Robust Asymptotic Growth in Stochastic Portfolio Theory under Long-Only Constraints
RRobust Asymptotic Growth in Stochastic Portfolio Theoryunder Long-Only Constraints ∗ David Itkin † Martin Larsson ‡ September 21, 2020
Abstract
We consider the problem of maximizing the asymptotic growth rate of an investor underdrift uncertainty in the setting of stochastic portfolio theory (SPT). As in the work of Kardarasand Robertson [15] we take as inputs (i) a Markovian volatility matrix c ( x ) and (ii) an invariantdensity p ( x ) for the market weights, but we additionally impose long-only constraints on theinvestor. Our principal contribution is proving a uniqueness and existence result for the classof concave functionally generated portfolios and developing a finite dimensional approximation,which can be used to numerically find the optimum. In addition to the general results outlinedabove, we propose the use of a broad class of models for the volatility matrix c ( x ), which can becalibrated to data and, under which, we obtain explicit formulas of the optimal unconstrainedportfolio for any invariant density. Keywords : Stochastic Portfolio Theory; Knightian Model Uncertainty; Robust Growth; Long-Only Constraints
MSC 2010 Classification:
In this paper we tackle the problem of maximizing an investor’s long-term growth in the setting ofStochastic Portfolio Theory (SPT) subject to model uncertainty and long-only constraints. SPTwas introduced by Fernholz in [9, 10] as a descriptive theory with the goal of explaining observablemarket phenomena. An important observation in SPT, and the main motivation of our paper, isthat the ranked relative market capitalizations have remained remarkably stable over time acrossdifferent US equity markets (see Chapter 5 in [10]). Many financial models have been developedto capture this phenomenon and can mostly be grouped into two types; (i) ranked-based modelssuch as the Atlas models developed in [3, 9, 12] and (ii) volatility-stabilized models discussed in[2, 6, 11]. ∗ Acknowledgments: We would like to thank Scott Robertson for helpful discussions. The first author also ac-knowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). † Department of Mathematical Sciences, Carnegie Mellon University, Wean Hall, 5000 Forbes Ave, Pittsburgh,Pennsylvania 15213, USA, [email protected] . ‡ Department of Mathematical Sciences, Carnegie Mellon University, Wean Hall, 5000 Forbes Ave, Pittsburgh,Pennsylvania 15213, USA, [email protected] . a r X i v : . [ q -f i n . M F ] S e p n this work, we allow for Knightian uncertainty regarding the dynamics of the relative marketcapitalizations and study a robust growth optimization problem in this setting. Portfolio opti-mization under model uncertainty has previously been studied in the SPT literature, such as in [8]where the authors study optimal arbitrages and in [7] where Cover’s theorem on universal portfoliosestablished in [5] is extended to the SPT framework. In [16], the authors study a robust long-termgrowth maximization problem under model uncertainty in a very general setting, of which SPT isa special case. The authors of [16] solve for the robust optimal growth rate and trading strategy,which they find to be the principal eigenvalue and eigenfunction, respectively, of a certain operator.They carry out this analysis by fixing the volatility structure of the market weights, but allow fordrift uncertainty. The paper [4] studies a similar problem, but additionally allows for uncertaintywith regards to the volatility specification.Recently, the authors of [16], in [15], studied an ergodic version of the problem from [16] wherein addition to fixing the volatility structure they took as an input an invariant measure. In thispaper, we borrow the setup from [15] by considering a financial market with d market weights andwe take as inputs two estimable quantities; (1) a Markovian instantaneous volatility matrix c ( x ) forthe relative market capitalizations and (2) an invariant density p ( x ) capturing the aforementionedstability of the ranked market weights. The authors of [15] maximize an investor’s asymptoticgrowth rate over an admissible class of probability measures; that is they study the quantity λ := sup V ∈V inf P ∈ Π g ( V ; P )where V ∈ V is an investor’s wealth process, P ∈ Π is an admissible probability measure and g ( V ; P ) is the asymptotic growth rate. All of these objects are precisely defined in Section 2. Itwas found in [15] that the optimal strategy is functionally generated in the sense of [9] and charac-terized as the solution of a variational problem. However, in high-dimensional market models thevariational problem is difficult to solve, even numerically, and it is difficult to establish qualitativeor quantitative properties of the optimal strategy.Our first contribution is in Section 3 where we propose the use of a very general form for thevolatility matrix c ( x ) given by c ij ( x ) = − f ij ( x − ij ) f i ( x i ) f j ( x j ) i (cid:54) = j (cid:80) k (cid:54) = i f ik ( x − ik ) f i ( x i ) f k ( x k ) i = j ≤ i, j ≤ d, where x − ij is a d − x by removing the i th and j th components.The mild conditions required for the functions f i and f ij are precisely stated in Assumption 3.1.This specification generalizes the volatility structure seen in the aforementioned volatility-stabilizedmodels (which can be obtained here by letting f i be the identity function and f ij ≡ σ for every i, j and some constant σ >
0) and allows us to obtain explicit formulas for both the optimal portfolioand λ .From these explicit expressions we observe that the optimal strategy requires heavy short-sellingand, as such, may not be implementable by money mangers due to various institutional and riskconsiderations. The main contribution of the paper is in Section 5 where we solve an analogousrobust asymptotic growth problem under long-only constraints . A priori, one may consider severaldifferent long-only constrained robust optimization problems; in descending order of generalitythey are to admit (1) arbitrary long-only portfolios, (2) long-only portfolios in feedback form,(3) functionally generated long-only portfolios, and (4) portfolios generated by concave generating2unctions. In this paper we consider problem (4) and believe it to be the natural problem to considerin this setting, both due to well-posedness of the mathematical problem under the level of generalityconsidered in this setup and due to practical considerations regarding the implementation of theoptimal strategy. We defer the discussion of this subtle point until Section 7, where we provide adetailed explanation for this choice.Similarly to [15] we obtain an existence and uniqueness result and are able to characterize theoptimal strategy via a constrained variational problem. Although we are unable to obtain explicitformulas in this case, the constrained variational problem is susceptible to a finite dimensionalapproximation and numerical implementation. We outline the procedure to exploit this approxima-tion scheme in Section 8.2 and provide numerical simulations of the constrained and unconstrainedoptimal strategies in Section 8.3.The paper is organized as follows. Section 2 introduces the problem and summarizes the resultsof [15]. In Section 3 we introduce the general specification of the instantaneous volatility matrix c ( x )and obtain explicit formulas for the optimal strategy and growth rate in the unconstrained case.Section 4 contains several examples and using these examples we observe that the optimal strategyoften requires heavy short-selling. Section 5 is dedicated to studying problem (4) mentioned aboveand contains our main results. In Section 6 we solve problem (2) in the two dimensional case andin Section 7 we discuss in detail the various differences between the problems (1)-(4). Section 8 isdevoted to applications. We connect the results of the previous sections with rank-based modelsin Section 8.1. In Section 8.2, we develop a finite dimensional approximation to the constrainedvariational problem in Section 5 using novel results regarding exponentially concave functions. Theperformance of the various optimal strategies are then illustrated using numerical simulations inSection 8.3. Finally, in Appendix A we prove Proposition 3.5, which establishes sufficient conditionsfor Assumption 2.4, regarding the inputs c and p , to hold. We consider a financial market in the context of SPT, where for d ≥ market weights X = { X i } di =1 are the assets. Under the assumption that no market weights vanish they take values inthe open simplex ∆ d − := (cid:40) x ∈ R d : x i > i ∈ { , . . . , d } , d (cid:88) i =1 x i = 1 (cid:41) . (1)As such, we work on the measurable space Ω = C ([0 , ∞ ); ∆ d − ) with the Borel σ -algebra F . Wedenote the coordinate process by X and consider the filtration F as the right-continuous enlargementof the natural filtration generated by X . One way to interpret the market weights is to view them asthe relative capitalizations of a collection of stocks; that is, if { S i } di =1 represent the capitalizationsof d stocks, then X i = S i / ( S + · · · + S d ). Note that the coordinate process X takes values in ∆ d − by definition.As defined in (1), ∆ d − is a relatively open d − R d and it is beneficial totake this viewpoint as it simplifies computations, such as the ones carried out in Section 4. However,by making the transformation x d = 1 − (cid:80) d − i =1 x i we can view ∆ d − as an open subset E of R d − .The analysis in this paper requires us to compute derivatives of functions defined on the simplexand it is convenient to take the latter viewpoint to make this precise.3 efinition 2.1. Let k ≥ , γ ∈ (0 , and U ⊆ R be given. We denote by C k,γ (∆ d − ; U ) the set ofall functions φ : ∆ d − → U such that the associated function ψ : E → U given by ψ ( x , . . . , x d − ) = φ ( x , . . . , x d − , − (cid:80) d − i =1 x i ) satisfies ψ ∈ C k,γ ( E ; U ) . Every function φ ∈ C ,γ (∆ d − ; U ) can be extended to a C ,γ function on an open set in R d containing ∆ d − . The chain rule then yields ∂ i ψ ( x , . . . , x d − ) = ∂ i φ ( x ) − ∂ d φ ( x ) , x ∈ ∆ d − , i = 1 , . . . , d − . Note that this expression does not depend on the chosen extension of φ . Using this observation, itis easy to verify that all expressions in the sequel involving derivatives of functions on the simplexcan be unambiguously computed using an arbitrary such extension.Following the setup of [15] we consider a family of probability measures, under which X is acontinuous semimartingale, and any member may drive the dynamics of X . We allow for driftuncertainty and as inputs take an instantaneous volatility matrix c and invariant density p whichsatisfy the following standing assumption: Assumption 2.2.
For a fixed constant γ ∈ (0 , we have(i) c ∈ C ,γ (∆ d − ; S d + ) is such that c ( x ) has rank d − and ∈ Ker ( c ( x )) for every x ∈ ∆ d − ,(ii) p ∈ C ,γ (∆ d − ; (0 , ∞ )) is such that (cid:82) ∆ d − p = 1 . Our goal is to find the optimal growth rate in the worst-case model and the portfolio generatingit under a class of admissible probability measures:
Definition 2.3.
Given inputs ( c, p ) satisfying Assumption 2.2 we define the set Π consisting of allprobability measures P on (Ω , F ) for which the following hold:(i) X is a P -semimartingale with covariation process (cid:104) X (cid:105) = (cid:82) · c ( X t ) dt ; P -a.s.(ii) For all Borel measurable functions h on ∆ d − with (cid:82) ∆ d − h + p < ∞ we have that lim T →∞ T (cid:90) T h ( X t ) dt = (cid:90) ∆ d − hp ; P -a.s.(iii) The laws of { X t } t ≥ under P are tight. With regards to investment strategies, we will consider portfolios π = ( π , . . . , π d ) which areprocesses that are X -integrable with respect to each P ∈ Π and satisfy π t + · · · + π dt = 1 , t ≥ . (2)The component π i represents the proportion of wealth invested in market weight i . If π it ≥ t ≥ i ∈ { , . . . , d } we say that π is long-only . We stress that we only consider strategiesthat are fully invested in the market and do not incorporate a bank account in the model. This iscaptured by (2) and is standard in the SPT literature. The wealth process induced by a portfolio π is given by dV πt V πt := d (cid:88) i =1 π it X it dX it ; V π = 1 (3)4nd we let V be the set of all such wealth processes. Via a standard change of numraire we interpret V π as the wealth represented in units of the market portfolio.Now we define the asymptotic growth rate (in probability) corresponding to a wealth process V ∈ V and a measure P ∈ Π as g ( V ; P ) := sup (cid:26) γ ∈ R : lim T ↑∞ P (cid:0) T − log V T ≥ γ (cid:1) = 1 (cid:27) . Our main goal is to identify the optimal robust asymptotic growth rate λ := sup V ∈V inf P ∈ Π g ( V ; P ) (4)together with the growth-optimal portfolio that achieves this growth rate. In [15], the authors wereable to identify λ under additional assumptions on the inputs c and p . To formulate this assumptionwe define the operator L to be the infinitesimal generator of X under driftless dynamics; that is Lf ( x ) := 12 d (cid:88) i,j =1 c ij ( x ) ∂ ij f ( x ) (5)for all f ∈ C ( R d ). Next, denote by c − div c ( x ) any vector y that satisfies the linear equation c ( x ) y = div c ( x ) (6)where div c i ( x ) = (cid:80) j ∂ j c ij ( x ) for every i = 1 , . . . , d . By Assumption 2.2 (i) we have that div c ( x ) ∈R ( c ( x )) so for each x ∈ ∆ d − the equation (6) has a solution which is unique up to multiples of .The equations in the sequel will not depend on the particular choice of solution to (6). However,occasionally c − div c can be represented as the gradient of a function, in which case we use thisrepresentative. Assumption 2.4.
