Portfolio Optimization and Optimal Martin- Martin-gale Measures

This section is supposed to provide the reader with the necessary background in portfolio optimization. We recall and slightly generalize previous results already given in Niethammer[103] and explain changes when jumps are included into our setting. In the next chapters, we then turn to new results in the continuous and the L´evy case. We close our theoretical consideration by an examination of signed and absolutely continuous martingale measures.

Afterwards we shortly compare both models and indicate a link to credit models. Finally, a method to estimate a covariance matrix and jump com-ponents of the stocks is provided.

The section is organized as follows: we start with some background on the market model, i.e. with the risky and riskless assets we can invest in.

We then define the set of allowable trading strategies and continue with the optimization problem. Afterwards we solve the problem under some general assumptions. Moreover, an application to the exponential and isoelastic type functions and its connection are given. Some notes on the corresponding literature close the chapter.

1.2.1 The Market Model

We start recalling the usual semimartingale setting. In particular, we restrict ourselves to continuous semimartingales and to exponential L´evy processes.

Let (Ω,F, P) be a probability space, T ∈ (0,∞) a time horizon, and F = (Ft)_t∈[0,T] a filtration satisfying the usual conditions, i.e. right-continuity and completeness. This enables us to use right-continuous with left-limits (RCLL) versions for all (P,F)-semimartingales representing our stocks. As only special semimartingales are considered, so that a Doob-M´eyer- decom-position holds, we simply call them semimartingales. All expectations and spaces without a subscript are defined with respect to the measureP. L^p is always meant with respect to Ω,FT, and P: L^p :=L^p(Ω,FT, P). Through-out the thesis an R^N+1-valued (P,F)-semimartingale (S,1) is given, where S= (S_t)_t∈[0,T]with unique decompositionS =S₀+M+Ainto a local mar-tingale M and a predictable process of bounded variation A. S represents a vector of N risky assets and 1 stands for a riskless asset with constant discounted price, i.e. the riskless asset serves as a num´eraire. As mentioned, we either propose S to be a continuous semimartingale or an exponential L´evy process. Note, the latter does not necessarily imply that S is locally bounded, as assumed e.g. inDelbaen et al. [36]:

Assumption 1.2.1. S is continuous.

Assumption 1.2.2. S is an exponential L´evy process:

St=S0e^Xˇt, t∈[0, T], S0 =diag(S₀⁽¹⁾, ..., S₀^(N)), S₀⁽ⁱ⁾∈R, (1.11)

andXˇ an R^N-valued L´evy process on (Ω,F, P). The filtration Fis supposed to be the completion of the filtration induced by Xˇ and E(kS_tk) < ∞ is assumed for all t∈[0, T].

In both cases, a bold S, S, will denote an N ×N-matrix with S_t,ii = S_t⁽ⁱ⁾ and zero elsewhere.

We next exploit the L´evy-Itˆo-decomposition, described in Section 1.1, to characterize the process of Assumption 1.2.2, i.e. the above L´evy process ˇX has the following form:

Xˇ_t=bt+σW_t+ Z t

0

Z

kxk>1

xNXˇ(dx, ds) + Z t

0

Z

kxk≤1

xN˜Xˇ(dx, ds) (1.12) where ˜N_Xˇ(dx, dt) = N_Xˇ(dx, dt)−ν(dx)dt. ν is the corresponding L´evy-measure,NXˇ is a Poisson random measure with intensity measureν(dx)dt, and W is a Brownian motion. Further, the L´evy measure ν is a Radon measure onR^N\{0}and verifies:

Z

kxk≤1kxk²ν(dx)<∞, Z

kxk>1

ν(dx)<∞

Except from Chapter 4 (see Remark 1.2.2), we propose the following as-sumption:

Assumption 1.2.3. Medenotes the set of all probability measuresQ equiv-alent toP such thatS_tis a localQ-martingale. It is assumed to be nonempty.

IfMe is a singleton, we call the market complete, otherwise incomplete. If our risky assetS contains jumps, the market is usually no longer complete, see e.g. Example 4.1.6. Note further, if there are no jumps, the above L´evy model corresponds to the Brownian model (a continuous semimartingale model) by settingµ=b−¹₂σ2, dS =Sµdt+SσdW,whereσ⁽ⁱ⁾₂ =Pd

j=1σ_ij². In the Brownian case, conditions such that Assumption 1.2.3 holds are easily given: firstly, the number of stocks should be less or equal to the number of random influences (dimension of the Brownian motion, N ≤ d). Secondly, the covariance matrix σ should have almost surely full rank. To generalize the model, one may further replace the constants µ and σ by functions in t and ω. Then, we need some integrability conditions in addition, e.g. one may assume that σ and µ are uniformly bounded, see [103, Section 1.2.2].

