Continuity of Utility Maximization under Weak Convergence
CCONTINUITY OF UTILITY MAXIMIZATION UNDER WEAKCONVERGENCE
ERHAN BAYRAKTAR , YAN DOLINSKY , AND JIA GUO
Abstract.
In this paper we find tight sufficient conditions for the continuityof the value of the utility maximization problem from terminal wealth withrespect to the convergence in distribution of the underlying processes. We alsoestablish a weak convergence result for the terminal wealths of the optimalportfolios. Finally, we apply our results to the computation of the minimal ex-pected shortfall (shortfall risk) in the Heston model by building an appropriatelattice approximation.
Contents
1. Introduction 12. Preliminaries and Main Results 32.1. On the verification of Assumption 2.2 and Assumption 2.3(ii) 73. The necessity of Assumptions 2.4,2.5,2.6 83.1. On the necessity of Assumption 2.4 83.2. On the necessity of Assumption 2.5 93.3. On the necessity of Assumption 2.6 124. The Lower Semi–Continuity under Weak Convergence 125. The Upper Semi–Continuity under Weak Convergence 175.1. Proof of Theorem 2.2 196. Lattice Based Approximations of the Heston Model 196.1. Discretization 206.2. Verification of Assumption 2.5 227. Approximations of the Shortfall Risk in the Heston Model 257.1. Numerical Results 28References 321.
Introduction
A basic problem of mathematical finance is the problem of an economic agent,who invests in a financial market so as to maximize the expected utility of histerminal wealth. The problem of utility maximization problem is going back to theseminal work by R. Merton [34, 35] and continuing e.g. in [7, 26, 28, 11, 9, 21, 40, 4].
Mathematics Subject Classification.
Key words and phrases.
Incomplete Markets, Utility Maximization, Weak Convergence .E. Bayraktar is supported in part by the National Science Foundation under grant DMS-1613170 and in part by the Susan M. Smith Professorship.Y. Dolinsky is supported in part by the Israeli Science Foundation under Grant 160/17. a r X i v : . [ q -f i n . M F ] J un This paper deals with the following question: Given a utility function and asequence of financial markets with underlying assets ( S ( n ) ) n ∈ N that are convergingweakly to S , under which conditions do the values of the utility maximizationproblems (from terminal wealth) converge to the corresponding value for the modelgiven by S ?Our paper is motivated by the following economic applications. First, in a general(incomplete) financial market the problem of utility maximization does not admitan explicit solution and so numerical schemes come naturally into the picture.Another important motivation is the practical limitations of calibration. Namely,we want to understand whether the utility maximization problem is stable undersmall misspecifications of the law of the asset prices.To the best of our knowledge, the continuity under weak convergence was studiedonly in a complete market setup (see [20, 39, 41]). In this work we consider thisconvergence question for general incomplete market models and continuous (as afunction of the terminal wealth) random utility functions.We divide the proof of our main result, namely Theorem 2.1, into two mainsteps identifying when we have lower and upper-semi-continuity respectively. Weshow that for the lower semi–continuity to hold, it is sufficient (in addition to sometechnical assumptions) that the approximating sequence ( S ( n ) ) n ∈ N has vanishingjump activity. The formal condition is given in Assumption 2.4. The main idea isto prove that an admissible integral of the form (cid:82) γdS can be approximated in theweak sense by admissible integrals of the form (cid:82) γ ( n ) dS ( n ) , n ∈ N . The assumptionon the jump activity is essential for the admissibility of the approximating sequence.We demonstrate the necessity of this assumption with an example; see Section 3.1.We would like to emphasize that the concavity of the utility function is not necessaryin this step.The second step, namely, the upper semi–continuity is more delicate. We provethat if the utility function is concave and the state price densities in the limitmodel can be approximated by state price densities in the approximating sequence(see Assumption 2.5) then upper semi–continuity holds. The proof relies on theoptional decomposition theorem. In Sections 3.2–3.3 we discuss the necessity ofour assumptions. Example 3.3 in Section 3.2 is surprising and quite interestingin its own right. In this example we construct a sequence of complete marketmodels (binomial models) which converge weakly to an incomplete market model(a stochastic volatility model).In addition to the convergence of the values, we prove a weak convergence forthe terminal wealths of the optimal portfolios; see Theorem 2.2. An open questionis whether there is a convergence of the optimal trading strategies, i.e. of the inte-grands. In a complete market setup convergence of the optimal trading strategieswere obtained in [20, 39]. The proof was based on an explicit characterization of theoptimal trading strategies. In the incomplete market setup we do not have explicitformulas for the optimal portfolios. Hence, the problem is much more complicatedand requires additional machinery (see Remark 2.7).We apply our continuity results in order to construct an approximating sequenceverifying all our assumptions for the Heston model in Section 6. Our method isbased on recombining trinomial trees and so, for technical reasons we truncate themodel in such a way that the volatility is bounded. The novelty of our constructionis that the approximating sequence lies on a grid and satisfies the assumptions required for the continuity of the value of the utility maximization problem fromterminal wealth. The grid structure enables efficient numerical computations forstochastic control problems via dynamic programming.Our last contribution, which is the subject of Section 7, is the implementationof the constructed approximating models for the numerical computations in theHeston model. For the shortfall risk measure we show that the truncation error canbe controlled, see Lemma 7.1, so our result applies to the non-truncated Hestonmodel. It is well known (see [10, 17, 13, 38]) that in the Heston model the super–replication price is prohibitively high and lead to buy–and–hold strategies. Namely,the cheapest way to super–hedge a European call option is to buy one stock at theinitial time and keep that position till maturity. That is why the computationof shortfall risk is important. This cannot be done analytically and so numericalschemes come into picture.A closely related topic to the one studied in the present paper is the stability ofthe utility maximization problem under market parameters and the investor pref-erences. Since the work [25] which dealt with complete markets, large progress wasmade in the study of the stability of the utility maximization problem in incom-plete markets (see, for instance, [31, 30, 27, 32, 3, 1, 33, 36]). The main differencefrom our setup is that in these papers the stochastic base is fixed while in oursetup each financial model is defined on its own probability space. As a result,while the above cited papers deal with the stability of the models with respect tosmall perturbations, we are able to obtain numerical approximations using discretemodels.The rest of the paper is organized as follows. In the next section we introducethe setup and formulate the main results. In Section 3 we discuss Assumptions2.4,2.5,2.6 and demonstrate their necessity. In Section 4 we prove the lower semi–continuity. In Section 5 we prove the upper semi–continuity. In Section 5.1 weestablish Theorem 2.2. Section 6 is devoted to the construction of an approximatingsequence for the Heston model. In Section 7 we provide a detailed numerical analysisfor shortfall risk minimization.2. Preliminaries and Main Results
We consider a model of a security market which consists of d risky assets which wedenote by S = ( S (1) t , ..., S ( d ) t ) ≤ t ≤ T , where T < ∞ is the time horizon. We assumethat the investor has a bank account that, for simplicity, bears no interest. Theprocess S is assumed to be a continuous semi–martingale on a filtered probabilityspace (Ω , F , ( F St ) ≤ t ≤ T , P ) where the filtration ( F St ) ≤ t ≤ T is the usual filtrationgenerated by S . Namely, the filtration {F St } Tt =0 is the minimal filtration which iscomplete, right continuous and satisfies F t ⊃ σ { S u : u ≤ t } . Without a loss ofgenerality we take F := F ST .A (self–financing) portfolio π is defined as a pair π = ( x, γ ) where the constant x is the initial value of the portfolio and γ = ( γ ( i ) ) ≤ i ≤ d is a predictable S –integrableprocess specifying the amount of each asset held in the portfolio. The correspondingportfolio value process is given by V πt := x + (cid:90) t γ u dS u , t ∈ [0 , T ] . Observe that the continuity of S implies that the wealth process { V πt } Tt =0 iscontinuous as well. We say that a trading strategy π is admissible if V πt ≥ ∀ t ≥ . For any x > A ( x ) the set of all admissible trading strategies.Denote by M ( S ) the set of all equivalent (to P ) local martingale measures. Weassume that M ( S ) (cid:54) = ∅ . This condition is intimately related to the absence ofarbitrage opportunities on the security market. See [12] for a precise statement andreferences.Next, we introduce our utility maximization problem. Consider a continuousfunction U : (0 , ∞ ) × D ([0 , T ]; R d ) → R . As usual, D ([0 , T ]; R d ) denotes the spaceof all RCLL (right continuous with left limits) functions f : [0 , T ] → R d equippedwith the Skorokhod topology (for details see [5]). Assumption 2.1. (i) For any s ∈ D ([0 , T ]; R d ) the function U ( · , s ) is non–decreasing.(ii) For any x > we have E P [ U ( x, S )] > −∞ . We extend U to R + × D ([0 , T ]; R d ) by U (0 , s ) := lim v ↓ U ( v, s ). In view ofAssumption 2.1(i) the limit exists (might be −∞ ).For a given initial capital x > u ( x ) := sup π ∈A ( x ) E P [ U ( V πT , S )] , where we set −∞ + ∞ = −∞ . Namely, for a random variable X which satisfies E P [max( − X, ∞ we set E P [ X ] := −∞ .Let us notice that Assumption 2.1(ii) implies u ( x ) > −∞ . Assumption 2.2.
The function u : (0 , ∞ ) → R ∪ {∞} is continuous. Namely, forany x > we have u ( x ) = lim y → x u ( y ) where a priori the joint value can be equalto ∞ . Next, for any n , let S ( n ) = ( S n, t , ..., S n,dt ) ≤ t ≤ T be a RCLL semi–martingaledefined on some filtered probability space (Ω n , F ( n ) , ( F ( n ) t ) ≤ t ≤ T , P n ) where thefiltration ( F ( n ) t ) ≤ t ≤ T satisfies the usual assumptions (right continuity and com-pleteness). For the n –th model we define A n ( x ) as the set of all pairs π n = ( x, γ ( n ) )such that γ ( n ) is a predictable S ( n ) –integrable process and the resulting portfoliovalue process V π n t := x + (cid:90) t γ ( n ) u dS ( n ) u ≥ , t ∈ [0 , T ] , is non-negative. Set, u n ( x ) := sup π n ∈A n ( x ) E P n [ U ( V π n T , S ( n ) )] . We assume the weak convergence S ( n ) ⇒ S on the space D ([0 , T ]; R d ) equippedwith the Skorokhod topology (for details see Chapter 4 in [5]). Moreover, we assumethe following uniform integrability assumptions. Assumption 2.3. (i) For any x > the family of random variables { U − ( x, S ( n ) ) } n ∈ N is uniformlyintegrable where U − := max( − U, .(ii) For any x > the family of random variables { U + ( V π n T , S ( n ) ) } n ∈ N ,π n ∈A n ( x ) isuniformly integrable, where U + := max( U, . Remark 2.1.
The verification of Assumption 2.2 and Assumption 2.3(ii) can be adifficult task. In Section 2.1 we provide quite general and easily verifiable conditionswhich are sufficient for the above assumptions to hold true.
Due to the admissibility requirements we will need the following assumptionwhich bounds the uncertainty of the jump activity. This assumption will be dis-cussed in details in Section 3.1.
Assumption 2.4.
For any n ∈ N consider the non-decreasing RCLL process givenby D ( n ) t := sup ≤ u ≤ t | S ( n ) u − S ( n ) u − | , t ∈ [0 , T ] where | · | denotes the Euclidean normin R d . For any n , there exists an adapted (to ( F ( n ) t ) ≤ t ≤ T ) left continuous process { J ( n ) t } Tt =0 , n ∈ N such that inf ≤ t ≤ T (cid:16) J ( n ) t − D ( n ) t (cid:17) ≥ a.s. and J ( n ) T → inprobability. Remark 2.2.
