Robust Utility Maximizing Strategies under Model Uncertainty and their Convergence
aa r X i v : . [ q -f i n . P M ] S e p Robust Utility Maximizing Strategies underModel Uncertainty and their Convergence
Jörn Sass ∗ and Dorothee Westphal † Department of Mathematics, Technische Universität KaiserslauternSeptember 5, 2019
Abstract
In this paper we investigate a utility maximization problem with drift uncertainty ina continuous-time Black–Scholes type financial market. We impose a constraint on theadmissible strategies that prevents a pure bond investment and we include uncertainty bymeans of ellipsoidal uncertainty sets for the drift. Our main results consist in finding anexplicit representation of the optimal strategy and the worst-case parameter and provinga minimax theorem that connects our robust utility maximization problem with the cor-responding dual problem. Moreover, we show that, as the degree of model uncertaintyincreases, the optimal strategy converges to a generalized uniform diversification strategy.
Keywords:
Portfolio optimization, Drift uncertainty, Minimax theorems, Diversification
1. Introduction
Model uncertainty is a challenge that is inherent in many applications of mathematical models.Optimization procedures in general take place under a particular model. This model, however,might be misspecified due to statistical estimation errors, incomplete information, biases, andfor various other reasons. In that sense, any specified model must be understood as an ap-proximation of the unknown “true” model. Difficulties arise since a strategy which is optimalunder the approximating model might perform rather bad for the true model specifications. Anatural way to deal with model uncertainty is to consider worst-case optimization.The optimization problem that we address is a utility maximization problem in a continuous-time financial market. The most basic utility maximization problem in a Black–Scholes marketis the Merton problem of maximizing expected utility of terminal wealth. It can be written inthe form V ( x ) = sup π ∈A ( x ) E (cid:2) U ( X πT ) (cid:3) , where U : R + → R is a utility function, X πT denotes the terminal wealth achieved when usingstrategy π , and A ( x ) is the class of admissible strategies starting with initial capital x . ∗ [email protected] † [email protected] µ of asset returns. This is a rather unrealistic assumptionsince drift parameters are notoriously difficult to estimate. To obtain strategies that are robustwith respect to a possible misspecification of the drift we consider the worst-case optimizationproblem V ( x ) = sup π ∈A ( x ) inf µ ∈ K E µ (cid:2) U ( X πT ) (cid:3) . Here, we write E µ [ · ] for the expectation with respect to a measure P µ under which the driftof the asset returns is µ ∈ R d , with d denoting the number of risky assets in the market.The set K ⊆ R d is called the uncertainty set . Our aim is to study the structure of optimalstrategies, as well as their asymptotic behavior as the uncertainty set K increases. Since forlarge uncertainty, investors usually do not invest in the risky assets at all, we restrict the classof admissible strategies by imposing a constraint that prevents a pure bond investment. Wefocus on ellipsoidal uncertainty sets K , see (3.2).Our main result is an explicit representation of the optimal strategy and the worst-casedrift parameter for the robust utility maximization problem with constrained strategies andellipsoidal uncertainty sets. Moreover, by using this explicit representation, a minimax theoremof the form sup π ∈A ( x ) inf µ ∈ K E µ (cid:2) U ( X πT ) (cid:3) = inf µ ∈ K sup π ∈A ( x ) E µ (cid:2) U ( X πT ) (cid:3) is proven. Additionally, we show that the optimal strategy converges to a generalized uniformdiversification strategy. In case of K being a ball, this is the equal weight strategy, corre-sponding to uniform diversification. In that sense, our results help to explain the popularity ofuniform diversification strategies by the presence of uncertainty in the model.Model uncertainty, also called Knightian uncertainty , has been addressed in numerous papers.Gilboa and Schmeidler [7] and Schmeidler [18] formulate rigorous axioms on preference relationsthat account for risk aversion and uncertainty aversion. A robust utility functional in their senseis 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 [3]give a continuous-time extension of this multiple-priors utility. Optimal investment decisionsunder such preferences are investigated in Quenez [15] and Schied [16], building up on Kramkovand Schachermayer [8, 9]. An extension of those results by means of a duality approach is givenin Schied [17]. Papers addressing drift uncertainty in a financial market are Garlappi et al. [6]and Biagini and Pınar [2], among others. The latter also focuses on ellipsoidal uncertainty sets.Uncertainty about both drift and volatility is investigated in a recent paper by Pham et al. [14].Pflug et al. [13] study a one-period risk minimization problem under model uncertainty andshow convergence of the optimal strategy to the uniform diversification strategy. Our resultsgeneralize these findings to a continuous-time utility maximization problem.The paper is organized as follows. In Section 2 we state our financial market model andintroduce the robust utility maximization problem. Our main results are given in Section 3,where we solve our optimization problem for power and logarithmic utility. The main idea is tosolve the dual problem explicitly and to show then that the solution forms a saddle point of the2roblem. We give representations of the optimal strategy and the worst-case drift parameterand prove a minimax theorem. In Section 4 we study the asymptotic behavior of the optimalstrategy and the worst-case parameter as the degree of uncertainty goes to infinity. We showthat the optimal strategy converges to a generalized uniform diversification strategy, where byuniform diversification we mean the equal weight or /d strategy for the investment in the riskyassets. Furthermore, we analyze the influence of the investor’s risk aversion on the speed ofconvergence and investigate measures for the performance of the optimal robust strategies. Forbetter readability, all proofs are collected in Appendix A. Notation.
We use the notation I d for the identity matrix in R d × d as well as e i , i = 1 , . . . , d ,for the i -th standard unit vector in R d , and d for the vector in R d containing a one in everycomponent. We shortly write R + = (0 , ∞ ) . By h· , ·i we denote the scalar product on R d × R d with h x, y i = x ⊤ y for x, y ∈ R d . If x ∈ R d is a vector, k x k denotes the Euclidean norm of x .
2. Robust Utility Maximization Problem
We consider a continuous-time financial market with one risk-free and various risky assets. By
T > we denote some finite investment horizon. Let (Ω , F , F , P ) be a filtered probability spacewhere the filtration F = ( F t ) t ∈ [0 ,T ] satisfies the usual conditions. All processes are assumed tobe F -adapted. The risk-free asset S is of the form S t = e rt , t ∈ [0 , T ] , where r ∈ R is thedeterministic risk-free interest rate. Aside from the risk-free asset, investors can also invest in d ≥ risky assets. Their return process R = ( R , . . . , R d ) ⊤ is defined by d 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 we assume that σ has full rank equal to d .We introduce model uncertainty 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 thought ofas an estimate for the drift that was for instance obtained from historical stock prices. Changingthe drift from ν to some µ ∈ K can be expressed by a change of measure. For this purpose,define the process ( Z µt ) t ∈ [0 ,T ] by Z µt = exp (cid:16) θ ( µ ) ⊤ W t − k θ ( µ ) k t (cid:17) , where θ ( µ ) = σ ⊤ ( σσ ⊤ ) − ( µ − ν ) . We 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 we obtain 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 . We can thus rewrite the return dynamics as d 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 , and see that a change of measure from P to P µ corresponds to changing the drift in the returndynamics from ν to µ . We thus shortly write E µ [ · ] for the expectation under measure P µ and E [ · ] = E ν [ · ] for the expectation under our reference measure P = P ν .3n 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 > can then bedescribed by the stochastic differential equation d 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 . We require trading strategies to be F R -adapted, where F R = ( F Rt ) t ∈ [0 ,T ] for F Rt = σ (( R s ) s ∈ [0 ,t ] ) . The 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) . Our robust portfolio optimization problem can then be formulated as V ( x ) = sup π ∈A ( x ) inf µ ∈ K E µ (cid:2) U ( X πT ) (cid:3) , (2.1)where U : R + → R is a utility function. In the following, we investigate problem (2.1) for power and logarithmic utility. We use thenotation U γ : R + → R for γ ∈ ( −∞ , , where U γ ( x ) = x γ γ for γ = 0 denotes power utilityand U ( x ) = log( x ) is the logarithmic utility function. First, we make the observation thatfor a large degree of model uncertainty the trivial strategy π ≡ becomes optimal both forlogarithmic and for power utility. Proposition 2.1.
Let γ ∈ ( −∞ , and K ⊆ R d . If r d ∈ K , then the strategy ( π t ) t ∈ [0 ,T ] with π t = 0 for all t ∈ [0 , T ] is optimal for the optimization problem sup π ∈A ( x ) inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) . (2.2)This observation implies that as the level of uncertainty about the true drift parameterexceeds a certain threshold, it will be optimal for investors to not invest anything in the stocks. Remark 2.2.
Proposition 2.1 is in line with a result in Biagini and Pınar [2] where the authorsalso consider an increasing degree of uncertainty. In Øksendal and Sulem [11, 12] the authorsobtain a similar result for optimality of π ≡ . They consider a jump diffusion model with aworst-case approach where the market chooses a scenario from a fixed but very comprehensiveset of probability measures. In contrast, it is shown in Zawisza [20] that, if the model allowsfor stochastic interest rate r , the optimal strategy does not invest exclusively in the bond.Investing everything in the risk-free asset is a very extreme reaction to model uncertainty. Weare interested in finding less conservative strategies that still take into account the increasingrisk coming with a higher degree of model uncertainty. For that purpose, we introduce aconstraint on our strategies that prevents investors from solely investing in the bond. Considerfor some h > the admissibility set 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) . h = 1 would imply that investors are not allowed to invest anything in the risk-freeasset. They must then distribute all of their wealth among the risky assets. For instance, aconstraint of the form h π t , d i = h > typically applies for some mutual funds when investorsare required to invest a certain amount in risky assets. Remark 2.3.
