Robust Utility Maximization in a Multivariate Financial Market with Stochastic Drift
RRobust Utility Maximization in a Multivariate
Financial Market with Stochastic Drift
Jörn Sass ∗ and Dorothee Westphal † Department of Mathematics, Technische Universität KaiserslauternSeptember 30, 2020
Abstract
We study a utility maximization problem in a financial market with a stochastic driftprocess, combining a worst-case approach with filtering techniques. Drift processes aredifficult to estimate from asset prices, and at the same time optimal strategies in portfoliooptimization problems depend crucially on the drift. We approach this problem by settingup a worst-case optimization problem with a time-dependent uncertainty set for the drift.Investors assume that the worst possible drift process with values in the uncertainty set willoccur. This leads to local optimization problems, and the resulting optimal strategy needsto be updated continuously in time. We prove a minimax theorem for the local optimizationproblems and derive the optimal strategy. Further, we show how an ellipsoidal uncertaintyset can be defined based on filtering techniques and demonstrate that investors need tochoose a robust strategy to be able to profit from additional information.
Keywords:
Portfolio optimization; drift uncertainty; robust strategies; stochastic filtering;minimax theorems
Financial market models are usually prone to statistical estimation errors, incomplete infor-mation and other reasons for model misspecifications. Especially the drift of asset prices isnotoriously difficult to estimate from historical data. Drift processes tend to fluctuate ran-domly over time, and even for estimating a constant drift with a reasonable degree of precisionone needs very long time series, an observation already made by Merton [16]. At the same time,trading strategies in portfolio optimization problems depend crucially on the drift. Strategiesthat are determined based on a misspecified model can therefore perform rather badly in thetrue financial market setting, see Chopra and Ziemba [3] and Kan and Zhou [9].There are two main approaches to deal with these problems. On the one hand, it is crucialto approximate the true model as accurately as possible using all the information available. ∗ [email protected] † [email protected] a r X i v : . [ q -f i n . P M ] S e p hen estimating the hidden drift process the best estimate in a mean-square sense is theconditional mean of the drift given the available information, the so-called filter . Observationsusually include the stock returns but can also involve external sources of information like news,company reports or ratings. In fact, Merton [16] points out that due to the difficulty of esti-mating expected returns, sources of information other than time series data of market returnsare needed to improve estimates. Filtering techniques thus are a way to reduce uncertaintyabout model parameters. On the other hand, model uncertainty can be approached by settingup worst-case optimization problems. Instead of working with just one particular model, onespecifies a range of possible models and tries to optimize the objective, given that for anychosen strategy the worst of all possible models will occur. This leads to robust strategies, i.e.strategies that are less vulnerable to the specific choice of the model.In this paper we combine a worst-case approach with filtering techniques for a utility maxi-mization problem in a financial market with stochastic drift. This is a follow-up paper on Sassand Westphal [20] where a worst-case utility maximization problem for a financial market withconstant drift is investigated. In [20] we work with a Black–Scholes market and address anoptimization problem of the form sup π ∈A h ( x ) inf µ ∈ K E µ (cid:2) U ( X πT ) (cid:3) , (1.1)where U : R + → R is a utility function, X πT denotes the terminal wealth achieved when usingstrategy π , and A h ( x ) is a class of constrained admissible strategies with initial capital x .The expectation E µ [ · ] is with respect to a measure under which the drift of the asset returnsis constantly equal to µ ∈ R d , with d denoting the number of risky assets in the market. By K ⊆ R d we denote a fixed ellipsoid and speak of the uncertainty set . The main result in [20]is a representation of the optimal strategy for (1.1) in the case of power or logarithmic utilityand a corresponding minimax theorem.In the present paper we generalize the results from Sass and Westphal [20] to a financial mar-ket with a stochastic drift process and time-dependent uncertainty sets K . This is motivatedby the idea that information about the hidden drift process, as e.g. obtained from filteringtechniques, might change over time. A surplus of information should then be reflected in asmaller uncertainty set. More precisely, we assume that under the reference measure returnsfollow the dynamics d R t = ν t d t + σ d W t , where the reference drift ( ν t ) t ∈ [0 ,T ] is adapted to the investor filtration ( G t ) t ∈ [0 ,T ] representingthe investor’s information. This is justified by a separation principle where one performs afiltering step before solving the optimization problem, i.e. ( ν t ) t ∈ [0 ,T ] represents the investor’sfilter for the drift process. We introduce a time-dependent uncertainty set ( K t ) t ∈ [0 ,T ] thatis a set-valued stochastic process adapted to ( G t ) t ∈ [0 ,T ] , meaning that the investor knows therealization of K t at time t . In our case, K t is an ellipsoid in R d .It is not obvious how to set up a worst-case optimization problem in this time-dependentsetting. The problem lies in the fact that the realization of the uncertainty sets ( K t ) t ∈ [0 ,T ] is not known initially but gets revealed over time. A worst-case drift process ( µ t ) t ∈ [0 ,T ] ischaracterized by being the worst one with the property that µ t ∈ K t for all t ∈ [0 , T ]. However,optimization with respect to this worst-case drift process is not feasible for an investor sinceit is not known initially. Instead, it makes sense to consider the following local approach. Forany fixed t ∈ [0 , T ], the current uncertainty set K t is known. Given this K t , investors take2odel uncertainty into account by assuming that in the future the worst possible drift processhaving values in K t will be realized, i.e. the worst drift process from the class K ( t ) = (cid:8) µ ( t ) = ( µ ( t ) s ) s ∈ [ t,T ] (cid:12)(cid:12) µ ( t ) s ∈ K t and µ ( t ) s is G t -measurable for each s ∈ [ t, T ] (cid:9) . Investors then solve at each time t ∈ [0 , T ] the local optimization problemsup π ( t ) ∈A h ( t,X πt ) inf µ ( t ) ∈K ( t ) E µ ( t ) h U (cid:0) X π ( t ) T (cid:1) (cid:12)(cid:12)(cid:12) G t i , (1.2)leading to an optimal strategy ( π ( t ) , ∗ s ) s ∈ [ t,T ] . In our continuous-time setting this decision willbe revised as soon as K t changes, possibly continuously in time. The realized optimal strategyof the investor is then given by π ∗ t = π ( t ) , ∗ t for any t ∈ [0 , T ].The focus of this paper lies in carrying over the results for the robust utility maximizationproblem with constant drift from Sass and Westphal [20] to the more general model describedabove. We determine the optimal strategy for (1.2) and prove a local minimax theorem inanalogy to [20, Thm. 3.12]. We then show how the time-dependent uncertainty set ( K t ) t ∈ [0 ,T ] can be defined based on the filter m t = E [ µ t | G t ] for various investor filtrations ( G t ) t ∈ [0 ,T ] . Theconstruction is motivated by confidence regions. Finally, we compare the optimal strategies fordifferent investor filtrations ( G t ) t ∈ [0 ,T ] and investigate which effect a surplus of information hason their performance. By means of a numerical simulation we demonstrate that investors doneed to account for model uncertainty by choosing a robustified strategy π ∗ . When investorsrely on the respective filter only, adding more information leads to a smaller worst-case ex-pected utility since the naive strategy that relies only on the filter is very vulnerable to modelmisspecifications. Investors need to robustify their strategy by taking model uncertainty intoaccount to be able to profit from additional information. This effect can also be understoodas an overconfidence of experts as studied empirically by Heath and Tversky [8].Model uncertainty, also called Knightian uncertainty in reference to the seminal book byKnight [11], has been addressed in numerous papers. Gilboa and Schmeidler [7] and Schmei-dler [25] formulate rigorous axioms on preference relations that account for risk aversion anduncertainty aversion. A robust utility functional in their sense is a mapping X inf Q ∈Q E Q (cid:2) U ( X ) (cid:3) , where U is a utility function and Q a convex set of probability measures. Chen and Epstein [2]give a continuous-time extension of this multiple-priors utility. Optimal investment decisionsunder such preferences are investigated in Quenez [19] and Schied [23], building up on Kramkovand Schachermayer [12, 13]. An extension of those results by means of a duality approach isgiven in Schied [24]. Pflug et al. [17] study risk minimization under model uncertainty. Papersaddressing drift uncertainty in a financial market are Garlappi et al. [6] and Biagini andPınar [1], among others. The latter also focuses on ellipsoidal uncertainty sets. Uncertaintyabout both drift and volatility is investigated in a recent paper by Pham et al. [18].Filtering techniques play a crucial role in utility maximization problems under partial infor-mation. There are essentially two models for the drift process that lead to finite-dimensionalfilters. In the first one the drift is modelled as an Ornstein–Uhlenbeck process, in the secondone as a continuous-time Markov chain. The filters are the well-known Kalman and Wonhamfilter, respectively, see e.g. Elliott et al. [4] and Liptser and Shiryaev [14].3he paper is organized as follows. Since this is a follow-up paper on Sass and Westphal [20]that generalizes results for a financial market with constant drift to one with stochastic driftwe recap the main results of [20] in Section 2. For the convenience of the reader the latersections then refer to Section 2. In Section 3 we set up the generalized financial market modelwith stochastic drift process and state our local worst-case optimization problem. Section 4solves this problem in several steps. We provide representations of the optimal strategy thatwill be realized by an investor whose information about the drift process changes continuouslyin time and of the worst-case drift process. Further, we prove a minimax theorem for the localoptimization problems. In Section 5 we explain how filtering techniques can be used to setup time-dependent uncertainty sets, motivated by confidence regions. We also compare theperformance of the optimal strategies for different investor filtrations by means of a numericalsimulation. This paper builds up on Sass and Westphal [20] and generalizes results for a financial marketwith constant drift to a model with a stochastic drift process. For the convenience of thereader we recap the main results of [20] in this section. Our follow-up results then refer to thissection.