Let us notice that Assumption 2.4 and the weak convergence S ( n ) ⇒ S imply that S is a continuous process. Indeed, from Assumption 2.4 we have sup ≤ u ≤ T | S ( n ) u − S ( n ) u − | → . This together with Theorem 13.4 in [5] yields that S is continuous. Now, we ready to formulate our first result (lower semi–continuity) which willbe proved in Section 4.
Proposition 2.1.
Under Assumptions 2.1–2.2, Assumption 2.3(i) and Assumption2.4 we have u ( x ) ≤ lim inf n →∞ u n ( x ) , ∀ x > . Next, we treat upper semi–continuity.
Assumption 2.5.
Recall the set M ( S ) of all equivalent local martingale measures.Denote by M ( S ( n ) ) , n ∈ N the set of all equivalent local martingale measures forthe n –th model. For any Q ∈ M ( S ) there exists a sequence of probability measures Q n ∈ M ( S ( n ) ) , n ∈ N such that under P n the joint distribution of (cid:16) { S ( n ) t } Tt =0 , d Q n d P n (cid:17) on the space D ([0 , T ]; R d ) × R converges to the joint distribution of (cid:0) { S t } Tt =0 , d Q d P (cid:1) under P . We denote this relation by (2.1) (cid:18)(cid:18) S ( n ) , d Q n d P n (cid:19) ; P n (cid:19) ⇒ (cid:18)(cid:18) S, d Q d P (cid:19) ; P (cid:19) . Remark 2.3.
The verification of Assumption 2.5 requires a comfortable represen-tation of the corresponding local martingale measures. This is the case for tree basedapproximations of diffusion processes. In Section 6.2 we illustrate the verificationof Assumption 2.5 for tree based approximations of the Heston model.We do notice that in order to verify Assumption 2.5 it is sufficient to establish(2.1) for a dense subset of (cid:8) d Q d P : Q ∈ M ( S ) (cid:9) . This simplification will be used inSection 6.2. Remark 2.4.
In the complete market setup, Assumption 2.5 is equivalent to thejoint convergence of the underlying assets and of the Radon–Nikodym derivativeswith respect to the unique risk neutral probability measure. In this case (see [39])Assumption 2.5 guarantees the convergence of both the values and the optimal strate-gies.
Assumption 2.6.
For any s ∈ D ([0 , T ]; R d ) , the function U ( · , s ) is concave. Assumption 2.6 says that the investor can not gain from additional randomiza-tion.The following upper semi–continuity result will be proved in Section 5.
Proposition 2.2.
Under Assumption 2.1(i), Assumption 2.3(ii) and Assumptions2.5,2.6 we have u ( x ) ≥ lim sup n →∞ u n ( x ) , ∀ x > . We now combine the statements of the above propositions and state them as themain theorem of our paper:
Theorem 2.1.
Under Assumptions 2.1–2.3,2.4,2.5,2.6 we have (2.2) u ( x ) = lim n →∞ u n ( x ) , ∀ x > . Proof.
Follows from Proposition 2.1 and Proposition 2.2. (cid:3)
Remark 2.5.
Observe that in view of Assumption 2.3 we have −∞ < lim inf n →∞ u n ( x ) ≤ lim sup n →∞ u n ( x ) < ∞ , ∀ x > . We conclude that the joint value in (2.2) is finite.
Next, we establish the weak convergence for the optimal terminal wealths.
Theorem 2.2.
Assume that Assumptions 2.1–2.3,2.4,2.5,2.6 hold true. Moreover,assume that for any s ∈ D ([0 , T ]; R d ) the function U ( · , s ) is strictly concave. Let x > and ˆ π n ∈ A n ( x ) , n ∈ N be a sequence of asymptotically optimal portfolios,namely (2.3) lim n →∞ (cid:16) u n ( x ) − E P n [ U ( V ˆ π n T , S ( n ) )] (cid:17) = 0 . Then (cid:16) S ( n ) , V ˆ π n T (cid:17) ⇒ (cid:0) S, V ˆ πT (cid:1) , where ˆ π ∈ A ( x ) is the unique portfolio that satisfies u ( x ) = E P [ U ( V ˆ πT , S )] . The proof of Theorem 2.2 will be given in Section 5.1.
Remark 2.6.
It is well known (see Theorem 2.2 in [28]) that for a utility functionwhich is strictly concave there exists a unique optimizer. Although in [28] the au-thors do not consider a random utility, their argument can be without much effortextended to our setup.
Remark 2.7.
A natural question is whether Theorem 2.2 can be applied for estab-lishing a convergence result for optimal trading strategies (the integrands). It seemsthat this problem is closely related to the robustness of martingale representationswhich was studied in [24].The main result in [24] (Theorem A) deals with case where the underlying pro-cesses are martingales with respect to the same filtration, this is not satisfied inour setup. Section 4 in [24] deals with the weak convergence setup, however theobtained results are limited to some particular cases for which there are explicitrepresentation for the integrands.
On the verification of Assumption 2.2 and Assumption 2.3(ii).
The following result provides a simple and quite general condition which impliesAssumption 2.2.
Lemma 2.1.
Assume that Assumption 2.1 holds true and there exist continuousfunctions m , m : [0 , → R + with m (0) = m (0) = 0 (modulus of continuity)and a non-negative random variable ζ ∈ L (Ω , F , P ) such that for any λ ∈ (0 , and v > U ((1 − λ ) v, S ) ≥ (1 − m ( λ )) U ( v, S ) − m ( λ ) ζ. Then Assumption 2.2 holds true.Proof.
In view of the fact that u is a non-decreasing function (follows from As-sumption 2.1(i)) it sufficient to prove that for any x > α ↓ u ((1 − α ) x ) ≥ lim α ↓ u ((1 + α ) x ) . For any β, y > y, { γ t } Tt =0 ) → ( βy, { βγ t } Tt =0 ) is a bijection between A ( y )and A ( βy ). Thus,lim α ↓ u ((1 − α ) x ) ≥ lim α ↓ (cid:18)(cid:18) − m (cid:18) − − α α (cid:19)(cid:19) u ((1 + α ) x ) − m (cid:18) − − α α (cid:19) E P [ ζ ] (cid:19) = lim α ↓ u ((1 + α ) x ) . (cid:3) Remark 2.8.
We notice that the power and the log utility satisfy the assumptionsof Lemma 2.1. On the other hand for these utility functions Assumption 2.2 isstraightforward.A “real” application of Lemma 2.1 is the case which corresponds to the utilityfunction given by (3.1). In this case, if v ≥ S T − λ then U ((1 − λ ) v, S ) = U ( v, S ) = 0 .If v < S T − λ then | U ((1 − λ ) v, S ) − U ( v, S ) | ≤ λv ≤ λ − λ S T . Thus, for m ( λ ) := 0 , m ( λ ) := λ − λ and ζ := S T the assumptions of Lemma 2.1 hold true (provided that E P [ S T ] < ∞ ). Next, we treat Assumption 2.3(ii).
Lemma 2.2.
Suppose there exist constants
C > , < γ < and q > − γ whichsatisfy the following.(I) For all ( v, s ) ∈ (0 , ∞ ) × D ([0 , T ]; R d ) , (2.4) U ( v, s ) ≤ C (1 + v γ ) . (II) For any n ∈ N there exists a local martingale measure Q n ∈ M ( S ( n ) ) such that (2.5) sup n ∈ N E Q n (cid:20)(cid:18) d P n d Q n (cid:19) q (cid:21) < ∞ . Then Assumption 2.3(ii) holds true.Proof.
Let p = qq − . Clearly p > γ . Thus in view of (2.4), in order to prove thatAssumption 2.3(ii) holds true, it suffices to show that for any x > n ∈ N sup π n ∈A n ( x ) E P n [( V π n T ) /p ] < ∞ . For any n ∈ N and π n ∈ A n ( x ), { V π n t } Tt =0 is a Q n super–martingale. Hence, fromthe Holder inequality (observe that p + q = 1) we getsup n ∈ N sup π n ∈A n ( x ) E P n [( V π n T ) /p ]= sup n ∈ N sup π n ∈A n ( x ) E Q n (cid:20) ( V π n T ) /p d P n d Q n (cid:21) ≤ sup n ∈ N sup π n ∈A n ( x ) ( E Q n [ V π n T ]) /p sup n ∈ N (cid:18) E Q n (cid:20)(cid:18) d P n d Q n (cid:19) q (cid:21)(cid:19) /q ≤ x /p sup n ∈ N (cid:18) E Q n (cid:20)(cid:18) d P n d Q n (cid:19) q (cid:21)(cid:19) /q < ∞ , and the result follows. (cid:3) The necessity of Assumptions 2.4,2.5,2.6
On the necessity of Assumption 2.4.
Let us explain by example whyAssumption 2.4 is essential for the lower semi–continuity to hold.
Example 3.1.
Naive discretization does not work .Let d = 1 . Consider a random utility which corresponds to shortfall risk minimiza-tion for a call option with strike price K > . Namely, we set (3.1) U ( v, s ) := − (( s T − K ) + − v ) + . We have, u ( x ) = − inf π ∈A ( x ) E P (cid:104)(cid:0) ( S T − K ) + − V πT (cid:1) + (cid:105) . Consider the Black–Scholes model S t = S e σW t − σ t/ , t ∈ [0 , T ] where σ > is a constant volatility and W = { W t } Tt =0 is a Brownian motion (under P ).We take the naive discretization and define the processes S ( n ) , n ∈ N , by S ( n ) t := S kTn , kT /n ≤ t < ( k + 1) T /n.
Let F ( n ) the usual filtration which is generated by S ( n ) . Namely, F ( n ) t := σ (cid:110) S Tn , ..., S kTn , N (cid:111) , kT /n ≤ t < ( k + 1) T /n where N is the collection of all null sets. We also set P n := P .It is easy to see that S ( n ) ⇒ S and Assumptions 2.1–2.3 hold true (for Assump-tion 2.2 see Remark 2.8).Next, we check Assumption 2.4. Fix n . Recall the processes D ( n ) , J ( n ) fromAssumption 2.4. First, observe that if J ( n ) is an adapted left continuous process,then for all k < n J ( n ) ( k +1) Tn is F ( n ) kTn measurable. Notice that for or all k < n , ess sup (cid:18) S ( n ) ( k +1) Tn − S ( n ) kTn |F ( n ) kTn (cid:19) = ∞ a.s.As usual ess sup( Y |G ) is the minimal random variable (which may take the value ∞ ) that is G measurable and ≥ Y a.s. These two simple observations yield that there is no (finite) adapted left continuous process { J ( n ) t } Tt =0 which satisfy J ( n ) ( k +1) Tn ≥ D ( n ) ( k +1) Tn . Thus, Assumption 2.4 is not satisfied.In [37] (see Section 6.1.2) it was proved that for the processes S ( n ) , n ∈ N definedabove and the initial capital x := E P [( S T − K ) + ] (i.e. the Black–Scholes price) wehave lim inf n →∞ inf π n ∈A n ( x ) E P (cid:104)(cid:0) ( S T − K ) + − V π n T (cid:1) + (cid:105) > . Clearly, the fact that x is the Black–Scholes price implies that inf π ∈A ( x ) E P (cid:104)(cid:0) ( S T − K ) + − V πT (cid:1) + (cid:105) = 0 . We get u ( x ) = 0 > lim sup n →∞ u n ( x ) , and as a result Proposition 2.1 does not hold true. Example 3.2.