The admissibility set A h ( x ) might seem unnecessarily restrictive at first glance.Instead of fixing h π t , d i = h one might want to consider utility maximization among the largerclass of strategies π with h π t , d i ≥ h . However, we are mainly interested in the asymptoticbehavior of the optimal strategies as the level of uncertainty increases. It is intuitively clearthat, when uncertainty is large, investors seek to invest as little as possible in the risky assets.Therefore, we consider optimization among strategies in A h ( x ) and use our results to showthat enlarging the class of admissible strategies asymptotically does not change the value ofthe optimization problem, see Section 4.2.
3. A Duality Approach
In this section we solve for power or logarithmic utility U γ and for specific uncertainty sets K the optimization problem sup π ∈A h ( x ) inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) . (3.1) Remark 3.1.
In the situation with logarithmic utility and uncertainty sets that are balls insome p -norm, p ∈ [1 , ∞ ) , it is possible to carry over methods from a one-period risk minimiza-tion problem as in Pflug et al. [13] to our continuous-time robust utility maximization problem.If K = { µ ∈ R d | k µ − ν k p ≤ κ } , then for every ε > there exists a κ > such that for all κ ≥ κ the strategy π ∗ ( κ ) that is optimal for sup π ∈A h ( x ) π deterministic inf µ ∈ K E µ (cid:2) log( X πT ) (cid:3) satisfies (cid:13)(cid:13)(cid:13)(cid:13) T Z T (cid:16) π ∗ s ( κ ) − hd d (cid:17) d s (cid:13)(cid:13)(cid:13)(cid:13) q < ε, where q ∈ (1 , ∞ ] with p + q = 1 . See Westphal [19, Thm. 3.4] for a proof. This shows that theoptimal strategy among the deterministic ones converges, as model uncertainty increases, to auniform diversification strategy π u with π ut = hd d for every t ∈ [0 , T ] . Hence, as uncertaintyabout the true drift parameter goes to infinity, investors split the proportion h of their moneymore and more evenly among all risky assets.This approach has several drawbacks. Firstly, we can follow the ideas from Pflug et al. [13]in continuous time only for logarithmic utility and uncertainty sets K that are balls in p -norm.Secondly, we have to restrict to the class of deterministic strategies to be able to use theirmethods. However, it is by no means clear in the first place that an optimal strategy to ourproblem should be a deterministic one. In fact, in many worst-case optimization problems it iseven beneficial to use randomized strategies, see Delage et al. [4]. And lastly, the above resultdoes not yield an explicit solution to the robust optimization problem, it only gives asymptoticresults for large levels of uncertainty. To overcome these problems we follow here a differentapproach that works for both power and logarithmic utility and that results in an explicitsolution of the optimization problem. 5e 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) . (3.2)Here, κ > , ν ∈ R d , and Γ ∈ R d × d is symmetric and positive definite. For Γ = I d we simplyget a ball in the Euclidean norm with radius κ and center ν . Another special case discussed inthe literature is Γ = σσ ⊤ , see e.g. Biagini and Pınar [2]. The value of κ determines the size ofthe ellipsoid. Higher values of κ correspond to more uncertainty about the true drift. To solve the optimization problem (3.1) we first address the non-robust constrained utilitymaximization problem under a fixed parameter µ ∈ R d . We repeatedly make use of a specificmatrix that we introduce in the following lemma. Lemma 3.2.
Consider the matrix D = − . . . ... − ∈ R ( d − × d . Then, given that σ ∈ R d × m has rank d , Dσ has rank d − . The matrix D defined in the lemma above comes up naturally in calculations when usingthe constraint h π t , d i = h in the form π dt = h − P d − i =1 π it . This can be seen as a reduction ofthe problem from d dimensions to d − dimensions. For better readability of the calculationsbelow we introduce the following notation. Definition 3.3.
We 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 , where D ∈ R ( d − × d is as given in Lemma 3.2 and e d is the d -th standard unit vector in R d .Note that we assume σ ∈ R d × m to have full rank, hence by the previous lemma we knowthat Dσ has full rank, in particular Dσσ ⊤ D ⊤ = Dσ ( Dσ ) ⊤ is nonsingular. Using this notationwe give the optimal strategy for the constrained optimization problem given a fixed drift µ . Proposition 3.4.
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 3.3. In the proof the d -dimensional constrained problem is reduced to a ( d − -dimensionalunconstrained problem. Using the form of the optimal strategy in the ( d − -dimensionalmarket yields the following representation for the optimal expected utility from terminal wealth.6 orollary 3.5. 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 − . The previous results give a representation of the optimal strategy and the optimal expectedutility of terminal wealth under the constraint h π t , d i = h , given that the drift parameter µ isknown. Of course, both the strategy and the terminal wealth then depend on µ . However, weaim at solving the robust utility maximization problem sup π ∈A h ( x ) inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) . For that purpose, we address in a next step the question what the worst possible parameter µ would be for the investor, given that she reacts optimally, i.e. by applying the strategy fromProposition 3.4. This corresponds to solving the dual problem inf µ ∈ K sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) . Note here that we do not know yet whether equality holds between our original problem and thecorresponding dual problem. In general the solution of the dual problem may not be of greathelp. In the following, after deriving the solution to the dual problem, we prove a minimaxtheorem that establishes the desired equality. Results from the literature, e.g. from Quenez [15],cannot be applied here as we discuss in Remark 3.9 below.
From Corollary 3.5 we have a representation of the optimal expected utility of terminal wealth,depending on the transformed parameters e r , e µ and e σ . Note that for any γ ∈ ( −∞ , , mini-mizing this expression in µ is equivalent to minimizing 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) . We now plug in the representations of e r , e µ and e σ from the corollary and obtain (1 − h ) r + he ⊤ d µ −
12 (1 − γ ) k hσ ⊤ e d k + 12(1 − γ ) (cid:0) Dµ − h (1 − γ ) Dσσ ⊤ e d (cid:1) ⊤ ( Dσσ ⊤ D ⊤ ) − (cid:0) Dµ − h (1 − γ ) Dσσ ⊤ e d (cid:1) . µ . We see that many terms do not depend on µ . The minimization is therefore equivalent to the minimization of he ⊤ d µ + 12(1 − γ ) (cid:16) µ ⊤ D ⊤ ( Dσσ ⊤ D ⊤ ) − Dµ − h (1 − γ )( Dσσ ⊤ e d ) ⊤ ( Dσσ ⊤ D ⊤ ) − Dµ (cid:17) = 12(1 − γ ) µ ⊤ D ⊤ ( Dσσ ⊤ D ⊤ ) − Dµ + h (cid:16) e ⊤ d µ − ( Dσσ ⊤ e d ) ⊤ ( Dσσ ⊤ D ⊤ ) − Dµ (cid:17) = 12(1 − γ ) µ ⊤ Aµ + hc ⊤ µ (3.3)on the ellipsoid K , where A and c were introduced in Definition 3.3. To make this minimizationproblem easier, we apply a transformation to the elements µ ∈ K . For that purpose, note thatsince Γ ∈ R d × d is assumed to be symmetric and positive definite, there exists some nonsingular τ ∈ R d × d such that Γ = τ τ ⊤ . Then we can rewrite the constraint ( µ − ν ) ⊤ Γ − ( µ − ν ) ≤ κ as κ ≥ ( µ − ν ) ⊤ ( τ τ ⊤ ) − ( µ − ν ) = ( µ − ν ) ⊤ ( τ ⊤ ) − τ − ( µ − ν ) = (cid:0) τ − ( µ − ν ) (cid:1) ⊤ (cid:0) τ − ( µ − ν ) (cid:1) . Hence, for an arbitrary µ ∈ K we define ρ := τ − ( µ − ν ) so that µ = ν + τ ρ and k ρ k ≤ κ . Wecan then rewrite (3.3) as − γ ) µ ⊤ Aµ + hc ⊤ µ = 12(1 − γ ) (cid:0) ( τ ρ ) ⊤ Aτ ρ + 2 ν ⊤ Aτ ρ + ν ⊤ Aν (cid:1) + hc ⊤ τ ρ + hc ⊤ ν = 12(1 − γ ) ρ ⊤ τ ⊤ Aτ ρ + (cid:16) − γ Aν + hc (cid:17) ⊤ τ ρ + 12(1 − γ ) ν ⊤ Aν + hc ⊤ ν. Minimizing (3.3) in µ ∈ K is therefore equivalent to minimizing the function g : B κ (0) → R with g ( ρ ) = 12(1 − γ ) ρ ⊤ τ ⊤ Aτ ρ + (cid:16) hc + 11 − γ Aν (cid:17) ⊤ τ ρ in ρ and then setting µ = ν + τ ρ . The behavior of g is determined to a large extent by thematrix A from Definition 3.3. So we analyze properties of A next. Lemma 3.6.
The matrix A is symmetric and positive semidefinite with ker( A ) = span( { d } ) . We immediately deduce that also τ ⊤ Aτ ∈ R d × d is symmetric and positive semidefinite with ker( τ ⊤ Aτ ) = span( { τ − d } ) . Having collected these properties of the matrix A and of τ ⊤ Aτ enables us to find the parameter ρ that minimizes g ( ρ ) on the set B κ (0) . Lemma 3.7.