In what follows write (cid:96) := (cid:0) ∇ log p + c − div c (cid:1) . We then assume(i) (cid:82) ∆ d − (cid:96) (cid:62) c(cid:96)p < ∞ ,(ii) (cid:82) ∆ d − (div pc(cid:96) ) + < ∞ ,(iii) There exists a non-explosive solution ˜ P to the generalized martingale problem associated tothe operator L R given by L R f := Lf + (cid:96) (cid:62) c ∇ f, f ∈ C ( R d ) . In Appendix A we state the precise formulation of (iii) and develop sufficient conditions forwhen this condition holds. Under these assumptions it follows that ˜ P belongs to Π and under thislaw the dynamics of X t are given by dX t = c(cid:96) ( X t ) dt + σ ( X t ) dW t (7)for some Brownian motion W and where σ ( x ) is the unique positive definite square root of c ( x ).To state the main results of [15], which identify both λ and the growth-optimal portfolio, weneed the notion of functionally generated portfolios and the master formula developed in [9].5 efinition 2.5. Let π be a portfolio and let G : R d → (0 , ∞ ) be a function that is continuous on ∆ d − such that log G ( X ) is a semimartingale. If we have the representation log V πT = log G ( X T ) + Γ T (8) for some finite variation process Γ with Γ = 0 then we call π a functionally generated portfoliowith generating function G and drift process Γ and we write π G for π .If a portfolio is of the form π t = π ( X t ) for some deterministic function π ( x ) , we say that it isin feedback form. Theorem 2.6 (Master Formula) . Let G : R d → (0 , ∞ ) be a function that is continuous on ∆ d − such that G ( X ) is a semimartingale. Assume there exist locally bounded measurable functions g i : ∆ d − → R for i = 1 , . . . , d and a finite variation process Q with Q (0) = 0 such that d log G ( X t ) = d (cid:88) i =1 g i ( X t ) dX it + dQ t . (9) Then G functionally generates the portfolio π G with weights given by π iG ( x ) x i = g i ( x ) + 1 − d (cid:88) j =1 x j g j ( x ); i = 1 , . . . , d. (10) Remark 2.7.
If the function G from the previous theorem is C then it follows that the assumptionsof the theorem are satisfied with g i ( x ) = ∂ i log G ( x ) and dQ t = LG ( X t ) dt where L is given by (5) .Thus the corresponding portfolio weights become π iG ( x ) x i = ∂ i log G ( x ) + 1 − d (cid:88) j =1 x j ∂ j log G ( x ); i = 1 , . . . , d. (11) and we have the representation log V πT = log G ( X T ) + Γ T where Γ T = (cid:82) T − LGG ( X t ) dt . We are now ready to state the main results from [15].
Theorem 2.8.
Under Assumptions 2.2 and 2.4 we have that λ = 12 (cid:90) ∆ d − ∇ ˆ u (cid:62) c ∇ ˆ up (12) where ˆ u : ∆ d − → R is the unique (up to an additive constant) minimizer of the variational problem min φ ∈ C (∆ d − ) (cid:90) ∆ d − ( ∇ φ − (cid:96) ) (cid:62) c ( ∇ φ − (cid:96) ) p. (13) Moreover the optimal portfolio is functionally generated by ˆ G := exp(ˆ u ) and it is the growth-optimalportfolio under the unique measure (up to initial distribution) P ˆ u characterized by X having thesemimartingale decomposition dX t = c ∇ ˆ u ( X t ) dt + σ ( X t ) dW t (14) under P ˆ u . P ˆ u can be interpreted as the worst-case measure since the growth-optimality of π ˆ G under P ˆ u yields λ = sup V ∈V inf P ∈ Π g ( V ; P ) = g ( V π ˆ G ; P ˆ u ) = sup V ∈V g ( V ; P ˆ u ) . In the case that (cid:96) = ∇ log R is a gradient the solution to (13) is clearly given (up to an additiveconstant) by ˆ u = log R and the measure P ˆ u is equal to the measure ˜ P ; that is it solves the generalizedmartingale problem for L R in Assumption 2.4 (iii). If (cid:96) is not a gradient then the authors in [15]proved that ˆ u solves the Euler-Lagrange equationdiv( pc ∇ ˆ u ) = div( pc(cid:96) ) . (15)The measures ˜ P and P ˆ u may differ in this case. The theory developed by Kardaras and Robertson provides a theoretical way to perform asymptoticgrowth maximization, but when the number of assets d is large, solving the variational problem (13)or equivalently the PDE (15) can be intractable. In this section we propose the use of a specific, butfairly general, volatility structure which generalizes the volatility structures previously considered inthe SPT literature, while still providing closed form solutions. Take functions f i : (0 , → (0 , ∞ ), f ij : R d − → (0 , ∞ ) and set c ij ( x ) := − f ij ( x − ij ) f i ( x i ) f j ( x j ) i (cid:54) = j (cid:80) k (cid:54) = i f ik ( x − ik ) f i ( x i ) f k ( x k ) i = j ≤ i, j ≤ d (16)where we write x − ij for the d − x ∈ ∆ d − by removing the i th and j th coordinate. We impose the following assumption on the functions: Assumption 3.1.
For all i , f i is C and satisfies lim x ↓ f i ( x ) = 0 and lim x ↑ f i ( x ) < ∞ . For all i (cid:54) = j , f ij is C , nonnegative, bounded and satisfies f ij = f ji . Theorem 3.2.
With these specifications it follows that c − div c ( x ) = ∇ log d (cid:89) i =1 f i ( x i ) . (17) Proof.
It suffices to show that c ( x ) ∇ log (cid:81) di =1 f i ( x i ) = div c ( x ). Writing c i ( x ) for the i th row ofthe matrix c ( x ), we compute for every i ∈ { , . . . , d } that c i ( x ) ∇ log d (cid:89) i =1 f i ( x i ) = d (cid:88) j =1 j (cid:54) = i f ij ( x − ij ) f i ( x i ) f j ( x j ) (cid:18) ∂ i f i ( x i ) f i ( x i ) − ∂ j f j ( x j ) f j ( x j ) (cid:19) = d (cid:88) j =1 j (cid:54) = i f ij ( x − ij ) ∂ i f i ( x i ) f j ( x j ) − d (cid:88) j =1 j (cid:54) = i f ij ( x − ij ) f i ( x i ) ∂ j f j ( x j )= div c i ( x ) . This gives the result. 7ow applying Theorem 2.8 we get the following result for this tractable class of models.
Corollary 3.3.
Let c be given by (16) . Under Assumptions 2.2 and 2.4 on the inputs c, p thesolution to (13) is given by ˆ φ ( x ) = (log p ( x ) + (cid:80) di =1 log f i ( x i )) and the corresponding growth rateis λ = 18 (cid:90) ∆ d − ∇ log (cid:32) p ( x ) d (cid:89) i =1 f i ( x i ) (cid:33) (cid:62) c ( x ) ∇ log (cid:32) p ( x ) d (cid:89) i =1 f i ( x i ) (cid:33) p ( x ) dx. Moreover the optimal portfolio is functionally generated with generating function ˆ G = exp( ˆ φ ) andcorresponding portfolio weights π i ˆ G ( x ) x i = 12 ∂ i log p ( x ) + 12 ∂ i log f i ( x i ) + 1 − d (cid:88) j =1 x j (cid:0) ∂ j log p ( x ) + ∂ j log f j ( x j ) (cid:1) . (18) Proof.
This follows immediately from Theorems 2.8 and 3.2.Next we establish necessary and sufficient conditions for c to be nondegenerate in the sense ofAssumption 2.2 (i). Proposition 3.4.
For each x ∈ ∆ d − let G x be a graph on { , . . . , d } where we create an edgebetween elements i and j if f ij ( x − ij ) > . Then the matrix c ( x ) satisfies Assumption 2.2 (i) if andonly if the graph G x is connected. Proof.
Fix an x ∈ ∆ d − . First we note that v (cid:62) c ( x ) v = (cid:88) i>j ζ ij , where ζ ij := f ij ( x − ij ) f i ( x i ) f j ( x j )( v i − v j ) for every v ∈ R d . (19)( ⇐ ) We see that v (cid:62) c ( x ) v = 0 ⇐⇒ ζ ij = 0 for all i (cid:54) = j ⇐⇒ f ij ( x − ij )( v i − v j ) = 0 for all i (cid:54) = j. From the last condition if follows that if f ij ( x − ij ) > ζ ij = 0 ⇐⇒ v i = v j . Now fix anarbitrary i, j ∈ { , . . . , d } with i (cid:54) = j . Since G x is connected we can find a path of the graph G x i (cid:55)→ i (cid:55)→ · · · (cid:55)→ i m (cid:55)→ j connecting i to j . Thus by the definition of the graph it must be that v i = v i = · · · = v i m = v j tohave v (cid:62) c ( x ) v = 0. Since i, j was arbitrary it follows that v (cid:62) c ( x ) v = 0 ⇐⇒ v i = v j for all i, j ∈ { , . . . , d } so that v ∈ span( ).( ⇒ ) Suppose by way of contradiction that G x is not connected. Then we can find a nonemptysubset A of vertices of G x such that A c is nonempty and there are no edges between A and A c .Setting v := (cid:88) i ∈ A e i − (cid:88) j ∈ A c e j , where { e i } di =1 are the standard basis vectors in R d , we see from (19) that v (cid:62) c ( x ) v = 0 which is therequired contradiction. 8ext we present sufficient conditions on the inputs p and f i for i = 1 , . . . , d so that Assump-tion 2.4 holds. Proposition 3.5.
Let c be given by (16) where the functions f i , f ij satisfy Assumption 3.1 and let R ( x ) = (cid:118)(cid:117)(cid:117)(cid:116) p ( x ) d (cid:89) i =1 f i ( x i ) . If lim x → ∂ ∆ d − R ( x ) = 0 , LR/R is bounded from below and we have that (cid:90) ∆ d − (cid:18) | LRR | + | L (log R ) | (cid:19) p < ∞ then conditions (i)–(iii) in Assumption 2.4 hold. The proof of this result is located in Appendix A.
We now consider a few examples.
Take f i ( x i ) = ( x i ) b i for some parameters b i ≥ p ( x ) = 1 B ( a ) d (cid:89) i =1 ( x i ) a i − be the Dirichlet density with parameters a = ( a , . . . , a d ) with a i > i ∈ { , . . . , d } . Here B ( a ) = (cid:81) di =1 Γ( a i )Γ( (cid:80) di =1 a i )is the generalized Beta function.Define γ i := a i + b i −
1. Under the condition that γ i > i we have that Assumption 2.4holds. Indeed using the notation of Proposition 3.5 we have that R ( x ) = (cid:81) di =1 ( x i ) γ i / so thatlim x → ∂ ∆ d − R ( x ) = 0. A direct computation in this case yields LR ( x ) = 12 R ( x ) (cid:88) i>j f ij ( x − ij )( x i ) b i ( x j ) b j (cid:18) γ i ( γ i − x i ) + γ j ( γ j − x j ) − γ i γ j x i x j (cid:19) . Since γ i , γ j >
1, the coefficients in the first two terms are strictly positive and the condition that b i , b j ≥ −∞ . Hence we see that lim x → ∂ ∆ d − LR ( x ) /R ( x ) > −∞ and so by continuity it follows that LR/R is bounded from below on ∆ d − . Next we see that LR ( x ) R ( x ) p ( x ) = 12 B ( a ) (cid:88) i>j (cid:18) f ij ( x − ij ) (cid:89) k (cid:54)∈{ i,j } ( x k ) a k − (cid:16) γ i ( γ i − x i ) γ i − ( x j ) γ j + γ j ( γ j − x i ) γ i ( x j ) γ j − − γ i γ j ( x i ) γ i − ( x j ) γ j − (cid:17) (cid:19) which is integrable since γ i − > − i . Lastly we compute that L (log R )( x ) = − d (cid:88) i,j =1 i (cid:54) = j γ i f ij ( x − ij )( x i ) b i − ( x j ) b i so we see that | p ( x ) L (log R )( x ) | = 14 B ( a ) d (cid:88) i,j =1 i (cid:54) = j γ i f ij ( x − ij ) γ i ( x i ) γ i − ( x j ) γ j which is again integrable. Hence, by Proposition 3.5 we conclude that Assumption 2.4 holds in thiscase.Further assume that f ij ( x − ij ) = α ij for some constants α ij satisfying the graph condition ofProposition 3.4. Then, thanks to Corollary 3.3 the robust growth-optimal portfolio is characterizedby ˆ φ = (cid:80) di =1 γ i log( x i ). By the master formula Theorem 2.6, the portfolio weights are given by π i ˆ G ( x ) = 12 γ i + x i − d (cid:88) j =1 γ j . (20)Furthermore we have λ = 18 B ( a ) d (cid:88) i,j =1 i (cid:54) = j α ij (cid:18) ( γ i ) B ( a + ( b i − e i + b j e j ) − γ i γ j B ( a + ( b i − e i + ( b j − e j ) (cid:19) where { e i } di =1 are the standard basis vectors in R d . Indeed, a calculation gives (cid:96) (cid:62) c(cid:96) ( x ) = 14 d (cid:88) i,j =1 i (cid:54) = j α ij ( x i ) b i ( x j ) b j (cid:18) ( γ i ) ( x i ) − γ i γ j x i x j (cid:19) so that multiplying by p , integrating and using the fact that (cid:82) ∆ d − (cid:81) di =1 ( x i ) r i − dx = B ( r ) forany constants r i > Remark 4.1.