In the one-dimensional model with constant coefficients, we essentially need that σ 6= 0. This is also a sufficient condition to guarantee no-arbitrage in a one-dimensional exponential L´evy model, see Kunita [84, Lemma 3.2].

If σ = 0, we assume that positive and negative jumps appear to guarantee thatMe6=∅:

ν((−∞,0))6= 06=ν((0,∞))

Otherwise the stock is strictly increasing or strictly decreasing, if jumps have the same sign asb. In this quite pathological case if the driftbhas opposite sign of the jumps, we also have no arbitrage. In any case no arbitrage implies thatσ >0 or ν(R\{0})6= 0.

Finally, it should be noted that in the Brownian case the underlying filtration is set equal to completion of the filtration induced by the Brownian motion F = (σ(W_s; s ≤t))_t.If N < d this leads to incompleteness in the market. This is not the case if the filtration is equal to the one induced by the assetF= (σ(S_s; s≤t))_t.Without loss of generality the number of stocksN can be set equal tod. The market is complete. This approach coincides with the above L´evy model without jumps: (σ(Ss; s≤ t))t = (σ( ˇXs; s ≤ t))t. Incompleteness is only caused by the jumps.

1.2.2 Trading Strategies and Optimization Problems

In this section, at first we introduce some spaces of (signed) martingale measures. This enables us to specify suitable classes of trading strategies.

Afterwards, we continue with the optimization problem, its static version, and the corresponding dual problem. Utility optimization problems are solved for the exponential case (U(x) =u_exp,α(x) =−e^−αx, α >0) and the isoelastic-type case (U(x) :=up,α(x) =−|1−^αx_p |^p, α >0, p >1).

Exponential utility functions are frequently discussed, in particular for pricing issues they are closely related to utility indifference pricing, see Del-baen et al. [36] and the references therein. Isoelastic type utility functions up are particularly interesting, when p = 2m for some m ∈ N. For p = 2, we obtain

u_2,α(x) =αx−α²

4 x²−1.

Depending on the risk aversion parameter α, one maximizes the expected terminal wealth while penalizing potential risk from a large second moment (comparable to a non-centralized squared standard deviation). As already indicated in our introductory section, this is deeply related to the Markowitz problem of identifying efficient portfolios, see e.g. Markowitz[91, 92], and mean variance hedging, see e.g. Duffie and Richardson [40] or Zhou [121].

It is well-known, that empirical studies of stock times series show that stock returns rather follow a skewed and heavy tailed distribution than a simple Gaussian distribution, which is uniquely identified by its mean and its second moment. In particular, in the presence of jumps, the second moment may be insufficient to measure risk, higher moments can contain useful information about the risk of large downwards jumps. Similarly, to the case p = 2, for p = 2m the utility function u_2m,α rewards large odd moments up to order 2m−1 and penalizes the even moments up to order

2m. For instance,

u_4,α(x) =αx+ α³

16x³−3α²

8 x²− α⁴

256x⁴−1

additionally rewards a non-centralized version of the skewness and penalizes a non-centralized version of the kurtosis.

Without loss of generality we can always set α= 1,see [36] and Section 4.5, (6). We setu_p:=u_p,1, u_exp :=u_exp,1.

(Signed) Martingale Measures

The optimization problem, (1.13) below, is closely related to an optimiza-tion problem over a set of (signed) martingale measures: the optimizaoptimiza-tion problem will be solved via an optimization over these measures - the dual problem, see below. Moreover, if we do not allow for consumption, in prin-ciple we have to consider signed martingale measures. Fortunately, in many situations (dealt with in Chapter 2/3), we will see that the optimizer of the dual problem is always an equivalent martingale measure. The dual problem can be reduced to a problem over equivalent martingale measures.

