B 1Introduction Robustutilitymaximisationinmarketswithtransactioncosts

https://doi.org/10.1007/s00780-019-00389-0

Robust utility maximisation in markets with transaction costs

Huy N. Chau¹·Miklós Rásonyi¹

Received: 12 March 2018 / Accepted: 8 February 2019 / Published online: 23 April 2019

© The Author(s) 2019

Abstract We consider a continuous-time market with proportional transaction costs.

Under appropriate assumptions, we prove the existence of optimal strategies for in- vestors who maximise their worst-case utility over a class of possible models. We consider utility functions defined on either the positive axis or the whole real line.

Keywords Utility maximisation·Transaction cost·Model uncertainty Mathematics Subject Classification (2010) 91G10·49J27

JEL Classification G11

1 Introduction

In this paper, the existence of solutions to the utility maximisation problem from ter- minal utility is studied in the presence of model ambiguity. We assume that investors prepare for the worst-case scenario in the sense that they take the infimum of util- ity functionals over the class of possible models before maximising over admissible investment strategies.

The literature on robust optimisation typically assumes that uncertainty is mod- elled by a family of prior measuresP on some canonical space in which trajectories

Both authors were supported by the “Lendület” grant LP2015-6 of the Hungarian Academy of Sciences and by the NKFIH (National Research, Development and Innovation Office, Hungary) grant KH 126505.

B

chau@renyi.hu M. Rásonyi rasonyi@renyi.hu

1 Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13-15, 1053 Budapest, Hungary

of the processes lie. Starting with Quenez [29] and Schied [34], the case in whichPis dominated by a reference measureP_∗has received ample treatment. In diffusion set- tings, this corresponds to uncertainty in the drift. Such an approach is not completely convincing since market participants may also be uncertain about the volatilities.

More recently, the non-dominated problem has also been studied in various con- texts. For instance, Tevzadze et al. [35] investigated a compact set of possible drift and volatility coefficients and tackled the robust problem by solving an associated Hamilton–Jacobi–Bellman equation. In Matoussi et al. [24], where volatility coeffi- cients are uncertain over a compact set and the drift is known, the theory of BSDEs is applied. Existence results in a fairly general class of models are available only in dis- crete time; see Nutz [27], Blanchard and Carassus [8], Neufeld and Šiki´c [26], Bartl [2], Bartl et al. [3] and Rásonyi and Meireles-Rodrigues [31]. A minimax result was established for bounded utilities in frictionless continuous-time markets in Denis and Kervarec [14].

As far as we know, our existence results below are the first to apply in a broad class of continuous-time models. We now summarise the principal ideas underlying our arguments. First, we work under proportional transaction costs. In this setting, strategies can be identified with finite-variation processes which we endow with a suitable convergence structure. Second, instead of a family of measures, we consider a parametrised family of stochastic processes on a fixed filtered probability space.

Necessarily, instead of one portfolio value, we need to consider a family of possible values corresponding to the respective parameters. Third, the latter fact forces us to take the family of strategies as our domain of optimisation (unlike most of the opti- mal investment literature since Kramkov and Schachermayer [21], which prefers to optimise over a set of random variables, the terminal values of possible portfolios).

Fourth, we exploit that the boundedness of terminal portfolio values in an appropri- ate sense implies boundedness of the strategies themselves (again, in an appropriate sense); this is false in continuous-time frictionless markets, but true in our setting.

Fifth, we profit from a method first developed in Rásonyi [30] that verifies the super- martingale property of a putative optimiser, based on a lemma of Delbaen and Owari [13]. Because of the fourth point above, our techniques do not seem to be applica- ble in the continuous-time frictionless setting. See, however, the companion paper by Rásonyi and Meireles-Rodrigues [31] which treats discrete-time frictionless markets.

The robust model in this paper, similarly to those introduced in Biagini and Pınar [6], Neufeld and Nutz [25], Lin and Riedel [22], assumes that there is a parametrisa- tion for the uncertain dynamics of risky assets. However, as we shall see below, no specific assumption is made about the parametrisation and an arbitrary index set is permitted. From a practical point of view, this approach is particularly tractable and easily implemented when it comes to calibration. For example, when estimating drift and volatility parameters for diffusion price processes, the results only give guesses (hopefully with some confidence sets) about the true values. Thus it is reasonable to parametrise ambiguity by considering suitable ranges which contain possible values for the coefficients being estimated.

From a mathematical point of view, the treatment of robust models in the present paper simplifies technical issues, as will become apparent from the proofs. Working on the same (filtered) probability space, instead of considering a family of measures,

gives us more flexibility by avoiding the canonical setting with problems concerning null events, filtration completion, etc. Measurable selection arguments, see Bouchard and Nutz [9], Biagini et al. [4] or Nutz [27], are not needed any more. Our approach can still incorporate most of the relevant model classes, and their laws on the path space do not need to be equivalent; see Sect.2.