In the case when d = 2 the condition γ i > for i = 1 , is both necessary andsufficient for Assumption 2.4 to hold as can be checked by Feller’s test for explosion (see e.g.Theorem 5.29 in [14] for the statement of Feller’s test). Remark 4.2.
It is worth noting that before studying the ergodic robust asymptotic growth problemin [15], the authors in [16] had studied under a similar framework the asymptotic growth problemwhere the only input was the covariation matrix c . They found that the optimal growth rate is givenby the negative of the principal eigenvalue λ ∗ of the operator L and the optimal generating functionis log η ∗ where η ∗ is the eigenfunction corresponding to − λ ∗ ; that is η ∗ and λ ∗ satisfy Lη ∗ = − λ ∗ η ∗ . ote that one has the following minimax representation for the principal eigenvalue of an operator(see Theorem 4.4.7 in [18] ) − λ ∗ = inf µ ∈P (∆ d − ) sup u ∈ C (∆ d − ) u> (cid:90) ∆ d − − Luu ( x ) µ ( dx ) (21) where P (∆ d − ) is the space of probability measures on ∆ d − .It will be shown in Lemma 5.4 that for any positive C function G and any measure P ∈ Π thegrowth rate is given by g ( V ; P ) = (cid:90) ∆ d − − LGG p.
By comparing this representation of the growth rate with (21) , we might expect that, in many cases,by minimizing the robust optimal-growth rate over densities p for which Assumption 2.4 holds, wewould recover the eigenvalue problem of [16]. Indeed, we can verify this in the following example.Taking b = in the Dirichlet example above it can easily be verified that − λ ∗ = 12 d (cid:88) i,j =1 i (cid:54) = j α ij with eigenfunction η ∗ ( x ) = (cid:81) di =1 x i . Thus, by taking a = (2 , , . . . , we do indeed recover theworst-case model from [16]. In [12], the authors study a class of models called hybrid Atlas models, under which the rankedmarket weights have an invariant density p . Let p n be a Dirichlet density with parameter a n ∈ R d for n = 1 , . . . , d ! and let the c be as in the Dirichlet example above. In Corollary 5 from [12] theranked market weights have invariant density given by p ( x ) := d ! (cid:88) n =1 ξ n p n ( x )for some constants ξ n ≥ (cid:80) d ! n =1 ξ n = 1. In [12] the authors derived the invariant measurefrom a model specification and the parameters a n and ξ n are possible to calibrate to empirical data.Let γ n = b + a n − . It is clear from a similar argument to the one used in the Dirichlet examplethat Assumption 2.4 is satisfied if γ in > n ∈ { , . . . , d ! } and i ∈ { , . . . , d } . Then itfollows that ∂ i ˆ φ ( x ) = (cid:96) i ( x ) = d ! (cid:80) n =1 γ in ξ n p n ( x )2 x i d ! (cid:80) n =1 ξ n p n ( x ) ; i = 1 , . . . , d
11o that in this case π i ˆ G ( x ) = d ! (cid:80) n =1 ξ n p n ( x ) (cid:32) γ in + (cid:32) − d (cid:80) j =1 γ jn (cid:33) x i (cid:33) d ! (cid:80) n =1 ξ n p n ( x ) . Let f i ( x i ) = ( x i ) a i (1 − x i ) b i for some constants a i , b i > p be the density of the logit-normaldistribution with parameters µ ∈ R d and Σ ∈ S d ++ ; that is we have p ( x ) = 1 (cid:112) π | det Σ | (cid:81) di =1 x i (1 − x i ) exp (cid:18) −
12 (logit( x ) − µ ) (cid:62) Σ − (logit( x ) − µ ) (cid:19) where logit( x ) = (cid:18) log (cid:18) x − x (cid:19) , . . . , log (cid:18) x d − x d (cid:19)(cid:19) . The logit-normal distribution has the property that if Y is logit-normal with parameters µ, Σ thenlogit( Y ) is d -variate normal with mean vector µ and covariance matrix Σ.We will now show that under these specifications Assumption 2.4 is satisfied by virtue of Propo-sition 3.5. Using the notation from Proposition 3.5 we have in this case that R ( x ) ∝ d (cid:89) i =1 ( x i ) ( a i − (1 − x i ) ( b i − exp (cid:18) −
14 (logit( x ) − µ ) (cid:62) Σ − (logit( x ) − µ ) (cid:19) so we see that lim x → ∂ ∆ d − R ( x ) = 0. A direct calculation shows that (cid:12)(cid:12)(cid:12)(cid:12) LRR ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) d (cid:88) i =1 ( x i ) a i − (1 − x i ) b i − | log R ( x ) | (cid:46) d (cid:88) i =1 ( x i ) a i − (1 − x i ) b i − and that LR/R is bounded from below since a i , b i >
2. It follows that both | LR/R | p and | log R | p are integrable so by Proposition 3.5 we have that Assumption 2.4 is satisfied.Hence, by Corollary 3.3 we conclude thatˆ φ ( x ) = −
12 (logit( x ) − µ ) (cid:62) Σ − (logit( x ) − µ )+ 12 d (cid:88) i =1 ( a i −
1) log x i + ( b i −
1) log(1 − x i )and by the master formula Theorem 2.6 that π i ˆ G ( x ) = 12 (cid:32) a i − b i − x i − x i − Σ − (logit( x ) − µ ) i − x i x i − x i d (cid:88) j =1 (cid:32) a j − b j − x j − x j − Σ − (logit( x )) − µ ) j − x j (cid:33)(cid:33) . A striking feature of the optimal strategy from Corollary 3.3 is that it does not depend on thefunctions f ij . In Example 4.1, where the functions f ij are constant, they have the interpretation ofcorrelations, so it is puzzling from a financial perspective that they do not appear in the optimalportfolio. However, the correlations do affect the growth rate. This indicates that under thisframework, asset correlations play a role in determining the achievable growth rate in the market,but under the worst case measure P ˆ u (which is equal to ˜ P under the specification (16) thanks toTheorem 3.2) cannot be explicitly exploited to obtain additional growth beyond what is given bythe other inputs; namely the invariant density p ( x ) and the functions f i ( x i ) which establish theproduct form in the definition of c ( x ). From a numerical perspective this is an attractive property.Indeed, one does not need to estimate the correlations, which is often a difficult task, to determinethe optimal strategy. Nevertheless, this is an unexpected and interesting property of the solutionfor which we have not been able to obtain a deeper financial explanation.A second significant feature of the solution is that the optimal portfolio requires heavy short-selling in certain regions of the simplex. As an illustration take the Dirichlet example of Section 4.1and set γ i = γ for some γ > i . Then the portfolio weights in (20) become π i ˆ G ( x ) = 12 (cid:0) γ + x i (2 − dγ ) (cid:1) . Whenever x i > γdγ − , the portfolio takes a short position in asset i . In particular, we see that forlarge d the region where π is long-only is very small. Since many investors have restrictions onshort-selling it is desirable to tackle this problem under long-only constraints . Motivated by the substantial short-selling required by the optimal unconstrained portfolio, our nextgoal is to introduce long-only constraints. We will do so by restricting to a large and well-behavedclass of long-only functionally generated portfolios – those generated by concave functions. It iswell-known that concave functions generate long-only portfolios (see Proposition 3.1.15 in [9] forthe C case and Theorem 3.7 in [13] for the general case).Another, different, attempt would have been to optimize over all long-only portfolios; howeverwithout imposing any additional structure it is unclear how to proceed. Indeed, the techniquesemployed in [15] heavily rely on the functionally generated structure of the candidate optimalstrategy. In Section 6 we are able to explicitly solve the problem among all portfolios in feedbackform in the d = 2 case. We observe that the resulting portfolio is functionally generated by a functionwhich is (typically) not C . The absence of this feature along with the lack of an explicit candidatesolution causes difficulty in generalizing the results to higher dimension; a further discussion ofthese points is conducted in Section 7.In our analysis of portfolios generated by concave functions it will be convenient to emphasizethe logarithm of the generating function. This is visible already in Theorem 2.8, but now becomesessential. As such, we introduce the concept of exponentially concave functions :13 efinition 5.1. We say that a function φ : ∆ d − → R is exponentially concave if e φ is concaveand denote the set of all such functions by E . In this section we restrict our attention to the class of wealth processes produced by concaveportfolio generating functions, V E := { V π G ∈ V : log G ∈ E} . We consider the problem of characterizing λ EC := sup V ∈V E inf P ∈ Π g ( V ; P ) . (22)Recalling the measure ˜ P from Assumption 2.4 (iii), under which the coordinate process X hasdynamics given by (7), we are ready to state our main result. Theorem 5.2 (Main Theorem) . Under Assumptions 2.2 and 2.4 we have that λ EC = (cid:90) ∆ d − (cid:16) (cid:96) (cid:62) c(cid:96) − ( (cid:96) − ∇ ˆ φ ) (cid:62) c ( (cid:96) − ∇ ˆ φ ) (cid:17) p (23) where ˆ φ is the unique (up to an additive constant) solution of inf φ ∈E (cid:90) ∆ d − ( (cid:96) − ∇ φ ) (cid:62) c ( (cid:96) − ∇ φ ) p. (24) Moreover, the robust growth-optimal strategy is functionally generated by the concave function ˆ G =exp ˆ φ . It satisfies g ( V π ˆ G ; P ) ≥ λ EC for every P ∈ Π and g ( V π ˆ G ; ˜ P ) = λ EC . The remainder of this section will be dedicated to proving this theorem. To accomplish thiswe will find the growth-optimal portfolio π ˆ G under the measure ˜ P using the tractable properties ofthis measure. π ˆ G is the candidate robust optimal portfolio and the growth rate it achieves under ˜ P will serve as an upper bound for λ EC . Then, approximating ˆ G by C functions ˆ G n and using thefact that portfolios generated by C functions achieve the same growth rate under every admissiblemeasure (established in Lemma 5.4 below) we will be able to lower bound λ EC by the same quantity.To start, we establish an explicit form for the growth rate of portfolios under ˜ P . Lemma 5.3.
Let π = π ( x ) be a portfolio in feedback form. Then we have that g ( V π ; ˜ P ) = 12 (cid:90) ∆ d − (cid:32) (cid:96) (cid:62) c(cid:96) ( x ) − (cid:18) (cid:96) ( x ) − π ( x ) x (cid:19) (cid:62) c ( x ) (cid:18) (cid:96) ( x ) − π ( x ) x (cid:19)(cid:33) p ( x ) dx where π ( x ) x = ( π ( x ) x , . . . , π d ( x ) x d ) . Proof.