The paper [20] deals with a continuous-time financial market with one risk-free and variousrisky assets. Let
T > , F , F , P ) be a filteredprobability space where the filtration F = ( F t ) t ∈ [0 ,T ] satisfies the usual conditions. All processesare assumed to be F -adapted. The risk-free asset S is of the form S t = e rt , t ∈ [0 , T ], where r ∈ R is the deterministic risk-free interest rate. Aside from the risk-free asset, investors canalso invest in d ≥ R = ( R , . . . , R d ) > is defined byd R t = ν d t + σ d W t , R = 0 , where W = ( W t ) t ∈ [0 ,T ] is an m -dimensional Brownian motion under P with m ≥ d . Further, ν ∈ R d and σ ∈ R d × m , where it is assumed that σ has full rank equal to d .Model uncertainty is introduced by assuming that the true drift of the stocks is only knownto be an element of some set K ⊆ R d with ν ∈ K and that investors want to maximize theirworst-case expected utility when the drift takes values within K . The value ν can be thoughtof as an estimate for the drift that was for instance obtained from historical stock prices.Changing the drift from ν to some µ ∈ K can be expressed by a change of measure. For thispurpose, let the process ( Z µt ) t ∈ [0 ,T ] be defined by Z µt = exp (cid:16) θ ( µ ) > W t − k θ ( µ ) k t (cid:17) , where θ ( µ ) = σ > ( σσ > ) − ( µ − ν ). One can then define a new measure P µ by setting d P µ d P = Z µT .Note that since θ ( µ ) is a constant, the process ( Z µt ) t ∈ [0 ,T ] is a strictly positive martingale.Therefore, P µ is a probability measure that is equivalent to P and it follows from Girsanov’sTheorem that the process ( W µt ) t ∈ [0 ,T ] , defined by W µt = W t − θ ( µ ) t , is a Brownian motionunder P µ . The return dynamics can therefore be rewritten asd R t = ν d t + σ d W t = ν d t + σ (cid:0) d W µt + θ ( µ ) d t (cid:1) = µ d t + σ d W µt , P to P µ corresponds to changing the drift in the returndynamics from ν to µ . In the following, let E µ [ · ] denote the expectation under measure P µ and E [ · ] = E ν [ · ] the expectation under the reference measure P = P ν .An investor’s trading decisions are described by a self-financing trading strategy ( π t ) t ∈ [0 ,T ] with values in R d . The entry π it , i = 1 , . . . , d , is the proportion of wealth invested in asset i attime t . The corresponding wealth process ( X πt ) t ∈ [0 ,T ] given initial wealth x > X πt = X πt (cid:16) r d t + π > t ( µ − r d ) d t + π > t σ d W µt (cid:17) , X π = x , for any µ ∈ K . Trading strategies are required to be F R -adapted, where F R = ( F Rt ) t ∈ [0 ,T ] for F Rt = σ (( R s ) s ∈ [0 ,t ] ). The basic admissibility set is defined as A ( x ) = (cid:26) ( π t ) t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) π is F R -adapted , X π = x , E µ (cid:20)Z T k σ > π t k d t (cid:21) < ∞ for all µ ∈ K (cid:27) . The paper Sass and Westphal [20] considers investors with power or logarithmic utility, usingthe notation U γ : R + → R for γ ∈ ( −∞ , U γ ( x ) = x γ γ for γ = 0 denotes power utilityand U ( x ) = log( x ) is the logarithmic utility function. Investors with a robust approach to theportfolio optimization problem would try to maximizeinf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) among the admissible strategies. It is quite straightforward to show that as soon as r d ∈ K ,the strategy ( π t ) t ∈ [0 ,T ] with π t = 0 for all t ∈ [0 , T ] is optimal in the class of admissible strategies A ( x ). This observation is proven in Sass and Westphal [20, Prop. 2.1] and implies that asthe level of uncertainty exceeds a certain threshold, it will be optimal for investors to notinvest anything in the stocks and everything in the risk-free asset. For finding less conservativestrategies that still take into account model uncertainty a constraint on the admissible strategiesis introduced that prevents a pure bond investment. Consider for some h > A h ( x ) = (cid:8) π ∈ A ( x ) (cid:12)(cid:12) h π t , d i = h for all t ∈ [0 , T ] (cid:9) . Taking h = 1 would imply that investors are not allowed to invest anything in the risk-free asset.They must then distribute all of their wealth among the risky assets. Sass and Westphal [20]study the case where the uncertainty set is an ellipsoid in R d centered around the referenceparameter ν , i.e. K = (cid:8) µ ∈ R d (cid:12)(cid:12) ( µ − ν ) > Γ − ( µ − ν ) ≤ κ (cid:9) . Here, κ > ν ∈ R d , and Γ ∈ R d × d is symmetric and positive definite. For Γ = I d one simplygets a ball in the Euclidean norm with radius κ and center ν . Another special case discussedin the literature is Γ = σσ > , see e.g. Biagini and Pınar [1]. The value of κ determines the sizeof the ellipsoid. Higher values of κ correspond to more uncertainty about the true drift. Therobust utility maximization problem over the constrained strategies π ∈ A h ( x ) can then bewritten in the form sup π ∈A h ( x ) inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) . (2.1)5 .2 Solution of the non-robust problem To solve the optimization problem in (2.1) Sass and Westphal [20] first address the non-robust constrained utility maximization problem under a fixed parameter µ ∈ R d . For betterreadability the following notation is introduced. Definition 2.1.
Denote with D the matrix D = −
1. . . ...0 1 − ∈ R ( d − × d and define the matrix A ∈ R d × d and the vector c ∈ R d by A = D > ( Dσσ > D > ) − D,c = e d − D > ( Dσσ > D > ) − Dσσ > e d = ( I d − Aσσ > ) e d . The following result gives the optimal strategy for the non-robust problem and can be foundin Sass and Westphal [20, Prop. 3.4].