Compactness plays an important role in proving the existence of optimisers. Usu- ally, the utility maximisation problem is transformed into an “abstract” version with random variables (the terminal wealth of admissible portfolios), and then convex compactness results inL⁰, in particular, Komlós-type arguments, are applied suc- cessfully; see Kramkov and Schachermayer [21]. Unfortunately, the robust setting is unlikely to be lifted to “abstract” versions, since the uncertainty produces a whole collection of wealth processes. As a result, Komlós-type arguments on the spaceL⁰ cannot be employed. Furthermore, the candidate dual problem in this setting does not, in general, admit a solution (see Bartl [2, Remark 2.3]) so that the usual approach of getting optimisers from solutions of dual problems seems inapplicable. Therefore, we are forced to work on the primal problem directly.

We are using two Komlós-type arguments. The first one is performed on the space of finite-variation processes (strategies), which gives a candidate for the optimiser, and the second is used in an Orlicz space context, to handle possible losses of trading when establishing the supermartingale property of the optimal wealth process, relying on Delbaen and Owari [13]. A crucial observation is that the utility of a portfolio is a sequentially upper semicontinuous function of the strategies (when the latter are equipped with a convenient convergence structure); see Guasoni [15] where the optimisation problem was viewed in a similar manner.

The paper is organised as follows. Section2introduces the robust market model and technical assumptions. Sections3and4study the existence of solutions for the robust utility maximisation problem when the utility functions are defined onR₊ andR, respectively. Ramifications are discussed in Sect. 5. Some preliminaries on finite-variation processes and on Orlicz space theory are presented in Sect.5.

2 The market model

Let(Ω,F, (Ft)_t∈[0,T], P )be a filtered probability space, where the filtration is as- sumed to be right-continuous andF0coincides with theP-completion of the trivial sigma-algebra. We define byL⁰the set of all a.s.-equivalence classes of random vari- ables and its positive cone byL⁰₊.

LetΘbe a (non-empty) set, which is interpreted as the parametrisation of uncer- tainty. We consider a financial market consisting of a riskless assetS_t⁰=1 for all t∈ [0, T]and a risky asset, whose dynamics is unknown. To describe the latter, we consider a family(S_t^θ)_t∈[0,T],θ∈Θ, of adapted, positive processes with continuous trajectories which represent the possible price evolutions. No condition is imposed onΘnor, for the moment, on the dynamics of the risky asset.

Remark 2.1 We now comment on the difference between our concept of model am- biguity and that of most previous papers, where a family of priors is considered on a canonical space.

Working on a given probability space and filtration amounts to fixing the infor- mation structure of the problem; the information flow is normally generated by a particular driving process (such as a multidimensional Brownian motion). Possible prices are then functionals of a parameter (finite- or infinite-dimensional, see Exam- ples2.2and2.4below) and the driving noise. Strategies are functionals adapted to the given information flow.

Considering a family of probabilities, one has greater liberty in the sense that no common driving noise is required, but the choice of strategies is limited; they must be adapted functionals on the canonical space, i.e., they are functions of the price process. In our modelling, the controls are adapted to an information flow that may be strictly bigger than the natural filtration of any possible price process.

In a strictly formal sense, none of two the approaches is more general than the other; see also examples in [31]. Intuitively, the standard setting is the more general one, while ours seems more easily tractable and fits better with a practical calibration and/or statistical inference framework.

We illustrate by the following examples that the present setting is useful and con- tains interesting models from previous studies.

Example 2.2 In the robust Black–Scholes market model, the risky asset satisfies the SDE

dS_t^{(μ,σ )}=S_t^{(μ,σ )}(μ dt+σ dW_t), S₀^{(μ,σ )}=s₀>0,

whereμ,σ are constants andW is a standard Brownian motion. The uncertainty is modelled by

Θ= {θ=(μ, σ )∈R²:μ≤μ≤μ, σ≤σ≤σ},

whereμ≤μ, 0< σ ≤σ are given constants. The classical Black–Scholes model corresponds to the case μ=μand σ =σ. It is observed that the laws of S^μ1,σ1, S^μ2,σ2 are singular when σ1=σ2. If only volatility uncertainty is considered, the family of laws is mutually singular. See [22,6] about treatments for similar models.

Remark 2.3 In the domain of robust finance, measurable selection techniques are of- ten used; see e.g. [27]. This requires a certain measurability of the family of laws corresponding to various models. In our present approach, however, this is not a necessity. Let e.g.Θbe a non-Borelian (or even non-analytic) subset of Θ in Ex- ample2.2above. Theorems3.6and4.7apply to the family of modelsS^θ,θ∈Θ, too.

Example 2.4 In the above example,Θwas a subset of a finite-dimensional Euclidean space. One may easily fabricate similar examples whereΘ is infinite-dimensional.