Let h ( x ) = π ( x ) x . Under the measure ˜ P , the dynamics of X t are given by (7). ApplyingItˆo’s lemma to (3) we see that the normalized log wealth is given by1 T log V πT = 1 T (cid:90) T (cid:18) h (cid:62) c(cid:96) − h (cid:62) ch (cid:19) ( X t ) dt + 1 T (cid:90) T σ ( X t ) h ( X t ) dW t
14 1 T (cid:90) T (cid:96) (cid:62) c(cid:96) ( X t ) dt + 1 T (cid:90) T σ(cid:96) ( X t ) dW t − T (cid:90) T
12 ( (cid:96) − h ) (cid:62) c ( (cid:96) − h )( X t ) dt (25) − T (cid:90) T σ ( (cid:96) − h )( X t ) dW t where as before σ ( x ) is the unique positive-definite square root of c ( x ). We claim thatlim T →∞ (cid:32) T (cid:90) T (cid:96) (cid:62) c(cid:96) ( X t ) dt + 1 T (cid:90) T σ(cid:96) ( X t ) dW t (cid:33) = 12 (cid:90) ∆ d − (cid:96) (cid:62) c(cid:96)p ; ˜ P -a.s.Indeed, the first term converges to the required limit by Assumption 2.4 (i) and the ergodic property,while the second term N T := (cid:82) T σ(cid:96) ( X t ) dW t is a local martingale with quadratic variation (cid:104) N (cid:105) T = (cid:82) T (cid:96) (cid:62) c(cid:96) ( X t ) dt . By the ergodic property lim T →∞ T − α (cid:104) N (cid:105) T = 0 for any α >
1, so it follows thatlim T →∞ T − N T = 0; ˜ P -a.s. (see for example Lemma 1.3.2 in [9]).Next let M T = (cid:82) T σ ( (cid:96) − h )( X t ) dW t . First suppose that (cid:82) ∆ d − ( (cid:96) − h ) (cid:62) c ( (cid:96) − h ) p < ∞ . Then bythe ergodic property lim T →∞ T − (cid:104) M (cid:105) T = (cid:90) ∆ d − ( (cid:96) − h ) (cid:62) c ( (cid:96) − h ) p ; ˜ P -a.s.so by the same argument as for N T it follows that lim T →∞ T − M T = 0. Hence from (25) we seethat lim T →∞ T − log V πT = 12 (cid:90) ∆ d − ( (cid:96) (cid:62) c(cid:96) − ( (cid:96) − h ) (cid:62) c ( (cid:96) − h )) p ; ˜ P -a.s.which establishes the required growth rate.Now suppose that (cid:82) ∆ d − ( (cid:96) − h ) (cid:62) c ( (cid:96) − h ) p = ∞ , so that in particular (cid:104) M (cid:105) T → ∞ ; ˜ P -a.s. as T → ∞ . By the Dambis–Dubins–Schwarz Theorem there exists a standard Brownian motion B ,such that M T = B (cid:104) M (cid:105) T . We then havelim sup T →∞ T log V πT = lim sup T →∞ (cid:32) T (cid:90) T (cid:96) (cid:62) c(cid:96) ( X t ) dt + 1 T (cid:90) T σ(cid:96) ( X t ) dW t − T (cid:18) B (cid:104) M (cid:105) T + 12 (cid:104) M (cid:105) T (cid:19)(cid:33) = 12 (cid:90) ∆ d − (cid:96) (cid:62) c(cid:96)p ( x ) dx − lim inf T →∞ (cid:104) M (cid:105) T T (cid:18) B (cid:104) M (cid:105) T (cid:104) M (cid:105) T + 12 (cid:19) = 12 (cid:90) ∆ d − (cid:96) (cid:62) c(cid:96)p ( x ) dx −
12 lim inf T →∞ (cid:104) M (cid:105) T T (26)where we used the strong law of Brownian motion in the last step. Setting K N = { x ∈ ∆ d − : ( (cid:96) − h ) (cid:62) c ( (cid:96) − h )( X t ) ≤ N } we have (cid:104) M (cid:105) T T = 1 T (cid:90) T ( (cid:96) − h ) (cid:62) c ( (cid:96) − h )( X t ) dt ≥ T (cid:90) T ( (cid:96) − h ) (cid:62) c ( (cid:96) − h ) I K N ( X t ) dt. (cid:96) − h ) (cid:62) c ( (cid:96) − h ) I K N is a bounded function, so the ergodic property yieldslim inf T →∞ (cid:104) M (cid:105) T T ≥ (cid:90) ∆ d − ( (cid:96) − h ) (cid:62) c ( (cid:96) − h ) p I K N . Now taking N → ∞ we see from (26) thatlim sup T →∞ T log V πT ≤ (cid:90) ∆ d − ( (cid:96) (cid:62) c(cid:96) − ( (cid:96) − h ) (cid:62) c ( (cid:96) − h )) p = −∞ . It follows that g ( V ; ˜ P ) = −∞ in this case, completing the proof.Next, using this lemma we are able to establish a useful characterization of the growth rate forthe portfolio π G whenever G is twice continuously differentiable. Lemma 5.4.
Let G ∈ C (∆ d − ; (0 , ∞ )) be given. Then we have for every P ∈ Π that g ( V π G ; P ) = (cid:90) ∆ d − − LGG p = 12 (cid:90) ∆ d − (cid:0) (cid:96) (cid:62) c(cid:96) − ( (cid:96) − ∇ φ ) (cid:62) c ( (cid:96) − ∇ φ ) (cid:1) p (27) where φ = log G and the operator L is given by (5) . Proof.
We fix P ∈ Π and note that by Remark 2.7 we have the representationlog V π G T = log G ( X T ) + (cid:90) T − LGG ( X t ) dt since G is C . The ergodic property yieldslim T →∞ T (cid:90) T − LGG ( X t ) dt = (cid:90) ∆ d − − LGG p ; P -a.s.so it just remains to examine the term log G ( X T ). We claim that T − log G ( X T ) → T → ∞ . Indeed for any δ >
0, by Assumption 2.3 (iii) we can find a compact set K δ such that P ( X T ∈ K δ ) ≥ − δ for every T ≥
0. We then see for every δ, (cid:15) > P (cid:18) | φ ( X T ) | T > (cid:15) (cid:19) ≤ P (cid:18) | φ ( X T ) | T > (cid:15) ; X T ∈ K δ (cid:19) + δ. By continuity φ is bounded on K δ , so the first term on the right hand side vanishes when we send T → ∞ . Thus we obtain lim T →∞ P ( T − | log G ( X t ) | > (cid:15) ) ≤ δ for every δ, (cid:15) >
0. Now sending δ → g ( V π G ; P ) = (cid:90) ∆ d − − LGG p (28)for every P ∈ Π. To establish the second equality in (27) we note that by Lemma 5.3 together withthe master formula Theorem 2.6 we have g ( V π G ; ˜ P ) = 12 (cid:90) ∆ d − (cid:0) (cid:96) (cid:62) c(cid:96) − ( (cid:96) − ∇ φ ) (cid:62) c ( (cid:96) − ∇ φ ) (cid:1) p. (29)Comparing (28) when P = ˜ P with (29) completes the proof.16 emark 5.5. For any G ∈ C ∞ c (∆ d − ; (0 , ∞ )) it follows from integration by parts that (cid:90) ∆ d − − LGG p = 12 (cid:90) ∆ d − (cid:0) (cid:96) (cid:62) c(cid:96) − ( (cid:96) − ∇ φ ) (cid:62) c ( (cid:96) − ∇ φ ) (cid:1) p. However it is not immediately clear why the equality would extend to the class of C functions. Itis the probabilistic Assumption (2.4) (iii), which indirectly influences the boundary behaviour of c and p and allows us to conclude that this integration by parts formula holds for all C functions. Before proceeding we will make the following standing convention to simplify the notation andarguments to come. Every exponentially concave function φ is concave and is therefore continuousand almost everywhere differentiable. Though there may exist a Lebesgue null set N such that thesuperdifferential ∂φ ( x ) is larger than a singleton for x ∈ N , we will denote by ∇ φ any version of ∂φ .With this notation established, we note that Lemma 5.3 together with the master formulaindicates that maximizing the growth rate under ˜ P over all portfolios generated by concave functionsis equivalent to the minimization problem (24).To study this problem we introduce the space H c,p = (cid:40) v : ∆ d − → R d (cid:12)(cid:12)(cid:12) (cid:107) v (cid:107) H c,p := (cid:90) ∆ d − v (cid:62) cvp < ∞ (cid:41) (cid:30) ∼ (30)where we say that v ∼ w if and only if there exists a measurable function h : ∆ d − → R such that v ( x ) = w ( x ) + h ( x ) for almost every x . It is clear that this is a Hilbert space when equipped withthe inner product ( v, w ) H c,p := (cid:90) ∆ d − v (cid:62) cwp by the nondegeneracy of cp guaranteed by Assumption 2.2. With this new notation the minimizationproblem (24) becomes inf ∇ φ ∈ ∂ E (cid:107) (cid:96) − ∇ φ (cid:107) H c,p (31)where ∂ E ⊆ H c,p consists of those maps that arise as supergradients of exponentially concavefunctions, ∂ E := { v ∈ H c,p : v = ∇ φ for some φ ∈ E} . (32)To prove existence and uniqueness for (31) we first need a technical lemma regarding exponentiallyconcave functions. We start with a definition: Definition 5.6.
Let U ⊆ R d and let T : U ⇒ R d be a multi-valued map taking non-empty values.We say that T is multiplicatively cyclically monotone (MCM) if for every m ∈ N and for all { x i } mi =0 ⊆ U with x = x m (called a cycle) we have for all values y i ∈ T ( x i ) that(a) (cid:104) y i , x i +1 − x i (cid:105) ≥ − for all i = 0 , . . . , m − ,(b) (cid:81) m − i =0 (1 + (cid:104) y i , x i +1 − x i (cid:105) ) ≥ . It turns out that superdifferentials of exponentially concave functions satisfy MCM and, con-versely, that multi-valued maps possessing the MCM property are subsets of superdifferentials ofexponentially concave functions. This was proven in [17], however we will need a slight refinementof this result, which only assumes that the property holds outside a Lebesgue null set.17 emma 5.7. (1) Let φ : ∆ d − → R be an exponentially concave function. Then the superdiffer-ential ∂φ is MCM.(2) Let T : ∆ d − → R d be a multi-valued map taking non-empty values. Suppose there existsa Lebesgue null set N ⊆ ∆ d − such that T is MCM on ∆ d − \ N . Then there exists anexponentially concave function φ : ∆ d − → R such that T ( x ) ⊆ ∂φ ( x ) for every x ∈ ∆ d − \ N . Proof.
We refer the reader to Proposition 4 in [17] for the proof of (1) and note that the proof of(2) presented here is a minor modification of that proof as well. Fix x ∈ ∆ d − \ N and define for z ∈ ∆ d − Φ( z ) := inf m − (cid:89) j =0 (1 + (cid:104) y j , x j +1 − x j (cid:105) where the infimum is taken over all m ∈ N , { x j } mj =1 , and { y j } m − j =0 such that x j ∈ ∆ d − \ N for j ∈ { , . . . , m − } , x m = z and y j ∈ T ( x j ) for j ∈ { , . . . , m − } . Since Φ is the pointwise infimumof a family of affine functions it is concave on ∆ d − . By condition (i) of MCM we see that Φ isnonnegative on ∆ d − \ N and by condition (ii) it is clear that Φ( x ) = 1. Hence by concavity andcontinuity of Φ we must have that Φ is strictly positive everywhere on ∆ d − . Now let z ∈ ∆ d − \ N and q ∈ ∆ d − be given and choose α such that α > Φ( z ). Then by definition of Φ we can find an m ≥ { x j } mj =1 ⊆ ∆ d − \ N with x m = z and y j ∈ T ( x j ) for j = 0 , . . . , m − m − (cid:89) j =0 (1 + (cid:104) y j , x j +1 − x j (cid:105) ) < α. Set x m +1 = q and let y m ∈ T ( z ) be arbitrary. We then see by definition of Φ and the fact that z (cid:54)∈ N that Φ( q ) ≤ m (cid:89) j =0 (1 + (cid:104) y j , x j +1 − x j (cid:105) ) < α (1 + (cid:104) y m , q − z (cid:105) ) . Sending α ↓ Φ( z ) shows that Φ( q ) − Φ( z ) ≤ (cid:104) Φ( z ) y m , q − z (cid:105) . Since this holds for every q ∈ ∆ d − and we have that Φ > d − we see by definition ofsuperdifferential that Φ( z ) y m ∈ ∂ Φ( r ) and hence y m ∈ ∂ log Φ( z ) for every z ∈ ∆ d − \ N . Since y m ∈ T ( z ) was arbitrary it follows that T ( z ) ⊆ ∂ log Φ( z ) for every z ∈ ∆ d − \ N . Setting φ = log Φcompletes the proof.With these preparations in hand we prove some properties regarding the set ∂ E . Theorem 5.8.