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 theminimum of the function g : B κ (0) → R with g ( ρ ) = 12(1 − γ ) ρ ⊤ τ ⊤ Aτ ρ + (cid:16) hc + 11 − γ Aν (cid:17) ⊤ τ ρ on the domain B κ (0) = { ρ ∈ R d | k ρ k ≤ κ } is attained by the vector ρ ∗ = − 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 , where ψ ( κ ) ∈ (0 , κ ] is uniquely determined by k ρ ∗ k = κ . Theorem 3.8.
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 ] . Remark 3.9.
The preceding theorem solves the problem inf µ ∈ K sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) . (3.4)This is the corresponding dual problem to our original optimization problem sup π ∈A h ( x ) inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) , (3.5)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 the context of our portfolio optimization problem, aminimax theorem has been shown in Quenez [15], building up on the theory by Kramkov andSchachermayer [8]. However, due to our constraint h π t , d i = h for all t ∈ [0 , T ] , we cannotcarry over the results from Quenez [15] directly. In the following, we will however use ourknowledge about the optimal strategy for (3.4) to show that it indeed also solves (3.5) and thatin this case, the supremum and the infimum can be interchanged. The following representation of π ∗ is useful for proving our minimax theorem. Lemma 3.10.
The strategy π ∗ from Theorem 3.8 satisfies π ∗ t = − hψ ( κ ) k τ − d k Γ − ( µ ∗ − ν ) for all t ∈ [0 , T ] . π ∗ that is optimal for the parameter µ ∗ . Inthe following we show that, vice versa, µ ∗ is also the worst possible drift parameter, given thatan investor applies strategy π ∗ . It then follows that the point ( π ∗ , µ ∗ ) is a saddle point of ourproblem, 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 our minimax theorem.Note that the inequality sup π ∈A h ( x ) inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) ≤ inf µ ∈ K sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) always holds when interchanging the supremum and the infimum, see for example Ekeland andTemam [5, Ch. VI, Prop. 1.1]. For the reverse inequality the saddle point property is needed. Proposition 3.11.
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 ∗ . The above proposition establishes an equilibrium result and a direct connection between theoptimization problems sup π ∈A h ( x ) inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) (3.6)and inf µ ∈ K sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) . (3.7)The strategy π ∗ is the best strategy that an investor can choose when the drift of stocks is µ ∗ . On the other hand, µ ∗ is also the parameter the market has to choose to minimize theinvestor’s expected utility of terminal wealth, given that the investor applies strategy π ∗ . Thepoint ( π ∗ , µ ∗ ) therefore constitutes a saddle point, which enables us to show that in our settingthe solution to both optimization problems (3.6) and (3.7) is the same. Theorem 3.12.
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 3.8. The previous theorem establishes duality between our original robust utility maximizationproblem (3.6) and the dual problem (3.7) where supremum and infimum are interchanged.Additionally, we now also know the solution to our original problem. The optimal strategy forour constrained robust utility maximization problem is given in a nearly explicit way. Notethat the parameter µ ∗ in Theorem 3.8 is not given explicitly since the parameter ψ ( κ ) is definedin an implicit way. However, finding ψ ( κ ) numerically can be done in a straightforward wayby a numerical root search of a monotone function. For this reason, determining µ ∗ and π ∗ numerically does not pose any problems. 10 emark 3.13. One can think of other reasonable sets K for modelling uncertainty about thedrift parameter µ . Our duality approach can also be applied to the optimization problem with K = (cid:8) µ ∈ R d (cid:12)(cid:12) ⊤ d µ = b (cid:9) for some b ∈ R . The motivation for this uncertainty set is that one has an estimate for theperformance of a stock index, and therefore the overall average performance of the stocks, butnot for the single stocks themselves. In that case, one can show that the optimal strategy forthe optimization problem inf µ ∈ K sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) is ( π ∗ t ) t ∈ [0 ,T ] with π ∗ t = hd d for all t ∈ [0 , T ] . The worst-case parameter µ ∗ can be determinedexplicitly given the eigenvalues and eigenvectors of the matrix A . Further, one can show aminimax theorem in analogy to Theorem 3.12. The optimal strategy is here just a uniformdiversification strategy given the constraint on the bond investment. In the next section weshow how this fits into the framework of our results for ellipsoidal uncertainty sets when we letthe degree of uncertainty κ go to infinity.
4. Asymptotic Behavior as Uncertainty Increases
In this section we consider again the setting with ellipsoidal uncertainty sets as in (3.2) andinvestigate what happens as the degree of uncertainty increases. Since K is an ellipsoid, weincrease the degree of uncertainty about the true drift parameter by increasing the radius κ . We analyze the optimal strategy π ∗ and the corresponding worst-case drift µ ∗ in more detail.The only quantity in the representation of µ ∗ from Theorem 3.8 that depends on κ is ψ ( κ ) . Lemma 4.1.
It holds lim κ →∞ ψ ( κ ) κ = 1 . From this lemma we gain insights into the asymptotic behavior of µ ∗ . To underline thedependence on the degree of uncertainty, we write µ ∗ = µ ∗ ( κ ) and π ∗ = π ∗ ( κ ) in the following. Proposition 4.2.
It holds lim κ →∞ κ τ − (cid:0) µ ∗ ( κ ) − ν (cid:1) = − v = − k τ − d k τ − d and lim κ →∞ κ µ ∗ ( κ ) = − τ v = − k τ − d k d . Hence, asymptotically the direction of the worst-case parameter is − d . This means that, as κ tends to infinity, the worst drift which the market can choose for an investor who applies theoptimal strategy π ∗ , is a drift vector where all entries are the same and negative. We have thefollowing result for the asymptotic behavior of the investor’s optimal strategy. Theorem 4.3.
For any t ∈ [0 , T ] it holds lim κ →∞ π ∗ t ( κ ) = h ⊤ d Γ − d Γ − d . π ∗ ( κ ) converges as the degree of uncertainty κ goes to infinity. An interesting special case is Γ = I d , i.e. when K is simply a ball with radius κ . In that case we have lim κ →∞ π ∗ t ( κ ) = hd d for any t ∈ [0 , T ] , hence the optimal strategy converges to a uniform diversification strategy,given by hd d at each point in time. Hence, when forced to invest a total fraction of h > inthe risky assets, then in the limit for κ going to infinity investors will diversify their portfoliouniformly. For general Γ we shall speak of a generalized uniform diversification strategy. We use the above results to show that, as uncertainty κ goes to infinity, our robust optimizationproblem yields the same optimal value as a slightly different optimization problem with a moregeneral class of admissible strategies. Recall that we have so far considered for h > the set 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) as the class of admissible strategies. Requiring h π t , d i ≥ h instead of h π t , d i = h obviouslyenlarges this set. In the following, we show for logarithmic utility that maximizing worst-caseexpected utility among bounded strategies in this larger set asymptotically leads to the samevalue as our original problem. We write K = K ( κ ) for the uncertainty ellipsoid with radius κ . Proposition 4.4.
Define for h > the admissibility set 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) and let M > . Then there exists a κ M > such that for all κ ≥ κ M it holds sup π ∈A ′ h ( x ) k π k≤ M inf µ ∈ K ( κ ) E µ (cid:2) log( X πT ) (cid:3) ≤ sup π ∈A h ( x ) inf µ ∈ K ( κ ) E µ (cid:2) log( X πT ) (cid:3) . Here we use k π k ≤ M as a short notation for k π t k ≤ M for all t ∈ [0 , T ] . For power utility, the result is slightly weaker. We first give a lemma that states some usefulequalities concerning the matrix A and vector c from Definition 3.3. Lemma 4.5.
For the matrix A and the vector c we have Aσσ ⊤ A = A, c ⊤ σσ ⊤ A = 0 and c ⊤ d = 1 . The next proposition gives a result similar to Proposition 4.4 for power utility. We definea different enlarged admissibility set A h ( x ) in this case. The reason is that, in contrast tothe logarithmic utility case, we cannot ensure that we can restrict to deterministic strategiesin A ′ h ( x ) . Proposition 4.6.
Let γ = 0 and h > and define the admissibility set A h ( x ) = [ h ′ ≥ h A h ′ ( x ) . Then there exists a κ ′ > such that for all κ ≥ κ ′ it holds sup π ∈A h ( x ) inf µ ∈ K ( κ ) E µ (cid:2) U γ ( X πT ) (cid:3) = sup π ∈A h ( x ) inf µ ∈ K ( κ ) E µ (cid:2) U γ ( X πT ) (cid:3) . π with h π t , d i as small as possible. Even if the class of admissible strategiesis enlarged, the optimal value will for large uncertainty be attained by a strategy from A h ( x ) .This is in line with the intuition from Proposition 2.1, where we have seen that as uncertaintyexceeds a certain threshold, investors prefer to not invest anything into the risky assets. As the class of admissible strategies we now take again 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) for some h > . We have seen in Section 4.1 that the optimal strategy π ∗ ( κ ) for our robustoptimization problem with ellipsoidal uncertainty sets K converges as the level of uncertainty κ goes to infinity. If the uncertainty set K is a ball, then the limit is a uniform diversificationstrategy hd d . In the following, we illustrate this convergence by an example and investigatewhich influence the risk aversion parameter γ has on the speed of convergence. Note that forour class of utility functions, the value − γ is equal to the Arrow–Pratt measure of relativerisk aversion. The smaller γ is, the more risk-averse is the investor. Example 4.7.