For instance, letΘconsist of all pairs of predictable processes(μ_t, σ_t)such that for allt∈ [0, T],μt∈ [μ, μ]a.s. andσt∈ [σ , σ]a.s., and consider the SDEs

dS_t^{(μ,σ )}=S_t^{(μ,σ )}(μtdt+σtdWt), S₀^{(μ,σ )}=s0>0, for each(μ, σ )∈Θ.

The following example extends the robust Black–Scholes model and allows an external economic factor.

Example 2.5 This is a factor model which is inspired by [20], but much simplified.

LetΘ⊆R^2×2be a set. The risky asset is governed by the SDE dS^θ_t =S_t^θ

m(Y_t^θ)+σ (θ¹¹Y_t^θ+θ²¹)

dt+σ dW_t¹

, S₀^θ=s₀>0, and the factor process evolves according to

dY_t^θ=

g(Y_t^θ)+ ρ, θ^1·Y_t^θ+θ^2·

dt+ρ₁dW_t¹+ρ₂dW_t², Y₀^θ=y₀, wherem,g are suitable functions,W=(W¹, W²)is a two-dimensional Brownian motion andρ=(ρ1, ρ2)∈R²a fixed parameter. The bracket·,· denotes the scalar product inR². Note that the original setting of [20] cannot be directly transferred to the present one as it involves a family of weak solutions of SDEs which are not necessarily realisable on our given stochastic basis.

The risky asset is traded under proportional transaction costsλ∈(0,1). More pre- cisely, investors have to pay a higher (ask) priceS^θ when buying the risky asset, but receive a lower (bid) price(1−λ)S^θ when selling it.

LetV denote the family of nondecreasing, right-continuous functions on[0, T] which are 0 at time 0. Let V denote the set of tripletsH =(H^↑, H^↓, H0), where H_t^↑,H_t^↓,t∈ [0, T], are optional processes such thatH^↑(ω), H^↓(ω)∈V for each ω∈Ω andH0∈R(deterministic). The space V can be equipped with a convergence structure; see Sect.A.1below for details.

Each trading strategy corresponds to an elementH∈V. In this formulation,H^↑ denotes the cumulative amount of transfers from the riskless asset to the risky one andH^↓represents the transfers in the opposite direction;H₀encodes the amount of initial transfer from the riskless asset to the risky one. Therefore the portfolio position in the risky asset at timet equalsφ_t:=H₀+H_t^↑−H_t^↓,t∈ [0, T],φ₀₋:=0.

For any x∈R, we denote x⁺:=max{0, x}, x⁻:=max{0,−x}. For an initial capitalx∈R, the dynamics of the cash account of an investor following the strategy Hevolves according to

W_t^x(θ, H ):=x−H₀⁺S₀^θ+H₀⁻S₀^θ(1−λ)− _t

0

S_u^θdH_u^↑+ _t

0

(1−λ)S^θ_udH_u^↓, fort∈ [0, T]. The liquidation value is defined by

W_t^x,liq(θ, H ):=W_t^x(θ, H )+φ⁺_t (1−λ)S_t^θ−φ_t⁻S^θ_t. (2.1) We introduce the definition of consistent price systems, which play a similar role to martingale measures in frictionless markets; see [19,17,16].

Definition 2.6 For eachθ∈Θ, aλ-consistent price system (λ-CPS) for the modelθ is a pair(S˜^θ, Q^θ)of a probability measureQ^θ≈P and a (càdlàg)Q^θ-local martin-

galeS˜^θ such that

(1−λ)S_t^θ≤ ˜S_t^θ≤S_t^θ a.s., for eacht∈ [0, T]. (2.2) Aλ-strictly consistent price system (λ-SCPS) is a CPS such that the inequalities in (2.2) are strict.

We impose the existence of consistent price systems for every modelS^θ. In Sect.3, we need the following assumption in order to be able to use the results of [12].

Assumption 2.7 For eachθ∈Θ and for all 0< μ < λ, the price processS^θ admits aμ-CPS.

This assumption is fulfilled if for every θ ∈Θ, the process S^θ satisfies the no- arbitrage condition forμ-transaction cost for allμ >0; see [17]. See Example4.6 for a risky asset violating Assumption2.7.

Clearly, if 0< μ < λ, then aμ-CPS is also aλ-SCPS.

Lemma 2.8 If(S˜^θ, Q^θ)is aλ-strictly consistent price system, the random variable δ(θ ):= inf

t∈[0,T]min{ ˜S_t^θ−(1−λ)S_t^θ, S_t^θ− ˜S_t^θ} (2.3) is almost surely strictly positive andE^Qθ[δ(θ )]<∞.

Proof The argument follows that of [19, Lemma 3.6.4].