The set ∂ E defined by (32) is a closed, convex and bounded set in H c,p . Moreoverthe set ∂ E ∩ C := { v ∈ H c,p : v = ∇ φ for some C exponentially concave function φ } is dense in ∂ E . roof. It is established in Proposition 5 of [1] that E , the set of exponentially concave functions,forms a convex set. From this it follows that ∂ E is convex.To prove closedness, suppose that {∇ φ n } n ∈ N is a sequence in ∂ E converging to some v in H c,p .Then lim n →∞ (cid:90) ∆ d − ( ∇ φ n − v ) (cid:62) c ( ∇ φ n − v ) p = 0so that there is a subsequence ( n k ) k ∈ N such that ∇ φ n k converges to v almost everywhere as k →∞ . We will verify that v possesses the MCM property and so must be the supergradient of anexponentially concave function by Lemma 5.7.Let N be the Lebesgue null set where the convergence does not take place. For any cycle { x j } mj =0 ⊆ ∆ d − we have that (cid:104)∇ φ n k ( x j ) , x j +1 − x j (cid:105) ≥ − (cid:81) m − j =0 (1+ (cid:104)∇ φ n k ( x j ) , x j +1 − x j (cid:105) ) ≥ v satisfies MCM on ∆ d − \ N . By Lemma 5.7 (2) we have that v is a version of the superdifferential ofan exponentially concave function; that is there exists a φ ∈ E such that v = ∇ φ , which establishesthat v ∈ ∂ E .To handle the boundedness claim, we will first show that ∂ E ∩ C is bounded and obtain that ∂ E is bounded by proving the density assertion. Fix ∇ φ ∈ ∂ E ∩ C and set G = exp φ . By concavityof G we have that − LG/G is a nonnegative function. Using Lemma 5.4 we get0 ≤ (cid:90) ∆ d − − LGG p = 12 (cid:90) ∆ d − (cid:0) (cid:96) (cid:62) c(cid:96) − ( (cid:96) − ∇ φ ) (cid:62) c ( (cid:96) − ∇ φ ) (cid:1) p = ( ∇ φ, (cid:96) ) H c,p − (cid:107)∇ φ (cid:107) H c,p . From this inequality and Cauchy–Schwarz we deduce12 (cid:107)∇ φ (cid:107) H c,p ≤ ( ∇ φ, (cid:96) ) H c,p ≤ (cid:107)∇ φ (cid:107) H c,p (cid:107) (cid:96) (cid:107) H c,p . This yields the bound (cid:107)∇ φ (cid:107) H c,p ≤ (cid:107) (cid:96) (cid:107) H c,p , which is finite by Assumption 2.4 (i). Thus ∂ E ∩ C isbounded.Now given a general ∇ φ ∈ ∂ E we will approximate it by members of ∂ E ∩ C in the followingway. Let G = exp φ , which is a positive concave function. Viewing ∆ d − as an open set in R d − , let ψ ∈ C ∞ c ( R d − ) be such that ψ ≥
0, supp( ψ ) ⊆ ∆ d − , diam(supp( ψ )) = 1 / (cid:82) R d − ψ = 1. Wedefine ψ n ( x ) := n d − ψ ( nx ). For each t ∈ (0 ,
1) define a concave function G t on a t -neighbourhoodof ∆ d − by G t ( x ) := G ((1 − t ) x + t ¯ x ); dist( x, ∆ d − ) < t, where ¯ x = d . We can then define the functions G n := G /n ∗ ψ n on ∆ d − . They are all positive,concave and smooth. It follows that G n → G pointwise on ∆ d − . Indeed, fix x ∈ ∆ d − and choosea compact set K such that x ∈ K and supp( ψ N ( x − · )) ⊆ K for some N large enough. Using thefact that every concave function is locally Lipschitz continuous there is a constant L > | G ( y ) − G ( z ) | ≤ L | y − z | for all y, z ∈ K . Then for all n ≥ N we estimate that | G ( x ) − G n ( x ) | ≤ (cid:90) B (0 , | G ( x ) − G ((1 − n − y ) + n − ¯ x ) | ψ n ( x − y ) dy ≤ L (cid:90) B (0 , (cid:0) (1 − n − ) | x − y | + n − | x − ¯ x | ψ n ( x − y ) (cid:1) dy L (cid:18) n − n + 1 n (cid:19) n →∞ −−−−→ . By Theorem 25.7 in [19] it follows that ∇ G n → ∇ G locally uniformly wherever G is differentiable.As a consequence we obtain that φ n → φ and ∇ φ n → ∇ φ almost everywhere where φ n = log G n .By construction we have that ∇ φ n ∈ ∂ E ∩ C for every n . We have aleady shown that ∂ E ∩ C is norm bounded so we can find a subsequence ( n k ) k ∈ N such that ∇ φ n k converges weakly in H c,p .However, since ∇ φ n → ∇ φ pointwise it follows that this weak limit must be ∇ φ . From here weconclude that ∇ φ is in the weak closure of ∂ E ∩ C . However, since ∂ E ∩ C is a convex set, its weakand strong closures coincide (see e.g. Corollary 2 in Chapter II of [20]) so that ∇ φ ∈ ∂ E ∩ C , where ∂ E ∩ C denotes the strong closure of ∂ E ∩ C in H c,p . This establishes that ∂ E = ∂ E ∩ C whichproves the density claim. The boundedness of ∂ E follows from the boundedness of ∂ E ∩ C . Corollary 5.9.
There exists a unique solution ˆ φ to (31) . Proof.
By the previous theorem we have that ∂ E is a closed convex set in H c,p so the result followsfrom the Hilbert space projection theorem.Now we are ready to prove our main theorem. Proof of Theorem 5.2.
From Lemma 5.3 together with the master formula we know that g ( V G ; ˜ P ) = 12 (cid:90) ∆ d − ( (cid:96) (cid:62) c(cid:96) − ( (cid:96) − ∇ φ ) (cid:62) c ( (cid:96) − ∇ φ )) p for any function G = exp φ where φ ∈ E . By Corollary 5.9 we have that there is a unique solution ∇ ˆ φ to (31) so combining these results yields the upper bound λ EC ≤ sup V ∈V E g ( V ; ˜ P ) = 12 (cid:90) ∆ d − ( (cid:96) (cid:62) c(cid:96) − ( (cid:96) − ∇ ˆ φ ) (cid:62) c ( (cid:96) − ∇ ˆ φ )) p. For the lower bound we use an approximation argument. By Theorem 5.8 we can find C exponentially concave functions ˆ φ n such that ∇ ˆ φ n → ∇ ˆ φ in H c,p as n → ∞ . But then by Lemma 5.4we see by setting G n := exp ˆ φ n that λ EC ≥ inf P ∈ Π g ( V π ˆ Gn ; P ) = 12 (cid:90) ∆ d − ( (cid:96) (cid:62) c(cid:96) − ( ∇ ˆ φ n − (cid:96) ) (cid:62) c ( ∇ ˆ φ n − (cid:96) )) p Sending n → ∞ yields the lower bound and completes the proof. Remark 5.10.
We can establish an inequality for λ EC in terms of the norm of the optimum ∇ ˆ φ .To derive this bound, we define the function g : [0 , → R by g ( t ) = ( t ∇ ˆ φ, (cid:96) ) H c,p − (cid:107) t ∇ ˆ φ (cid:107) H c,p ,which is maximized at t = 1 by the optimality of ˆ φ . It follows that ≤ g (cid:48) (1) = ( ∇ ˆ φ, (cid:96) ) H c,p − (cid:107)∇ ˆ φ (cid:107) H c,p from which obtain that ( ∇ ˆ φ, (cid:96) ) H c,p ≥ (cid:107)∇ ˆ φ (cid:107) H c,p . Applying this to (23) yields λ EC ≥ (cid:107)∇ ˆ φ (cid:107) H c,p . Long-Only Feedback Portfolios and the d = 2 Case In the d = 2 case more can be said about long-only portfolios. In this case we can phrase everythingin terms of a one dimensional problem since X = 1 − X and π = 1 − π for every portfolio π .Moreover if π = π ( x , x ) is a portfolio in feedback form then it is always functionally generated.Indeed, by defining the function ϕ π ( x ) := π ( x, − x ) x (1 − x ) − − x (33)we observe that π ( x, − x ) x = 1 + (1 − x ) ϕ π ( x ) π ( x, − x )1 − x = 1 − xϕ π ( x )for every x ∈ (0 , π is functionally generated by G ( x, − x ) = G ( x ) = exp (cid:18)(cid:90) xθ ϕ π ( y ) dy (cid:19) (34)where θ ∈ (0 ,
1) is arbitrary. Additionally from the above representation of the portfolio weightswe see that π is long-only if and only if − − x ≤ ϕ π ( x ) ≤ x (35)for every x ∈ (0 , c is of the form c ( x, − x ) = (cid:20) c ( x, − x ) − c ( x, − x ) − c ( x, − x ) c ( x, − x ) (cid:21) for some nonnegative function c . Moreover it is easy to check that c − div c = ∇ log c and so inparticular (cid:96) ( x ) = ∇ log( p ( x ) c ( x )).To state the next theorem we define ˜ p ( x ) = p ( x, − x ), ˜ c ( x ) = c ( x, − x ) and ˜ (cid:96) ( x ) = (log ˜ p ˜ c ) (cid:48) ( x ). Theorem 6.1.
Let Ξ be the set of all long-only portfolios in feedback form and set λ long := sup π ∈ Ξ inf P ∈ Π g ( V π ; P ) . Then we have that λ long = 12 (cid:90) (cid:18) ˜ (cid:96) ˜ c − (cid:16) ϕ ˆ π − ˜ (cid:96) (cid:17) ˜ c (cid:19) ˜ p ( x ) dx (36) where ˆ π ( x ) = if ˜ (cid:96) ( x ) > /x if ˜ (cid:96) ( x ) < − / (1 − x ) x + x (1 − x )˜ (cid:96) ( x ) otherwise, π ( x ) = 1 − ˆ π ( x ) and ϕ ˆ π is given by (33) . Here the optimal portfolio ˆ π itself has a simple interpretation: wherever the unconstrainedoptimal solution was long-only one performs that strategy. As soon as we enter the region wherewe were to short x we do not invest in it and hold all of our wealth in x . Similarly, when we hitthe region where we were to short x we do not invest in it and hold all of our wealth in x .To prove this theorem we will need the following lemma, which does not assume that d = 2 andis also used in the proof of Proposition 8.8. Lemma 6.2.
Let Assumptions 2.2 and 2.4 hold. Then (cid:90) ∆ d − x i x j | c ij ( x ) | p ( x ) dx < ∞ for all i, j ∈ { , . . . , d } . Proof.