We consider a market with d = 8 risky assets. The volatility matrix has theform σ = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Investors use strategies from A h ( x ) with h = 1 . Further, we take Γ = I d and ν = d as parameters of the uncertainty ellipsoid. We then compute the constant optimal portfoliocomposition π ∗ ( κ ) based on different values of γ and for all κ ∈ (0 , . , and plot the resultin Figure 4.1 against κ . For any fixed level of uncertainty κ , the optimal composition π ∗ ( κ ) isplotted as a stacked plot where every color corresponds to one stock.For small values of κ , the optimal strategy π ∗ is negative in some components. This leads to anoverall investment larger than one on the positive side. As κ becomes larger, the compositiongets closer and closer to the uniform diversification vector. When comparing the differentsubplots one sees that the convergence is faster for higher values of γ . This might be surprisingat first glance since one expects a more risk-averse investor to choose a “safer” strategy soonerthan a less risk-averse investor does. However, the effect becomes more intuitive when keepingin mind that we address a robust optimization problem where an investor is confronted withthe worst possible drift parameter in the uncertainty set. An investor with a high, positivevalue of γ would, in the non-robust problem, invest in the assets with the allegedly highestdrift. In the worst-case market this undiversified strategy would allow the market to choose avery extreme drift parameter with high absolute values for exactly these assets. This impliesthat a less risk-averse investor is much more prone to the market’s choice of a drift parameter.To make up for this, the optimal robust strategy converges very fast, so that even for smallvalues of uncertainty κ , the investor is already driven into the diversified uniform strategy.13 . . . . . − κ (a) γ = − . . . . . − κ (b) γ = −
10 0 . . . . . − κ (c) γ = − . . . . . . − κ (d) γ = 00 0 . . . . . − κ (e) γ = 0 . . . . . . − κ (f ) γ = 0 . Figure 4.1:
Optimal portfolio composition π ∗ plotted against κ for different values of γ . The modelparameters are given in Example 4.7. For any γ , we observe convergence against a uniformdiversification strategy. For larger values of γ , convergence appears to take place fasterthan for smaller values of γ . .4. Measures of robustness performance We have seen that introducing uncertainty in our utility maximization problem leads to morediversified strategies. The question arises what an investor gains from using robust strategiesand what downside comes with behaving in a robust way in situations where it is not necessary.These two antithetic effects can be rated by the measures cost of ambiguity and reward fordistributional robustness that have been studied in a different context in Analui [1, Sec. 3.4].For our robust maximization problem, the center ν of the uncertainty ellipsoid can be seenas an estimation for the true drift of the stocks. If an investor was sure that the estimationwas correct, she would simply maximize E ν [ U γ ( X πT )] . From Proposition 3.4 we know that theoptimal strategy is then of the form (ˆ π t ) t ∈ [0 ,T ] with ˆ π t = 11 − γ Aν + hc (4.1)for all t ∈ [0 , T ] . In the presence of uncertainty, the solution to our utility maximization problemis the strategy ( π ∗ t ) t ∈ [0 ,T ] with π ∗ t = 11 − γ Aµ ∗ + hc (4.2)for all t ∈ [0 , T ] , see Theorem 3.12. We now define measures for the robustness performancethat consider the difference in the corresponding certainty equivalents when using ˆ π or π ∗ . Definition 4.8.
We define the cost of ambiguity as COA = U − γ (cid:0) E ν (cid:2) U γ ( X ˆ πT ) (cid:3)(cid:1) − U − γ (cid:0) E ν (cid:2) U γ ( X π ∗ T ) (cid:3)(cid:1) and the reward for distributional robustness as RDR = U − γ (cid:0) E µ ∗ (cid:2) U γ ( X π ∗ T ) (cid:3)(cid:1) − U − γ (cid:0) E µ ∗ (cid:2) U γ ( X ˆ πT ) (cid:3)(cid:1) . The cost of ambiguity captures how big the loss in the certainty equivalent is when usingthe robust strategy π ∗ , given that the estimation ν for the drift was actually correct. Notethat ˆ π is the best strategy given drift ν and that U − γ is a strictly increasing function, hence COA is non-negative. The reward for distributional robustness reflects how much an investoris rewarded when using the robust strategy π ∗ compared to the “naive” strategy ˆ π , assumingthat indeed the worst possible drift parameter µ ∗ is the true one. We see that also RDR isnon-negative since π ∗ maximizes expected utility given µ ∗ . Remark 4.9.
A different definition of
COA and
RDR is possible where one measures the differ-ence in expected utility rather than the difference of the certainty equivalents. The asymptoticbehavior of the reward for distributional robustness for large uncertainty is then heavily affectedby the parameter γ of the investor’s utility function. In particular, as κ goes to infinity, thereward for distributional robustness goes to zero if γ > and to infinity if γ < . Proposition 4.10.
Independently of γ ∈ ( −∞ , it always holds COA ≥ RDR . Furthermore,
COA and
RDR converge as κ goes to infinity. We write COA( κ ) and RDR( κ ) to emphasize the dependence on the degree of uncertainty. Proposition 4.11. As κ goes to infinity, COA( κ ) converges to a non-negative limit and RDR( κ ) goes to zero. COA and
RDR in dependence on the level of uncertainty κ . We consider a market with d = 8 stocks, where the underlying market parameters are thosefrom Example 4.7. The figure shows COA and
RDR plotted against κ for different values of γ .Note that the scaling in the second row of subfigures is different from the scaling in the firstrow. The absolute values of COA and
RDR become smaller as γ increases.We observe that the qualitative behavior of COA and
RDR is the same for any value of therisk aversion coefficient γ . For any fixed γ and κ , RDR is always less than
COA , a propertythat we have proven in Proposition 4.10. As κ increases, COA goes to a finite positive limit,whereas
RDR tends to zero, as we have shown in Proposition 4.11. . . . κ (a) γ = − . . . κ (b) γ = − . . . . κ (c) γ = − .
10 2 400 . . . κ (d) γ = 0 0 2 400 . . . κ (e) γ = 0 . . . . κ (f ) γ = 0 . Legend
COARDR
Figure 4.2:
The behavior of
COA and
RDR plotted against uncertainty radius κ for different valuesof the risk aversion coefficient γ . The parameters are those from Example 4.7. A. Proofs
Proof of Proposition 2.1.
Let µ ∈ K and π ∈ A ( x ) and recall that X πt = x exp (cid:18)Z t (cid:16) r + π ⊤ s ( µ − r d ) − k σ ⊤ π s k (cid:17) d s + Z t π ⊤ s σ d W µs (cid:19) , where W µ is a Brownian motion under P µ . We consider the case γ = 0 first. When applyingthe logarithm U = log to terminal wealth X πT , we obtain log( X πT ) = log( x ) + Z T (cid:16) r + π ⊤ t ( µ − r d ) − k σ ⊤ π t k (cid:17) d t + Z T π ⊤ t σ d W µt . For any admissible π , the stochastic integral in the above equation is a martingale under P µ ,hence it vanishes in expectation. The expected logarithmic utility of terminal wealth under16easure P µ is then E µ (cid:2) log( X πT ) (cid:3) = log( x ) + E µ (cid:20)Z T (cid:16) r + π ⊤ t ( µ − r d ) − k σ ⊤ π t k (cid:17) d t (cid:21) . Since the vector r d is an element of the set K , we immediately see that inf µ ∈ K E µ (cid:2) log( X πT ) (cid:3) ≤ E r d (cid:2) log( X πT ) (cid:3) ≤ log( x ) + rT, so we can deduce that the trivial strategy π ≡ is optimal for (2.2), since π ≡ leads toexpected utility of terminal wealth log( x ) + rT under each of the measures P µ .For power utility, i.e. γ = 0 , the argumentation is similar. Since r d ∈ K , we have inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) ≤ x γ γ e γrT E r d (cid:20) exp (cid:18) − γ Z T k σ ⊤ π t k d t + γ Z T π ⊤ t σ d W r d t (cid:19)(cid:21) and we can rewrite E r d (cid:20) exp (cid:18) − γ Z T k σ ⊤ π t k d t + γ Z T π ⊤ t σ d W r d t (cid:19)(cid:21) = E r d (cid:20) exp (cid:18) γ Z T π ⊤ t σ d W r d t − γ Z T k σ ⊤ π t k d t (cid:19) exp (cid:18) − γ (1 − γ ) Z T k σ ⊤ π t k d t (cid:19)(cid:21) . Note that the term exp (cid:18) − γ (1 − γ ) Z T k σ ⊤ π t k d t (cid:19) is less or equal than one if γ > and greater or equal than one if γ < . Thus, in both cases, inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) ≤ x γ γ e γrT E r d (cid:20) exp (cid:18) γ Z T π ⊤ t σ d W r d t − γ Z T k σ ⊤ π t k d t (cid:19)(cid:21) . But the exponential local martingale in the expression above has expectation less or equal thanone, so inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) ≤ x γ γ e γrT . So again, as for logarithmic utility, the trivial strategy π ≡ is optimal for (2.2) if r d ∈ K ,since the zero strategy leads exactly to expected power utility x γ γ e γrT . Proof of Lemma 3.2.