Let

M^θ:= {dQ^θ/dP : (S˜^θ, Q^θ)is aλ-CPS}. For a consistent price system(S˜^θ, Q^θ), we define the process

V_t^x(θ, H ):=W_t^x(θ, H )+φ_tS˜_t^θ,

without emphasising the dependence ofV on the specific consistent price system. It is easy to check thatW_t^x,liq(H )≤V_t^x(θ, H )a.s., for eacht∈ [0, T].

3 Utility function onR+

Assumption 3.1 The utility functionU:(0,∞)→Ris nondecreasing and concave.

Define the convex conjugate ofUby V (y):=sup

x>0

U (x)−xy

, y >0.

Admissible strategies are defined in a natural way, thanks to the domain of the utility function.

Definition 3.2 A strategy H=(H^↑, H^↓, H₀)∈V is admissible for initial capital x >0 and for the modelθ∈Θif for eacht∈ [0, T],

W_t^x,liq(θ, H )≥0 a.s.

Denote byA^θ(x)the set of all admissible strategies forθ. Set A^θ₀(x):= {H∈A^θ(x): φT =H0+H_T^↑−H_T^↓=0}

andA(x)=

θ∈ΘA^θ₀(x).

Remark 3.3 For eachH∈A(x)andθ∈Θ,

W_T^x,liq(θ, H )=W_T^x(θ, H )=V_T^x(θ, H )

due toφ_T =0. We also see from (2.1) that at time 0< t < T, the liquidation value is neither concave nor convex inH. However, the conditionφ_T =0 recovers concav- ity of the liquidation value with respect toH at time T. This is crucial for finding maximisers in the subsequent analysis.

Letx >0. Note thatA(x)= ∅since it contains the identically zero strategy. Our investors want to find the optimiser for

u(x):= sup

H∈A(x) θinf∈ΘE

U

W_T^x,liq(θ, H ) . (3.1)

It is worth noting that maximising inHis a concave problem; however, minimising overΘis not a convex problem.

For eachθ∈Θandx >0, we denote

C^θ(x):= {X∈L⁰₊: X≤W_T^x,liq(θ, H )for someH∈A^θ(x)}.

For eachy >0, the set of supermartingale deflatorsB^θ(y)consists of the strictly positive processesY =(Y_t⁰, Y_t¹)_t∈[0,T],Y₀⁰=y, such thatY¹/Y⁰∈ [(1−λ)S^θ, S^θ] andW^x(θ, H )Y⁰+φY¹is a (càdlàg) supermartingale for allH∈A^θ(x). Also, we define

D^θ(y):= {Y_T⁰:(Y⁰, Y¹)∈B^θ(y)}. The primal and dual value functions for theθ-model are

u^θ(x):= sup

f∈C^θ(x)

E[U (f )], v^θ(y):= inf

h∈D^θ(y)

E[V (h)].

The next lemma states that the setsC^θ(x)andD^θ(y)are polar to each other. It follows directly from [12, Proposition 2.9].

Lemma 3.4 Fixx, y >0. Let Assumption2.7be in force. A random variableX∈L⁰₊ satisfiesX∈C^θ(x)if and only ifE[XY] ≤xyfor allY ∈D^θ(y). A random variable Y∈L⁰₊satisfiesY∈D(y)if and only ifE[XY] ≤xyfor allX∈C^θ(x).

We impose a technical assumption.

Assumption 3.5 The dual value functionv^θ(y), y >0, is finite for allθ∈Θ.

Theorem 3.6 Letx >0. Under Assumptions2.7,3.1,3.5, the robust utility maximi- sation problem (3.1) admits a solution, i.e., there isH^∗∈A(x)satisfying

u(x)= inf

θ∈ΘE U

W_T^x,liq(θ, H^∗) .

WhenUis bounded from above, the same conclusion holds assuming only that there exists (at least) oneθ˜∈Θfor which there exists aλ-SCPS.

Proof IfUis constant, there is nothing to prove. Otherwise, by adding a constant to U, we may assume thatU (∞) >0>limx→0U (x).

Notice thatU (∞) >0 and

u^θ(x)≥U (x) (3.2)

imply lim infx→∞u^θ(x)/x≥0. From Lemma3.4, trivially,

u^θ(x)≤v^θ(y)+xy (3.3)

for ally >0. Fixingy, we obtain lim sup_x→∞u^θ(x)/x≤y, and sending y to zero gives

xlim→∞

u^θ(x)

x =0. (3.4)

After these preparations, we turn to the main arguments. Assumption3.5, (3.3) and (3.2) imply thatu^θ(x)is finite for eachθand so isu(x). LetHⁿ∈A(x),n∈N, be a maximising sequence, i.e.,

θinf∈ΘE U

W_T^x,liq(θ, Hⁿ) ↑u(x) asn→ ∞.