Consider the portfolio π i given by π ii ( x ) = 1, π ji ( x ) = 0 for all j (cid:54) = i . For such a portfoliowe have that log V π i T = log X iT . By Lemma 5.3 we have that ˜ P -a.s.lim T →∞ T − log X T = lim T →∞ T − log V π i T = 12 (cid:107) (cid:96) (cid:107) H c,p − (cid:107) (cid:96) − h i (cid:107) H c,p where h i ( x ) = π i ( x ) /x .Now if (cid:107) (cid:96) − h i (cid:107) H c,p = ∞ then we would have that lim T →∞ T − log X T = −∞ ; ˜ P -a.s., whichwould in turn imply that lim T →∞ X iT = 0; ˜ P -a.s. This contradicts the egodicity of the process X T so the norm must be finite. But then by the triangle inequality we see that (cid:107) h i (cid:107) H c,p ≤ (cid:107) (cid:96) − h i (cid:107) H c,p + 2 (cid:107) (cid:96) (cid:107) H c,p < ∞ . Expanding out we have that (cid:107) h i (cid:107) H c,p = (cid:90) ∆ d − x i ) c ii ( x ) p ( x ) dx which proves the claim when i = j . To handle the general case let a ( x ) = diag(1 /x ) c ( x )diag(1 /x )which is a positive semidefinite matrix for every x ∈ ∆ d − . By properties of positive semidefinitematrices we have for all i, j that | c ij ( x ) | x i x j = | a ij ( x ) | ≤
12 ( a ii ( x ) + a jj ( x )) = 12 (cid:18) c ii ( x )( x i ) + c jj ( x )( x j ) (cid:19) . It follows from this bound that (cid:90) ∆ d − x i x j | c ij ( x ) | p ( x ) dx ≤ (cid:90) ∆ d − (cid:18) c ii ( x )( x i ) + c jj ( x )( x j ) (cid:19) p ( x ) dx < ∞ completing the proof. 22 roof of Theorem 6.1. By (33), we have that ˆ π ( x ) x = 1 + (1 − x ) ϕ ˆ π ( x ) where ϕ ˆ π ( x ) = /x if ˜ (cid:96) ( x ) > /x − / (1 − x ) if ˜ (cid:96) ( x ) < − / (1 − x )˜ (cid:96) ( x ) otherwise. (37)We obtain (in a similar way to the proof of Theorem 5.2) the upper bound λ long ≤ sup π ∈ Ξ g ( V π ; ˜ P ) = 12 (cid:90) ∆ d − (cid:96) (cid:62) c(cid:96)p −
12 inf π ∈ Ξ (cid:90) ∆ d − (cid:18) (cid:96) ( x ) − π ( x ) x (cid:19) (cid:62) c ( x ) (cid:18) (cid:96) ( x ) − π ( x ) x (cid:19) p ( x ) dx = 12 (cid:90) ˜ (cid:96) ˜ c ˜ p ( x ) dx −
12 inf π ∈ Ξ (cid:90) ( ϕ π ( x ) − ˜ (cid:96) ( x )) ˜ c ˜ p ( x ) dx (38)where we performed the change of variables x = 1 − x . By virtue of (35), which is a pointwiseconstraint, the infimum in (38) can be computed by pointwise minimization of the integrand.Because ˜ c ˜ p ( x ) > − / (1 − x ) ≤ y ≤ /x ( y − ˜ (cid:96) ( x )) for every x ∈ (0 , x the minimum is achieved at ˆ y = ϕ ˆ π ( x ). This establishes the upper bound λ long ≤ (cid:90) (cid:18) ˜ (cid:96) ˜ c − (cid:16) ϕ ˆ π − ˜ (cid:96) (cid:17) ˜ c (cid:19) ˜ p ( x ) dx. To obtain the lower bound we will again approximate the generating function of ˆ π by C generating functions. Extend ϕ ˆ π to R by setting it to be identically zero outside of (0 ,
1) and let ψ ∈ C ∞ c ( R ) be such that ψ ≥
0, supp( ψ ) = [0 ,
1] and (cid:82) R ψ = 1. Define the mollifiers ψ n ( x ) = n ψ ( n x ) and set ϕ n ( x ) := nn +1 ( ϕ ˆ π I (1 /n, − /n ) ∗ ψ n )( x ). We have that ϕ n ∈ C ∞ c ( R ) and claimthat ϕ n → ϕ ˆ π pointwise. To show this we fix x ∈ (0 ,
1) and let (cid:15) > ϕ ˆ π we can find an N large enough so that(i) if | x − y | < /N then | ϕ ˆ π ( x ) − ϕ ˆ π ( y ) | ≤ (cid:15)/ | ϕ ˆ π ( x ) | / ( N + 1) ≤ (cid:15)/ x − /N ≥ /N Then for all n ≥ N we see that | ϕ n ( x ) − ϕ ˆ π ( x ) | ≤ nn + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R ( ϕ ˆ π ( y ) I (1 /n, − /n ) ( y ) − ϕ ˆ π ( x )) ψ n ( x − y ) dy (cid:12)(cid:12)(cid:12)(cid:12) + | ϕ ˆ π ( x ) | n + 1 ≤ (cid:90) x +1 /n x − /n | ϕ ˆ π ( y ) − ϕ ˆ π ( x ) | n ψ ( n ( x − y )) dy + | ϕ ˆ π ( x ) | n + 1 (iii) ≤ (cid:15) (cid:15) (cid:15) (i) and (ii)which proves the pointwise convergence. Next we claim that − − x ≤ ϕ n ( x ) ≤ x (39)23or every x ∈ (0 ,
1) and n ∈ N . We fix n and estimate that ϕ n ( x ) = nn + 1 (cid:90) ( x +1 /n ) ∧ (1 − /n )( x − /n ) ∨ /n ϕ ˆ π ( y ) n ψ ( n ( x − y )) dy (40) ≤ nn + 1 (cid:90) ( x +1 /n ) ∧ (1 − /n )( x − /n ) ∨ /n y n ψ ( n ( x − y )) dy ≤ nn + 1 1( x − /n ) ∨ /n If x − /n ≥ /n then nn + 1 1( x − /n ) ∨ /n = n ( n + 1)( x − /n ) = nnx + x − /n − /n ≤ x . Conversely if x < /n + 1 /n then nn + 1 1( x − /n ) ∨ /n = n n + 1 ≤ x so that in any case we have ϕ n ( x ) ≤ /x . For the lower bound we proceed from (40) to obtain ϕ n ( x ) ≥ − nn + 1 (cid:90) ( x +1 /n ) ∧ (1 − /n )( x − /n ) ∨ /n − y n ψ ( n ( x − y )) dy ≥ − nn + 1 1( x + 1 /n ) ∧ (1 − /n ) . A similar case analysis shows that ϕ n ( x ) ≥ − / (1 − x ) for every x ∈ (0 , ϕ n satisfies (39)and is C we see by Lemma 5.4 that λ long ≥ inf P ∈ Π g ( V π Gn ; P ) = g ( V π Gn ; ˜ P ) = 12 (cid:90) (˜ (cid:96) ˜ c − (˜ (cid:96) − ϕ n ) ˜ c )˜ p ( x ) dx (41)where G n ( x ) = exp( (cid:82) xθ ϕ n ( y ) dy ) for arbitrary θ ∈ (0 , . By (39) we deduce the bound(˜ (cid:96) − ϕ n ) ˜ c ˜ p ( x ) ≤ max (cid:8) (˜ (cid:96) ( x ) − /x ) ˜ c ˜ p ( x ) , (˜ (cid:96) ( x ) + 1 / (1 − x )) ˜ c ˜ p ( x ) (cid:9) ≤ (cid:96) ˜ c ˜ p ( x ) + 2 x ˜ c ˜ p ( x ) + 2(1 − x ) ˜ c ˜ p ( x ) . Assumption 2.4 (i) and Lemma 6.2 guarantee that the function on the right hand side is integrableso by the Lebesgue dominated convergence theorem we can send n → ∞ in (41) to obtain λ long ≥ (cid:90) (cid:18) ˜ (cid:96) ˜ c − (cid:16) ϕ ˆ π − ˜ (cid:96) (cid:17) ˜ c (cid:19) ˜ p ( x ) dx. This completes the proof.We have been able to explicitly solve the robust optimal growth problem over all portfoliosin feedback form in the two dimensional case. Next we investigate the relationship between thisproblem and the concave problem studied in Section 5. The following example shows that theoptimal portfolio ˆ π from Theorem 6.1 need not be generated by a concave function.24 xample 6.3. Let κ : (0 , → (0 , be given by κ ( x ) = exp(sin( x )) . Choose a smooth, nonnegativecutoff function ψ : (0 , → [0 , such that ψ ( x ) = 1 for x ∈ (1 / , / and ψ ( x ) = 0 for x ∈ (0 , / ∪ (3 / , . Then define ˜ c ( x ) = κ ( x ) ψ ( x ) + x (1 − x )(1 − ψ ( x )); ˜ p ( x ) = 1 Z ( ψ ( x ) + x (1 − x )(1 − ψ ( x ))) where Z = (cid:82) ( ψ ( x ) + x (1 − x )(1 − ψ ( x ))) dx . The inputs c and p from the Dirichlet example inSection 4.1 with parameters b = b = 1 and a = a = 2 satisfy ˜ c ( x ) = c ( x, − x ); ˜ p ( x ) = Kp ( x, − x ) for x ∈ (0 , / ∪ (3 / , and for some constant K > . Thus, since Assumption 2.4 concerns thebehaviour of c and p near of boundary of the simplex it follows that the above choice of ˜ c and ˜ p satisfy Assumption 2.4 because c and p do.By Theorem 6.1 the optimal porfolio is generated by ˆ G ( x ) = exp( (cid:82) xθ ϕ ˆ π ( y ) dy ) where θ ∈ (0 , is arbitrary and ϕ ˆ π is given by (37) . Note that for x ∈ (1 / , / we have ˜ (cid:96) ( x ) = sin( x ) sothat in this interval the inequalities − / (1 − x ) ≤ ˜ (cid:96) ( x ) ≤ /x are satisfied. Thus, it follows that ϕ ˆ π ( x ) = sin( x ) for x ∈ (1 / , / . For such x we now compute that ˆ G (cid:48)(cid:48) ( x ) = ˆ G ( x )( ϕ ˆ π ( x ) + ϕ ˆ π ( x )) = ˆ G ( x ) (cid:18)
14 sin ( x ) + 12 cos( x ) (cid:19) > . This shows that ˆ G is not concave on (1 / , / . The next theorem gives a necessary and sufficient condition for the optimal portfolio in feedbackform to be generated by a concave function, so that it is also the solution to the less general robustoptimal growth problem considered is Section 5.
Proposition 6.4.
Let A = { x ∈ (0 ,
1) : − − x < ˜ (cid:96) ( x ) < x } . The portfolio ˆ π from Theorem 6.1 isfunctionally generated by a concave function if and only if ˜ (cid:96) ( x ) + ˜ (cid:96) (cid:48) ( x ) ≤ for every x ∈ A . Remark 6.5.
Since A is an open set we can write A = ∪ n ∈ N I n for some pairwise disjoint openintervals { I n } n ∈ N . The condition on ˜ (cid:96) in the statement of the previous proposition is then equivalentto √ ˜ c ˜ p being a concave function on each I n . Proof.