In the case of u_p(x) = −|1−^x_p|^p, p > 1, the optimization problem will be set up in an L^p-framework, for p ∈ (1,∞). So for the conjugate q = (1− ¹_p)⁻¹ ∈(1,∞) ofp, we will therefore impose sufficient conditions such that the space of all (signed) local martingale measures with L^q-integrable densities is nonempty, i.e. M^q_i 6=∅,where

M^q_i ={Q|dQ=Z_TdP, Z ∈ D^q_i} ⊂L^q(P), q ∈[1,∞), i=e, a, s with

De^q = {Z ∈ U^q|E(ZT) = 1, ZT >0, SZ ∈M_loc}, Da^q = {Z ∈ U^q|E(Z_T) = 1, Z_T ≥0, SZ ∈M_loc}, Ds^q = {Z ∈ U^q|E(Z_T) = 1, SZ ∈M_loc},

M_loc is the space of local martingales, and U^q is the class of uniformly integrable martingales ˜M withE^1p(|M˜T|^p)<∞. We further set M¹i =:Mi. In the case ofi=e,M^qe 6=∅further impliesMe6=∅and so an arbitrage free market. This last case is sufficient when considering continuous processes or those with nicely behaved jumps (as discussed in Chapter 2/3). Here the optimal dual solutions over M^qs are already inM^qe.

To treat the exponential problem u_exp(x) =−e^−x,we introduce P_f(P) :={Q∈M˜a:H(Q|P)<∞)}

with H(Q|P) = E_P(^dQ_dP log(^dQ_dP)) ifQ ≪ P and ∞ otherwise. M˜a denotes the set of absolutely continuous (local) martingale measures, see also Section

1.2.4:

M˜a := Ma, ifS is locally bounded

M˜a := {Q∈ Ma: S is aQ-martingale}, otherwise,

In the latter case, S is guaranteed to be a “true” martingale to verify the assumptions given inFrittelli[52]. Further, ˜Meis analogously defined by replacingMa by Me. With this definitions at hand, the measure inP_f(P) that minimizes the relative entropyH(Q|P) can be identified and is called the minimal entropy martingale measureQ_min due to [52]. It is further the dual solution to the exponential problem. Furthermore, ifP_f(P)∩M˜e 6=∅, Q_minis in ˜Me(see again [52]). Finally, we will see that it makes no difference to consider ˜MeorMe in our settings. Later, we will therefore consider the following set only:

De^log={Z ∈ D¹e, E(Z_TlogZ_T)<∞}.

Remark 1.2.1 (Notation). The density process can be identified with its terminal value and so with the induced measure. Density, density process, and the induced measure are often used synonymously in the literature. If the notation is clear from the context, we therefore writeZ instead of Z_T and add a superscript toZ when denoting a density process:

M^qi,Z :={Z_j : Z_j =Z_T^(j), Z^(j) ∈ D^qi}, i=a, e, s.

Remark 1.2.2. In Chapter 4, it is possible that Me = ∅, nevertheless we solve the portfolio optimization problem with respect tou_p(x) =−|1−^x_p|^p, ifM^qs 6=∅ orM^qa6=∅,respectively.

Trading Strategies and Consumption

A self-financing strategy (˜x, ϑ) is given by the initial wealth ˜xand the num-ber ϑ_t = (ϑ¹_t, ..., ϑ^N_t )^′ of stocks held at time t∈[0, T]. We require that our strategies are predictable and satisfy an integrability condition. Throughout Chapter 2 and Chapter 3 we exclusively consider the following class (see also Grandits and Rheinl¨ander [60]):

Definition 1.2.3. The set of L^p-trading or p-integrable strategies is defined as follows:

A^p :=L^p(M)∩L^p(A)

where

L^p(M) ={ϑ∈P^N| kϑkL^p(M)<∞}, L^p(A) ={ϑ∈P^N| kϑkL^p(A)<∞}

with kϑkL^p(M) := k(RT

0 ϑ^′_td[M]_tϑ_t)¹²kL^p, kϑkL^p(A) := kRT

0 |ϑ^′_tdA_t|kL^p and P^N is the set of all predictable R^N-valued processes. For the exponential utility function U(x) =−e^−αx, α >0, we consider

A^exp,α={ϑ∈ \

p>1

A^p :Ee^{−αR0Tϑ′tdSt} <∞}, A^exp:=A^exp,1.

We will potentially allow for consumption:

Definition 1.2.4. C^p denotes the class of non-decreasing, adapted, RCLL processes with C0= 0, CT ∈L^p.

We furthermore define a class of strategies including consumption that requires an integrability assumption on the terminal value only. It is quite similar to a class of strategies given inSchachermayer[117]:

Definition 1.2.5. We say a pair (ϑ, C) belongs to the class A^(p,c), ifϑis a predictable,S-integrable process, Cis an RCLL, non-decreasing and adapted process, and

Yt(ϑ, C, x) :=x+ Z t

0

ϑudSu−Ct

satisfies: Y_T(ϑ, C,0) ∈L^p and Z_tY_t(ϑ, C,0) is a supermartingale for every Z ∈ D^qa.