Discrete approximations with vanishing growth rates dowork .Consider a setup where for any n , S ( n ) is a pure jump process of the form S ( n ) t = m n (cid:88) i =1 S ( n ) τ ( n ) i I τ ( n ) i ≤ t<τ ( n ) i +1 + S ( n ) T I t = T where m n ∈ N and τ ( n )1 < τ ( n )2 < ... < τ ( n ) m n +1 = T are stopping times withrespect to {F ( n ) t } Tt =0 .Assume that there exists a deterministic sequence a n > , n ∈ N such that lim n →∞ a n = 0 and | S ( n ) τ ( n ) i +1 − S ( n ) τ ( n ) i | ≤ a n | S ( n ) τ ( n ) i | a.s , ∀ i, n. Then Assumption 2.4 holds true with the processes J ( n ) t := a n (cid:32) m n (cid:88) i =1 max ≤ j ≤ i | S ( n ) τ ( n ) j | I τ ( n ) i On the necessity of Assumption 2.5. A natural question to ask is whetherAssumption 2.5 can be replaced by a simpler one.In [22] the authors analyzed when weak convergence implies the convergence ofoption prices. Roughly speaking, the main result was that under contiguity proper-ties of the sequences of physical measures with respect to the martingale measuresthere is a convergence of prices of derivative securities. The contiguity assumption(for the exact definition see [22]) is simpler than Assumption 2.5 and deals only withthe approximating sequence. The main advantage of such assumption that it doesnot require establishing weak convergence (unlike Assumption 2.5). However, thisclassical result assumes that the limit model is complete. In general, in incompletemarkets “strange phenomena” can happen as we will demonstrate in Example 3.3. In Example 3.3 we construct a sequence of binomial (discrete) martingales S ( n ) considered with their natural filtrations that converge weakly to a continuous mar-tingale S (the contiguity assumption trivially holds true). Surprisingly, the limitingmodel, which is given by the martingale S , is a fully incomplete market (see Defini-tion 2.1 in [13]) and the set of all equivalent martingale measures is dense in the setof all martingale measures (for a precise formulation see Lemma 8.1 in [13]). Weuse this construction to illustrate that Assumption 2.5 is the “right” assumption tomake.The cornerstone of our construction is the following result which was establishedin [6] (see Theorem 8 there). For the reader’s convenience we provide a shortself-contained proof. Lemma 3.1. Let ξ i = ± , i ∈ N be i.i.d. and symmetric. Define the processes W ( n ) t , ˆ W ( n ) t , t ∈ [0 , T ] by W ( n ) t := (cid:113) Tn (cid:80) ki =1 ξ i , kTn ≤ t < ( k +1) Tn , ˆ W ( n ) t := (cid:113) Tn (cid:80) ki =1 (cid:81) ij =1 ξ j , kTn ≤ t < ( k +1) Tn where (cid:80) i =1 ≡ . Then, we have the weak convergence ( W ( n ) , ˆ W ( n ) ) ⇒ ( W, ˆ W ) , where W and ˆ W are independent Brownian motions.Proof. We apply the martingale invariance principle given by Theorem 2.1 in [43].For any n define the filtration {G ( n ) t } Tt =0 by G ( n ) t = σ { ξ , ..., ξ k } for kT /n ≤ t < ( k + 1) T /n . Observe that W ( n ) , ˆ W ( n ) are martingales with respect to the filtration G ( n ) . Thus it remains to establish (2)–(3) in [43]. Clearly,sup ≤ t ≤ T | W ( n ) t − W ( n ) t − | = sup ≤ t ≤ T | ˆ W ( n ) t − ˆ W ( n ) t − | = (cid:114) Tn , and so the maximal jump size goes to zero as n → ∞ . Moreover, [ W ( n ) ] t =[ ˆ W ( n ) ] t = kT /n for kT /n ≤ t < ( k + 1) T /n . Thus, [ W ( n ) ] t → t and [ ˆ W ( n ) ] t → t as n → ∞ .It remains to show that for all t ∈ [0 , T ](3.2) [ W ( n ) , ˆ W ( n ) ] t → . Indeed, let n ∈ N and kT /n ≤ t < ( k + 1) T /n . Clearly,[ W ( n ) , ˆ W ( n ) ] t = Tn k (cid:88) i =1 i − (cid:89) j =1 ξ j , kTn ≤ t < ( k + 1) Tn , where (cid:81) i =1 ≡ 1. Observe that the random variables (cid:81) mj =1 ξ j = ± m ∈ N arei.i.d. and symmetric. Thus, E (cid:18)(cid:16) [ W ( n ) , ˆ W ( n ) ] t (cid:17) (cid:19) = T n k ≤ T tn and (3.2) follows. This completes the proof. (cid:3) Example 3.3. Binomial models can converge weakly to fully incompletemarkets. Let d = 1 . For any n ∈ N define the stochastic processes { ν ( n ) t } Tt =0 and { S ( n ) t } Tt =0 by ν ( n ) t := (cid:81) ki =1 (cid:16) (cid:113) Tn ξ i (cid:17) , kTn ≤ t < ( k +1) Tn ,S ( n ) t := (cid:81) ki =1 (cid:18) ν ( n ) ( i − Tn , ln n ) (cid:113) Tn (cid:81) ij =1 ξ j (cid:19) , kTn ≤ t < ( k +1) Tn , where ξ i = ± , i ∈ N are i.i.d. and symmetric. Let P n be the correspondingprobability measure.We assume that n is sufficiently large so that S ( n ) and min( ν ( n ) , ln n ) are strictlypositive. Let F ( n ) be the filtration which is generated by S ( n ) , F ( n ) t := σ (cid:110) S Tn , ..., S kTn (cid:111) , kT /n ≤ t < ( k + 1) T /n. Observe that F ( n ) t = σ { ξ , ..., ξ k } for kT /n ≤ t < ( k + 1) T /n . Moreover, theconditional support of supp (cid:18) S ( n ) ( k +1) Tn | S ( n ) Tn , ..., S ( n ) kTn (cid:19) consists of exactly two points,and so the physical measure P n is the unique martingale measure for S ( n ) .From Theorems 4.3–4.4 in [15] and Lemma 3.1 we obtain the weak convergence ( S ( n ) , ν ( n ) ) ⇒ ( S, ν ) where ( S, ν ) is the (unique strong) solution of the SDE dS t = ν t S t d ˆ W t , S = 1(3.3) dν t = ν t dW t , ν = 1 where W and ˆ W are independent Brownian motions (under P ).Namely, for the complete binomial models S ( n ) , n ∈ N we have the weak conver-gence S ( n ) ⇒ S where S is the distribution of the stochastic volatility model givenby (3.3). This is a specific case of the Hull–White model which was introduced in[23]. From Theorem 3.3 in [42] it follows that { S t } Tt =0 is a true martingale. Hence, E P [ S T ] = S = 1 = S ( n )0 = E P n [ S ( n ) T ] . This together with Theorem 3.6 in [5] gives that the random variables { S ( n ) T } n ∈ N are uniformly integrable.Let us observe that Assumption 2.5 does not hold true. Indeed, for any n wehave the equality M ( S ( n ) ) = { P n } . Hence, (( S, 1) ; P ) is the only cluster point forthe distributions (cid:16)(cid:16) S ( n ) , d Q n d P n (cid:17) ; P n (cid:17) , Q n ∈ M ( S ( n ) ) . Since the set M ( S ) is not asingleton then clearly Assumption 2.5 is not satisfied.Next, let K > . Consider a call option with strike price K and the utility func-tion given by (3.1). Obviously, Assumption 2.1(i) and Assumption 2.3(ii) ( U + ≡ )are satisfied. We want to demonstrate that Proposition 2.2 does not hold true.For any n ∈ N let V n be the unique arbitrage free price of the above call optionin the (complete) model given by S ( n ) . From the weak convergence S ( n ) ⇒ S andthe uniform integrability of { S ( n ) T } n ∈ N we get lim n →∞ V n = E P (cid:104) ( S T − K ) + (cid:105) < S = 1 . In particular there exists (cid:15) > such that for sufficiently large n we have V n < − (cid:15) .Thus, lim n →∞ u n (1 − (cid:15) ) = 0 . On the other hand, the model given by S is a fully incomplete market (see Definition2.1 and Example 2.5 in [13]). In [13, 38] it was proved that in fully incompletemarkets the super–replication price is prohibitively high and lead to buy–and–holdstrategies. Namely, the super–hedging price of a call option is equal to the initialstock price S = 1 . Thus u (1 − (cid:15) ) < and so Proposition 2.2 does not hold. On the necessity of Assumption 2.6.Example 3.4. Non-concave utility. Let d = 1 . Assume that the investor utility function is given by U ( v, s ) := min(2 , max( v, , and depends only on the wealth. We notice that the function U does not satisfyAssumption 2.6.For any n ∈ N consider the binomial model given by S ( n ) t := k (cid:89) i =1 (cid:18) ξ i n (cid:19) , kTn ≤ t < ( k + 1) Tn , where ξ i = ± , i ∈ N are i.i.d. and symmetric. Namely, P n is the unique martingalemeasure for the n –th model. Clearly, for the constant process S ≡ we have theweak convergence S ( n ) ⇒ S . Thus, Assumption 2.1(i), Assumption 2.3(ii) andAssumption 2.5 are satisfied.Next, consider the initial capital x := 1 . Observe that for any n , there is a set A n ∈ σ { ξ , ..., ξ n } with P n ( A n ) = 1 / . Thus, from the completeness of the binomialmodels we get that there exists π n ∈ A n (1) such that V π n T = 2 I A n . In particular, u n (1) ≥ E P n [min(2 , max(2 I A n , / , n ∈ N . On the other hand, trivially u (1) = 1 , which means that Proposition 2.2 does nothold true. The paper [41] studies the continuity of the value of the utility maximizationproblem from terminal wealth (under convergence in distribution) in a completemarket. The author does not assume that the utility function is concave. Themain result says that if the limit probability space is atomless and the atoms inapproximating sequence of models are vanishing (see Assumption 2.1 in [41]) thencontinuity holds. Clearly, this is not satisfied in the Example 3.4 above where thefiltration generated by the limit process is trivial.An open question is to understand whether the continuity result from [41] canbe extended to the incomplete case.4. The Lower Semi–Continuity under Weak Convergence In this section we prove Proposition 2.1. We start by establishing a generalresult.For any M > n ∈ N introduce