First suppose that ˜ (cid:96) ( x ) + ˜ (cid:96) (cid:48) ( x ) ≤ x ∈ A . Note that ˆ π is functionally generatedby ˆ G ( x ) := exp( (cid:82) xθ ϕ ˆ π ( y ) dy ) for arbitrary θ ∈ (0 ,
1) where ϕ ˆ π is given by (37). Now fix x ∈ B := A ∪ { y : ˜ (cid:96) ( y ) > /y } ∪ { y : ˜ (cid:96) ( y ) > − / (1 − y ) } . Since B is an open set, it is clear that ϕ ˆ π is differentiable at x and we have that ϕ ˆ π ( x ) + ( ϕ ˆ π ) (cid:48) ( x ) ≤ (cid:96) together with the fact that the functions 1 /x and − / (1 − x ) satisfy theabove relationship (with equality). Next we claim thatlim sup h ↓ ˆ G (cid:48) ( x + h ) − ˆ G (cid:48) ( x ) h ≤ . (43)25ndeed, an application of the chain rule together with (42) yields ˆ G (cid:48)(cid:48) ( x ) ≤ x ∈ B , which inturn implies (43) for x ∈ B . Next fix x ∈ { y : ˜ (cid:96) ( y ) = 1 /y } . Let { h n } n ∈ N be a sequence converging to0 which achieves the limsup in (43) for our choice of x . Since ˆ G (cid:48) ( x ) = ˆ G ( x ) ϕ ˆ π ( x ) > G (cid:48) ( x + h n ) > h n small enough. For such h n , ˆ G (cid:48) ( x + h n ) = ˆ G ( x + h n ) min { / ( x + h n ) , ˜ (cid:96) ( x + h n ) } .Assume first that min { / ( x + h n ) , ˜ (cid:96) ( x + h n ) } = 1 / ( x + h n ) for infinitely many n . Denoting thesubsequence where this occurs by h n k we see thatlim k →∞ ˆ G (cid:48) ( x + h n k ) − ˆ G (cid:48) ( x ) h n k = lim k →∞ ˆ G ( x + h n k ) x + h nk − ˆ G ( x ) x h n k = (cid:18) ˆ G ( y ) 1 y (cid:19) (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) y = x = ˆ G ( x ) (cid:18) ϕ ˆ π ( x ) x − x (cid:19) = 0where we used the fact that ϕ ˆ π ( x ) = 1 /x . Since the original sequence converged to the limsup wesee that (43) holds in this case. If instead we have that min { / ( x + h n ) , ˜ (cid:96) ( x + h n ) } = ˜ (cid:96) ( x + h n ) forinfinitely many n then by considering the subsequence h n m where this occurs we again see thatlim m →∞ ˆ G (cid:48) ( x + h n m ) − ˆ G (cid:48) ( x ) h n m = lim m →∞ ˆ G ( x + h n m )˜ (cid:96) ( x + h n m ) − ˆ G ( x )˜ (cid:96) ( x ) h n m = ( ˆ G ˜ (cid:96) ) (cid:48) ( x ) = ˆ G ( x ) (cid:16) ϕ ˆ π ( x )˜ (cid:96) ( x ) + ˜ (cid:96) (cid:48) ( x ) (cid:17) = ˆ G ( x )(˜ (cid:96) ( x ) + ˜ (cid:96) (cid:48) ( x )) ≤ ϕ ˆ π ( x ) = ˜ (cid:96) ( x ) in the last equality and the assumption on ˜ (cid:96) togetherwith continuity of ˜ (cid:96) + ˜ (cid:96) (cid:48) in the inequality. This establishes that (43) holds for x ∈ { y : ˜ (cid:96) ( y ) = 1 /y } .A similar argument yields that (43) holds for x ∈ { y : ˜ (cid:96) ( y ) = − / (1 − y ) } which proves that (43)is satisfied for all x ∈ (0 , G (cid:48) is nonincreasing, which is equivalent to ˆ G beingconcave and completes the proof of the forward direction.Conversely, assume that there exists an x ∈ A such that ˜ (cid:96) ( x ) + ˜ (cid:96) (cid:48) ( x ) >
0. Then a similarcalculation to the one above yields ˆ G (cid:48)(cid:48) ( x ) >
0. Thus, we can find an (cid:15) > G (cid:48) isincreasing on ( x − (cid:15), x + (cid:15) ). It follows that ˆ G is not concave. Example 6.6 (The Dirichlet Case Revisited) . Revisiting the Dirichlet example from Section 4.1with specifications f i ( x i ) = ( x i ) b i and p ( x , x ) = ( x ) a − ( x ) a − where γ i = a i + b i − > for i = 1 , we see that here ˜ (cid:96) ( x ) = 12 (cid:18) γ x − γ − x (cid:19) . Setting θ = γ − γ + γ − and θ = γ γ + γ − it follows that ˆ π = x < θ γ + x (1 − γ − γ ) θ ≤ x ≤ θ x > θ . In particular we see that the unconstrained portfolio is long-only if and only if 1 < γ i ≤ i = 1 ,
2. Moreover, we can explicitly compute the growth rate achieved by this portfolio in termsof the incomplete Beta function. Additionally, it can be verified that (cid:112) ˜ c ˜ p ( x ) = ( x ) γ / ( x ) γ / is concave on (0 ∨ θ , θ ∧
1) so by Remark 6.5 together with Proposition 6.4 it follows that ˆ π isgenerated by a concave function. 26 Discussion of Long-Only Constraints
As previously mentioned, there are (at least) four possible long-only problems that could have beenconsidered. Namely, optimizing over(1) arbitrary long-only portfolios,(2) long-only portfolios in feedback form,(3) functionally generated long-only portfolios,(4) long-only portfolios generated by concave functions.The only procedure we are aware of to determine the optimum is to find a suitable class ofportfolios such that the following two properties are satisfied: • (Growth Rate Invariance) Each portfolio in the class has the same asymptotic growth rateunder every measure P ∈ Π. • (Approximation) The growth rate of the optimal portfolio can be approximated by the growthrates of portfolios in the chosen class.In [15], the authors used the class of portfolios generated by C functions to tackle the unconstrainedproblem. In this paper we were able to use the class ∂ E ∩ C (defined in the statement of Theorem5.8) to carry out a tractable analysis for problem (4).A seemingly natural choice would be to consider problem (3). At this point we would like toestablish that not all long-only functionally generated portfolios are generated by concave functions,so that problems (3) and (4) are truly different. Indeed, even convex functions can generate long-only portfolios as the following example shows. Example 7.1.
Set G ( x ) = exp( (cid:107) x (cid:107) ) . An application of the master formula Theorem 2.6 yields π iG ( x ) x i = 1 + ( x i − (cid:107) x (cid:107) ); i = 1 , . . . , d, so that π G is long-only. For problem (3) one would expect the candidate optimum to solve the variational problem (13)over all weakly differentiable functions φ under the constraint π ie φ ( x ) ≥ x ∈ ∆ d − and i ∈ { , . . . , d } . One can establish existence and uniqueness of the solution ˆ φ to this mathematicalproblem and the corresponding objective function value would serve as an upper bound for theoptimal robust asymptotic growth rate in this setting. However, the method for proving the lowerbound in Theorem 5.2 relies on approximating ∇ ˆ φ in the H c,p sense by gradients of C functionsthat generate long-only portfolios. In the general context of functionally generated portfolios, itis not clear that this is possible. Additionally, without establishing regularity properties of theoptimizer ˆ φ , it is not clear if ˆ φ ( X t ) is even a semimartingale, in which case the candidate optimalportfolio would not be functionally generated in the sense of Definition 9.In regards to problem (2), portfolios in feedback form do not in general have the growth rateinvariance property. Indeed, consider the case when (cid:96) is not a gradient so that the measures ˜ P and P ˆ u , defined in Assumption 2.4 (iii) and Theorem 2.8 respectively, are distinct and both belong toΠ. It is established in Theorem 2.8 that π ˆ G is growth-optimal under P ˆ u where ˆ G = exp ˆ u and itfollows from Lemma 5.3 that the portfolio π (cid:96) given by π i(cid:96) ( x ) := x i (cid:96) i ( x ) + x i − x i (cid:96) ( x ) (cid:62) x ; i = 1 , . . . , d
27s the growth-optimal portfolio under ˜ P . Moreover, since the objective function value for the varia-tional problem (13) is strictly positive at the optimum ˆ u , we see that g ( V π (cid:96) ; ˜ P ) > g ( V π ˆ G ; ˜ P ). Thus,if the portfolio π (cid:96) achieved the same growth rate in all admissible models it would outperform π ˆ G contradicting Theorem 2.8. It follows that the class of feedback portfolios is too large to establishgrowth-rate invariance under the admissible measures. The special structure of functionally gener-ated portfolios is needed. We note that this analysis only holds for d > (cid:96) is always a gradient. Moreover, as we have seen in Section 6, problems (2) and (3) areequivalent when d = 2 since every portfolio π in feedback form is functionally generated by thefunction ϕ π given by (33). As a result, the d = 2 case was explicitly solvable and we were able toget around the aforementioned difficulties. However, without explicit formulae for the solution tothe constrained variational problem in higher dimensions, problems (2) and (3) remain open andare beyond the scope of this paper.In view of the obstructions for problems (2) and (3), problem (1) seems out of reach. As such,we believe problem (4) is the natural one to consider in this context. Moreover, in Section 8.2 wewill see that the variational problem (31) is susceptible to a finite dimensional approximation dueto properties of exponentially concave functions and the fact that ∂E is a bounded set in H c,p .Thus, problem (4) can be numerically solved and implemented in practice. We are interested in exploring the case when the model from Section 3 is rank-based and whetheror not the optimal portfolios also share this property. Rank-based models play an important role instochastic portfolio theory due to the observed empirical stability of the capital distribution curveof the ranked market weights. The asymptotic growth rate in the setting of rank-based models wasstudied in [15] and we connect our model to their results as well as extending them to the long-onlyproblem.
Definition 8.1.
For x ∈ ∆ d − , we denote its rank vector by x () = ( x (1) , . . . , x ( d ) ) where x (1) ≥ x (2) ≥ · · · ≥ x ( d ) and { x ( k ) : k = 1 , . . . , d } = { x i : i = 1 , . . . , d } .We define the ordered simplex ∆ d − , ≤ := (cid:8) x ∈ ∆ d − : x ≥ x ≥ · · · ≥ x d (cid:9) so that x () ∈ ∆ d − , ≤ for every x ∈ ∆ d − . The procedure undertaken in [15] is to define ( c, p ) on the ordered simplex and then symmetri-cally extend them to the entire simplex. Then under Assumptions 2.2 and 2.4 they show that theoptimal strategy ˆ φ given by Theorem 2.8 is a function of the ranked weights.We take the converse route by establishing when the instantaneous covariation matrix c givenby (16) (defined on the entire simplex) is permutation invariant and under what conditions theoptimal strategy ˆ φ is a function of the ranked weights. Theorem 8.2. (1) The matrix c given by (16) is permutation invariant if and only if(i) f i = f j for all i, j = 1 , . . . , d ii) f ij = f kl for all i (cid:54) = j, k (cid:54) = l .(2) Under Assumptions 2.2 and 2.4 on c and p the optimal strategy ˆ φ from Corollary 3.3 is afunction of the ranked weights if and only if c satisfies ( i ) above and p is permutation invariant. Proof.
First assume c is permutation invariant. That is, given any x ∈ ∆ d − we have that c ij ( x ) = c σ − ( i ) σ − ( j ) ( x σ ) for any permutation x σ of x and all i, j . This reads that we must have − f ij ( x − ij ) f i ( x i ) f j ( x j ) = − f σ − ( i ) σ − ( j ) ( x − ij ) f σ − ( i ) ( x i ) f σ − ( j ) ( x j ); i (cid:54) = j. (44)By comparing like terms we see that ( i ) and ( ii ) must hold. For the converse direction we see from(44) if ( i ) and ( ii ) hold then c ij ( x ) = c σ − ( i ) σ − ( j ) ( x σ ) for any permutation x σ of x and all i (cid:54) = j .The fact that c ii ( x ) = − (cid:80) j c ij ( x ) for every i completes the proof of the first item.To prove (2) we note that a functionally generated portfolio is rank-based if and only if theportfolio generating function is permutation invariant. By Corollary 3.3 we see thatˆ G ( x ) = (cid:118)(cid:117)(cid:117)(cid:116) p ( x ) d (cid:89) i =1 f i ( x i )and from this expression it is clear that ˆ G is permutation invariant if and only if p is and f i = f j for all i, j . Remark 8.3.
Note that if (ii) from (1) holds then the backward direction of (2) holds by The-orem 3.5 in [15]. However, the remarkable structure of this class of models makes it so that theoptimal strategy is ranked-based if ( i ) holds even if ( ii ) does not hold; in this case c is not permu-tation invariant, but ˆ G is. Remark 8.4.
In the case that (1) holds we see that the optimal growth rate has the representation λ = 12 (cid:90) ∆ d − (cid:96) (cid:62) c(cid:96)p = d !2 (cid:90) ∆ d − , ≤ (cid:96) (cid:62) c(cid:96)p. Next we turn our attention to the long-only problem.
Proposition 8.5.
Assume that c satisfies (1) in Theorem 8.2 and that p is permutation invariant.Then the solution ˆ φ to (31) guaranteed by Corollary 5.9 is permutation invariant, the optimalportfolio generated by ˆ G = exp ˆ φ is rank-based and the growth rate is given by λ EC = d !2 (cid:90) ∆ d − , ≤ (cid:16) (cid:96) (cid:62) c(cid:96) − ( ∇ ˆ φ − (cid:96) ) (cid:62) c ( ∇ ˆ φ − (cid:96) ) (cid:17) p. (45) Proof.