Let us fix for the momentθ∈Θand aμ-CPS(S˜^θ, Q^θ)with 0< μ < λ. First, we prove that the process

V_t^x(θ, Hⁿ)=W_t^x(θ, Hⁿ)+φ_tⁿS˜_t^θ is aQ^θ-supermartingale for alln. Indeed, Itô’s formula gives

dV_t^x(θ, Hⁿ)= −S_t^θdH_t^n,↑+(1−λ)S_t^θdH_t^n,↓+ ˜S_t^θdφ_tⁿ+φ_tⁿ₋dS˜_t^θ

=(S˜_t^θ−S_t^θ) dH_t^n,↑+

(1−λ)S_t^θ− ˜S_t^θ

dH_t^n,↓+φ_tⁿ₋dS˜_t^θ. Admissibility ofHⁿimplies

(H₀ⁿ)⁺(S^θ₀− ˜S₀^θ)+ _t

0

(S_u^θ− ˜S_u^θ) dH_u^n,↑+ _t

0

S˜_u^θ−(1−λ)S_u^θ dH_u^n,↓

+ ^t

0

φ_uⁿ₋dS˜_u^θ ₋

≤x+(H₀ⁿ)⁻

S₀^θ(1−λ)− ˜S₀^θ +

t 0

φⁿ_u−dS˜_u^θ ₊

. (3.5)

In particular, we obtain t

0

φ_uⁿ₋dS˜_u^θ ₋

≤x+(H₀ⁿ)⁻S₀^θ(1−λ) (3.6)

for every t ∈ [0, T], and therefore _t

0φ_uⁿ₋dS˜_u^θ, t ∈ [0, T], is a Q^θ-supermar- tingale; see [1, Corollary 3.5]. It follows that V_t^x(θ, Hⁿ), t ∈ [0, T], is also a Q^θ-supermartingale.

We claim that sup_n(H₀ⁿ)⁻is finite. If this were not the case, then along a subse- quencenk,k∈N, we should have(H₀^nk)⁻→ ∞,k→ ∞, and(H₀^nk)⁺=0,k∈N. TakingQ^θ-expectations in (3.5), we should get

0≤x+ lim

k→∞(H₀^nk)⁻

S₀^θ(1−λ)− ˜S₀^θ

= −∞,

a contradiction. Hence the supremum is indeed finite.

Furthermore, from the supermartingale property of _t

0φ_uⁿ₋dS˜^θ_u,t ∈ [0, T], and from (3.6),

sup

n

E^Qθ

T 0

φ_uⁿ₋dS˜^θ_{u +}

≤x+sup

n

(H₀ⁿ)⁻S^θ₀(1−λ) follows. Using (2.3), we deduce from (3.5) that

sup

n

E^Qθ

(H₀ⁿ)⁺δ(θ )+ _T

0

δ(θ )

dH_u^n,↑+dH_u^n,↓

≤sup

n

E^Qθ

(H₀ⁿ)⁺(S₀^θ− ˜S₀^θ)+ _T

0

(S_u^θ− ˜S_u^θ) dH_u^n,↑+S˜_u^θ−(1−λ)S_u^θ dH_u^n,↓

<∞.

Apply Lemma A.1 with the choice dQ/dQ^θ :=δ(θ )/E^Qθ[δ(θ )]. It implies that there exist convex weightsα_jⁿ≥0,j =n, . . . , M(n), withM(n)

j=n α_jⁿ=1 such that H˜ⁿ:=M(n)

j=n Hⁿ→H^∗in V. Since the utility function is concave, we obtain that H˜ⁿ,n∈N, is also a maximising sequence as

θinf∈ΘE U

W_T^x,liq(θ,H˜ⁿ) ≥ inf

θ∈ΘE U

W_T^x,liq(θ, Hⁿ) −→u(x) asn→ ∞.

We now prove that the sequenceU⁺(W_T^x,liq(θ,H˜ⁿ)),n∈N, is uniformly inte- grable for eachθ∈Θ. Suppose by contradiction that the sequence is not uniformly integrable for someθ. Then one can find disjoint setsA_n∈F,n∈N, and a constant α >0 such that

E U⁺

W_T^x,liq(θ,H˜ⁿ)

1An ≥α forn≥1.

Setwⁿ=_n

i=1W_T^x,liq(θ,H˜ⁱ)1_{

W_T^x,liq(θ,H˜ⁱ)≥u0}1A_i, where u0 is chosen such that it satisfiesU (u0)=0. It is immediate that

E[U (wⁿ)] = n

i=1

E U⁺

W_T^x,liq(θ,H˜ⁱ)

1A_i ≥nα.