the set Γ ( n ) M of all simple predictableintegrands of the from γ ( n ) t = k (cid:88) i =1 β i I t i Let γ be a predictable process (with respect to ( F St ) ≤ t ≤ T ) with | γ | ≤ M for some constant M . Then there exists a sequence γ ( n ) ∈ Γ ( n ) M , n ∈ N such thatwe have the weak convergence (4.1) (cid:32) { S ( n ) t } Tt =0 , (cid:26)(cid:90) t γ ( n ) u dS ( n ) u (cid:27) Tt =0 (cid:33) ⇒ (cid:32) { S t } Tt =0 , (cid:26)(cid:90) t γ u dS u (cid:27) Tt =0 (cid:33) on the space D ([0 , T ]; R d ) × D ([0 , T ]; R ) .Proof. On the space (Ω , F , ( F St ) ≤ t ≤ T , P ), let Γ M be the set of all integrands of theform(4.2) γ t = k (cid:88) i =1 β i I t i Proof of Proposition 2.1 .The proof will be done in two steps. Step I: For any x > A ( x ) ⊂ A ( x ) be the set of all admissible portfolios π = ( x, γ ) such that γ is predictable, uniformly bounded and of bounded variation.In this step we show that for any x > x > u ( x ) ≤ sup π ∈A ( x ) E P [ U ( V πT , S )] . A priori the left hand side and the right hand side of (4.6) can be both equal to ∞ .Let ¯ π = ( x , ¯ γ ) ∈ A ( x ) be an arbitrary portfolio. By applying the density argu-ment given by Theorem 3.4 in [2] we obtain that there exists an adapted continuousprocess of bounded variation ˜ γ = { ˜ γ t } Tt =0 such thatsup ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t ˜ γ u dS u − (cid:90) t ¯ γ u dS u (cid:12)(cid:12)(cid:12)(cid:12) ≤ x − x π := ( x , ˜ γ ) satisfies(4.7) V ˜ πt ≥ V ¯ πt + x − x ≥ x − x , t ∈ [0 , T ] . Next, for the continuous process ˜ γ define the stopping times θ n := T ∧ inf { t : | ˜ γ t | = n } , n ∈ N and the trading strategies ˜ γ ( n ) t := I t ≤ θ n ˜ γ t , t ∈ [0 , T ] . Set ˜ π n = ( x , ˜ γ ( n ) ). Clearly, | ˜ γ ( n ) | ≤ n and from (4.7) we have V ˜ π n t = V ˜ πt ∧ θ n ≥ x − x , t ∈ [0 , T ] . Hence, ˜ π n ∈ A ( x ). Observe that θ n ↑ T a.s., and solim n →∞ V ˜ π n T = lim n →∞ V ˜ πθ n = V ˜ πT . This together with Fatou’s Lemma, Assumption 2.1 (notice that V ˜ π n T ≥ x − x > U is continuous and (4.7) gives E P [ U ( V ¯ πT , S )] ≤ E P [ U ( V ˜ πT , S )] ≤ lim inf n →∞ E P [ U ( V ˜ π n T , S )] . Since ¯ π ∈ A ( x ) was arbitrary we complete the proof of (4.6). Step II: In view of (4.6) and Assumption 2.2, in order to prove Proposition 2.1 itsufficient to show that for any initial capital x > 0, 0 < (cid:15) < x and π ∈ A ( x − (cid:15) )there exists a sequence π n ∈ A n ( x ), n ∈ N which satisfies(4.8) lim inf n →∞ E P n [ U ( V π n T , S ( n ) )] ≥ E P [ U ( V πT , S )] . Let 0 < (cid:15) < x and π = ( x − (cid:15), γ ) ∈ A ( x − (cid:15) ). Let M > | γ | ≤ M .Lemma 4.1 provides the existence of simple integrands γ ( n ) ∈ Γ ( n ) M , n ∈ N whichsatisfy (4.1).For a given n , the portfolio ( x, γ ( n ) ) might fail to be admissible and so a modifi-cation is needed. Recall Assumption 2.4 and the stochastic processes J ( n ) , n ∈ N .For any n ∈ N introduce the stopping time(4.9) Θ n := T ∧ inf (cid:26) t : x + (cid:90) t − γ ( n ) u dS ( n ) u < (cid:15) + M dJ ( n ) t (cid:27) , and define the portfolio π n = ( x, ¯ γ ( n ) ) by ¯ γ ( n ) t := I t ≤ Θ n γ ( n ) t . Let us show that V π n t ≥ (cid:15) for all t ∈ [0 , T ]. Indeed, V π n t = x + (cid:90) t ∧ Θ n γ ( n ) u dS ( n ) u ≥ x + (cid:90) t ∧ Θ n − γ ( n ) u dS ( n ) u − M d | S ( n ) t ∧ Θ n − − S ( n ) t ∧ Θ n |≥ (cid:15) + M dJ ( n ) t ∧ Θ n − M d | S ( n )Θ − S ( n )Θ − | ≥ (cid:15) as required. The first inequality follows from | γ ( n ) | ≤ M . The second inequalityfollows from the fact that on the time interval [0 , Θ n ) se have x + (cid:82) ·− γ ( n ) u dS ( n ) u ≥ (cid:15) + M dJ ( n ) · . The last inequality is due to J ( n )Θ ≥ | S ( n )Θ − S ( n )Θ − | . We conclude that π n ∈ A n ( x ) and(4.10) V π n T = x + (cid:90) Θ n γ ( n ) u dS ( n ) u ≥ (cid:15). Next, we apply the Skorokhod representation theorem. Recall that the processes { J ( n ) t } Tt =0 , n ∈ N are non–negative, non decreasing and J ( n ) T → (cid:32) { J ( n ) t } Tt =0 , { S ( n ) t } Tt =0 , (cid:26)(cid:90) t γ ( n ) u dS ( n ) u (cid:27) Tt =0 (cid:33) ⇒ (cid:32) , { S t } Tt =0 , (cid:26)(cid:90) t γ u dS u (cid:27) Tt =0 (cid:33) on the space D ([0 , T ]; R ) × D ([0 , T ]; R d ) × D ([0 , T ]; R ). For any n ∈ N the integrand γ ( n ) is of the form (4.4). Hence the integrand γ ( n ) and the corresponding stochastic integral (cid:82) · γ ( n ) u dS ( n ) u are determined pathwise by S ( n ) . Since γ is of bounded variation then we have (cid:90) t γ u dS u = γ t S t − γ S − (cid:90) t S u dγ u , where the last term is the pathwise Riemann–Stieltjes integral. We conclude that γ and the corresponding stochastic integral (cid:82) · γ u dS u are determined pathwise by S . Thus, from the Skorokhod representation theorem and (4.11) it follows that wecan redefine the stochastic processes γ ( n ) , S ( n ) , J ( n ) , n ∈ N and γ, S on the sameprobability space such that (4.5) holds true,(4.12) sup ≤ t ≤ T J ( n ) t → ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t γ ( n ) u dS ( n ) u − (cid:90) t γ u dS u (cid:12)(cid:12)(cid:12)(cid:12) → n , n ∈ N on the common probabilityspace. With abuse of notations we denote by P and E the probability and theexpectation on the common probability space, respectively.First, we argue that(4.14) lim n →∞ P (Θ n = T ) = 1 . Recall, the admissible portfolio π = ( x − (cid:15), γ ). From (4.13) it follows thatlim inf n →∞ inf ≤ t ≤ T (cid:18) x + (cid:90) t γ ( n ) u dS ( n ) u (cid:19) = x + inf ≤ t ≤ T (cid:90) t γ u dS u ≥ (cid:15). In particular lim n →∞ P (cid:18) inf ≤ t ≤ T (cid:18) x + (cid:90) t γ ( n ) u dS ( n ) u (cid:19) > (cid:15) (cid:19) = 1 . This together with (4.12) gives (4.14).Finally, from Fatou’s Lemma, the continuity of U , Assumption 2.1(i), Assump-tion 2.3(i) (recall that V π n T ≥ (cid:15) ), (4.5), (4.10) and (4.13)–(4.14) we obtainlim inf n →∞ E P n [ U ( V π n T , S ( n ) )] = lim inf n →∞ E (cid:34) U (cid:32) x + (cid:90) Θ n γ ( n ) u dS ( n ) u , S ( n ) (cid:33)(cid:35) ≥ E (cid:34) U (cid:32) x + (cid:90) T γ u dS u , S (cid:33)(cid:35) ≥ E P [ U ( V πT , S )] , and (4.8) follows. (cid:3) The Upper Semi–Continuity under Weak Convergence In this section we prove Proposition 2.2. Proof. Let x > 0. From Assumption 2.3(ii) it follows that for any n ∈ N u n ( x ) < ∞ . Hence, we can choose a sequence ˆ π n ∈ A n ( x ), n ∈ N which satisfy (2.3).Without loss of generality (by passing to a subsequence) we assume that the limitlim n →∞ E P n [ U ( V ˆ π n T , S ( n ) )] exists. We will prove that there exists ˆ π ∈ A ( x ) suchthat(5.1) E P [ U ( V ˆ πT , S )] ≥ lim n →∞ E P n [ U ( V ˆ π n T , S ( n ) )]and this will give Proposition 2.2. The proof will be done in two steps. Step I: Choose Q ∈ M ( S ) (recall that we assume M ( S ) (cid:54) = ∅ ) and set Z := d Q d P .From Assumption 2.5 it follows that there exists a sequence Q n ∈ M ( S ( n ) ), n ∈ N for which (2.1) holds true. For any n , { V ˆ π n t } Tt =0 is a Q n super–martingale. Hence, E P n (cid:18) V ˆ π n T d Q n d P n (cid:19) = E Q n [ V ˆ π n T ] ≤ V ˆ π n = x. We conclude that the sequence (cid:16) V ˆ π n T d Q n d P n ; P n (cid:17) , n ∈ N is tight. This together with(2.1) yields that the sequence (cid:16)(cid:16) S ( n ) , d Q n d P n , V ˆ π n T d Q n d P n (cid:17) ; P n (cid:17) , n ∈ N is tight on thespace D ([0 , T ]; R d ) × R . From Prohorov’s theorem it follows that there existsa subsequence (cid:16)(cid:16) S ( n ) , d Q n d P n , V ˆ π n T d Q n d P n (cid:17) ; P n (cid:17) (for simplicity the subsequence is stilldenoted by n ) which converge weakly. From (2.1) we obtain that(5.2) (cid:18)(cid:18) S ( n ) , d Q n d P n , V ˆ π n T d Q n d P n (cid:19) ; P n (cid:19) ⇒ ( S, Z, Y ) , where Y is some random variable. In particular we have the weak convergence(5.3) (cid:16)(cid:16) S ( n ) , V ˆ π n T (cid:17) ; P n (cid:17) ⇒ (cid:18) S, YZ (cid:19) . The random vector ( S, Z, Y ) is defined on a new probability space ( ˜Ω , ˜ F , ˜ P ), whichmight be different from the original probability space (Ω , F , P ). We redefine thefiltration F S (the usual filtration which is generated by S ) and the sets M ( S ) , A ( · )(as before, these sets defined with respect to F S ) on the new probability space( ˜Ω , ˜ F , ˜ P ).Set (notice that YZ ≥ V := E ˜ P (cid:18) YZ | F ST (cid:19) where a priori V can be equal to ∞ with finite probability. In order to prove (5.1)it is sufficient to show that there exists ˆ π ∈ A ( x ) such that(5.5) V ˆ πT ≥ V a.s.Indeed, if (5.5) holds true (in particular V < ∞ a.s.), then from the Jensen in-equality, the continuity of U , Assumption 2.1(i), Assumption 2.3(ii), Assumption2.6 and (5.3) we obtain(5.6) E P [ U ( V ˆ πT , S )] ≥ E ˜ P [ U ( V, S )] ≥ E ˜ P (cid:20) U (cid:18) YZ , S (cid:19)(cid:21) ≥ lim n →∞ E P n [ U ( V ˆ π n T , S ( n ) )] as required.This brings us to the second step. Step II: In this step we establish (5.5). In view of the optional decompositiontheorem (Theorem 