (apart from Yi, i= p,exp). Before discussing the convergence of the primal problem, we need to ensure that the solution found in Section 1.2.3, actually possesses the correct integrability. We still have to prove thatX0 ∈ O^p for allp >1. Under the assumptions of Theorem 2.2.1, we have E((M^Qmin)^2p_T )<∞, hence ([103]):

Elog^pZ_min = E(M_T^Qmin−1

2[M^Qmin]_T)^p

≤ 2^p−1E((M_T^Qmin)^p+ 1

2^p[M^Qmin]^p_T)

≤ K(p)·E((M_T^Qmin)^2p)<∞ (2.12) by the inequalities of Burkholder-Davis-Gundy and Doob andK(p) a posi-tive constant dependent onp. To establish the three items in Section 1.2.6 (implying convergence of the terminal values), it remains to show the as-sumptions in Lemma 1.2.18 and 1.2.19. Suppose now that Assumption 2.1.1 is satisfied. As in Santacroce [114] (proof of Theorem 1) we have for a positive constantK and some ˜µ >0 that

sup

1<q≤q0

E(|Z_T^(q)|^1+˜µ)< K. (2.13) The assumptions of Lemma 1.2.19 can be proven due to the explicit form given in Theorem 2.2.1 and 2.2.2. These theorems are given for general p.

The proof of the assumptions in Lemma 1.2.19 works exactly by going a long the line of the proof inNiethammer[103];Kohlmann and Niethammer [76] by just replacing 2m byp.

If the tradeoff process is deterministic, we know that Q_q for q ∈ (1, q₀] and Q_min almost surely coincide. We have almost sure and L¹-convergence of the primal solutions. As Z_q = ˆZ_T ∈ L^v for all q > 1 and for all v ≥1 (since ˆK_T is deterministic), X₀^(p)(˜x) ∈ L^v for all p > 1 and for all v ≥ 1.

Hence, we find thatX₀^(p)(˜x) converges in all L^v, v ≥1 (this obviously also holds forv 6=p).

Finally, we have

Theorem 2.2.3. In our model with an Ft-adapted continuous semimartin-galeS =S₀+M+A, let one of the following assumptions be satisfied:

1. Assumption 2.1.1

2. The terminal value of the mean variance tradeoff process (KˆT = h−R_·

0λˆ^′_udM_uiT) is deterministic.

Then, the solution of p-th problem converges in L¹ to the solution of the

exponential problem, i.e.: Moreover, the values of the dual problems converge:

mlim→∞φ_p(m)(y_m, Z_q) =φ_exp(y, Z_min),

for an arbitrary sequence p(m) → ∞. Hence the value functions of the primal problem converge for an arbitrary sequence p(m)→ ∞:

mlim→∞E(u_p(m)(X₀^(p(m))(˜x))) = lim

m→∞Vp(m)(˜x) =Vexp(˜x) =E(u_exp(X₀^(exp)(˜x))).

If the second assumption holds true, the dual problem of thep-th and the ex-ponential problem have the same solution up to the constantYi(˜x), the den-sity of the minimal entropy martingale measure times Yi(˜x) for i=p, exp.

The terminal values in (2.14) converge P-almost surely and in L^v for all v≥1.

Note, Assumption 1 holds in a Brownian setting with uniformly bounded coefficients, Assumption 2 is satisfied if the coefficients are in addition deterministic. Finally, the static and dynamic value functions coincide:

Vp =Vi,stat=Vj,dyn, i=a, s, j= 0, C for large enough p.

2.3 Convergence to the Optimal Portfolio for an

Im Dokument Portfolio Optimization and Optimal Martingale Measures in Markets with Jumps (Seite 83-87)