In addition, for any h ∈ D^θ(1), the supermartingale property shows that E[hwⁿ] ≤nx. Consequently, we obtainwⁿ∈C^θ(nx)by Lemma3.4. We compute

u^θ(nx)

nx ≥E[U (wⁿ)]

nx ≥ α x >0,

and passing to the limit whenn→ ∞contradicts (3.4). ThusU⁺(W_T^x,liq(θ,H˜ⁿ)), n∈N, is indeed uniformly integrable.

Since H˜ⁿ →H^∗ in V, W_T^x,liq(θ,H˜ⁿ)→W_T^x,liq(θ, H^∗) almost surely by Re- markA.2. So Fatou’s lemma and uniform integrability imply

lim sup

n→∞

θinf∈ΘE U

W_T^x,liq(θ,H˜ⁿ) ≤ inf

θ∈Θlim sup

n→∞ E U

W_T^x,liq(θ,H˜ⁿ)

≤ inf

θ∈ΘE U

W_T^x,liq(θ, H^∗) ,

which proves thatH^∗is an optimiser. It remains to check thatH^∗∈A(x). For each θ,W_t^x,liq(θ, H^∗)≥0 a.s., for Lebesgue-almost everyt, by RemarkA.2; so we get admissibility ofH^∗sincet→W_t^x,liqis a.s. right-continuous.

In the case whereU is bounded from above, it is enough to perform the first part of the above argument forθ, obtain˜ H^∗ and then simply invoke Fatou’s lemma to

complete the proof.

Remark 3.7 In the classical theory where there is no uncertainty, i.e., whenΘcon- tains only one element, the existence result holds assuming the finiteness ofu(x)only.

This condition, however, does not suffice to find optimisers in the robust problem. In- deed, the finiteness ofu(x)makes the robust problem well posed, compactness gives a candidate for the optimiser, but this is still not enough to prove that the candidate is indeed an optimiser. To complete the proof, it is necessary to have upper semiconti- nuity of the expected utility when considered as a function of the strategy variable. In [27], a counterexample (in whichu(x)is finite, but one could not find an optimiser) is given in the nondominated case. The author’s argument exploits precisely the lack of upper semicontinuity property in one model. Furthermore, [27] gives a sufficient

condition to have upper semicontinuity, namely the integrability of the positive part of the utility function under every possible model; see [27, Theorem 2.2] and also [8] for further developments. In our approach, upper semicontinuity follows from the finiteness of the dual value function for every model.

4 Utility functions onR

Assumption 4.1 The utility function U:R→R is bounded from above, nonde- creasing, concave andU (0)=0. Define the convex conjugate ofUby

V (y):=sup

x∈R

U (x)−xy

, y >0.

We also assume that

x→−∞lim U (x)

x = ∞, (4.1)

lim sup

y→∞

V (2y)

V (y) <∞. (4.2)

Remark 4.2 Under (4.1), the functionV takes finite values andV (y) >0 forylarge enough; hence (4.2) makes sense. The conditionU (0)=0 is used only to simplify calculations. Condition (4.1) is mild and so is (4.2); indeed, as shown in [32, Corol- lary 4.2(i)], for every utility functionUwith reasonable asymptotic elasticity, its con- jugateV satisfies (4.2). The studies [11,23] assumed a smoothU which is strictly concave on its entire domain; we need neither smoothness nor strict concavity ofU.

As discussed in [7,33], the choice of admissible trading strategies is a delicate issue in the context of utility maximisation with utility functions defined on the real line.

A common approach is to consider strategies whose wealth processes are bounded uniformly from below by a constant. This choice, however, turns out to be restrictive and fails to contain optimisers. In frictionless markets, [33] proved that for a utility function having reasonable asymptotic elasticity, the optimal investment process is a supermartingale under each martingale measureQsuch thatE[V (dQ/dP )]is finite.

We thus use the supermartingale property to define admissibility, just like in [28,10].

To begin with, we define

M^θ_V= {Q^θ:(S˜^θ, Q^θ)is aλ-consistent price system,E[V (dQ^θ/dP )]<∞}, the set of local martingale measures in consistent price systems for theθ-model with finite generalised relative entropy.

Definition 4.3 We define

A^θ(x):= {H∈V:φ_T =0, V^x(θ, H )is aQ^θ-supermartingale

for eachλ-consistent price system(S˜^θ, Q^θ)withQ^θ∈M^θV} and setA(x):=

θ∈ΘA^θ(x).

The optimisation problem becomes u(x)= sup

H∈A(x) θinf∈ΘE

U

W_T^x,liq(θ, H ) .

Assumption 4.4 For eachθ∈Θ, the price processS^θ admits aλ-SCPS(Q^θ,S˜^θ) such thatQ^θ∈M^θ_V.

Remark 4.5 Unlike in [11,12,23] and in Sect.3, we do not impose in the present section the existence of consistent price systems for every transaction cost coefficient 0< μ < λ; we only stipulate Assumption4.4. The following example shows that it is quite possible to have CPSs for relatively largeλwithout having them for arbitrarily smallμ. In this example, there is an obvious arbitrage, in the language of [17], which persists (ceases) with sufficiently small (large) transaction costs.