3.2 in [29]) it is sufficient to show that the super-hedging pricewhich is given by sup ˆ Q ∈M ( S ) E ˆ Q [ V ] is less or equal than x . From (5.4) we obtainsup ˆ Q ∈M ( S ) E ˆ Q [ V ] = sup ˆ Q ∈M ( S ) E ˜ P (cid:34) YZ d ˆ Q d ˜ P (cid:35) . Hence, it remains to prove that for any ˆ Q ∈ M ( S )(5.7) x ≥ E ˜ P (cid:34) YZ d ˆ Q d ˜ P (cid:35) . From Assumption 2.5 we get a sequence ˆ Q n ∈ M ( S ( n ) ), n ∈ N for which(5.8) (cid:32)(cid:32) S ( n ) , d ˆ Q n d P n (cid:33) ; P n (cid:33) ⇒ (cid:32)(cid:32) S, d ˆ Q d P (cid:33) ; P (cid:33) . This together with (5.2) yields that the sequence (cid:32)(cid:32) S ( n ) , d Q n d P n , V π n T d Q n d P n , d ˆ Q n d P n (cid:33) ; P n (cid:33) , n ∈ N , is tight on the space D ([0 , T ]; R d ) × R . From Prohorov’s Theorem and (5.2) thereis a subsequence which converge weakly(5.9) (cid:32)(cid:32) S ( n ) , d Q n d P n , V ˆ π n T d Q n d P n , d ˆ Q n d P n (cid:33) ; P n (cid:33) ⇒ ( S, Z, Y, X )for some random variable X .Once again, the random vector ( S, Z, Y, X ) is defined on a new probability space( ˜˜Ω , ˜˜ F , ˜˜ P ), on which we redefine the filtration F S and the sets M ( S ) , A ( · ).Observe that d ˆ Q d P is determined by S . Hence, there exists a measurable function g : D ([0 , T ]; R d ) → R such that d ˆ Q d P = g ( S ) P a.s, i.e. E P | d ˆ Q d P − g ( S ) | = 0. From(5.8)–(5.9) we get that the distribution of ( S, X ) equals to (cid:16)(cid:16) S, d ˆ Q d P (cid:17) ; P (cid:17) . Thus, E ˜˜ P | X − g ( S ) | = 0. We conclude that X = g ( S ) ˜˜ P a.s.Finally, from Fatou’s Lemma, (5.9) and the fact that { V ˆ π n t } Tt =0 is a ˆ Q n super–martingale it follows that E ˜ P (cid:32) YZ d ˆ Q d ˜ P (cid:33) = E ˜ P (cid:18) YZ g ( S ) (cid:19) = E ˜˜ P (cid:18) Y g ( S ) Z (cid:19) = E ˜˜ P (cid:18) Y XZ (cid:19) ≤ lim inf n →∞ E P n V ˆ π n T d Q n d P n d ˆ Q n d P n d Q n d P n = lim inf n →∞ E ˆ Q n [ V ˆ π n T ] ≤ x, from which we get (5.7). (cid:3) Next, we prove Theorem 2.2. Proof of Theorem 2.2. In order to prove the statement it is sufficient toshow the for any sub–sequence of laws (cid:16) S ( n ) , V ˆ π n T (cid:17) there is a further subsequencewhich converge weakly to (cid:0) S, V ˆ πT (cid:1) .Following the same arguments as in the proof of Proposition 2.2 we obtain thatfor any subsequence of laws (cid:16) S ( n ) , V ˆ π n T (cid:17) there is a further sequence which satisfies(5.3). Moreover, there exists ˆ π ∈ A ( x ) such that (5.5)–(5.6) hold true.From (5.6), Theorem 2.1 and the fact that ˆ π n ∈ A n ( x ) are asymptotically opti-mal we get u ( x ) ≥ E P [ U ( V ˆ πT , S )] ≥ E ˜ P (cid:20) U (cid:18) E ˜ P (cid:18) YZ | F ST (cid:19) , S (cid:19)(cid:21) ≥ E ˜ P (cid:20) U (cid:18) YZ , S (cid:19)(cid:21) ≥ u ( x ) . We conclude that all the above inequalities are in fact equalities. This equality to-gether with (5.5) and the assumption that U is strictly concave and strictly increas-ing in the first variable (follows from Assumption 2.1(i) and the strict concavity)imply that V ˆ πT = E ˜ P (cid:18) YZ | F ST (cid:19) = YZ and ˆ π ∈ A ( x ) is the unique optimal portfolio. This completes the proof. (cid:3) We end this section with a remark on how our results can be generalized. Remark 5.1. Consider the case where the filtration F := F S,Y is the usual fil-tration generated by S and another RCLL process R = ( R (1) t , ..., R ( m ) t ) ≤ t ≤ T . Theprocess R can be viewed as a collection of non traded assets.For the approximate model we take ( S ( n ) , R ( n ) ) and a filtration which satisfiesthe usual assumptions and makes both S ( n ) and R ( n ) adapted. Once again R ( n ) =( R n, t , ..., R n,mt ) ≤ t ≤ T is the collection of non traded assets. Consider a continuousutility function U : (0 , ∞ ) × D ([0 , T ]; R d ) × D ([0 , T ]; R m ) → R and assume theweak convergence ( S ( n ) , R ( n ) ) ⇒ ( S, R ) and an analogous assumptions to those inSection 2. Of course, as before the martingale measures are with respect to thetraded assets. Then, by using similar arguments as in Sections 4–5 we can extendthe main results Theorems 2.1-2.2 to this setup as well. Lattice Based Approximations of the Heston Model Consider the Heston model [19] given by d ˆ S t = ˆ S t ( µdt + √ ˆ ν t dW t ) ,d ˆ ν t = κ ( θ − ˆ ν t ) dt + σ √ ˆ ν t d ˜ W t , where µ ∈ R , κ, θ, σ > W , ˜ W are two standard Brownianmotions with a constant correlation ρ ∈ ( − , S , ˆ ν > κθ > σ which guarantees that ˆ ν does not touchzero (see [8]).For technical reasons our approximations require that the volatility will lie in aninterval of the form [ σ, σ ] for some 0 < σ < σ . Thus, we modify the Heston modelas following. Fix two barriers 0 < σ < σ and define the function h ( z ) := max( σ , min( z, σ )) , z ∈ R . Consider the SDE dS t = S t ( µdt + (cid:112) h ( ν t ) dW t )(6.1) dν t = κ ( θ − h ( ν t )) dt + σ (cid:112) h ( ν t ) d ˜ W t where the initial values are S := ˆ S , ν := ˆ ν . Observe that √ h, h are Lipschitzcontinuous, and so (6.1) has a unique solution.We expect that if σ is small and σ is large then the value of the utility maxi-mization problem in the Heston model will be close to the one in the model givenby (6.1). For the shortfall risk measure we provide an error estimate in Lemma 7.1.6.1. Discretization. In this section we construct discrete time lattice based ap-proximations for the model given by (6.1). The novelty of our constructions is thatthe approximating sequence satisfies Assumptions 2.4,2.5.It is more convenient to work with a transformed system of equations driven byindependent Brownian motions. Therefore, we setΦ t := ln S t , Ψ t := ν t σ − ρ Φ t . From Itˆo’s formula we obtain that d Φ t = µ Φ (Φ t , Ψ t ) dt + σ Φ (Φ t , Ψ t ) dW t d Ψ t = µ Ψ (Φ t , Ψ t ) dt + σ Ψ (Φ t , Ψ t ) d ˆ W t where µ Φ ( y, z ) := µ − h ( σ ( ρy + z )) / , σ Φ ( y, z ) := (cid:112) h ( σ ( ρy + z )) ,µ Ψ ( y, z ) := κσ ( θ − h ( σ ( ρy + z ))) − ρµ Φ ( y, z ) , σ Ψ := (cid:112) (1 − ρ ) σ Φ , and ˆ W := ˜ W − ρW √ − ρ is a Brownian motion independent of W .Next, we define lattice based approximations for the process (Φ , Ψ). Choose˜ σ ≥ σ . For any n ∈ N define the stochastic processes Φ ( n ) t , Ψ ( n ) t , t ∈ [0 , T ] byΦ ( n ) t := Φ + ˜ σ (cid:113) Tn (cid:80) ki =1 ξ i , kTn ≤ t < ( k +1) Tn Ψ ( n ) t := Ψ + ˜ σ (cid:113) Tn (cid:80) ki =1 ˆ ξ i , kTn ≤ t < ( k +1) Tn where ξ i , ˆ ξ i ∈ {− , , } . Observe that the processes Φ ( n ) − Φ , Ψ ( n ) − Ψ lie onthe grid ˜ σ (cid:113) Tn {− n, − n, ..., n } .Let F ( n ) t , t ≤ T be the piece wise constant filtration generated by the processesΦ ( n ) , Ψ ( n ) . Namely, F ( n ) t := σ (cid:110) ξ , ..., ξ k , ˆ ξ , ..., ˆ ξ k (cid:111) , kT /n ≤ t < ( k + 1) T /n. It remains to define the probability measure P n . First since W and ˆ W are inde-pendent Brownian motions we require that for all a, b ∈ {− , , } and k ≥ P n (cid:18) ξ k = a, ˆ ξ k = b |F ( n ) ( k − Tn (cid:19) := P n (cid:18) ξ k = a |F ( n ) ( k − Tn (cid:19) P n (cid:18) ˆ ξ k = b |F ( n ) ( k − Tn (cid:19) . In order to match the drift and the volatility, we set, P n (cid:18) ξ k = ± |F ( n ) ( k − Tn (cid:19) := σ (cid:32) Φ ( n )( k − Tn , Ψ ( n )( k − Tn (cid:33) σ ± (cid:113) Tn µ Φ (cid:32) Φ ( n )( k − Tn , Ψ ( n )( k − Tn (cid:33) σ , P n (cid:18) ξ k = 0 |F ( n ) ( k − Tn (cid:19) := 1 − σ (cid:32) Φ ( n )( k − Tn , Ψ ( n )( k − Tn (cid:33) ˜ σ , and P n (cid:18) ˆ ξ k = ± |F ( n ) ( k − Tn (cid:19) := σ (cid:32) Φ ( n )( k − Tn , Ψ ( n )( k − Tn (cid:33) σ ± (cid:113) Tn µ Ψ (cid:32) Φ ( n )( k − Tn , Ψ ( n )( k − Tn (cid:33) σ , P n (cid:18) ˆ ξ k = 0 |F ( n ) ( k − Tn (cid:19) := 1 − σ (cid:32) Φ ( n )( k − Tn , Ψ ( n )( k − Tn (cid:33) ˜ σ . Observe that for sufficiently large n , the right hand side of the above equations alllie in the interval [0 , Proposition 6.1. For any n ∈ N (sufficiently large) consider the financial marketgiven by S ( n ) := e Φ ( n ) and the filtration F ( n ) defined above. Then, the followingholds true.(I) We have the weak convergence S ( n ) ⇒ S to the modified Heston model.