The integrability assumption of A^(p,c) is less restrictive than A^p ×C^p. Hence, it is quite natural to adjust the trading strategies for the expo-nential problem. Consumption can be ignored as the utility function is strictly increasing. If S is locally bounded, Delbaen et al. [36] or also Schachermayer [117] propose the following class of trading strategies.

WithG_·(ϑ) :=R_·

0ϑ^′_udS_u they define M^exp,α = {ϑ∈P^N :

E(e^−αGT(ϑ))<∞,∀Q∈P_f(P), G_T(ϑ) is a Q-martingale} and

M^∞,α = {ϑ∈P^N :E(e^−αGT(ϑ))<∞,

∀Q∈P_f(P), G(ϑ) is a Q-supermartingale}.

It takes some effort to show that the optimal strategy is in M^exp,α/M^∞,α, we therefore also consider the following larger class of strategies:

M^min,α={ϑ∈P^N :E(e^−αG(ϑ))<∞, G(ϑ) is aQmin-martingale}

As we just consider the caseα= 1,we setMⁱ :=M^i,1, i= exp,min,∞.In Chapter 2 and 3 we stick to A^exp as it is the natural, canonical extension ofA^p.However, when we discuss the class A^(p,c),Mⁱ, i= exp,min,∞ will become more appropriate, see Section 1.2.5 and Section 4.5.

Remark 1.2.6. [·] denotes the right continuous quadratic variation process andh·iits predictable compensator process. Furthermore, in short [M, N]_t= hM^c, N^cit+P

0≤s≤t∆N_s∆M_s, whereN^c(M^c) is the unique continuous local martingale part of the local martingale N(M), see [110, p. 221]. Hence, hMi= [M], ifM is continuous. In the discontinuous case, it is essential to use [·] as h·i might not exist. Moreover, by the Burkholder-Davis-Gundy-inequality (BDG) (seeJacod[67, 2.34]/Theorem 1.1.23), the above classes defined using [·] then exclude doubling strategies (see the proof of Lemma 1.2.8 showing thatY Z is aP-martingale). For further undefined notations and the standard theorems concerning the theory of integration with respect to semimartingales, we refer toJacod [67] or Jacod and Shiryaev[68].

Optimization Problem

Self-financing strategies in the above classes plus a consumption process (ϑ, C) are then sufficient to define a class of wealth processes:

WC(˜x) ={Y|Yt= ˜x+ Z t

0

ϑ^′_udSu−Ct, ϑ∈ A, C ∈C}, x˜∈R, where C^p = C and A = A^p (if we consider u_p) resp. A^exp (if we consider u_exp). We setW0(˜x) :=W(˜x),if consumption is not allowed, i.e.C ≡0. In Chapter 4, we will further consider the following class of wealth processes:

WAT(˜x) ={Y|Y_t= ˜x+ Z _t

0

ϑ^′_udS_u−C_t,(ϑ, C)∈ A^(p,c)}, x˜∈R.

The following definitions are implied:

Definition 1.2.7. (i) An FT-measurable random variable X is called su-perreplicable/superhedgeable with respect to Wi(˜x), i = C, AT and initial wealth x˜ if there exists a Y ∈ Wi(˜x) such that X = Y_T. The correspond-ing tradcorrespond-ing strategyϑ and the consumption process C is called superhedging strategy (ϑ, C).

(ii) AnFT-measurable random variableX is called replicable/hedgeable with initial wealth x˜ if there exists a Y ∈ W0(˜x) such that X =Y_T. The corre-sponding trading strategyϑ is called hedging strategy.

On the above classes of wealth processes, the following dynamic opti-mization problems are considered:

V(˜x)_i,dyn≡ sup

Y∈Wi(˜x)

E[U(Y_T)], i= 0, C, AT,exp,min,∞, x˜∈R (1.13)

where U is a concave, not necessarily increasing function and Wi, i = exp,min,∞is defined asW0by replacingA^pbyMⁱ,i= exp,min,∞. Note, replication with respect to Wi(˜x), i = min,∞ can be defined analogously.

However, it will not be important for our studies. Replication in Definition 1.2.7 is therefore always meant with respect toW0(˜x).