Since d ≤ m and σ ∈ R d × m has rank d , the rows of σ are independentvectors in R m . Now Dσ ∈ R ( d − × m and due to the specific form of D , the i -th row of Dσ is σ i, · − σ d, · , i = 1 , . . . , d − . Here, σ i, · denotes the i -th row of matrix σ . Now from theindependence of σ , · , . . . , σ d, · it follows for any a , . . . , a d − ∈ R that if d − X i =1 a i ( σ i, · − σ d, · ) = d − X i =1 a i σ i, · − d − X i =1 a i σ d, · , then a = · · · = a d − = 0 . Hence, the rows of Dσ are independent, and rank( Dσ ) = d − .17 roof of Proposition 3.4. Let π ∈ A h ( x ) . Then π dt = h − P d − i =1 π it for all t ∈ [0 , T ] . Theterminal wealth under strategy π can be written as X πT = x exp (cid:18) rT + Z T (cid:16) π ⊤ t ( µ − r d ) − k σ ⊤ π t k (cid:17) d t + Z T π ⊤ t σ d W µt (cid:19) . Now note that π ⊤ t ( µ − r d ) = d − X i =1 π it ( µ i − r ) + (cid:18) h − d − X i =1 π it (cid:19) ( µ d − r )= h ( µ d − r ) + d − X i =1 π it ( µ i − µ d ) = h ( e ⊤ d µ − r ) + e π ⊤ t Dµ, (A.1)where e π t := π d − t for all t ∈ [0 , T ] . With the same notation we can also rewrite π ⊤ t σ = d − X i =1 π it σ i, · + (cid:18) h − d − X i =1 π it (cid:19) σ d, · = hσ d, · + d − X i =1 π it ( σ i, · − σ d, · ) = he ⊤ d σ + e π ⊤ t Dσ, (A.2)where σ i, · denotes the i -th row of matrix σ .In the case γ = 0 we now apply the power function to terminal wealth and get E µ (cid:2) ( X πT ) γ (cid:3) = x γ e γrT E µ (cid:20) exp (cid:18) γ Z T (cid:16) π ⊤ t ( µ − r d ) − k σ ⊤ π t k (cid:17) d t + γ Z T π ⊤ t σ d W µt (cid:19)(cid:21) . (A.3)Here, we can plug in (A.2) in the stochastic integral. The integral then splits up into Z T γπ ⊤ t σ d W µt = Z T γhe ⊤ d σ d W µt + Z T γ e π ⊤ t Dσ d W µt . We then perform a change of measure d e P d P µ = Z T = exp (cid:18)Z T γhe ⊤ d σ d W µt − Z T k γhσ ⊤ e d k d t (cid:19) . With all these considerations, (A.3) becomes E µ (cid:2) ( X πT ) γ (cid:3) = x γ e γrT E µ (cid:20) exp (cid:18) γ Z T (cid:16) π ⊤ t ( µ − r d ) − k σ ⊤ π t k (cid:17) d t + γ Z T π ⊤ t σ d W µt (cid:19)(cid:21) = x γ e γrT e E (cid:20) exp (cid:18) γ Z T (cid:16) π ⊤ t ( µ − r d ) − k σ ⊤ π t k + 12 γ k hσ ⊤ e d k (cid:17) d t + Z T γ e π ⊤ t Dσ d W µt (cid:19)(cid:21) . Note that, under e P , the process ( f W µt ) t ∈ [0 ,T ] with f W µt = W µt − Z t γhσ ⊤ e d d s is a Brownian motion by Girsanov’s Theorem. Hence, we substitute Z T γ e π ⊤ t Dσ d W µt = Z T γ e π ⊤ t Dσ d f W µt + Z T γ h e π ⊤ t Dσσ ⊤ e d d t γ Z T (cid:16) π ⊤ t ( µ − r d ) − k σ ⊤ π t k + 12 γ k hσ ⊤ e d k (cid:17) d t + Z T γ e π ⊤ t Dσ d W µt = γ Z T (cid:16) π ⊤ t ( µ − r d ) − k σ ⊤ π t k + 12 γ k hσ ⊤ e d k + γh e π ⊤ t Dσσ ⊤ e d (cid:17) d t + Z T γ e π ⊤ t Dσ d f W µt . By using (A.1) and (A.2) the integrand in the Lebesgue integral above can be written as he ⊤ d µ − hr + e π ⊤ t Dµ − k hσ ⊤ e d + ( Dσ ) ⊤ e π t k + 12 γ k hσ ⊤ e d k + γh e π ⊤ t Dσσ ⊤ e d = he ⊤ d µ − hr + e π ⊤ t (cid:0) Dµ + γhDσσ ⊤ e d (cid:1) −
12 (1 − γ ) k hσ ⊤ e d k − h e π ⊤ t Dσσ ⊤ e d − k ( Dσ ) ⊤ e π t k = e π ⊤ t (cid:0) Dµ − h (1 − γ ) Dσσ ⊤ e d (cid:1) − k ( Dσ ) ⊤ e π t k + he ⊤ d µ − hr −
12 (1 − γ ) k hσ ⊤ e d k . If we now substitute 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 − , (A.4)then the expected utility of terminal wealth is given by E µ (cid:2) U γ ( X πT ) (cid:3) = x γ γ e E (cid:20) exp (cid:18) γ Z T (cid:16)e r + e π ⊤ t ( e µ − e r d − ) − k e σ ⊤ e π t k (cid:17) d t + γ Z T e π ⊤ t e σ d f W t (cid:19)(cid:21) . (A.5)In the case γ = 0 we apply the logarithm to terminal wealth and get E µ (cid:2) log( X πT ) (cid:3) = log( x ) + rT + E µ (cid:20)Z T (cid:16) π ⊤ t ( µ − r d ) − k σ ⊤ π t k (cid:17) d t (cid:21) . Like in the case for power utility, we see that we can rewrite this expression as E µ (cid:2) log( X πT ) (cid:3) = log( x ) + e r T + E (cid:20)Z T (cid:16)e π ⊤ t (cid:0)e µ − e r d − (cid:1) − k e σ ⊤ e π t k (cid:17) d t (cid:21) , (A.6)where we use the same substitution with e r , e µ and e σ as in (A.4) for γ = 0 .In both cases γ = 0 and γ = 0 we realize that the expressions in (A.5) and (A.6) are againthe expected utility of terminal wealth in a financial market with d − risky assets where therisk-free interest rate is e r , the drift of the d − risky assets is given by e µ ∈ R d − , and thevolatility matrix is e σ ∈ R ( d − × m . So we have reduced the d -dimensional constrained problemto a ( d − -dimensional unconstrained problem. When trying to maximize the right-hand sideof (A.5), respectively (A.6), over all admissible strategies e π with values in R d − , we know thatthe optimal strategy is constant in time and has the form e π t = 11 − γ ( e σ e σ ⊤ ) − ( e µ − e r d − ) = 11 − γ ( Dσσ ⊤ D ⊤ ) − (cid:0) Dµ − h (1 − γ ) Dσσ ⊤ e d (cid:1) . (A.7)19ow note that π t = d X i =1 π it e i = d − X i =1 π it e i + (cid:18) h − d − X i =1 π it (cid:19) e d = d − X i =1 π it ( e i − e d ) + he d = D ⊤ e π t + he d . Plugging in the optimal e π t from (A.7) then yields π t = D ⊤ − γ ( Dσσ ⊤ D ⊤ ) − (cid:0) Dµ − h (1 − γ ) Dσσ ⊤ e d (cid:1) + he d = 11 − γ D ⊤ ( Dσσ ⊤ D ⊤ ) − Dµ + h (cid:0) I d − D ⊤ ( Dσσ ⊤ D ⊤ ) − Dσσ ⊤ (cid:1) e d = 11 − γ Aµ + hc for all t ∈ [0 , T ] . Proof of Lemma 3.6.
Note that
Dσσ ⊤ D ⊤ is symmetric. Hence, the same is true for its inverseand thus for D ⊤ ( Dσσ ⊤ D ⊤ ) − D . Also, Dσσ ⊤ D ⊤ = ( Dσ )( Dσ ) ⊤ is positive definite since σ ∈ R d × m has rank d and therefore by Lemma 3.2, Dσ has full row rank d − . It follows thatalso the inverse ( Dσσ ⊤ D ⊤ ) − is positive definite. So since x ⊤ Ax = x ⊤ D ⊤ ( Dσσ ⊤ D ⊤ ) − Dx = ( Dx ) ⊤ ( Dσσ ⊤ D ⊤ ) − ( Dx ) ≥ for any x ∈ R d , the matrix A is positive semidefinite. Furthermore, it is easy to check that ker( D ) = span( { d } ) and ker( D ⊤ ) = { } . Hence, it holds Ax = D ⊤ ( Dσσ ⊤ D ⊤ ) − Dx = 0 if and only if ( Dσσ ⊤ D ⊤ ) − Dx = 0 , which is equivalent to Dx = 0 . Hence we can deduce ker( A ) = ker( D ) = span( { d } ) . Proof of Lemma 3.7.