Proposition 2.2.
Let µ ∈ R d . Then the optimal strategy for the optimization problem sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) is the strategy ( π t ) t ∈ [0 ,T ] with π t = 11 − γ Aµ + hc for all t ∈ [0 , T ] , with A and c as in Definition 2.1. In the proof the d -dimensional constrained problem is reduced to a ( d − d − Corollary 2.3.
Let µ ∈ R d . Then the optimal expected utility from terminal wealth is sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) = x γ γ exp (cid:16) γT (cid:16)e r + 12(1 − γ ) (cid:0)e µ − e r d − (cid:1) > ( e σ e σ > ) − (cid:0)e µ − e r d − (cid:1)(cid:17)(cid:17) , γ = 0 , log( x ) + (cid:16)e r + 12 (cid:0)e µ − e r d − (cid:1) > ( e σ e σ > ) − (cid:0)e µ − e r d − (cid:1)(cid:17) T, γ = 0 , where e σ = Dσ, e r = (1 − h ) r + he > d µ −
12 (1 − γ ) k hσ > e d k , e µ = Dµ − h (1 − γ ) Dσσ > e d + e r d − . .3 The worst-case parameter In a next step one may ask what the worst possible parameter µ would be for the investor, giventhat she reacts optimally, i.e. by applying the strategy from Proposition 2.2. This correspondsto solving the dual problem inf µ ∈ K sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) . Note that at this point it is not clear whether equality holds between the original problemand the corresponding dual problem. The following result for the solution of the dual problemis given in Sass and Westphal [20, Thm. 3.8]. Let us decompose Γ = τ τ > for a nonsingularmatrix τ ∈ R d × d . Theorem 2.4.
Let λ < λ ≤ · · · ≤ λ d denote the eigenvalues of τ > Aτ , and let v = 1 k τ − d k τ − d , v , . . . , v d ∈ R d denote the respective orthogonal eigenvectors with k v i k = 1 for all i = 1 , . . . , d . Then inf µ ∈ K sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) = E µ ∗ (cid:2) U γ ( X π ∗ T ) (cid:3) , where µ ∗ = ν − τ d X i =1 (cid:18) λ i − γ + hψ ( κ ) k τ − d k (cid:19) − (cid:28) hτ > c + λ i − γ τ − ν, v i (cid:29) v i for ψ ( κ ) ∈ (0 , κ ] that is uniquely determined by k τ − ( µ ∗ − ν ) k = κ , and where ( π ∗ t ) t ∈ [0 ,T ] isdefined by π ∗ t = 11 − γ Aµ ∗ + hc for all t ∈ [0 , T ] . The preceding theorem solves the probleminf µ ∈ K sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) . (2.2)This is the corresponding dual problem to the original optimization problemsup π ∈A h ( x ) inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) , (2.3)but in general the values of these two problems do not coincide. There are, of course, spe-cial cases in which the supremum and the infimum do interchange. Those results are called minimax theorems in the literature. In a portfolio optimization setting that is similar to oursa minimax theorem has been shown in Quenez [19], building up on the theory by Kramkovand Schachermayer [12]. Due to the constraint h π t , d i = h for all t ∈ [0 , T ], the result fromQuenez [19] does not apply directly to our setting. It is possible, however, to use the knowledgeabout the optimal strategy for (2.2) to show that it indeed also solves (2.3) and that in thiscase, the supremum and the infimum can be interchanged.7 .4 A minimax theorem The following representation of π ∗ , given in Sass and Westphal [20, Lem. 3.10], is useful forproving a minimax theorem. Lemma 2.5.
The strategy π ∗ from Theorem 2.4 satisfies π ∗ t = − hψ ( κ ) k τ − d k Γ − ( µ ∗ − ν ) for all t ∈ [0 , T ] . The preceding lemma characterizes the strategy π ∗ that is optimal for the parameter µ ∗ .Vice versa, µ ∗ is also the worst possible drift parameter, given that an investor applies strategy π ∗ . This is shown in Sass and Westphal [20, Prop. 3.11]. Proposition 2.6.
The parameter µ that attains the minimum in inf µ ∈ K E µ (cid:2) U γ ( X π ∗ T ) (cid:3) is µ ∗ , i.e. µ ∗ is the worst possible parameter, given that an investor chooses strategy π ∗ . It then follows that the point ( π ∗ , µ ∗ ) is a saddle point of the problem, i.e. it holds E µ ∗ (cid:2) U γ ( X πT ) (cid:3) ≤ E µ ∗ (cid:2) U γ ( X π ∗ T ) (cid:3) ≤ E µ (cid:2) U γ ( X π ∗ T ) (cid:3) for all µ ∈ K and π ∈ A h ( x ). This property is essential for proving the following minimaxtheorem, given in Sass and Westphal [20, Thm. 3.12]. Theorem 2.7.
Let K = { µ ∈ R d | ( µ − ν ) > Γ − ( µ − ν ) ≤ κ } . Then sup π ∈A h ( x ) inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) = E µ ∗ (cid:2) U γ ( X π ∗ T ) (cid:3) = inf µ ∈ K sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) , where µ ∗ and π ∗ are defined as in Theorem 2.4. In the following we generalize the approach from Sass and Westphal [20] to a financial marketmodel where the drift is a stochastic process instead of a constant. To account for a changein information about the drift we also introduce time-dependence in the uncertainty set. Thebasic idea is that the available information in the market, for instance the observed assetreturns or external sources of information, are used to estimate the true drift based on filteringtechniques and to set up a corresponding uncertainty set K t at any time t ∈ [0 , T ]. Given K t ,investors then take model uncertainty into account by assuming that in the future the worstpossible drift process ( µ ( t ) s ) s ∈ [ t,T ] with values in K t will be realized. In our continuous-timesetting the decision about the uncertainty set will be revised as soon as the information aboutthe true drift changes, so in the extreme case continuously in time.8 .1 Reference model Before stating our generalized financial market, we make an observation that justifies the setupof the model.
Remark 3.1.