Static Problem, up

To find a solution of the above dynamic problems, (1.13), at first we have to find suitable static problems. We then show that the static solutionX0is the optimal terminal value of the dynamic problem by constructing an allowed (super)hedging strategy that attains X₀. We start with the isoelastic-type model as it fits nicely in the proposed L^p-framework. Adjustments to the exponential case are presented afterwards. For U(x) = u_p(x) = −|1−^x_p|^p and fixedp >1,we solve the following static problems:

V(˜x)_i,stat≡ sup

X∈O^p,∀Q∈M^qiEQ(X)≤˜x

E[U(X)], x˜∈R, i=a, s, (1.14) whereO^p := {X ∈L^p(FT, P) :EU(X)<∞},which is equal to L^p for the utility functionu_p with conjugateq = _p−^p₁.

We next show that allowing for consumption or not is crucial for the choice of the suitable static problem, i.e. if we need to choose M^qa orM^qs. Lemma 1.2.8. We have

(i) A^p× {0} ⊂ A^p×C^p ⊂ A^(p,c).

(ii) The value of the above static problem (1.14) is greater or equal than the value of the above dynamic problem (1.13), i.e.

V(˜x)_C,dyn ≤ V(˜x)_AT,dyn ≤ V(˜x)_a,stat, x˜∈R, (1.15) V(˜x)_0,dyn ≤ V(˜x)_s,stat ≤ V(˜x)_a,stat, x˜∈R. (1.16)

To show the reverse directions, we later need to ensure that we can find a (super)hedging strategy of the optimal value of (1.14), see Re-mark 1.2.10. The idea of the proof of Lemma 1.2.8 is as follows. We know that for an ϑ ∈ A^p : X = ˜x+RT

0 ϑ^′_udSu ∈ L^p(Ω,FT, P) (see e.g.

Grandits and Rheinl¨ander [60, Proof of Lemma 2.1]) and therefore E(Z_TX)<∞. Moreover, it is well known that then, by an application of the Burkholder-Davis-Gundy inequality (BDG), ZR_·

0ϑ^′_udSu is a P-martingale, hence E(Z_TX) = ˜x. If Z_T ≥0, we always have E(Z_TC_T) ≥ 0. Otherwise we have to exclude consumption. Finally, the supremum in (1.14) is not smaller than the one in (1.13):

Proof of Lemma 1.2.8. Let a local martingale ˜Mbe given. It is a martingale second inequality of (1.15). There are immediate for the classA^(p,c) : If X is superreplicable with initial capital ˜x within the class of strategies A^(p,c), then for all Z∈ D^qa

E[ZTX] = ˜x+E[ZTYT(ϑ, C,0)]≤x˜

andX= ˜x+Y_T(ϑ, C,0) ∈L^p.Finally, it is obvious thatA^p×C^p ⊂ A^(p,c). When inspecting the proof, it is immediate that BDG is essential to solve the dynamic problem via a transformation to a static version. If we used h·i instead of [·] in the definition of A^p, BDG in [67] would not necessarily hold. ϑ ∈ A^p would not guarantee that R_T

0 ϑ_udS_u ∈ L^p. This is however essential to extract that the value of the static maximization problem is not less than the value of the dynamic maximization problem.

Moreover, when we later consider the exponential problem, the proof automatically shows that the static problem dominates the dynamic problem w.r.t. A^exp for an arbitrarily chosen p by the definition of A^exp. For the reverse direction in both cases, i.e. that the static optimum actually can be reached by a (super)hedging strategy, see Remark 1.2.10.

Dual Problem, up

We have reduced the above dynamic problems to problems over random variables satisfying some constraints. We next need to solve the static prob-lem over random variables. This is done via a dual optimization probprob-lem over (signed) martingale measures. The exact relation is explained in Sec-tion 1.2.3. However, the intuiSec-tion is quite clear. In a complete discounted market, we know from pricing theory thatE_Q(X) is the today’s price for a terminal payoff X. Roughly speaking, the budget constraintEQ(X)≤x˜ is then equivalent to the possibility to choose a strategy leading to a wealth process with terminal wealth X and with initial wealth at most equal to ˜x.

As we have several martingale measures, we have several pricesE_Q(X),the budget constraint has to hold under all (signed) martingale measures. The number of constraints depends on the set of strategies that we are allowed to use, see Lemma 1.2.8: depending on a set of strategies, we have to choose the correct static problem (V(˜x)_i,stat, i =a, sin 1.14), which further leads to the dual problem, (see Section 1.2.3):

Z∈Dmin^qi,y≥0E( ˇU(yZT) + ˜xy), i=a, s, (1.17) where ˇU(y) = sup_x(U(x)−xy). In Chapter 2 and Chapter 3, it will be sufficient to discuss the case i=e(then equivalent to the case i=a) only, as the dual optimizer is always inD^qe. Claims satisfying the budget constraints with respect to the set M^qe can be shown to be hedged. Further budget constraints E_Q(X)≤x, Q˜ ∈ M^qs\M^qe are not necessary. The optimizer of all dynamic (i= 0, C) and static problem (i=a, s, e) are the same (provided the initial wealth is smaller than p). Consumption is irrelevant.