Recall that τ ⊤ Aτ has eigenvalue λ = 0 with a corresponding normedeigenvector of the form v = k τ − d k τ − d . Also, the other eigenvalues of τ ⊤ Aτ are positive,and due to symmetry we can assume that v , . . . , v d are orthogonal and form a basis of R d .Firstly, we show that the minimum of g is attained on the boundary of B κ (0) . For thatpurpose, we observe that the gradient of g is ∇ g ( ρ ) = 12(1 − γ ) 2 τ ⊤ Aτ ρ + τ ⊤ (cid:16) hc + 11 − γ Aν (cid:17) = 11 − γ τ ⊤ Aτ ρ + hτ ⊤ ( I d − Aσσ ⊤ ) e d + 11 − γ τ ⊤ Aν = τ ⊤ (cid:18) A (cid:16) − γ ( τ ρ + ν ) − hσσ ⊤ e d (cid:17) + he d (cid:19) = τ ⊤ (cid:18) D ⊤ ( Dσσ ⊤ D ⊤ ) − D (cid:16) − γ ( τ ρ + ν ) − hσσ ⊤ e d (cid:17) + he d (cid:19) . From the last representation of the gradient it becomes apparent that there is no ρ ∈ B κ (0) with ∇ g ( ρ ) = 0 , since τ ⊤ is nonsingular and the vector he d is not in the range of D ⊤ . Theminimum of the function on B κ (0) is therefore attained on the boundary.Let ρ ∈ B κ (0) be arbitrary. Since v , . . . , v d form a basis of R d , we can write ρ = P di =1 a i v i ,where a , . . . , a d ∈ R are uniquely determined. Since we know that a minimizer of the function20 must lie on the boundary of B κ (0) we obtain the constraint κ = k ρ k = d X i =1 a i (A.8)on the coefficients. Before doing the minimization, we first notice that for our minimizer, thecoefficient a will be less or equal than zero. This is because g (cid:18) d X i =1 a i v i (cid:19) = 12(1 − γ ) (cid:18) d X i =1 a i v i (cid:19) ⊤ τ ⊤ Aτ (cid:18) d X i =1 a i v i (cid:19) + (cid:16) hc + 11 − γ Aν (cid:17) ⊤ τ (cid:18) d X i =1 a i v i (cid:19) = 12(1 − γ ) d X i =1 d X j =1 a i a j v ⊤ i τ ⊤ Aτ v j + d X i =1 a i hc ⊤ τ v i + 11 − γ d X i =1 a i ( Aν ) ⊤ τ v i = 12(1 − γ ) d X i =1 a i λ i + d X i =1 a i hc ⊤ τ v i + 11 − γ d X i =1 a i ν ⊤ λ i ( τ ⊤ ) − v i = 12(1 − γ ) d X i =2 a i λ i + d X i =2 a i (cid:16) hc + λ i − γ Γ − ν (cid:17) ⊤ τ v i + a hc ⊤ τ v . For the third equality we have used that v i is an eigenvector of τ ⊤ Aτ to eigenvalue λ i and that v , . . . , v d are orthogonal. In the last step we have used λ = 0 . Next, one easily sees that c ⊤ τ v = e ⊤ d ( I d − Aσσ ⊤ ) ⊤ τ k τ − d k τ − d = 1 k τ − d k e ⊤ d ( d − σσ ⊤ A d ) = 1 k τ − d k , (A.9)since A d = 0 by Lemma 3.6. By plugging in this representation we deduce that, when lookingfor the minimizer of g , we can restrict to the parameters ρ with coefficient a ≤ . Hence, wecan rewrite the constraint (A.8) as a = − vuut κ − d X i =2 a i . We plug this representation of a , as well as (A.9), back in to obtain e g ( a , . . . , a d ) := g (cid:18) d X i =1 a i v i (cid:19) = 12(1 − γ ) d X i =2 a i λ i + d X i =2 a i (cid:16) hc + λ i − γ Γ − ν (cid:17) ⊤ τ v i − h k τ − d k vuut κ − d X i =2 a i , and minimize this expression in a , . . . , a d . Note that the domain of e g is { x ∈ R d − | k x k ≤ κ } .In the interior of this domain, the partial derivative of e g with respect to a k , k = 2 , . . . , d , isgiven by ∂ e g∂a k ( a , . . . , a d ) = 2 a k λ k − γ ) + (cid:16) hc + λ k − γ Γ − ν (cid:17) ⊤ τ v k − h k τ − d k q κ − P di =2 a i ( − a k )= λ k − γ + h k τ − d k q κ − P di =2 a i ! a k + (cid:16) hc + λ k − γ Γ − ν (cid:17) ⊤ τ v k . a k = − λ k − γ + h k τ − d k q κ − P di =2 a i ! − (cid:16) hc + λ k − γ Γ − ν (cid:17) ⊤ τ v k = − (cid:18) λ k − γ − h k τ − d k a (cid:19) − (cid:28) hτ ⊤ c + λ k − γ τ − ν, v k (cid:29) . (A.10)Note that this representation does not provide the coefficients a k explicitly since a here is afunction of ( a , . . . , a d ) . However, it is easy to check that the function [ − κ, ∋ a a + d X i =2 (cid:18) λ i − γ − h k τ − d k a (cid:19) − (cid:28) hτ ⊤ c + λ i − γ τ − ν, v i (cid:29) has a strictly negative derivative on [ − κ, . For a = − κ , the value of the function is greateror equal κ , for a tending to zero from below it converges to zero, hence there is a uniquevalue of a ∈ [ − κ, where the function has value κ . So (A.10) together with (A.8) uniquelydetermines a , . . . , a d .Moreover, the second partial derivatives of e g have the form ∂ e g∂a k ( a , . . . , a d ) = λ k − γ + h k τ − d k q κ − P di =2 a i + ha k k τ − d k (cid:0) κ − P di =2 a i (cid:1) / for k = 2 , . . . , d , and for k, l = 1 , . . . , d with k = l we obtain ∂ e g∂a l ∂a k ( a , . . . , a d ) = − ha k k τ − d k (cid:0) κ − P di =2 a i (cid:1) / ( − a l ) = ha k a l k τ − d k (cid:0) κ − P di =2 a i (cid:1) / . Hence, the Hessian of e g is of the form − γ e Λ + h k τ − d k q κ − P di =2 a i I d − + h k τ − d k (cid:0) κ − P di =2 a i (cid:1) / ( a , . . . , a d ) ⊤ ( a , . . . , a d ) , where e Λ ∈ R ( d − × ( d − is a diagonal matrix with diagonal entries λ , . . . , λ d > . Obviously,the first two summands on the right-hand side are positive-definite matrices. The last summandis positive semidefinite. So we conclude that the Hessian of e g is positive definite on the wholeinterior of the domain of e g . In particular, in the point ( a , . . . , a d ) defined via (A.10) togetherwith (A.8), there is a global minimum of the function e g .To conclude with, the minimum of the function g on B κ (0) is attained by ρ ∗ = P di =1 a i v i ,where a i = − (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) (A.11)for i = 1 , . . . , d , and where ψ ( κ ) = − a ∈ (0 , κ ] is uniquely determined by k ρ ∗ k = κ . Notethat (A.11) also holds for i = 1 since λ = 0 and c ⊤ τ v = k τ − d k by (A.9). Proof of Theorem 3.8.
For any fixed parameter µ ∈ R d , Proposition 3.4 gives the optimalstrategy for the optimization problem sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) . µ onthe set K = (cid:8) µ ∈ R d (cid:12)(cid:12) ( µ − ν ) ⊤ Γ − ( µ − ν ) ≤ κ (cid:9) is equivalent to minimizing the function g : B κ (0) → R from Lemma 3.7 in ρ and then setting µ = ν + τ ρ . The claim now follows fromLemma 3.7 together with the representation in Proposition 3.4. Proof of Lemma 3.10.
Throughout the proof, let a i = − (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) for i = 1 , . . . , d , so that τ − ( µ ∗ − ν ) = P di =1 a i v i . Due to the form of the a i we can write d X i =1 (cid:16) λ i − γ + hψ ( κ ) k τ − d k (cid:17) a i v i = − d X i =1 D hτ ⊤ c + λ i − γ τ − ν, v i E v i . Since the vectors v , . . . , v d form an orthonormal basis of R d and are eigenvectors to the eigen-values λ , . . . , λ d of the symmetric matrix τ ⊤ Aτ , the right-hand side equals − hτ ⊤ c − − γ d X i =1 h τ − ν, λ i v i i v i = − hτ ⊤ c − − γ d X i =1 h τ − ν, τ ⊤ Aτ v i i v i = − hτ ⊤ c − − γ d X i =1 h τ ⊤ Aν, v i i v i = − hτ ⊤ c − − γ τ ⊤ Aν.