(II) Assumption 2.4 holds true.Proof. Proof of (I). Let us prove that(6.2) (Φ ( n ) , Ψ ( n ) ) ⇒ (Φ , Ψ) . Clearly, (6.2) implies that S ( n ) ⇒ S .From the definition of P n we have(6.3) E P n (cid:18) Φ ( n ) kTn − Φ ( n ) ( k − Tn (cid:12)(cid:12) F ( n ) ( k − Tn (cid:19) = Tn µ Φ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) , (6.4) E P n (cid:18) Ψ ( n ) kTn − Ψ ( n ) ( k − Tn (cid:12)(cid:12) F ( n ) ( k − Tn (cid:19) = Tn µ Ψ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) , E P n (cid:18) (Φ ( n ) kTn − Φ ( n ) ( k − Tn ) (cid:12)(cid:12) F ( n ) ( k − Tn (cid:19) = Tn σ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) , E P n (cid:18) (Ψ ( n ) kTn − Ψ ( n ) ( k − Tn ) (cid:12)(cid:12) F ( n ) ( k − Tn (cid:19) = Tn σ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) and E P n (cid:18) (Φ ( n ) kTn − Φ ( n ) ( k − Tn )(Ψ ( n ) kTn − Ψ ( n ) ( k − Tn ) (cid:12)(cid:12) F ( n ) ( k − Tn (cid:19) = O ( n − ) . Thus, (6.2) follows from the the martingale convergence result Theorem 7.4.1 in[16]. Proof of II. The statement follows from applying Example 3.2 for m n = n , τ ( n ) i =( i − T /n and a n = e ˜ σ √ Tn − (cid:3) Verification of Assumption 2.5. We start with some preparations. Denoteby D the set of all stochastic processes Υ = { Υ t } Tt =0 of the form Υ = F (Φ) where F : D ([0 , T ]; R ) → D ([0 , T ]; R ) is a bounded, continuous function (we take theSkorokhod topology on the space D ([0 , T ]; R ) and F is a progressively measurablemap. Namely, for any t ∈ [0 , T ] and f (1) , f (2) ∈ D ([0 , T ]; R ), f (1)[0 ,t ] = f (2)[0 ,t ] impliesthat F t ( f (1) ) = F t ( f (2) ).Define the set M d ( S ) := (cid:26) Q : ∃ Υ ∈ D , d Q d P |F ST = e (cid:82) T − µ √ h ( νt ) dW t + (cid:82) T Υ t d ˆ W t − (cid:82) T µ h ( νt ) dt − (cid:82) T Υ t dt (cid:27) . From the Girsanov theorem it follows that M d ( S ) ⊂ M ( S ). Moreover, since Φ =ln S then the usual filtration which is generated by S equals to the usual filtrationwhich is generated by Φ. Hence standard arguments yield that M d ( S ) ⊂ M ( S ) isdense.Choose an arbitrary Υ = F (Φ) ∈ D and denote(6.5) Z t := e (cid:82) t − µ √ h ( νu ) dW u + (cid:82) t Υ u d ˆ W u − (cid:82) t µ h ( νu ) du − (cid:82) t Υ u du , t ∈ [0 , T ] . It is sufficient to prove that (recall Remark 2.3) there exists a sequence of probabilitymeasures Q n ∈ M ( S ( n ) ), n ∈ N , such that for the processes Z ( n ) t := d Q n d P n |F ( n ) t , t ∈ [0 , T ], we have the weak convergence(6.6) ( S ( n ) , Z ( n ) ) ⇒ ( S, Z ) . For any n ∈ N (sufficiently large) define the probability measure Q n by the followingrelations Q n (cid:18) ξ k = a, ˆ ξ k = b |F ( n ) ( k − Tn (cid:19) := Q n (cid:18) ξ k = a |F ( n ) ( k − Tn (cid:19) Q n (cid:18) ˆ ξ k = b |F ( n ) ( k − Tn (cid:19) , Q n (cid:18) ξ k = ± |F ( n ) ( k − Tn (cid:19) := σ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) ˜ σ (cid:16) e ± ˜ σ √ Tn (cid:17) , Q n (cid:18) ξ k = 0 |F ( n ) ( k − Tn (cid:19) := 1 − σ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) ˜ σ , (6.7)and Q n (cid:18) ˆ ξ k = ± |F ( n ) ( k − Tn (cid:19) := σ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) σ ± (cid:114) Tn F ( k − Tn (Φ ( n ) ) σ Ψ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) + µ Ψ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) σ , Q n (cid:18) ˆ ξ k = 0 |F ( n ) ( k − Tn (cid:19) := 1 − σ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) ˜ σ . Observe that (6.7) implies Q n ∈ M ( S ( n ) ). Lemma 6.1. We have the weak convergence (Φ ( n ) , Ψ ( n ) , Z ( n ) ) ⇒ (Φ , Ψ , Z ) . Proof. In order to prove the lemma it suffices to show that for any subsequencethere exists a further subsequence (still denoted by n ) such that(6.8) (Φ ( n ) , Ψ ( n ) , Z ( n ) ) ⇒ (Φ , Ψ , Z ) . Fix n ∈ N . By applying Taylor’s expansion we obtain that there exist uniformlybounded (in n ) processes E n, k , E n, k , k = 0 , , ..., n such that Q n (cid:18) ξ k |F ( n ) ( k − Tn (cid:19) P n (cid:18) ξ k |F ( n ) ( k − Tn (cid:19) = 1 − ˜ σξ k (cid:114) Tn 12 + µ Φ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) σ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) + E n, k n + o (1 /n )and Q n (cid:18) ˆ ξ k |F ( n ) ( k − Tn (cid:19) P n (cid:18) ˆ ξ k |F ( n ) ( k − Tn (cid:19) = 1 + ˜ σ ˆ ξ k (cid:114) Tn F ( k − Tn (Φ ( n ) ) σ Ψ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) + E n, k n + o (1 /n ) . We conclude that there exists a uniformly bounded (in n ) process E ( n ) k , k = 0 , ..., n such that Z ( n ) kTn − Z ( n ) ( k − Tn Z ( n ) ( k − Tn = Q n (cid:18) ξ k |F ( n ) ( k − Tn (cid:19) P n (cid:18) ξ k |F ( n ) ( k − Tn (cid:19) Q n (cid:18) ˆ ξ k |F ( n ) ( k − Tn (cid:19) P n (cid:18) ˆ ξ k |F ( n ) ( k − Tn (cid:19) − − (Φ ( n ) kTn − Φ ( n ) ( k − Tn ) 12 + µ Φ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) σ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) + (Ψ ( n ) kTn − Ψ ( n ) ( k − Tn ) F ( k − Tn (Φ ( n ) ) σ Ψ (cid:18) Φ ( n ) ( k − Tn , Ψ ( n ) ( k − Tn (cid:19) + E ( n ) k n + o (1 /n ) . (6.9)In particular, (cid:32) Z ( n ) kTn − Z ( n )( k − Tn Z ( n )( k − Tn (cid:33) is of order O (1 /n ). Since Z ( n ) is a martingale,then by taking conditional expectation we arrive to E P n (cid:18) [ Z ( n ) kTn ] |F ( n ) ( k − Tn (cid:19) = [ Z ( n ) ( k − Tn ] (1 + O (1 /n )) . By taking expectation we obtain E P n (cid:16) [ Z ( n ) kTn ] (cid:17) = E P n (cid:18) [ Z ( n ) ( k − Tn ] (cid:19) (1 + O (1 /n )) . This together with the Doob–Kolmogorov inequality gives(6.10) sup n ∈ N E P n (cid:18) sup ≤ t ≤ T [ Z ( n ) t ] (cid:19) ≤ n ∈ N E P n (cid:16) [ Z ( n ) T ] (cid:17) < ∞ . Next, define ˆ E ( n ) k := E P n (cid:18) E ( n ) k |F ( n ) ( k − Tn (cid:19) , k = 1 , ..., n and consider the martingaleˆ M ( n ) k := 1 n k (cid:88) i =1 ( E ( n ) i − ˆ E ( n ) i ) , k = 0 , , ..., n. Since E ( n ) , n ∈ N , are uniformly bounded then E P n (cid:16) max ≤ k ≤ n | ˆ M ( n ) k | (cid:17) ≤ E P n (cid:16) | ˆ M ( n ) n | (cid:17) = n (cid:80) ni =1 E P n (cid:20)(cid:16) E ( n ) i − ˆ E ( n ) i (cid:17) (cid:21) = O (1 /n ) . Thus,(6.11) max ≤ k ≤ n | ˆ M ( n ) k | → . Introduce the adapted (to F ( n ) ) processesΞ ( n ) t := (cid:82) t ˆ E ( n ) (cid:98) nu/T (cid:99) du, t ∈ [0 , T ] M ( n ) t := ˆ M ( n ) (cid:98) nt/T (cid:99) , t ∈ [0 , T ]where (cid:98)·(cid:99) is the integer part of · and ˆ E ( n )0 := E ( n )0 . Again, E ( n ) , n ∈ N , are uniformly bounded, and so Ξ ( n ) , n ∈ N , is tight. From(6.2) and (6.11) we conclude that the sequence (Φ ( n ) , Ψ ( n ) , Ξ ( n ) , M ( n ) ), n ∈ N , istight as well. Thus, from Prohorov’s Theorem, (6.2) and (6.11) it follows that forany subsequence there exists a further subsequence such that(6.12) (Φ ( n ) , Ψ ( n ) , Ξ ( n ) , M ( n ) ) ⇒ (Φ , Ψ , Ξ , { Ξ t } Tt =0 . From Theorems 4.3–4.4 in[15], (6.9), (6.12) and the equality E ( n ) k n = ˆ E ( n ) k n + ˆ M ( n ) k − ˆ M ( n ) k − we obtain that(Φ ( n ) , Ψ ( n ) , Ξ ( n ) , M ( n ) , Z ( n ) ) ⇒ (Φ , Ψ , Ξ , , ˆ Z )where ˆ Z is the solution of the SDE(6.13) d ˆ Z t ˆ Z t = − (cid:18) 12 + µ Φ (Φ t , Ψ t ) σ (Φ t , Ψ t ) (cid:19) d Φ t + Υ t σ Ψ (Φ t , Ψ t ) d Ψ t + d Ξ t T with the initial condition ˆ Z = 1.Finally, (6.10) implies that for any t ∈ [0 , T ] the random variables { Z ( n ) t } n ∈ N areuniformly integrable. This together with the fact that for any n , Z ( n ) is a martingalewith respect to the filtration generated by Φ ( n ) , Ψ ( n ) , Ξ ( n ) , M ( n ) , Z ( n ) gives that ˆ Z isa martingale with respect to the filtration generated by Φ , Ψ , Ξ , ˆ Z . Moreover, from(6.3)–(6.4) we get that { Φ t − (cid:82) t µ Φ (Φ u , Ψ u ) du } Tt =0 and { Ψ t − (cid:82) t µ Ψ (Φ u , Ψ u ) du } Tt =0 are martingales with respect to the filtration generated by Φ , Ψ , Ξ , ˆ Z . In particular, from L´evy’s Theorem it follows that the stochastic processes W and ˆ W which weredefine by W t := Φ t − (cid:82) t µ Φ (Φ u , Ψ u ) duσ Φ (Φ t , Ψ t ) , ˆ W t := Ψ t − (cid:82) t µ Ψ (Φ u , Ψ u ) duσ Ψ (Φ t , Ψ t )are (independent) Brownian motions with respect to the filtration generated byΦ , Ψ , Ξ , ˆ Z . We conclude that the drift of the right hand side of (6.13) is equal tozero. Namely, d ˆ Z t ˆ Z t = − (cid:18) 12 + µ Φ (Φ t , Ψ t ) σ (Φ t , Ψ t ) (cid:19) σ Φ (Φ t , Ψ t ) dW t + Υ t d ˆ W t = dZ t Z t , where the last equality follows from (6.5). Hence, ˆ Z = Z and (6.8) follows. (cid:3) Clearly, Lemma 6.1 implies (6.6). This gives us the following result. Proposition 6.2. Consider the set-up of Proposition 6.1. Assumption 2.5 holdstrue. We end this section by addressing condition (II) in Lemma 2.2. Remark 6.1. Consider the martingale measures Q n ∈ M ( S ( n ) ) , n ∈ N which weredefined before Lemma 6.1 for Υ ≡ . Since µ Φ , σ Φ , σ Φ are uniformly bounded, thenstandard arguments yield that for any q > (2.5) holds true. Approximations of the Shortfall Risk in the Heston Model In this section we focus on shortfall risk minimization for European call options(which corresponds to U given by (3.1)) in the Heston model. We start with thefollowing estimate. Lemma 7.1. For an initial capital x let ˆ R ( x ) be the shortfall risk in the Hestonmodel and let R ( x ) be the shortfall risk in the model given by (6.1). Then for any m ∈ N | ˆ R ( x ) − R ( x ) | ≤ O ( σ κθ/σ − ) + O (1 /σ m ) , where the O terms do not depend on x .Proof. Define the stopping timeΘ σ,σ := T ∧ inf { t : (cid:112) ˆ ν t / ∈ ( σ, σ ) } . Observe that on the event Θ σ,σ = T the processes ˆ S and S coincide. Hence,(7.1) | ˆ R ( x ) − R ( x ) | ≤ E P [( ˆ S T + S T ) I Θ σ,σ 0) and Assumption 2.6 triv-ially hold true. Moreover, from Remark 2.8 we obtain Assumption 2.2. Since thedrift and the volatility are uniformly bounded we get that the random variables { S ( n ) T } n ∈ N are uniformly integrable, which gives Assumption 2.3(i). In view ofPropositions 6.1,6.2 we conclude that our Assumptions