Example 4.6 Let us consider St=1+t+ 1

2πarctanWt, t∈ [0,1].

Ifλ <3/7, then(1−λ)S1>1=S0a.s.; therefore, there is no consistent price system.

Ifλ≥2/3, then

S_t(1−λ)≤3/4≤S_t, t∈ [0, T]. In other words,(P ,S˜≡3/4)is a consistent price system.

Theorem 4.7 Under Assumptions4.1 and 4.4, there exists a strategy H^∗∈A(x) such that

u(x)= inf

θ∈ΘE U

W_T^x,liq(θ, H^∗) .

Proof We adapt certain techniques of [30]. Our arguments bring novelties even in the case where Θ is a singleton (i.e., without model uncertainty). Define Φ^∗(x)= −U (−x)forx≥0. Its conjugate (in the sense of Sect.A.2below) is

Φ(y):=

0, if 0≤y≤β,

V (y)−V (β), ify > β,

whereβis the left derivative ofUat 0; see [5]. Note thatΦ,Φ^∗are Young functions andΦ is of class₂by (4.2).

LetHⁿ∈A(x),n∈N, be a maximising sequence, i.e.,

θinf∈ΘE U

W_T^x,liq(θ, Hⁿ) ↑u(x)≥U (x). (4.3) First, for allθ∈Θ, it holds that

sup

n

E U⁻

W_T^x,liq(θ, Hⁿ) <∞. (4.4)

Indeed, let us assume that there existsθ∈Θsuch that (4.4) does not hold, or equiv- alently, there exists a subsequencen_k=n^θ_k,k∈N, such that

E U⁻

W_T^x,liq(θ, H^nk) > k.

Let us denote byCan upper bound ofU; then E

U

W_T^x,liq(θ, H^nk) ≤C−E U⁻

W_T^x,liq(θ, H^nk) −→ −∞ ask→ ∞ which contradicts (4.3). Hence (4.4) indeed holds.

Consider aλ-strictly consistent price system(S˜^θ, Q^θ). Fenchel’s inequality gives U

V_T^x(θ, Hⁿ)

−V (dQ^θ/dP )≤(dQ^θ/dP )V_T^x(θ, Hⁿ) and therefore

(dQ^θ/dP )

V_T^x(θ, Hⁿ)₋

≤ U

V_T^x(θ, Hⁿ)

−V (dQ^θ/dP ) ₋

. (4.5) From (4.4) and (4.5), we deduce that

sup

n

E^Qθ

V_T^x(θ, Hⁿ)₋

<∞. (4.6)

Itô’s formula gives

dV_t^x(θ, Hⁿ)= −S_t^θdH_t^n,↑+(1−λ)S_t^θdH_t^n,↓+ ˜S_t^θdφ_tⁿ+φ_tⁿ₋dS˜_t^θ

=(S˜_t^θ−S_t^θ) dH_t^n,↑+

(1−λ)S_t^θ− ˜S_t^θ

dH_t^n,↓+φ_tⁿ₋dS˜_t^θ. This implies that

(H₀ⁿ)⁺(S₀^θ− ˜S₀^θ)+ _t

0

(S_u^θ− ˜S_u^θ) dH_u^n,↑+ _t

0

S˜_u^θ−(1−λ)S_u^θ dH_u^n,↓

+ ^t

0

φ_uⁿ₋dS˜_u^θ ₋

≤x+(H₀ⁿ)⁻

S₀^θ(1−λ)− ˜S₀^θ +

V_t^x(θ, Hⁿ)₋ +

_t

0

φ_uⁿ₋dS˜_u^θ ₊

.

In particular, _t

0

φ_uⁿ₋dS˜_u^θ ₋

≤x+(H₀ⁿ)⁻S₀^θ(1−λ)+

V_t^x(θ, Hⁿ)₋

. (4.7)

For each n, the process V^x(θ, Hⁿ) is a Q^θ-supermartingale; so there exists a Q^θ-martingale which dominates the right-hand side of (4.7) and also the left-hand side of the same expression. [1, Corollary 3.5] implies that_t

0φ_uⁿ₋dS˜_u^θ,t∈ [0, T], is

aQ^θ-supermartingale. We get sup_n(H₀ⁿ)⁻<∞in the same way as in the proof of Theorem3.6. Consequently, (4.6), (4.7) and the boundedness of(H₀ⁿ)⁻,n∈N, give

sup

n

E^Qθ

T 0

φ_tⁿ₋dS˜^θ_{t +}

<∞.