On the other hand, we get d X i =1 (cid:16) λ i − γ + hψ ( κ ) k τ − d k (cid:17) a i v i = 11 − γ d X i =1 a i λ i v i + hψ ( κ ) k τ − d k d X i =1 a i v i = 11 − γ d X i =1 a i τ ⊤ Aτ v i + hψ ( κ ) k τ − d k τ − ( µ ∗ − ν )= 11 − γ τ ⊤ A ( µ ∗ − ν ) + hψ ( κ ) k τ − d k τ − ( µ ∗ − ν ) . We have used here that v i is an eigenvector of τ ⊤ Aτ to the eigenvalue λ i for each i = 1 , . . . , d .In conclusion, − γ τ ⊤ Aµ ∗ = − hψ ( κ ) k τ − d k τ − ( µ ∗ − ν ) − hτ ⊤ c. Hence, by using the representation of π ∗ from Theorem 3.8 we obtain π ∗ t = 11 − γ Aµ ∗ + hc = ( τ ⊤ ) − (cid:16) − γ τ ⊤ Aµ ∗ + hτ ⊤ c (cid:17) = − hψ ( κ ) k τ − d k Γ − ( µ ∗ − ν ) for all t ∈ [0 , T ] . 23 roof of Proposition 3.11. Since π ∗ is a strategy that is constant in time and deterministic, wecan rewrite the expected utility of terminal wealth in the case γ = 0 as E µ (cid:2) U γ ( X π ∗ T ) (cid:3) = x γ γ E µ (cid:20) exp (cid:18) γrT + γT (cid:16) ( π ∗ ) ⊤ ( µ − r d ) − k σ ⊤ π ∗ k (cid:17) + γ ( π ∗ ) ⊤ σW T (cid:19)(cid:21) = x γ γ exp (cid:18) γrT + γT (cid:16) ( π ∗ ) ⊤ ( µ − r d ) − k σ ⊤ π ∗ k (cid:17) + 12 γ T k σ ⊤ π ∗ k (cid:19) . In the case γ = 0 we have E µ (cid:2) log( X π ∗ T ) (cid:3) = log( x ) + rT + T (cid:16) ( π ∗ ) ⊤ ( µ − r d ) − k σ ⊤ π ∗ k (cid:17) . Obviously, for any γ ∈ ( −∞ , the parameter µ ∈ K that minimizes the expressions above is theparameter that minimizes the term ( π ∗ ) ⊤ µ . For an arbitrary θ ∈ R d , θ = 0 , an easy calculationshows that the parameter µ ∈ R d that minimizes θ ⊤ µ such that ( µ − ν ) ⊤ Γ − ( µ − ν ) ≤ κ hasthe form e µ = ν − κ √ θ ⊤ Γ θ Γ θ. (A.12)Hence it is sufficient to show that the parameter µ ∗ is equal to e µ from (A.12) for θ = π ∗ . FromLemma 3.10 we recall π ∗ t = − hψ ( κ ) k τ − d k Γ − ( µ ∗ − ν ) . (A.13)Hence, ( π ∗ ) ⊤ Γ π ∗ = h ψ ( κ ) k τ − d k ( µ ∗ − ν ) ⊤ Γ − ( µ ∗ − ν ) = h κ ψ ( κ ) k τ − d k and q ( π ∗ ) ⊤ Γ π ∗ = hκψ ( κ ) k τ − d k . (A.14)When rearranging (A.13) for µ ∗ and plugging in (A.14) we obtain µ ∗ = ν − ψ ( κ ) k τ − d k h Γ π ∗ = ν − κ p ( π ∗ ) ⊤ Γ π ∗ Γ π ∗ . Comparing with e µ in (A.12) we conclude that µ ∗ is the parameter that minimizes ( π ∗ ) ⊤ µ overall µ ∈ K and therefore the worst possible parameter for the strategy π ∗ . Proof of Theorem 3.12.
For an arbitrary parameter µ ∈ K , let π ( µ ) = ( π t ( µ )) t ∈ [0 ,T ] denote thestrategy from A h ( x ) that is optimal, given that the drift parameter is µ . Then we know fromTheorem 3.8 that inf µ ∈ K sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) = inf µ ∈ K E µ (cid:2) U γ ( X π ( µ ) T ) (cid:3) = E µ ∗ (cid:2) U γ ( X π ∗ T ) (cid:3) . (A.15)On the other hand, Proposition 3.11 yields E µ ∗ (cid:2) U γ ( X π ∗ T ) (cid:3) = inf µ ∈ K E µ (cid:2) U γ ( X π ∗ T ) (cid:3) ≤ sup π ∈A h ( x ) inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) . (A.16)Furthermore, we also have sup π ∈A h ( x ) inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) ≤ inf µ ∈ K sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) 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) ≤ sup π ∈A h ( x ) inf µ ∈ K E µ (cid:2) U γ ( X πT ) (cid:3) ≤ inf µ ∈ K sup π ∈A h ( x ) E µ (cid:2) U γ ( X πT ) (cid:3) . (A.17)Consequently, all inequalities in (A.17) are equalities and the claim follows. Proof of Lemma 4.1.
As before, by acknowledging the dependence on κ , we write a i ( κ ) forthe coefficients of ρ ∗ = τ − (cid:0) µ ∗ − ν (cid:1) . We have already seen in the proof of Lemma 3.7 that a ( κ ) = − ψ ( κ ) . Hence, the constraint k τ − ( µ ∗ − ν ) k = κ implies κ = k τ − ( µ ∗ − ν ) k = d X i =1 a i ( κ ) = ψ ( κ ) + d X i =2 a i ( κ ) (A.18)due to orthonormality of the vectors v , . . . , v d . We rewrite (A.18) as (cid:16) ψ ( κ ) κ (cid:17) = 1 − d X i =2 (cid:16) a i ( κ ) κ (cid:17) . In the following, we show that the sum in the expression above goes to zero as κ goes to infinity.To prove this, take some i ∈ { , . . . , d } . We know that (cid:16) a i ( κ ) κ (cid:17) = 1 κ (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) , where the expression in the inner product does not depend on κ . For the other factor, recallthat ψ ( κ ) > and λ i > . Hence, λ i − γ + hψ ( κ ) k τ − d k > λ i − γ > and therefore κ (cid:18) λ i − γ + hψ ( κ ) k τ − d k (cid:19) − ≤ κ (cid:18) λ i − γ (cid:19) − , where the upper bound goes to zero as κ goes to infinity. Now we can deduce that lim κ →∞ (cid:16) a i ( κ ) κ (cid:17) = 0 , hence lim κ →∞ (cid:16) ψ ( κ ) κ (cid:17) = 1 . The claim now follows from the fact that ψ ( κ ) is positive for each κ . Proof of Proposition 4.2.
Using the same notation as before, as well as the result from theprevious lemma, we can deduce that κ τ − (cid:0) µ ∗ ( κ ) − ν (cid:1) = a ( κ ) κ v + d X i =2 a i ( κ ) κ v i = − ψ ( κ ) κ v + d X i =2 a i ( κ ) κ v i goes to − v as κ goes to infinity. The second claim follows immediately.25 roof of Theorem 4.3. Recall that by Lemma 3.10 we can write π ∗ t ( κ ) = − hψ ( κ ) k τ − d k Γ − (cid:0) µ ∗ ( κ ) − ν (cid:1) = − h k τ − d k κψ ( κ ) 1 κ Γ − (cid:0) µ ∗ ( κ ) − ν (cid:1) for any t ∈ [0 , T ] . We then obtain lim κ →∞ π ∗ t ( κ ) = h k τ − d k ( τ ⊤ ) − v = h k τ − d k ( τ τ ⊤ ) − d = h ⊤ d Γ − d Γ − d by combining the results from Lemma 4.1 and Proposition 4.2. Proof of Proposition 4.4.
Let π ′ ∈ A ′ h ( x ) with k π ′ k ≤ M . Then π ′ can be decomposed as π ′ t = π t + ε t d for all t ∈ [0 , T ] , where π = ( π t ) t ∈ [0 ,T ] ∈ A h ( x ) and ε t ≥ for all t ∈ [0 , T ] . Forany fixed µ ∈ K ( κ ) we rewrite the expected logarithmic utility given strategy π ′ as E µ (cid:2) log( X π ′ T ) (cid:3) = log( x ) + rT + E µ (cid:20)Z T (cid:16) ( π ′ t ) ⊤ ( µ − r d ) − k σ ⊤ π ′ t k (cid:17) d t (cid:21) = E µ (cid:2) log( X πT ) (cid:3) + E µ (cid:20)Z T ε t (cid:16) ⊤ d ( µ − r d ) − ε t k σ ⊤ d k − ⊤ d σσ ⊤ π t (cid:17) d t (cid:21) . In particular, we have inf µ ∈ K ( κ ) E µ (cid:2) log( X π ′ T ) (cid:3) ≤ E µ ∗ (cid:2) log( X π ′ T ) (cid:3) = E µ ∗ (cid:2) log( X πT ) (cid:3) + E µ ∗ (cid:20)Z T ε t (cid:16) ⊤ d (cid:0) µ ∗ ( κ ) − r d (cid:1) − ε t k σ ⊤ d k − ⊤ d σσ ⊤ π t (cid:17) d t (cid:21) , (A.19)where µ ∗ = µ ∗ ( κ ) is the worst-case parameter from Theorem 3.8. Our assumption k π ′ k ≤ M implies that also k π t k is bounded for every t ∈ [0 , T ] , and so is ⊤ d σσ ⊤ π t . Hence the secondsummand in (A.19) becomes non-positive when κ is big enough (depending on M ). That isbecause ε t ≥ for all t ∈ [0 , T ] and lim κ →∞ ⊤ d µ ∗ ( κ ) = ⊤ d ν − lim κ →∞ ψ ( κ ) ⊤ d τ v = ⊤ d ν − lim κ →∞ ψ ( κ ) d k τ − d k = −∞ . So there exists a κ M > such that inf µ ∈ K ( κ ) E µ (cid:2) log( X π ′ T ) (cid:3) ≤ E µ ∗ (cid:2) log( X πT ) (cid:3) for all κ ≥ κ M . Since κ M depends only on M but not on the strategy π ′ or its decomposition,we can further deduce sup π ∈A ′ h ( x ) k π k≤ M inf µ ∈ K ( κ ) E µ (cid:2) log( X πT ) (cid:3) ≤ sup π ∈A h ( x ) E µ ∗ (cid:2) log( X πT ) (cid:3) = sup π ∈A h ( x ) inf µ ∈ K ( κ ) E µ (cid:2) log( X πT ) (cid:3) for all κ ≥ κ M , which completes the proof. 26 roof of Lemma 4.5. Using the definition of A in Definition 3.3 we see that Aσσ ⊤ A = D ⊤ ( Dσσ ⊤ D ⊤ ) − Dσσ ⊤ D ⊤ ( Dσσ ⊤ D ⊤ ) − D = D ⊤ ( Dσσ ⊤ D ⊤ ) − D = A, and hence in particular c ⊤ σσ ⊤ A = e ⊤ d ( I d − σσ ⊤ A ) σσ ⊤ A = e ⊤ d ( σσ ⊤ A − σσ ⊤ A ) = 0 . Further, we also have c ⊤ d = e ⊤ d ( I d − σσ ⊤ A ) d = e ⊤ d d = 1 due to A d = 0 . Proof of Proposition 4.6.