are satisfied and so Theo-rem 2.1 holds true.Thus, fix n ∈ N and recall the discrete models introduced in Section 6.1. Thestock price process S ( n ) is piece wise constant and so the investor trades only atthe jump times kTn , k = 0 , ..., n. Notice that (cid:110)(cid:80) km =1 ξ m , (cid:80) km =1 ˆ ξ m (cid:111) nk =0 is a latticevalued Markov chain (with respect to P n ). Hence, we introduce the functions J ( n ) k ( i, j, λ ), k = 0 , ..., n such that J ( n ) k ( i, j, λ ) measures the shortfall risk at time kT /n given that (cid:80) km =1 ξ m = i , (cid:80) km =1 ˆ ξ m = j , and λ is the ratio of the portfoliovalue and the stock price. The stock price is recovered by S ( n ) kTn = S e ˜ σ √ Tn (cid:80) km =1 ξ m = S e i ˜ σ √ Tn . Clearly, if λ ≥ 1, then the shortfall risk is zero because we can buy the stock andhold it until maturity. Namely, J ( n ) k ( i, j, λ ) = 0 for λ ≥ 1. Hence, we assume that λ ∈ [0 , kT /n the investor decides about his investment pol-icy. Assume that the investor portfolio value is λS ( n ) kTn . We have a trinomial modelwith growth rates (cid:110) e − ˜ σ √ Tn , , e ˜ σ √ Tn (cid:111) . From the binomial representation theoremwe easily deduce that the set of replicable portfolios at time ( k + 1) T /n are of the form Λ( ξ k +1 ) S ( n ) ( k +1) Tn where Λ : {− , , } → R satisfies Λ(0) = λ andΛ( − 1) + Λ(1) e ˜ σ √ Tn e ˜ σ √ Tn = λ. Thus, if Λ( − 1) is known then we set(7.5) Λ(1) := 1 ∧ (cid:16) λ (1 + e − ˜ σ √ Tn ) − Λ( − e − ˜ σ √ Tn (cid:17) . We take a truncation in order to have Λ(1) ∈ [0 , A ( λ ) the set of all Λ( − ∈ [0 , 1] for which the right handside of (7.5) is non-negative.We arrive to the following recursive relations. Define J ( n ) k ( i, j, λ ) : {− k, − k, ..., k } × {− k, − k, ..., k } × [0 , → R + , k = 0 , , ..., n by J ( n ) n ( i, j, λ ) := U (cid:32) λS exp (cid:32) i ˜ σ (cid:114) Tn (cid:33) , S exp (cid:32) i ˜ σ (cid:114) Tn (cid:33)(cid:33) , and for k < n , J ( n ) k ( i, j, λ ) :=sup Λ( − ∈A ( λ ) E P n (cid:18) J ( n ) k +1 (cid:18) i + ξ m +1 , j + ˆ ξ m +1 , Λ( ξ m +1 ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) m =1 ξ m = i, k (cid:88) m =1 ˆ ξ m = j (cid:19) (7.6)where Λ(0) = λ and Λ(1) is given by (7.5). For k = 0 we have J ( n )0 ( x/S ) = u n ( x ) . Observe that the functions J ( n ) k ( i, j, λ ) are piece wise linear and continuous in λ , and so they can be represented by an array which consists of the slope valuesand the slope jump points. This together with the condition J ( n ) k ( i, j, 1) = 0 issufficient to recover the function. Of course the array will depend on time kT /n and the states i, j . Thus, theoretically, the dynamic programming given by (7.6)can be implemented using a computer. However, from practical point of view thenumber of the slope points of the function J ( n ) k grows exponentially (in n − k ),and so for large n it cannot be implemented. Hence, we need to introduce a gridstructure for the portfolio value as well.Thus, choose M ∈ N and consider the grid(7.7) GR := (cid:26) , M , M , ..., (cid:27) . For a given Λ( − ∈ GR we define two grid values for Λ(1). The first value is(7.8) Λ − (1) := 1 ∧ (cid:106)(cid:16) λ (1 + e − ˜ σ √ Tn ) − Λ( − e − ˜ σ √ Tn (cid:17) M (cid:107) M where, recall that (cid:98)·(cid:99) is the integer part of · . The second value is(7.9) Λ + (1) := 1 ∧ (cid:108)(cid:16) λ (1 + e − ˜ σ √ Tn ) − Λ( − e − ˜ σ √ Tn (cid:17) M (cid:109) + 1 M where (cid:100)·(cid:101) = min { n ∈ Z : n ≥ ·} . Define two value functions(7.10) J ( n ) k ( ± , i, j, λ ) : {− k, − k, ..., k } × {− k, − k, ..., k } × GR → R + , k = 0 , , ..., n as following. The terminal condition is J ( n ) n ( ± , i, j, λ ) := U (cid:32) λS exp (cid:32) i ˜ σ (cid:114) Tn (cid:33) , S exp (cid:32) i ˜ σ (cid:114) Tn (cid:33)(cid:33) . For k < n , J ( n ) k ( ± , i, j, λ ):= max Λ( − ∈A ( λ ) (cid:84) GR E P n (cid:18) J ( n ) k +1 (cid:18) ± , i + ξ m +1 , j + ˆ ξ m +1 , Λ ± ( ξ m +1 ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) m =1 ξ m = i, k (cid:88) m =1 ˆ ξ m = j (cid:19) where Λ ± ( − 1) = Λ( − ± (0) = λ and Λ ± (1) are given by (7.8)–(7.9).For k = 0 we obtain two values J ( n )0 (+ , x/S ) and J ( n )0 ( − , x/S ). Observe thatthe complexity of the above dynamic programming is polynomial in M, n . For theexact value u n ( x ) = J ( n )0 ( x/S ) we have the following simple lemma. Lemma 7.2. Assume that xS ∈ GR . Then J n ( x/S ) ∈ [ J ( n )0 ( − , x/S ) , J ( n )0 (+ , x/S )] . Proof. The inequality J ( n )0 ( − , x/S ) ≤ J ( n )0 ( x/S ) is obvious. Let us prove that J ( n )0 ( x/S ) ≤ J ( n )0 (+ , x/S ). Choose λ ∈ GR and ˜Λ( − , ˜Λ(1) ∈ [0 , 1] which satisfy(7.5). Define Λ( − 1) := 1 ∧ (cid:100) ˜Λ( − M (cid:101) M and let Λ + (1) be given by (7.9). Then itis straightforward to check that Λ( − ≥ ˜Λ( − 1) and Λ + (1) ≥ ˜Λ(1) . Hence, byapplying backward induction (on k ) and the fact that J ( n ) k ( i, j, λ ) is non-decreasingin λ we get that for any k , J ( n ) k ( · ) ≤ J ( n ) k (+ , · ) where we take the restrictionof J ( n ) k ( · ) to {− k, − k, ..., k } × {− k, − k, ..., k } × GR . For k = 0, we obtain J ( n )0 ( x/S ) ≤ J ( n )0 (+ , x/S ) as required. (cid:3) Remark 7.1. By using the fact that U is Lipschitz continuous in the first vari-able, it can be shown that the difference J ( n )0 (+ , x/S ) − J ( n )0 ( − , x/S ) is of order O ( n/M ) . In practice this difference goes to zero much faster (in M ). As we willsee in the following numerical results, already for M “close” to n the difference J ( n )0 (+ , x/S ) − J ( n )0 ( − , x/S ) becomes very small. Numerical Results. In this section we implement numerically the abovedescribed procedure. In Table 1 and in the corresponding Figure 1 we compute thefunctions defined in (7.10). To serve as a reference we also evaluate the function u ( x ) = − E P [(( S T − K ) + − x ) + ] , a lower bound, which corresponds to the valueof spending no extra effort in reducing the shortfall. J ( n )0 ( − , x/S ) J ( n )0 (+ , x/S ) u ( n )0 ( x )x=0 -24.5421 -24.0371 -24.6095x=5 -18.4702 -17.7050 -21.4086x=10 -12.3159 -11.6165 -18.2077x=15 -7.0529 -6.3398 -16.3018x=20 -2.7913 -2.2453 -14.3959x=25 -0.6802 -0.4201 -12.4901x=30 -0.0825 -0.0274 -10.5842x=35 -0.0043 -0.0004 -8.6783x=40 0 0 -7.1540x=45 0 0 -6.4423x=50 0 0 -5.7306x=55 0 0 -5.0190x=60 0 0 -4.3073x=70 0 0 -2.8840x=80 0 0 -2.0045x=90 0 0 -1.6418x=100 0 0 -1.2792 Table 1. Shortfall risk minimization for call options. Parametersused in computation: K = 90 , σ = 1 , ˜ σ = 5 , σ = 0 . σ =0 . , ρ = − . , κ = 1 . , θ = 0 . , µ = 0 . , S = 100 , T =1 , ν = 0 . , n = 400 , M = 400. Figure 1. Plot of the values reported in Table 1.In the next table we analyze the sensitivity of the problem to σ . The smaller thisparameter, the faster the algorithm takes. Although, Lemma 7.1 indicates an errorbound for large σ (which was obtained by an application of Markov’s inequality),we observe that we can in practice take σ = 1 for our parameters. σ = 0 . . σ = 0 . . σ = 0 . . σ = 1 σ = 2x=0 -15.3139 -23.1077 -22.0861 -24.5421 -24.5421x=10 -4.1129 -9.6884 -10.9334 -12.3159 -12.3159x=20 -0.1435 -4.5287 -1.9145 -2.7913 -2.7913 Table 2. Variation with respect to σ . Parameters are the same asin Table 1. The values in the parentheses represent P (Θ σ,σ < T )rounded to 4 decimals points. We did not indicate these valueswhen this probability is extremely close to 1.In Table 3 we analyze the sensitivity of solution to the grid size of the controlvariable defined in (7.7). We observe, as stated in Remark 7.1, that the we can actually take M = kn , where k < 1. In this table, we determine the range of k wecan choose. We observe that choosing n larger leads to more error reduction thanchoosing k larger. We have also checked this for values of k > Table 3. Variation with respect to M . x = 20. Other parametersare the same as in Table 1.Table 4 and the corresponding Figure 4 demonstrate the convergence with respectto n . We observe that the convergence rate is a power of n . We leave the rigorousdemonstration of this result for future work.M=n/4 J n ( − ,x/S ) − J n/ ( − ,x/S ) | J n/ ( − ,x/S ) | n=50 -9.2138 –n=100 -5.4667 0.4067n=200 -3.7184 0.3198n=400 -2.9834 0.1977n=800 -2.6675 0.1059n=1600 -2.6171 0.0189 Table 4. x = 20. Other parameters are the same as in Table 1. Figure 2. Plot of the values in Table 4. References [1] Julio Backhoff Veraguas and Francisco J. Silva. Sensitivity analysis for ex-pected utility maximization in incomplete Brownian market models. Math.Financ. Econ. , 12(3):387–411, 2018. ISSN 1862-9679. doi: 10.1007/s11579-017-0209-9. URL https://doi.org/10.1007/s11579-017-0209-9 .[2] Peter Bank and Dietmar Baum. Hedging and portfolio optimization in financialmarkets with a large trader. Math. Finance , 14(1):1–18, 2004. ISSN 0960-1627.[3] Erhan Bayraktar and Ross Kravitz. Stability of exponential utility maxi-mization with respect to market perturbations. Stochastic Process. Appl. , 123(5):1671–1690, 2013. ISSN 0304-4149. doi: 10.1016/j.spa.2012.12.007. URL https://doi.org/10.1016/j.spa.2012.12.007 .