Noting that(S˜^θ, Q^θ)is aλ-strictly consistent price system, we obtain from the above arguments that

sup

n

E^Qθ

(H₀ⁿ)⁺δ(θ )+δ(θ ) _T

0

(dH_t^n,↑+dH_t^n,↓)

≤sup

n

E^Qθ

(H₀ⁿ)⁺(S₀^θ− ˜S₀^θ)+ _T

0

(S_t^θ− ˜S_t^θ) dH_t^n,↑+S˜_t^θ−(1−λ)S_t^θ dH_t^n,↓

<∞.

LemmaA.1implies the existence of convex weightsα_jⁿ≥0,j=n, . . . , M(n), with M(n)

j=n αⁿ_j=1 such thatH˜ⁿ:=M(n)

j=n αⁿ_jHⁿ→H^∗in V. Since the utility function is concave,H˜ⁿ,n∈N, is also a maximising sequence.

We now prove that H^∗ ∈A(x); in other words, the process V^x(θ, H^∗) is a Q^θ-supermartingale, for eachQ^θ∈M^θ_V and eachθ∈Θ. To do so, it suffices to con- trol the negative part ofV^x(θ, H^∗). It should be emphasised that (4.6) is not enough for our purposes and a stronger statement using Orlicz space theory is needed (see Sect.A.2). Using concavity ofUand linearity ofV^x(θ,·), we get from (4.4) that

sup

n

E U⁻

V_T^x(θ,H˜ⁿ) <∞. (4.8) Applying LemmaA.3 to the sequence of random variables in (4.8), we obtain convex weightsα_jⁿ≥0,n≤j≤M(n), withM(n)

j=n α_jⁿ=1 such that Zⁿ:=

M(n)

j=n

α_jⁿ

V_T^x(θ,H˜ⁿ)₋

satisfy

L:=sup

n

Zⁿ

Φ^∗<∞. (4.9)

By the Fenchel inequality and (4.9), E^Qθ

sup

n

Zⁿ

=LE^Qθ

sup_nZⁿ L

≤LE

Φ dQ^θ

dP

+LE

Φ^∗

sup_nZⁿ L

<∞ (4.10)

for eachQ^θ∈M^θ_V. WhenL=0, we have thatE^Qθ[sup_nZⁿ] =0 trivially. Now we define

Hⁿ:=

M(n)

j=n

α_jⁿH˜ⁿ,

which is also a maximising sequence. Using the fact that the negative part of a su- permartingale is a submartingale, we getV_t^x(θ, Hⁿ)⁻≤E^Qθ[V_T^x(θ, Hⁿ)⁻|Ft]and thus

sup

n

V_t^x(θ, Hⁿ)⁻≤sup

n

E^Qθ[V_T^x(θ, Hⁿ)⁻|Ft]. Taking expectations on both sides of the above inequality, we obtain

E^Qθ

sup

n

V_t^x(θ, Hⁿ)⁻

≤E^Qθ

sup

n

E^Qθ[V_T^x(θ, Hⁿ)⁻|Ft]

≤E^Qθ

E^Qθ sup

n

V_T^x(θ, Hⁿ)⁻Ft

=E^Qθ

sup

n

V_T^x(θ, Hⁿ)⁻

≤E^Qθ

sup

n

Zⁿ

<∞,

using (4.10). Since the random variable sup_n(V_t^x(θ, Hⁿ))⁻is an upper bound of the sequenceV_t^x(θ, Hⁿ)⁻,n∈N, this proves uniform integrability of that sequence un- derQ^θ at any timet∈ [0, T].

Clearly,Hⁿ→H^∗ in V and therefore V_t^x(θ, Hⁿ)→V_t^x(θ, H^∗) a.s., for every t∈ [0, T] \ZwhereZhas Lebesgue measure 0; see RemarkA.2. Also,

V_t^x(θ, H^∗)₋

≤E^Qθ

sup

n

V_T^x(θ, Hⁿ)⁻Ft

, t∈ [0, T], (4.11) where the latter process is a martingale, and hence(V_t^x(θ, H^∗))_t∈[0,T]is uniformly integrable. Let 0≤s≤t < T be both in[0, T] \Z. Noting the supermartingale prop- erty, Fatou’s lemma yields

E^Qθ[V_t^x(θ, H^∗)|Fs] =E^Qθ

lim inf

n→∞ V_t^x(θ, Hⁿ)Fs

≤lim inf

n→∞ E^Qθ[V_t^x(θ, Hⁿ)|Fs]

≤lim inf

n→∞ V_s^x(θ, Hⁿ)=V_s^x(θ, H^∗).

The same argument works fort=T, too. Now it extends to arbitraryt∈ [0, T]by using Fatou’s lemma and (4.11). Finally, it extends to arbitrarys∈ [0, T]by the back- ward martingale convergence theorem and by right-continuity ofs→V_s^x(θ, H^∗).

This means thatV^x(θ, H^∗)is aQ^θ-supermartingale and thereforeH^∗∈A(x).