Suppose that the “true” dynamics of the d -dimensional return process R aregiven by d R t = µ t d t + σ d W t , R = 0 , for some stochastic drift process ( µ t ) t ∈ [0 ,T ] , an m -dimensional Brownian motion ( W t ) t ∈ [0 ,T ] , m ≥ d , and some σ ∈ R d × m with full rank. Assume further that the information of an investoris given by the investor filtration G = ( G t ) t ∈ [0 ,T ] . The investor’s best estimator for µ is thenthe conditional mean m t := E [ µ t | G t ] and one can rewrite the dynamics of the return processas d R t = m t d t + σ d V t , where the so-called innovations process ( V t ) t ∈ [0 ,T ] is a G -adapted Brownian motion. Forinstance, in the setting where an investor observes only the return process R , the process( m t ) t ∈ [0 ,T ] would be the Kalman filter.In the following, we set up our continuous-time financial market model working directly withthe innovations process and therefore assuming a G -adapted drift process. The separationprinciple that we use here by filtering first and then performing the optimization is a commonapproach for dealing with partial information.We fix an investment horizon T > , F , F , P ) wherethe filtration F = ( F t ) t ∈ [0 ,T ] satisfies the usual conditions. All processes are assumed to be F -adapted. We assume that an investor’s information is described by the investor filtration G = ( G t ) t ∈ [0 ,T ] with G t ⊆ F t for all t ∈ [0 , T ]. We consider, as before, a financial market withone risk-free and d ≥ S evolves asd S t = S t r d t, S = 1 , where r > R , . . . , R d of the riskyassets follow the dynamics d R t = ν t d t + σ d W t , R = 0 , (3.1)where R = ( R , . . . , R d ) > . Here, ( W t ) t ∈ [0 ,T ] is an m -dimensional Brownian motion under P , m ≥ d . Note that the volatility matrix σ ∈ R d × m in (3.1) is constant. Further, we assume that σ has full rank equal to d . In contrast to the volatility, the drift might change in the courseof time. We assume that ( ν t ) t ∈ [0 ,T ] is an R d -valued G -adapted stochastic process and think of( ν t ) t ∈ [0 ,T ] as an estimation for the true drift process given all available information. We speakof ( ν t ) t ∈ [0 ,T ] as the reference drift . As before, we are concerned with investors who are uncertain about the true drift. They areaware that ( ν t ) t ∈ [0 ,T ] in (3.1) might not be the true drift process. In utility maximizationproblems they want to maximize their worst-case expected utility, given that the true driftprocess is in a way “close” to ν . To model the uncertainty about the drift we specify theellipsoidal sets K t = (cid:8) µ ∈ R d (cid:12)(cid:12) ( µ − ν t ) > Γ − t ( µ − ν t ) ≤ κ t (cid:9) , t ∈ [0 , T ] , t ) t ∈ [0 ,T ] is a G -adapted stochastic process of symmetric and positive-definite matricesΓ t ∈ R d × d and ( κ t ) t ∈ [0 ,T ] is G -adapted with κ t > t ∈ [0 , T ]. The set K t is determinedat time t ∈ [0 , T ] by taking the available information about the true drift process into account,for example based on filtering techniques. The process ( K t ) t ∈ [0 ,T ] is a G -adapted set-valuedprocess, therefore the investor knows the realization of K t at time t ∈ [0 , T ].Given this K t , investors then take model uncertainty into account by assuming that in thefuture the worst possible drift process having values in K t will be realized. We denote thisworst-case future drift by ( µ ( t ) , ∗ s ) s ∈ [ t,T ] . This allows for some deterministic dynamics given K t ,i.e. the µ ( t ) , ∗ s for any s ∈ [ t, T ] are G t -measurable. The worst-case optimization problem thenleads to an optimal strategy ( π ( t ) , ∗ s ) s ∈ [ t,T ] , determined at time t . In our continuous-time settingthis decision will be revised as soon as K t changes, possibly continuously in time. The realizedworst-case drift process ( µ ∗ t ) t ∈ [0 ,T ] and optimal strategy ( π ∗ t ) t ∈ [0 ,T ] are then given by µ ∗ t = µ ( t ) , ∗ t , π ∗ t = π ( t ) , ∗ t for any t ∈ [0 , T ]. If µ ∗ and π ∗ are uniquely determined, then they are by construction G -adapted. This is not so much a game setting but rather a way how the investor determinesthe worst case. It is a mixture of using estimation methods and taking model uncertainty intoaccount.The optimization problem can be derived only locally for each t ∈ [0 , T ]. In detail, the setuplooks as follows. At time t ∈ [0 , T ] investors assume that the future drift process will be theworst one within the class K ( t ) = (cid:8) µ ( t ) = ( µ ( t ) s ) s ∈ [ t,T ] (cid:12)(cid:12) µ ( t ) s ∈ K t and µ ( t ) s is G t -measurable for each s ∈ [ t, T ] (cid:9) . For each µ = µ ( t ) ∈ K ( t ) we can construct a new measure by defining the R m -valued process( θ s ( µ )) s ∈ [0 ,T ] with θ s ( µ ) = ( , s < t,σ > ( σσ > ) − ( µ s − ν s ) , s ≥ t, and Z µs = exp (cid:18)Z s θ u ( µ ) > d W u − Z s k θ u ( µ ) k d u (cid:19) for s ∈ [0 , T ]. We then define the new probability measure P µ byd P µ d P = Z µT and note that, under P µ , the process ( W µs ) s ∈ [0 ,T ] with W µs = W s − Z s θ u ( µ ) d u for s ∈ [0 , T ] is a Brownian motion by Girsanov’s Theorem. Note that due to boundedness of K t the process θ ( µ ) is bounded and therefore ( Z µs ) s ∈ [0 ,T ] is a true martingale. The change ofmeasure causes a change in the drift on the interval [ t, T ] only. For our optimization problemsthis is the only relevant time interval since we condition on G t . For s ∈ [ t, T ] we can rewritethe dynamics of the asset returns asd R s = ν s d s + σ d W s = µ s d s + σ d W µs , which means that under P µ the future drift of the stocks is given by ( µ s ) s ∈ [ t,T ] . We write E µ [ · ] = E µ ( t ) [ · ] for expectation under the measure P µ .10 .3 Local optimization problem An investor’s behavior in the time interval [ t, T ] is described by a self-financing trading strategy π ( t ) = ( π ( t ) s ) s ∈ [ t,T ] . The class of admissible trading strategies, given that the investor has wealth x > t , is A ( t, x ) = (cid:26) π ( t ) = ( π ( t ) s ) s ∈ [ t,T ] (cid:12)(cid:12)(cid:12)(cid:12) π ( t ) is G -adapted , X πt = x, E µ ( t ) (cid:20)Z Tt k σ > π ( t ) s k d s (cid:21) < ∞ for all µ ( t ) ∈ K ( t ) (cid:27) . We will restrict these strategies by imposing, as in Sass and Westphal [20], a constraint thatprevents a pure bond investment. For any h > A h ( t, x ) = (cid:8) π ( t ) ∈ A ( t, x ) (cid:12)(cid:12) h π ( t ) s , d i = h for all s ∈ [ t, T ] (cid:9) . For an investor choosing strategy π = π ( t ) ∈ A ( t, X πt ) the terminal wealth can be written as X πT = X πt exp (cid:18)Z Tt (cid:16) r + π > s ( µ s − r d ) − k σ > π s k (cid:17) d s + Z Tt π > s σ d W µs (cid:19) . We are now able to state our utility maximization problem. At time t the local optimizationproblem reads sup π ( t ) ∈A h ( t,X πt ) inf µ ( t ) ∈K ( t ) E µ ( t ) h U γ (cid:0) X π ( t ) T (cid:1) (cid:12)(cid:12)(cid:12) G t i . (3.2)Here, U γ with γ ∈ ( −∞ ,
1) again denotes the power utility function U γ ( x ) = x γ γ if γ = 0, andlogarithmic utility U ( x ) = log( x ) if γ = 0. Remark 3.2.