[4] Sara Biagini and Marco Frittelli. A unified framework for utility maximizationproblems: an Orlicz space approach. Ann. Appl. Probab. , 18(3):929–966, 2008.ISSN 1050-5164. doi: 10.1214/07-AAP469. URL https://doi.org/10.1214/07-AAP469 .[5] Patrick Billingsley. Convergence of probability measures . Wiley Series in Prob-ability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., NewYork, second edition, 1999. ISBN 0-471-19745-9. doi: 10.1002/9780470316962.URL https://doi.org/10.1002/9780470316962 . A Wiley-Interscience Pub-lication. [6] Andrea Collevecchio, Kais Hamza, and Meng Shi. Bootstrap random walks. Stochastic Process. Appl. , 126(6):1744–1760, 2016. ISSN 0304-4149. doi:10.1016/j.spa.2015.11.016. URL https://doi.org/10.1016/j.spa.2015.11.016 .[7] J. Cox and C. Huang. Optimal consumption and portfolio policies when assetprices follow a diffusion process. Journal of Economic Theory , 49:33–83, 1989.[8] J.C. Cox, J.E. Ingersoll, and S.A. Ross. A theory of the trerm structure ofinterest rates. Econometrica , 53:385–407, 1985.[9] Jaksa Cvitani´c, Walter Schachermayer, and Hui Wang. Erratum to: Utilitymaximization in incomplete markets with random endowment [ MR1841719]. Finance Stoch. , 21(3):867–872, 2017. ISSN 0949-2984. doi: 10.1007/s00780-017-0331-9. URL https://doi.org/10.1007/s00780-017-0331-9 .[10] Jakˇsa Cvitani´c, Huyˆen Pham, and Nizar Touzi. Super-replication in stochasticvolatility models with portfolio constraints. Journal of Applied Probability , 36:523–545, 1999.[11] Jakˇsa Cvitani´c, Walter Schachermayer, and Hui Wang. Utility maximizationin incomplete markets with random endowment. Finance Stoch. , 5(2):259–272,2001. ISSN 0949-2984. doi: 10.1007/PL00013534. URL https://doi.org/10.1007/PL00013534 .[12] Freddy Delbaen and Walter Schachermayer. A general version of the funda-mental theorem of asset pricing. Math. Ann. , 300(3):463–520, 1994. ISSN0025-5831. doi: 10.1007/BF01450498. URL https://doi.org/10.1007/BF01450498 .[13] Yan Dolinsky and Ariel Neufeld. Super-replication in fully incomplete markets. Math. Finance , 28(2):483–515, 2018. ISSN 0960-1627. doi: 10.1111/mafi.12149. URL https://doi.org/10.1111/mafi.12149 .[14] R. M. Dudley. Distances of probability measures and random vari-ables. Ann. Math. Statist , 39:1563–1572, 1968. ISSN 0003-4851.doi: 10.1007/978-1-4419-5821-1 4. URL https://doi.org/10.1007/978-1-4419-5821-1_4 .[15] Darrell Duffie and Philip Protter. From discrete– to continuous–time finance:Weak convergence of the financial gain process. Math. Finance , 2:1–15, 1992.[16] Stewart N. Ethier and Thomas G. Kurtz. Markov processes . Wiley Series inProbability and Mathematical Statistics: Probability and Mathematical Sta-tistics. John Wiley & Sons, Inc., New York, 1986. ISBN 0-471-08186-8. doi:10.1002/9780470316658. URL https://doi.org/10.1002/9780470316658 .Characterization and convergence.[17] R¨udiger Frey and Carlos A. Sin. Bounds on European option prices under sto-chastic volatility. Math. Finance , 9(2):97–116, 1999. ISSN 0960-1627. doi: 10.1111/1467-9965.00064. URL https://doi.org/10.1111/1467-9965.00064 .[18] Anja G¨oing-Jaeschke and Marc Yor. A survey and some generalizations ofBessel processes. Bernoulli , 9(2):313–349, 2003. ISSN 1350-7265. doi: 10.3150/bj/1068128980. URL https://doi.org/10.3150/bj/1068128980 .[19] Steven L. Heston. A closed-form solution for options with stochastic volatilitywith applications to bond and currency options. Review of Financial Studies ,6:327–343, 1993.[20] H.He. Optimal consumption- portfolio policies: A convergence from discreteto continuous time models. Journal of Economical Theory , 55:340–363, 1991. [21] Ying Hu, Peter Imkeller, and Matthias M¨uller. Utility maximization in in-complete markets. Ann. Appl. Probab. , 15(3):1691–1712, 2005. ISSN 1050-5164. doi: 10.1214/105051605000000188. URL https://doi.org/10.1214/105051605000000188 .[22] Friedrich Hubalek and Walter Schachermayer. When does convergence of assetprice processes imply convergence of option prices? Math. Finance , 8(4):385–403, 1998. ISSN 0960-1627. doi: 10.1111/1467-9965.00060. URL https://doi.org/10.1111/1467-9965.00060 .[23] J. Hull and A. White. Pricing of options on assets with stochastic volatility. Journal of Finance , 42:281–300, 1987.[24] J. Jacod, S. Mlard, and P. Protter. Explicit form and robustness of martingalerepresentations. Annals of Probability , 28:1747–1780, 2000.[25] Ely`es Jouini and Clotilde Napp. Convergence of utility functions and conver-gence of optimal strategies. Finance Stoch. , 8(1):133–144, 2004. ISSN 0949-2984. doi: 10.1007/s00780-003-0106-3. URL https://doi.org/10.1007/s00780-003-0106-3 .[26] Ioannis Karatzas, John P. Lehoczky, Steven E. Shreve, and Gan-Lin Xu. Mar-tingale and duality methods for utility maximization in an incomplete mar-ket. SIAM J. Control Optim. , 29(3):702–730, 1991. ISSN 0363-0129. doi:10.1137/0329039. URL https://doi.org/10.1137/0329039 .[27] Constantinos Kardaras and Gordan ˇZitkovi´c. Stability of the utility maximiza-tion problem with random endowment in incomplete markets. Math. Finance ,21(2):313–333, 2011. ISSN 0960-1627. doi: 10.1111/j.1467-9965.2010.00433.x.URL https://doi.org/10.1111/j.1467-9965.2010.00433.x .[28] D. Kramkov and W. Schachermayer. The asymptotic elasticity of utility func-tions and optimal investment in incomplete markets. Ann. Appl. Probab. , 9(3):904–950, 1999. ISSN 1050-5164. doi: 10.1214/aoap/1029962818. URL https://doi.org/10.1214/aoap/1029962818 .[29] D. O. Kramkov. Optional decomposition of supermartingales and hedgingcontingent claims in incomplete security markets. Probab. Theory RelatedFields , 105(4):459–479, 1996. ISSN 0178-8051. doi: 10.1007/BF01191909.URL https://doi.org/10.1007/BF01191909 .[30] Kasper Larsen. Continuity of utility-maximization with respect to prefer-ences. Math. Finance , 19(2):237–250, 2009. ISSN 0960-1627. doi: 10.1111/j.1467-9965.2009.00365.x. URL https://doi.org/10.1111/j.1467-9965.2009.00365.x .[31] Kasper Larsen and Gordan ˇZitkovi´c. Stability of utility-maximization in in-complete markets. Stochastic Process. Appl. , 117(11):1642–1662, 2007. ISSN0304-4149. doi: 10.1016/j.spa.2006.10.012. URL https://doi.org/10.1016/j.spa.2006.10.012 .[32] Kasper Larsen and Hang Yu. Horizon dependence of utility optimizersin incomplete models. Finance Stoch. , 16(4):779–801, 2012. ISSN 0949-2984. doi: 10.1007/s00780-012-0171-6. URL https://doi.org/10.1007/s00780-012-0171-6 .[33] Kasper Larsen, Oleksii Mostovyi, and Gordan ˇZitkovi´c. An expansion in themodel space in the context of utility maximization. Finance Stoch. , 22(2):297–326, 2018. ISSN 0949-2984. doi: 10.1007/s00780-017-0353-3. URL https://doi.org/10.1007/s00780-017-0353-3 . [34] R.C. Merton. Lifetime portoflio selection under uncertainty: The continuoustime case. Rev. Econ. Stat , 51:247–257, 1969.[35] R.C. Merton. Optimum consumption and portfolio rules in a contiunous-timemodel. J. Econ. Th , 3:373–413, 1971.[36] O. Mostovyi and M. Sˆırbu. Sensitivity analysis of the utility maximizationproblem with respect to model perturbations. preprint, arXiv: 1705.08291 ,2018.[37] Sabrina Mulinacci. The efficient hedging problem for American options. Finance Stoch. , 15(2):365–397, 2011. ISSN 0949-2984. doi: 10.1007/s00780-010-0151-7. URL https://doi.org/10.1007/s00780-010-0151-7 .[38] Ariel Neufeld. Buy-and-hold property for fully incomplete markets when super-replicating markovian claims. International Journal of Theoretical and Ap-plied Finance , 21(08):1850051, 2018. doi: 10.1142/S0219024918500516. URL https://doi.org/10.1142/S0219024918500516 .[39] Jean-Luc Prigent. Weak convergence of financial markets . Springer Fi-nance. Springer-Verlag, Berlin, 2003. ISBN 3-540-42333-8. doi: 10.1007/978-3-540-24831-6. URL https://doi.org/10.1007/978-3-540-24831-6 .[40] Mikl´os R´asonyi and Lukasz Stettner. On utility maximization in discrete-timefinancial market models. Ann. Appl. Probab. , 15(2):1367–1395, 2005. ISSN1050-5164. doi: 10.1214/105051605000000089. URL https://doi.org/10.1214/105051605000000089 .[41] Christian Reichlin. Behavioral portfolio selection: asymptotics and stabilityalong a sequence of models. Math. Finance , 26(1):51–85, 2016. ISSN 0960-1627.doi: 10.1111/mafi.12053. URL https://doi.org/10.1111/mafi.12053 .[42] Carlos A. Sin. Complications with stochastic volatility models. Adv. in Appl.Probab. , 30(1):256–268, 1998. ISSN 0001-8678. doi: 10.1239/aap/1035228003.URL https://doi.org/10.1239/aap/1035228003 .[43] Ward Whitt. Proofs of the martingale FCLT. Probab. Surv. , 4:268–302, 2007.ISSN 1549-5787. doi: 10.1214/07-PS122. URL https://doi.org/10.1214/07-PS122 . DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MICHIGAN E-mail address : [email protected] DEPARTMENT OF STATISTICS, HEBREW UNIVERSITY E-mail address : [email protected] DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MICHIGAN E-mail address ::