However, if jumps become too irregular and we do not want to allow for consumption, the dual optimizer might be in M^qs\M^qa. We have to restrict the claims by budget constraints also including signed measures, otherwise a hedging strategy does not exist. If we include consumption, the spaceM^qa is large enough. The last two cases are discussed in Chapter 4.

Exponential Case

In the exponential problem we consider the utility functionu_exp(x) =−e^−x. Recall the approximatingp-th problem u_p(x) =−|1−^x_p|^p fits well into the above structure and can be nicely treated in an L^p-setting. Optimizers can be easily shown to have the right integrability. To treat the exponential problem in the above framework (in particular (1.14)), we need some ad-justments or switch to a slightly different static problem as e.g. dealt with inKabanov and Stricker[71], see also Section 1.2.5. We start with the dual side.

We have already mentioned that the dual optimizer is the so-called min-imal entropy martingale measure (MEMM), i.e. the density process ofQmin,

denoted byZ^(min) solves (up to the scaling constanty(˜x)) min

Z∈De^log,y≥0

E( ˇU(yZ_T) + ˜xy), (1.18) where U(x) = u_exp(x) and ˇU(y) = sup_x(U(x)−xy). Recall, if S is locally bounded andP_f(P)∩Meis non-empty, then the minimal entropy martingale measureQ_min (MEMM) exists in P_f,e(P), i.e. the unique measure inP_f(P) that minimizes H(Q|P). If S is not locally bounded, in ˜Ma only “true”

martingale measures are considered, i.e.S has to be aQ-martingale see Sec-tion 1.2.4. That does not restrict our approach as the following arguments show. Firstly, continuous processes are locally bounded. Secondly, in the one-dimensional L´evy settingHubalek and Sgarra[66] show that condi-tion C in Esche and Schweizer [46]; Fujiwara and Miyahara[54] is equivalent to the existence of the MEMM (see Section 3.2.1). We therefore assume that condition C holds, but condition C furthermore implies that S is a Q_min-martingale. Condition C essentially consists of a martingale condition that specifies how Q ∈ P_f(P) has to be chosen and guarantees sufficient integrability.

In Chapter 2 and 3, we impose a slight strengthening of condition C or the Reverse H¨older inequality for the continuous case. This implies, as we later see, that there exists an ˜ǫ > 0 such that the density of the minimal entropy martingale measureZ_minis inL^1+˜ǫ. We can setq= 1 + ˜ǫ, such that M^qe∩P_f(P) is nonempty. The dual problem overD^loge coincides with the dual problem over D^elog∩ D^eq. The static problem (1.14, i=e, U(x) = u_exp(x)) can be solved up to the integrability ofX₀:

V(˜x)_i,stat≡ sup

X∈O^p,∀Q∈M^qe∩P_f(P)EQ(X)≤x˜

E[U(X)], ˜x∈R.

Supposing necessary integrability ofX₀, the static problem again dominates the dynamic one as in the proof of Lemma 1.2.8. Consumption is irrelevant as the exponential utility function is increasing. A^exp is then one choice to find a hedging strategy. In fact, a later construction of an optimal portfolio ϑ^(exp) in A^exp gives the reverse direction (the dynamic problem also domi-nates the static one) and the necessary integrability of the optimal terminal value, i.e.X₀ ∈ O^p0, p₀= (1−_1+˜¹_ǫ)⁻¹.

In the continuous model, integrability can be shown directly. However in a L´evy setting with unbounded jumps, it might be difficult to find a hedging strategy inA^exp.In this case, the L^p-setting is hard to justify, we therefore consider claims withinL¹(Q_min) and Z_min does not have to be in L^1+˜ǫ.We

In the continuous model, integrability can be shown directly. However in a L´evy setting with unbounded jumps, it might be difficult to find a hedging strategy inA^exp.In this case, the L^p-setting is hard to justify, we therefore consider claims withinL¹(Q_min) and Z_min does not have to be in L^1+˜ǫ.We

Im Dokument Portfolio Optimization and Optimal Martingale Measures in Markets with Jumps (Seite 48-81)