In the case where K t = { µ ∈ R d | ( µ − ν ) > Γ − ( µ − ν ) ≤ κ } for all t ∈ [0 , T ],i.e. where our reference drift is simply a constant ν , and also the matrix Γ t = Γ as well as theradius κ t = κ are constant in time, we obtain the setting from Section 2 as a special case. In this section we solve (3.2) by computing the optimal strategy π ( t ) , ∗ and the worst-casedrift µ ( t ) , ∗ and prove a minimax theorem in analogy to Theorem 2.7. We proceed as in thesetting with constant drift in Section 2. Looking at the local optimization problem for a fixed t ∈ [0 , T ] enables us to reduce the drift uncertainty to K t and make use of our results forconstant drift. At the end of this section we explain which strategy will be realized by aninvestor whose information about the drift process changes continuously in time and how thisstrategy is naturally obtained from the solution to the local optimization problems. As a first step towards solving (3.2) we compute the optimal strategy for an investor given aparticular future drift µ ( t ) ∈ K ( t ) . 11 roposition 4.1. Let t ∈ [0 , T ] and µ ( t ) ∈ K ( t ) . Then the optimal strategy for the optimizationproblem sup π ( t ) ∈A h ( t,X πt ) E µ ( t ) h U γ (cid:0) X π ( t ) T (cid:1) (cid:12)(cid:12)(cid:12) G t i is the strategy ( π ( t ) s ) s ∈ [ t,T ] with π ( t ) s = 11 − γ Aµ ( t ) s + hc for all s ∈ [ t, T ] , where A ∈ R d × d and c ∈ R d are as introduced in Definition 2.1.Proof. The proof works along the lines of the proof of Proposition 2.2. We take an arbitrarystrategy π = π ( t ) ∈ A h ( t, X πt ) and recall that we can write the terminal wealth under strategy π as X πT = X πt exp (cid:18)Z Tt (cid:16) r + π > s ( µ ( t ) s − r d ) − k σ > π s k (cid:17) d s + Z Tt π > s σ d W µs (cid:19) . We now proceed exactly as in the proof of Proposition 2.2, replacing the constant µ by the G t -measurable ( µ ( t ) s ) s ∈ [ t,T ] , and perform the same transformation to a ( d − E µ ( t ) [ U γ ( X πT ) | G t ] equals the expected utility of terminal wealth, con-ditional on G t , in an unconstrained financial market with d − e µ s ) s ∈ [ t,T ] , the risk-free interest rate is ( e r s ) s ∈ [ t,T ] and the volatility matrix is e σ ∈ R ( d − × m . These transformed market parameters have the form e σ = Dσ, e r s = (1 − h ) r + he > d µ ( t ) s −
12 (1 − γ ) k hσ > e d k , e µ s = Dµ ( t ) s − h (1 − γ ) Dσσ > e d + e r s d − . Note that since the ( µ ( t ) s ) s ∈ [ t,T ] are G t -measurable, so are ( e r s ) s ∈ [ t,T ] and ( e µ s ) s ∈ [ t,T ] , in particularthe market parameters in the transformed market can be observed by the investor. In this( d − e π s = 11 − γ ( e σ e σ > ) − ( e µ s − e r s d − ) = 11 − γ ( Dσσ > D > ) − (cid:0) Dµ ( t ) s − h (1 − γ ) Dσσ > e d (cid:1) for every s ∈ [ t, T ]. For the logarithmic utility case, this is immediate, for power utility, thisneeds to be shown.Merton [15] yields the form of the optimal strategy in a Black–Scholes market with constantparameters. This result can be extended to a market where the risk-free interest rate as wellas drift and volatility of the stocks are not necessarily constant but still observable by theinvestor, see Westphal [26, App. B] for a complete proof. A similar result has been provenin Karatzas et al. [10] for complete markets with deterministic market coefficients and forincomplete markets with totally unhedgeable market coefficients.12ow we can return to our original market and obtain that the optimal strategy fulfills π ( t ) s = D > e π s + he d = D > − γ ( Dσσ > D > ) − (cid:0) Dµ ( t ) s − h (1 − γ ) Dσσ > e d (cid:1) + he d = 11 − γ D > ( Dσσ > D > ) − Dµ ( t ) s + h (cid:0) I d − D > ( Dσσ > D > ) − Dσσ > (cid:1) e d = 11 − γ Aµ ( t ) s + hc for all s ∈ [ t, T ], where we have used the notation for A and c from Definition 2.1. Note that( π ( t ) s ) s ∈ [ t,T ] is indeed admissible due to boundedness of K t .The preceding proposition states the form of the investor’s optimal strategy under the as-sumption that a specific future drift process ( µ ( t ) s ) s ∈ [ t,T ] is given. The explicit form can be usedto compute also the expected utility obtained when applying the optimal strategy. Corollary 4.2.
Let t ∈ [0 , T ] and µ ( t ) ∈ K ( t ) . Then the optimal expected utility from terminalwealth is sup π ( t ) ∈A h ( t,X πt ) E µ ( t ) h U γ (cid:0) X π ( t ) T (cid:1) (cid:12)(cid:12)(cid:12) G t i = ( X πt ) γ γ exp (cid:18) γ Z Tt (cid:16)e r s + 12(1 − γ ) (cid:0)e µ s − e r s d − (cid:1) > ( e σ e σ > ) − (cid:0)e µ s − e r s d − (cid:1)(cid:17) d s (cid:19) , γ = 0 , log( X πt ) + Z Tt (cid:16)e r s + 12 (cid:0)e µ s − e r s d − (cid:1) > ( e σ e σ > ) − (cid:0)e µ s − e r s d − (cid:1)(cid:17) d s, γ = 0 , where e σ = Dσ, e r s = (1 − h ) r + he > d µ ( t ) s −
12 (1 − γ ) k hσ > e d k , e µ s = Dµ ( t ) s − h (1 − γ ) Dσσ > e d + e r s d − . Proof.
The representation in the corollary follows, just like in the proof of Corollary 2.3, bythe fact that we have reduced our constrained utility maximization problem to a ( d − d − e π s = 11 − γ ( e σ e σ > ) − ( e µ s − e r s d − )for all s ∈ [ t, T ]. Plugging this optimal strategy in yields the expression from the corollary. In the following, we compute the worst-case future drift process that is determined at time t ∈ [0 , T ], i.e. the drift process µ ( t ) ∈ K ( t ) for whichsup π ( t ) ∈A h ( t,X πt ) E µ ( t ) h U γ (cid:0) X π ( t ) T (cid:1) (cid:12)(cid:12)(cid:12) G t i
13s minimized. Due to the previous corollary we see that this is equivalent to the minimizationof the integral Z Tt (cid:16)e r s + 12 (cid:0)e µ s − e r s d − (cid:1) > ( e σ e σ > ) − (cid:0)e µ s − e r s d − (cid:1)(cid:17) d s. (4.1)When plugging the representations for e µ , e r and e σ back in, we obtain an expression that dependson ( µ ( t ) s ) s ∈ [ t,T ] again. By the same calculations as in the setting with constant drift we deducethat minimizing (4.1) is equivalent to minimizing Z Tt (cid:16) − γ ) ( µ ( t ) s ) > Aµ ( t ) s + hc > µ ( t ) s (cid:17) d s. But the minimization of this integral is equivalent to a pointwise minimization of K t µ − γ ) µ > Aµ + hc > µ. Now it is straightforward to see that we can use the results from Section 2 to obtain the worst-case drift process ( µ ( t ) , ∗ s ) s ∈ [ t,T ] . Here, µ ( t ) , ∗ s is for any s ∈ [ t, T ] obtained as the minimizer ofthe above function on K t . Recall that the uncertainty set is an ellipsoid of the form K t = { µ ∈ R d | ( µ − ν t ) > Γ − t ( µ − ν t ) ≤ κ t } . We have assumed that Γ t is a symmetric positive-definite matrix in R d × d . In the following weuse the representation Γ t = τ t τ > t where τ t ∈ R d × d is a nonsingular matrix. Corollary 4.3.