Take an arbitrary strategy π ∈ A h ( x ) . Then there exists some h ′ ≥ h such that π ∈ A h ′ ( x ) and we know that inf µ ∈ K ( κ ) E µ (cid:2) U γ ( X πT ) (cid:3) ≤ inf µ ∈ K ( κ ) E µ (cid:2) U γ ( X π ′ T ) (cid:3) = E µ ′ (cid:2) U γ ( X π ′ T ) (cid:3) , where µ ′ = µ ′ ( κ ) is the minimizer of the function µ − γ ) µ ⊤ Aµ + h ′ c ⊤ µ on the uncertainty set K ( κ ) and π ′ = π ′ ( κ ) ≡ − γ Aµ ′ + h ′ c . In the following we show that forsufficiently large level of uncertainty E µ ′ (cid:2) U γ ( X π ′ T ) (cid:3) ≤ E µ ∗ (cid:2) U γ ( X π ∗ T ) (cid:3) (A.20)where µ ∗ = µ ∗ ( κ ) and π ∗ = π ∗ ( κ ) are the worst-case parameter and the optimal strategyfor the utility maximization among strategies in A h ( x ) . Note that for strategies π that aredeterministic and constant in time we can write E µ (cid:2) U γ ( X πT ) (cid:3) = x γ γ exp (cid:18) γT (cid:16) r + π ⊤ ( µ − r d ) − − γ k σ ⊤ π k (cid:17)(cid:19) for any µ ∈ K ( κ ) , hence for showing (A.20) it is sufficient to prove ( π ′ ) ⊤ ( µ ′ − r d ) − − γ k σ ⊤ π ′ k ≤ ( π ∗ ) ⊤ ( µ ∗ − r d ) − − γ k σ ⊤ π ∗ k . (A.21)Using the representation of π ′ we obtain ( π ′ ) ⊤ ( µ ′ − r d ) − − γ k σ ⊤ π ′ k = 11 − γ ( µ ′ ) ⊤ Aµ ′ + h ′ c ⊤ ( µ ′ − r d ) − − γ ) ( µ ′ ) ⊤ Aµ ′ − − γ h ′ ) c ⊤ σσ ⊤ c = 12(1 − γ ) ( µ ′ ) ⊤ Aµ ′ + h ′ c ⊤ µ ′ − h ′ r − − γ h ′ ) c ⊤ σσ ⊤ c. In the first step we have used A d = 0 , Aσσ ⊤ A = A and c ⊤ σσ ⊤ A = 0 , in the second step c ⊤ d = 1 , see Lemma 4.5. An analogous computation can be done for π ∗ and µ ∗ . We then seethat, since µ ′ minimizes µ − γ ) µ ⊤ Aµ + h ′ c ⊤ µ K ( κ ) , in particular it holds − γ ) ( µ ′ ) ⊤ Aµ ′ + h ′ c ⊤ µ ′ ≤ − γ ) ( µ ∗ ) ⊤ Aµ ∗ + h ′ c ⊤ µ ∗ = 12(1 − γ ) ( µ ∗ ) ⊤ Aµ ∗ + hc ⊤ µ ∗ + ( h ′ − h ) c ⊤ µ ∗ . Using again c ⊤ d = 1 it is easy to show that c ⊤ µ ∗ = c ⊤ µ ∗ ( κ ) goes to minus infinity as κ goesto infinity. Hence we can choose κ ′ > such that c ⊤ µ ∗ ≤ for all κ ≥ κ ′ . Note that κ ′ doesnot depend on π ′ . For all κ ≥ κ ′ we then have ( π ′ ) ⊤ ( µ ′ − r d ) − − γ k σ ⊤ π ′ k ≤ − γ ) ( µ ∗ ) ⊤ Aµ ∗ + hc ⊤ µ ∗ + ( h ′ − h ) c ⊤ µ ∗ − h ′ r − − γ h ′ ) c ⊤ σσ ⊤ c ≤ − γ ) ( µ ∗ ) ⊤ Aµ ∗ + hc ⊤ µ ∗ − hr − − γ h c ⊤ σσ ⊤ c = ( π ∗ ) ⊤ ( µ ∗ − r d ) − − γ k σ ⊤ π ∗ k , which proves (A.21) and hence (A.20). Since κ ′ was chosen independent of h ′ or π ′ , we deducein particular sup π ∈A h ( x ) inf µ ∈ K ( κ ) E µ (cid:2) U γ ( X πT ) (cid:3) ≤ E µ ∗ (cid:2) U γ ( X π ∗ T ) (cid:3) = sup π ∈A h ( x ) inf µ ∈ K ( κ ) E µ (cid:2) U γ ( X πT ) (cid:3) for all κ ≥ κ ′ . The reverse inequality holds trivially. Proof of Proposition 4.10.
Since both π ∗ and ˆ π are constant in time and deterministic, we canshow for γ = 0 that COA = x e rT (cid:18) exp (cid:16) T (cid:16) (ˆ π ) ⊤ ( ν − r d ) − − γ k σ ⊤ ˆ π k (cid:17)(cid:17) − exp (cid:16) T (cid:16) ( π ∗ ) ⊤ ( ν − r d ) − − γ k σ ⊤ π ∗ k (cid:17)(cid:17)(cid:19) (A.22)and RDR = x e rT (cid:18) exp (cid:16) T (cid:16) ( π ∗ ) ⊤ ( µ ∗ − r d ) − − γ k σ ⊤ π ∗ k (cid:17)(cid:17) − exp (cid:16) T (cid:16) (ˆ π ) ⊤ ( µ ∗ − r d ) − − γ k σ ⊤ ˆ π k (cid:17)(cid:17)(cid:19) . (A.23)For γ = 0 we obtain the same representations as in (A.22) and (A.23) with γ = 0 . We nowplug in the representations from (4.1), respectively (4.2), of the strategies π ∗ and ˆ π and use theproperties A d = 0 , c ⊤ σσ ⊤ A = 0 and Aσσ ⊤ A = A , see Lemma 4.5. We obtain COA x e rT = exp (cid:16) T (cid:16) hc ⊤ ( ν − r d ) + 11 − γ ν ⊤ Aν − − γ h c ⊤ σσ ⊤ c − − γ ) ν ⊤ Aν (cid:17)(cid:17) − exp (cid:16) T (cid:16) hc ⊤ ( ν − r d ) + 11 − γ ( µ ∗ ) ⊤ Aν − − γ h c ⊤ σσ ⊤ c − − γ ) ( µ ∗ ) ⊤ Aµ ∗ (cid:17)(cid:17) = L ( γ, κ ) exp (cid:16) T (cid:16) − hr − − γ h c ⊤ σσ ⊤ c + hc ⊤ ν + 12(1 − γ ) ν ⊤ Aν (cid:17)(cid:17) , L ( γ, κ ) = 1 − exp (cid:16) − T − γ ) ( µ ∗ − ν ) ⊤ A ( µ ∗ − ν ) (cid:17) . Analogously we get
RDR x e rT = L ( γ, κ ) exp (cid:16) T (cid:16) − hr − − γ h c ⊤ σσ ⊤ c + hc ⊤ µ ∗ + 12(1 − γ ) ( µ ∗ ) ⊤ Aµ ∗ (cid:17)(cid:17) . Hence, we can deduce in particular that
COARDR = exp (cid:16) T (cid:16) − γ ) ν ⊤ Aν + hc ⊤ ν (cid:17)(cid:17) exp (cid:16) T (cid:16) − γ ) ( µ ∗ ) ⊤ Aµ ∗ + hc ⊤ µ ∗ (cid:17)(cid:17) ≥ , since µ ∗ minimizes the function µ − γ ) µ ⊤ Aµ + hc ⊤ µ on the set K . Proof of Proposition 4.11.
Firstly, note that by the same reasoning as in the proof of Proposi-tion 3.11 we have (ˆ π ) ⊤ µ ∗ ≤ ( π ∗ ) ⊤ µ ∗ = ( π ∗ ) ⊤ ν − κ q ( π ∗ ) ⊤ Γ π ∗ , and that the right-hand side goes to −∞ as κ goes to infinity. It follows that lim κ →∞ E µ ∗ (cid:2) U γ ( X ˆ πT ) (cid:3) = lim κ →∞ E µ ∗ (cid:2) U γ ( X π ∗ T ) (cid:3) = ( −∞ , γ ≤ , , γ > , and therefore lim κ →∞ RDR( κ ) = 0 .For COA we observe that E ν [ U γ ( X π ∗ T )] converges to a finite value as κ goes to infinity, withthat limit being different from zero if γ = 0 . It follows that U − γ ( E ν [ U γ ( X π ∗ T )]) also converges.We thus deduce convergence of COA( κ ) . Since COA( κ ) ≥ for any κ , we know that the limitis non-negative. References [1]
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