We fix some t ∈ [0 , T ] and let λ t, < λ t, ≤ · · · ≤ λ t,d denote theeigenvalues of τ > t Aτ t , and v t, = 1 k τ − t d k τ − t d , v t, , . . . , v t,d ∈ R d the respective orthogonal eigenvectors with k v t,i k = 1 for all i = 1 , . . . , d . Then inf µ ( t ) ∈K ( t ) sup π ( t ) ∈A h ( t,X πt ) E µ ( t ) h U γ (cid:0) X π ( t ) T (cid:1) (cid:12)(cid:12)(cid:12) G t i = E µ ( t ) , ∗ h U γ (cid:0) X π ( t ) , ∗ T (cid:1) (cid:12)(cid:12)(cid:12) G t i , where µ ( t ) , ∗ s = ν t − τ t d X i =1 (cid:18) λ t,i − γ + hψ t ( κ t ) k τ − t d k (cid:19) − D hτ > t c + λ t,i − γ τ − t ν t , v t,i E v t,i for all s ∈ [ t, T ] , and where ψ t ( κ t ) ∈ (0 , κ t ] is uniquely determined by k τ − t ( µ ( t ) , ∗ s − ν t ) k = κ t .The strategy ( π ( t ) , ∗ s ) s ∈ [ t,T ] has the form π ( t ) , ∗ s = 11 − γ Aµ ( t ) , ∗ s + hc for all s ∈ [ t, T ] . roof. We have seen that the worst-case drift process ( µ ( t ) , ∗ s ) s ∈ [ t,T ] is the one where µ ( t ) , ∗ s is forany s ∈ [ t, T ] equal to the minimizer of the function µ − γ ) µ > Aµ + hc > µ over all µ ∈ K t . So we can do the minimization as in Section 2. We know that the matrix τ > t Aτ t ∈ R d × d is symmetric and positive definite withker( τ > t Aτ t ) = span( { τ − t d } ) . Now the representation of µ ( t ) , ∗ s follows as in Theorem 2.4. The form of the optimal strategy π ( t ) , ∗ then follows from Proposition 4.1.The preceding corollary shows that the probleminf µ ( t ) ∈K ( t ) sup π ( t ) ∈A h ( t,X πt ) E µ ( t ) h U γ (cid:0) X π ( t ) T (cid:1) (cid:12)(cid:12)(cid:12) G t i is solved by drift process ( µ ( t ) , ∗ s ) s ∈ [ t,T ] and strategy ( π ( t ) , ∗ s ) s ∈ [ t,T ] . Note that both the worst-casedrift process and the optimal strategy are constant on [ t, T ] and G t -measurable. This is due tothe setup of the model in which investors assume that the future drift process will take valuesin the ellipsoid K t only.The problem above is the dual to our original problemsup π ( t ) ∈A h ( t,X πt ) inf µ ( t ) ∈K ( t ) E µ ( t ) h U γ (cid:0) X π ( t ) T (cid:1) (cid:12)(cid:12)(cid:12) G t i . To ensure that µ ( t ) , ∗ and π ( t ) , ∗ are also a solution to this problem we have to show that µ ( t ) , ∗ is the worst drift process in the set K ( t ) , given that an investor chooses trading strategy π ( t ) , ∗ .In that case, the infimum and the supremum interchange and we can deduce that π ( t ) , ∗ and µ ( t ) , ∗ also establish a solution to our original robust optimization problem. We proceed as in Section 2 and note that the strategy π ( t ) , ∗ from the previous corollary satisfies π ( t ) , ∗ s = − hψ t ( κ t ) k τ − t d k Γ − t (cid:0) µ ( t ) , ∗ s − ν t (cid:1) for all s ∈ [ t, T ]. This can be proven by analogy with Lemma 2.5. This observation helps toprove the following proposition. Proposition 4.4.
The drift process ( µ ( t ) s ) s ∈ [ t,T ] that attains the minimum in inf µ ( t ) ∈K ( t ) E µ ( t ) h U γ (cid:0) X π ( t ) , ∗ T (cid:1) (cid:12)(cid:12)(cid:12) G t i is ( µ ( t ) , ∗ s ) s ∈ [ t,T ] , i.e. µ ( t ) , ∗ is the worst possible drift process, given that an investor chooses thestrategy π ( t ) , ∗ . roof. We take an arbitrary µ = µ ( t ) ∈ K ( t ) . Note that in case γ = 0 we can write E µ h U γ (cid:0) X π ( t ) , ∗ T (cid:1) (cid:12)(cid:12)(cid:12) G t i = ( X πt ) γ γ e γr ( T − t ) E µ (cid:20) exp (cid:18) γ Z Tt (cid:16) ( π ( t ) , ∗ s ) > ( µ s − r d ) − k σ > π ( t ) , ∗ s k (cid:17) d s + γ Z Tt ( π ( t ) , ∗ s ) > σ d W µs (cid:19)(cid:21) = ( X πt ) γ γ e γr ( T − t ) exp (cid:18) γ Z Tt (cid:16) ( π ( t ) , ∗ s ) > ( µ s − r d ) − − γ k σ > π ( t ) , ∗ s k (cid:17) d s (cid:19) . In case γ = 0 we have E µ h log (cid:0) X π ( t ) , ∗ T (cid:1) (cid:12)(cid:12)(cid:12) G t i = log( X πt ) + r ( T − t ) + Z Tt (cid:16) ( π ( t ) , ∗ s ) > ( µ s − r d ) − k σ > π ( t ) , ∗ s k (cid:17) d s. In both cases, the drift process ( µ s ) s ∈ [ t,T ] ∈ K ( t ) that minimizes this expression is the one thatminimizes Z Tt ( π ( t ) , ∗ s ) > µ s d s. Since ( π ( t ) , ∗ s ) s ∈ [ t,T ] is constant, we find the minimizer as the minimizer of ( π ( t ) , ∗ s ) > µ s . Recallthat π ( t ) , ∗ s = − hψ t ( κ t ) k τ − t d k Γ − t (cid:0) µ ( t ) , ∗ s − ν t (cid:1) . It follows that( π ( t ) , ∗ s ) > Γ t π ( t ) , ∗ s = h ψ t ( κ t ) k τ − t d k (cid:0) µ ( t ) , ∗ s − ν t (cid:1) > Γ − t (cid:0) µ ( t ) , ∗ s − ν t (cid:1) = h κ t ψ t ( κ t ) k τ − t d k . Knowing that ψ t ( κ t ) > q ( π ( t ) , ∗ s ) > Γ t π ( t ) , ∗ s = hκ t ψ t ( κ t ) k τ − t d k . The drift process µ ( t ) , ∗ s at time s can thus be rewritten in the form µ ( t ) , ∗ s = ν t − ψ t ( κ t ) k τ − t d k h Γ t π ( t ) , ∗ s = ν t − κ t q ( π ( t ) , ∗ s ) > Γ t π ( t ) , ∗ s Γ t π ( t ) , ∗ s . This is exactly the vector that minimizes ( π ( t ) , ∗ s ) > µ over all µ ∈ K t , see the proof of [20,Prop. 3.11]. Hence, µ ( t ) , ∗ is the drift process that minimizes the expected utility of terminalwealth for an investor who chooses strategy π ( t ) , ∗ .The previous proposition establishes an equilibrium result. By definition, the strategy π ( t ) , ∗ is optimal for the drift µ ( t ) , ∗ . Due to the proposition, it also holds that µ ( t ) , ∗ is the worst driftgiven that an investor chooses strategy π ( t ) , ∗ . Hence, we see that ( π ( t ) , ∗ , µ ( t ) , ∗ ) is a saddle pointof the optimization problem sup π ( t ) ∈A h ( t,X πt ) inf µ ( t ) ∈K ( t ) E µ ( t ) h U γ (cid:0) X π ( t ) T (cid:1) (cid:12)(cid:12)(cid:12) G t i . In particular, the supremum and infimum can be interchanged. We obtain the following mini-max theorem. 16 heorem 4.5.
Let t ∈ [0 , T ] . Then sup π ( t ) ∈A h ( t,X πt ) inf µ ( t ) ∈K ( t ) E µ ( t ) h U γ (cid:0) X π ( t ) T (cid:1) (cid:12)(cid:12)(cid:12) G t i = E µ ( t ) , ∗ h U γ (cid:0) X π ( t ) , ∗ T (cid:1) (cid:12)(cid:12)(cid:12) G t i = inf µ ( t ) ∈K ( t ) sup π ( t ) ∈A h ( t,X πt ) E µ ( t ) , ∗ h U γ (cid:0) X π ( t ) , ∗ T (cid:1) (cid:12)(cid:12)(cid:12) G t i , where µ ( t ) , ∗ and π ( t ) , ∗ are defined as in Corollary 4.3.Proof. The proof is analogous to the proof of Theorem 2.7.The previous theorem solves our original local optimization problem (3.2) for a fixed time t ∈ [0 , T ]. It shows that the best strategy for an investor in this robust optimization problemis the strategy ( π ( t ) , ∗ s ) s ∈ [ t,T ] with π ( t ) , ∗ s = 11 − γ Aµ ( t ) , ∗ s + hc for all s ∈ [ t, T ], where ( µ ( t ) , ∗ s ) s ∈ [ t,T ] is defined as in Corollary 4.3. The process ( µ ( t ) , ∗ s ) s ∈ [ t,T ] canbe interpreted as the worst possible realization of the future drift process from the investor’spoint of view at time t . The worst-case drift and optimal strategy in this setting are constanton [ t, T ]. This is due to the assumption of the investor that the future drift will take valuesin the set K t only, where K t is determined at time t using all available information, i.e. K t is G t -measurable.In our continuous-time setting it is likely that the information about the unobservable truedrift process changes continuously, therefore also the uncertainty set K t will be updated con-tinuously in time. At each time t ∈ [0 , T ], the investor will revise both the uncertainty set andthe optimization problem sup π ( t ) ∈A h ( t,X πt ) inf µ ( t ) ∈K ( t ) E µ ( t ) h U γ (cid:0) X π ( t ) T (cid:1) (cid:12)(cid:12)(cid:12) G t i . The strategy that is realized by the investor can then be found as ( π ∗ t ) t ∈ [0 ,T ] with π ∗ t = π ( t ) , ∗ t for any t ∈ [0 , T ]. It has the form π ∗ t = 11 − γ Aµ ∗ t + hc where ( µ ∗ t ) t ∈ [0 ,T ] is constructed via µ ∗ t = µ ( t ) , ∗ t for all t ∈ [0 , T ]. Note that the processes ( µ ∗ t ) t ∈ [0 ,T ] and ( π ∗ t ) t ∈ [0 ,T ] are uniquely determined, G -adapted and in general non-constant. In the special case where K t = K for all t ∈ [0 , T ],i.e. where our reference drift is simply a constant ν , and also the matrix Γ t = Γ as well as theradius κ t = κ are constant in time, also ( µ ∗ t ) t ∈ [0 ,T ] and ( π ∗ t ) t ∈ [0 ,T ] are constant in time. Theconstant values are the ones that we also get in the setting with constant drift and uncertaintyset in Theorem 2.4. 17 Construction of Uncertainty Sets via Filters
In the preceding sections we have seen how the duality approach from Sass and Westphal [20]carries over to a financial market where the drift is not necessarily constant. The generalizedmodel allows for local uncertainty sets of the form K t = (cid:8) µ ∈ R d (cid:12)(cid:12) ( µ − ν t ) > Γ − t ( µ − ν t ) ≤ κ t (cid:9) , t ∈ [0 , T ] . We have fixed an investor filtration G = ( G t ) t ∈ [0 ,T ] describing the investor’s information in thecourse of time. Our model then assumes that the processes ν = ( ν t ) t ∈ [0 ,T ] , Γ = (Γ t ) t ∈ [0 ,T ] and κ = ( κ t ) t ∈ [0 ,T ] are G -adapted. Recall that ν takes values in R d , Γ in the set of symmetric andpositive-definite matrices in R d × d and κ on the positive real line. We motivated the referencedrift ν as an estimation for the true drift, based on the information available to the investor.Here we want to make this more specific by considering the filter. The filter is the conditional distribution of µ given the available information G . We take ν tobe the conditional expectation of the drift given G , i.e. ν t = m t := E [ µ t | G t ] for every t ∈ [0 , T ].The conditional covariance matrix Q t := E (cid:2) ( µ t − m t )( µ t − m t ) > (cid:12)(cid:12) G t (cid:3) measures how close the estimator m t is to the true drift. Note that by construction both( m t ) t ∈ [0 ,T ] and ( Q t ) t ∈ [0 ,T ] are G -adapted processes. The key idea for constructing uncertaintysets based on the filter is to create confidence regions centered around m t , shaped by Q t forevery t ∈ [0 , T ].Let us assume that the drift process and the investor filtration are such that the filter isnormally distributed, more precisely µ t | G t ∼ N ( m t , Q t ) . By applying a simple transformation we deduce that( µ t − m t ) > ( Q t ) − ( µ t − m t )given G t is χ -distributed with d degrees of freedom. We fix some η ∈ (0 ,
1) and observe thata (1 − η )-confidence region can be obtained from1 − η = P (cid:16) ( µ t − m t ) > ( Q t ) − ( µ t − m t ) ≤ χ d, − η (cid:12)(cid:12)(cid:12) G t (cid:17) . Here, χ d, − η denotes the (1 − η )-quantile of the χ -distribution with d degrees of freedom. Thismotivates the choice of K t = (cid:8) µ ∈ R d (cid:12)(cid:12) ( µ − m t ) > ( Q t ) − ( µ − m t ) ≤ χ d, − η (cid:9) , t ∈ [0 , T ] , i.e. taking ν t = m t , Γ t = Q t and κ t = q χ d, − η for every t ∈ [0 , T ].If indeed µ t given G t is normally distributed, we additionally know that at any fixed time t ∈ [0 , T ] the probability that µ t ∈ K t , conditional on G t , is equal to 1 − η . Note that K t isstill a reasonable uncertainty set for µ t in the case where the assumption about the normaldistribution of the filter is not fulfilled. 18 .2 Comparison of different investor filtrations The preceding section explains how time-dependent uncertainty sets can be created based onfilters. We now apply this to a model with an unobservable Ornstein–Uhlenbeck drift processand unbiased, normally distributed expert opinions arriving at discrete points in time. Thesetting is based on Gabih et al. [5] as well as Sass et al. [21, 22]. Returns in this setting aremodelled as d R t = µ t d t + σ R d W Rt , where W R = ( W Rt ) t ∈ [0 ,T ] is an m -dimensional Brownian motion with m ≥ d and where weassume that σ R ∈ R d × m has full rank. The drift process µ is defined by the Ornstein–Uhlenbeckdynamics d µ t = α ( δ − µ t ) d t + β d B t , where α and β ∈ R d × d , δ ∈ R d and B = ( B t ) t ∈ [0 ,T ] is a d -dimensional Brownian motion thatis independent of W R . The matrices α and ββ > are assumed to be symmetric and positivedefinite. We further make the assumption that µ ∼ N ( m , Σ ) for some m ∈ R d and somesymmetric and positive-semidefinite matrix Σ ∈ R d × d , and that µ is independent of theBrownian motions W R and B , i.e. µ is independent of W R .Information about the drift process can be drawn from return observations. An additionalsource of information in this model are expert opinions that arrive at discrete points in timeand give an unbiased estimate of the state of the drift at that time point. We assume that theexpert opinions arrive at the information dates ( T k ) k ∈ I and that an expert opinion at time T k is of the form Z k = µ T k + (Γ k ) / ε k , where the matrices Γ k ∈ R d × d are symmetric and positive definite and the ε k are multivariate N (0 , I d )-distributed and independent of the Brownian motions in the market and of µ . Thesequence of information dates ( T k ) k ∈ I is also independent of the ( ε k ) k ∈ I and the Brownianmotions as well as of µ . In particular, given µ T k the expert opinion is multivariate N ( µ T k , Γ k )-distributed.The model then gives rise to various investor filtrations G = ( G t ) t ∈ [0 ,T ] . We consider thecases G = F N = ( F Nt ) t ∈ [0 ,T ] where F Nt = σ ( N P ) , G = F R = ( F Rt ) t ∈ [0 ,T ] where F Rt = σ (( R s ) s ∈ [0 ,t ] ) ∨ σ ( N P ) , G = F E = ( F Et ) t ∈ [0 ,T ] where F Et = σ (( T k , Z k ) T k ≤ t ) ∨ σ ( N P ) , G = F C = ( F Ct ) t ∈ [0 ,T ] where F Ct = σ (( R s ) s ∈ [0 ,t ] ) ∨ σ (( T k , Z k ) T k ≤ t ) ∨ σ ( N P )for the investor filtrations, where we write N P for the set of null sets under P , i.e. we workwith the filtrations that are augmented by null sets. We speak of the investor with filtration F H , H ∈ { N, R, E, C } , as the H -investor. Note that the N -investor observes neither returnsnor expert opinions and only has knowledge about the market parameters. The R -investorobserves only the return process, the E -investor only the discrete-time expert opinions, andthe C -investor the combination of both. Example 5.1.
Based on one realization of the model’s stochastic processes, fixing one infor-mation setting H ∈ { N, R, E, C } , we obtain one realization of the filter, leading to a time-dependent uncertainty set K H . For illustration purposes we plot in Figure 5.1 against time arealization of the different filters with resulting uncertainty sets in a market with d = 1 stock.19he various subplots are all based on the same realization of the drift process µ , returns R and expert opinions Z k . As a first case we consider in Figure 5.1a the degenerate informationsetting H = N , corresponding to an investor who observes neither the return process northe expert opinions. The only knowledge the investor has about the model are the modelparameters. The conditional mean is in this case constantly equal to the long-term mean δ ofthe drift process. The resulting uncertainty set converges very fast to a fixed interval centeredaround δ .For H = R , the uncertainty set moves up and down along with the conditional mean as canbe seen in Figure 5.1b. In Figures 5.1c and 5.1d we have equidistant information dates withexpert opinions. The corresponding uncertainty set jumps at information dates along with theconditional mean, due to the updates caused by an incoming expert opinion. It also becomesapparent from the plots that the conditional variance decreases at information dates, leadingto a shrinking uncertainty set.Neither of the information filtrations leads to a perfect uncertainty set in the sense that thetrue drift stays in that uncertainty set at any point in time. By the setup of the uncertainty setthere is always a positive probability that the true drift process moves out of the uncertaintyset at some point in time. . . . . − − . . (a) G = F N . . . . − − . . (b) G = F R µm H Z k K H . . . . − − . . (c) G = F E . . . . − − . . (d) G = F C Figure 5.1:
Uncertainty sets based on filters for various investor filtrations F H . Each subplot is basedon the same realization of the drift and return process and expert opinions. Based on thisrealization, the filter of the H -investor can be computed. The uncertainty set K H is thendetermined according to the filter realization.
20e now give a numerical example to illustrate the effect that the worst-case optimizationamong uncertainty sets created from filters has for the various investor filtrations consideredbefore. We create for a fixed realization of the drift process µ , of the return process R andthe expert opinions Z k a time-dependent uncertainty set for each of the corresponding filters.The aim is to compare the robust strategies that take into account model uncertainty with the“naive” strategies that rely on the respective drift estimates, only. Model parameters.
We want to apply our worst-case utility maximization problem, in par-ticular also imposing the constraint h π t , d i = h on the investor’s strategies. For that purposewe take a market with d = 2 stocks here. We fix an investment horizon of T = 1 and take h = 1. Moreover, we assume that investors start with an initial wealth of x = 1, use powerutility functions U γ with γ = 0 . η = 0 . α = ! volatility of drift process β = .
50 0 . .
25 0 . ! mean reversion level of drift process δ = . . ! initial mean of drift process m = . . ! initial variance of drift process Σ = .
01 00 0 . ! volatility of returns σ R = .
10 0 . .
05 0 . ! volatility of continuous expert σ J = .
10 0 . .
05 0 . ! Table 5.1:
Market parameters for numerical example.
Simulation study.
For the given model parameters we simulate a drift process, the returnprocess R and n = 10 discrete-time expert opinions arriving at deterministic and equidistantinformation dates on [0 , T ]. We then obtain a realization of the filters ( m H , Q H ) for any of theinformation settings H from above. As before, this leads to one time-dependent uncertaintyset for each of the investors.We can then determine the worst-case drift process ( µ ∗ t ) t ∈ [0 ,T ] and the optimal strategy( π ∗ t ) t ∈ [0 ,T ] that is realized by the investor who solves at each time point the local optimizationproblem sup π ( t ) ∈A h ( t,X πt ) inf µ ( t ) ∈K ( t ) E µ ( t ) h U γ (cid:0) X π ( t ) T (cid:1) (cid:12)(cid:12)(cid:12) F Ht i . µ ∗ t ) t ∈ [0 ,T ] and ( π ∗ t ) t ∈ [0 ,T ] are calculated from the solutions of the local optimizationproblems via π ∗ t = π ( t ) , ∗ t , µ ∗ t = µ ( t ) , ∗ t for all t ∈ [0 , T ]. The value of each investor’s worst-case optimization is then equal to E µ ∗ (cid:2) U γ ( X π ∗ T ) (cid:3) . (5.1)The quantity in (5.1) is the worst-case expected utility from the H -investor’s point of viewwhen using the robust strategy π ∗ . For comparison, we also compute E µ ∗ (cid:2) U γ ( X ˆ πT ) (cid:3) , E ν (cid:2) U γ ( X π ∗ T ) (cid:3) and E ν (cid:2) U γ ( X ˆ πT ) (cid:3) , where ν = m H is the conditional mean of the H -investor’s filter and ˆ π is the correspondingoptimal strategy given that the drift equals m H , i.e.ˆ π t = 11 − γ Am Ht + hc. We repeat this simulation 10 000 times where in each iteration a new drift process, a newreturn process and new expert opinions are simulated based on the parameters given above.Table 5.2 gives the sample mean of the various expected utilities over all simulations and inbrackets the corresponding sample standard deviation.
H n E µ ∗ (cid:2) U γ ( X π ∗ T ) (cid:3) E µ ∗ (cid:2) U γ ( X ˆ πT ) (cid:3) E ν (cid:2) U γ ( X π ∗ T ) (cid:3) E ν (cid:2) U γ ( X ˆ πT ) (cid:3) N (0.0000) (0.0000) (0.0000) (0.0000) R (0.1057) (0.3737) (2.4692) (732.4104) E
10 1.7055 (0.1117) (0.3870) (3.4208) (530.6829) C
10 1.7854 (0.4027) (0.3752) (134.5858) (19 288.2826)
Table 5.2:
Comparison of utility for different investors.
Observations.
When comparing the worst-case expected utility E µ ∗ [ U γ ( X π ∗ T )] among the in-vestors we see that the information setting H = N , which corresponds to only knowing themodel parameters, gives the lowest value. The observation of returns or of n = 10 expertopinions increases this value. The combination of return observation and discrete-time expertopinions yields a considerably larger worst-case expected utility.In the next column, E µ ∗ [ U γ ( X ˆ πT )] measures the expected utility when using strategy ˆ π , giventhat the true drift is actually the worst-case drift µ ∗ . The values are in any case smaller thanthe corresponding expected utility when using the robust strategy π ∗ . What is striking is thatthe information setting H = N , i.e. only knowledge of the model parameters, gives the bestexpected utility here. Adding more information, from return observations or expert opinions,and using the optimal strategy based on the filter leads to a smaller worst-case expected utility.This shows that for the worst-case optimization problem it is dangerous for investors to rely ontheir estimates of the drift, i.e. the conditional mean of the filter, only. They need to robustifytheir strategy by taking into account model uncertainty to be able to profit from any additionalinformation. This effect can be linked to overconfidence of experts as studied empirically byHeath and Tversky [8], seeing that more knowledge about the drift process leads to a worseexpected utility in the non-robust case due to taking more risky strategies.22he last two columns show the expected utility when using strategy π ∗ , respectively ˆ π , giventhat the true drift was actually the conditional mean ν = m H . Of course, when compared tothe expected utility given the worst-case drift µ ∗ , the expected utility given ν is much higher.Not surprisingly, the performance of ˆ π given drift ν is on average extremely good. However, wealso notice the very large sample standard deviation. In comparison to that, we see that therobust strategies π ∗ perform reasonably well given drift ν , even though they are tailored forthe worst-case drift in the respective uncertainty set. At the same time, the sample standarddeviation is much smaller than for strategy ˆ π . Conclusions.
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