The proof is very similar to the proof of Theorem 3.5 in [15]. Let ˆ φ be the unique solutionto (31). For every permutation σ of { , . . . , d } we define ˆ φ σ : ∆ d − → R via ˆ φ σ ( x ) = ˆ φ ( x σ ) where x iσ = x σ ( i ) . It follows that ˆ φ σ is exponentially concave for every σ and by convexity of E we seethat d ! (cid:80) σ ˆ φ σ is exponentially concave. Setting R ( x ) = p ( x ) / (cid:81) di =1 f i ( x i ) / and recalling thatin this framework (cid:96) = ∇ log R , it follows by the permutation invariance of c and p and a change ofvariables that (cid:90) ∆ d − ( ∇ φ − (cid:96) ) (cid:62) c ( ∇ φ − (cid:96) ) p = (cid:90) ∆ d − ( ∇ φ σ − (cid:96) ) (cid:62) c ( ∇ φ σ − (cid:96) ) p σ . Hence we see that (cid:107) d ! (cid:88) σ ∇ ˆ φ σ − (cid:96) (cid:107) H c,p ≤ d ! (cid:88) σ (cid:107)∇ ˆ φ σ − (cid:96) (cid:107) H c,p = (cid:107)∇ ˆ φ − (cid:96) (cid:107) H c,p = inf ∇ φ ∈ ∂ E (cid:107)∇ φ − (cid:96) (cid:107) H c,p . Hence, d ! (cid:80) σ ˆ φ σ is a minimizer for (31). By uniqueness it follows that (up to an additive constant)ˆ φ = d ! (cid:80) σ ˆ φ σ so that ˆ φ , and hence ˆ G , is permutation invariant. The integral representation of thegrowth rate in (45) follows by a change of variables. Next we turn our attention to establishing a method for finding the solution to (31). The class ofexponentially concave functions has nice properties that make the optimization problem susceptibleto a finite dimensional approximation which can be numerically implemented. Indeed, the next tworesults establish a large class of extreme points for the set ∂ E defined in (32). Definition 8.6.
Let C be a convex set. We say that f ∈ C is an extreme point if whenever we have f = αg + (1 − α ) h for some g, h ∈ C and α ∈ (0 , then we must have that g = h = f . For the next lemma set R d ++ = { a ∈ R d : a i > , i = 1 , . . . , d } . Lemma 8.7.
Let a, v, w ∈ R d ++ , and α ∈ (0 , be given such that log( a (cid:62) x ) = α log( v (cid:62) x ) + (1 − α ) log( w (cid:62) x ) (46) for every x ∈ D , where D is any nonempty relatively open set in ∆ d − . Then there exist constants c , c > such that a = c v = c w . Proof.
Since D is relatively open we can take directional derivatives to obtain that a i a (cid:62) x = α v i v (cid:62) x + (1 − α ) w i w (cid:62) x (47)for every x ∈ D and i = 1 , . . . , d . Taking a second derivative yields( a i ) ( a (cid:62) x ) = α ( v i ) ( v (cid:62) x ) + (1 − α ) ( w i ) ( w (cid:62) x ) (48)By squaring (47) and equating with (48) we obtain0 = α (1 − α ) (cid:18) v i v (cid:62) x − w i w (cid:62) x (cid:19) . Since α ∈ (0 ,
1) it follows that v i = v (cid:62) xw (cid:62) x w i . Now plugging this back into (47) gives a i a (cid:62) x = w i w (cid:62) x . a i = a (cid:62) xw (cid:62) x w i = a (cid:62) xv (cid:62) x v i Picking any ¯ x ∈ D and setting c = a (cid:62) ¯ xv (cid:62) ¯ x , c = a (cid:62) ¯ xw (cid:62) ¯ x completes the proof. Proposition 8.8.
The set E laff := (cid:40) ∇ (cid:32) n (cid:94) k =1 log w (cid:62) i x (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) n ∈ N , { w k } nk =1 ⊂ R d ++ (cid:41) is dense in ∂ E under the H c,p norm and every member of E laff is an extreme point in ∂ E . Proof.
We fix n ∈ N , and a collection { w k } nk =1 as in the definition of the set E laff . Set g k ( x ) :=log( w (cid:62) k x ) and g ( x ) := (cid:86) nk =1 g k ( x ). By monotonicity of the logarithm and the fact that the minimumof finitely many concave functions is concave, we see that g is exponentially concave. Now supposethat we have ∇ g = α ∇ ψ + (1 − α ) ∇ φ for some ∇ ψ, ∇ φ ∈ ∂ E . It follows that there exists a constant C > g = αφ + (1 − α ) ψ + C and we assume without loss of generality that C = 0.Define D k = (cid:26) x ∈ ∆ d − : g k ( x ) < min j (cid:54) = k g j ( x ) (cid:27) and assume without loss of generality that D k (cid:54) = ∅ for every k = 1 , . . . , n (since otherwise we couldexclude the corresponding g k from the minimum). If n = 1 we just set D = ∆ d − . It is clearthat each D k is convex and relatively open in ∆ d − . Additionally we have that g ( x ) = g k ( x ) on¯ D k and that ∆ d − = ∪ ni =1 ¯ D k where ¯ · denotes the closure in ∆ d − . Now fix an index k and let X be a random variable whose (essential) range is equal to ¯ D k . By convexity of ¯ D k we have that E [ X ] ∈ ¯ D k so we see that αψ ( E [ X ]) + (1 − α ) φ ( E [ X ]) = g ( E [ X ])= log E [ w (cid:62) i X ]= log E [ e log( w (cid:62) i X ) ]= log E [ e g ( X ) ]= log E [ e αψ ( X ) e (1 − α ) φ ( X ) ] ≤ α log E [ e ψ ( X ) ] + (1 − α ) log E [ e φ ( X ) ] ≤ αψ ( E [ X ]) + (1 − α ) φ ( E [ X ])where we used Hlder’s inequality with exponent 1 /α in the first inequality and concavity of e ψ and e φ together with Jensen’s inequality in the second one. Thus we see that we have equality all theway through and in particular0 = α (cid:16) log E [ e ψ ( X ) ; ¯ D k ] − ψ ( E [ X ; ¯ D k ]) (cid:17) + (1 − α ) (cid:16) log E [ e φ ( X ) ; ¯ D k ] − φ ( E [ X ; ¯ D k ]) (cid:17) . By Jensen’s inequality we have that log( E [ e ψ ( X ) ]) = E [ ψ ( X )]. Since we have equality in Jensen’sinequality if and only if e ψ is affine on the (essential) range of X we conclude that ψ ( x ) = log( u (cid:62) k x )31or all x ∈ ¯ D k and some u k ∈ R d ++ . By a similar argument φ ( x ) = log( v (cid:62) k x ) for every x ∈ ¯ D k andsome v k ∈ R d ++ . It follows thatlog( w (cid:62) k x ) = α log( u (cid:62) k x ) + (1 − α ) log( v (cid:62) k x ); x ∈ ¯ D k . Lemma 8.7 now implies that w k = c u k = c v k for some constants c , c >
0. As such, we concludethat g ( x ) = g k ( x ) = ψ ( x ) + α log c = φ ( x ) + (1 − α ) log c on D k for every k . It follows that ∇ g ( x ) = ∇ ψ ( x ) = ∇ φ ( x ) proving that ∇ g is an extreme point.To prove density we argue in a similar way as in the proof of Theorem 5.8. Fix ∇ φ ∈ ∂ E .Since e φ is a positive concave function and x (cid:55)→ e x is monotone we can find a sequence of vectors { v k } k ∈ N ⊆ R d ++ such that φ ( x ) = lim n →∞ h n ( x ) for all x ∈ ∆ d − where h n ( x ) = (cid:86) nk =1 log( v (cid:62) k x ).Since h n is a decreasing sequence of concave functions converging to a concave function it followsby Theorem 25.7 in [19] that h n → φ uniformly and ∇ h n → ∇ φ almost everywhere. As in theproof of Theorem 5.8, since { h n } n ∈ N is a bounded sequence in H c,p we can find a subsequence thatconverges weakly. The almost everywhere convergence ∇ h n → ∇ φ implies that this weak limitmust be ∇ φ . This, along with the fact that h n and ∇ h n are bounded functions for every n , impliesthat ∂ E = ∂ E laff w = ∂ E bw where · w denotes the weak closure in H cp and ∂ E b = ∇ φ ∈ ∂ E : sup x ∈ ∆ d − | φ ( x ) | < ∞ and esssup |∇ φ | < ∞ . Since ∂ E b is a convex set, its strong and weak closure coincide so we see from this that ∂ E b is densein ∂ E . As such to show that ∂ E laff (which is not a convex set) is dense in ∂ E it suffices to show thatwe can strongly approximate any member of ∂ E b .We now assume that ∇ φ ∈ ∂ E b is given and approximate φ, ∇ φ by h n , ∇ h n as before. Inparticular ∇ h n → ∇ φ weakly in H c,p . Setting A = sup x ∈ ∆ d − e φ ( x ), and a = inf x ∈ ∆ d − e φ ( x ) wehave by definition of ∂ E b that ∞ > A ≥ a >
0. Moreover, we can assume without loss of generalitythat h n ( x ) ≤ A + 1 (for example by setting v = ( A + 1) ) for every n . For a fixed x we have bydefinition that h n ( x ) = log( v (cid:62) k ( x ) x ) for some v k ( x ) where the index k depends on x . It follows bythe monotone convergence of e h n to e φ that( A + 1) ≥ v (cid:62) k ( x ) x ≥ e φ ( x ) ≥ a. In particular we see that 1 / ( v (cid:62) k ( x ) x ) ≤ /a and that v ik ( x ) ≤ ( A + 1) /x i for every i . Now fixing x inthe full measure set { x ∈ ∆ d − : φ and h n are differentiable at x for every n } we obtain ∂ i h n ( x ) = v ik ( x ) v (cid:62) k ( x ) x ≤ A + 1 ax i . Next we estimate that ∇ h n ( x ) (cid:62) c ( x ) ∇ h n ( x ) p ( x ) = d (cid:88) i,j =1 c ij ( x ) ∂ i h n ( x ) ∂ j h n ( x ) p ( x )32 ( A + 1) a d (cid:88) i,j =1 x i x j | c ij ( x ) | p ( x )for almost every x . By Lemma 6.2 the function on the right hand side is integrable so by dominatedconvergence we conclude that (cid:107)∇ φ (cid:107) H c,p = lim n →∞ (cid:107)∇ h n (cid:107) H c,p . Since we know that ∇ h n → ∇ φ weakly, it follows that ∇ h n → ∇ φ strongly as required. This completes the proof. In this section we make a specific choice for the model inputs c, p to simulate the behaviour of theoptimal portfolios encountered in the previous sections in a rank-based model. We make the choiceof the Dirichlet distribution p ( x ) = B ( a ) (cid:81) di =1 x a i − i as in Example 4.1, since it is one of the mostwidely used distributions on the simplex and, as mentioned in Example 4.2, it and its relatives havebeen used in the literature to model the stationarity of the ranked market weights. It is clear that p ( x ) is permutation invariant if and only if a i = a j for every i, j and so it suffices to parameterizethe density by a single one dimensional parameter a . Figure 1 plots on a log-log scale the orderstatistics of random samples drawn from a Dirichlet distribution with parameters a = 0 . , , and 2for d = 500 and d = 5000. These samples represent the capital distribution curve for the rankedmarket weights in the model described above. Capital Distribution Curve Simulation
Figure 1: Capital distribution curves from Dirichlet densitiesThe shape of the curves are similar to those observed empirically (c.f. Figure 5.1 in[9]), howeverthe largest market weight is not quite as large as seen in real financial markets. Nevertheless, sincewe are unaware of any better fitting distributions with an analytically available density, we choose33he Dirichlet distribution as the invariant density for the ranked market weights in our simulations.For the c matrix we make the choice f i ( x ) = x and f ij ( x ) = σ for every i, j ∈ { , . . . , d } and someconstant σ >
0. As previously mentioned this recovers the volatility structure from volatility-stabilized models introduced in [11].The unconstrained optimal strategy and the optimal portfolio if feedback form for d = 2 areavailable analytically from Theorems 2.8 and 6.1 respectively. In Example 6.6 we observed thatthe optimal concave portfolio is given by the optimal portfolio in feedback form, but for d > K, M ∈ N and select vectors a km ∈ R d ++ such that a km /d is uniformly distributed on thesimplex for k = 1 . . . , K and m = 1 , . . . , M . Then we set φ m ( x ) = min k ∈{ ,...,K } log( a (cid:62) km x )for m = 1 , . . . , M so that ∇ φ m is an extreme point of the set ∂ E by Proposition 8.8. Next we define ψ µ ( x ) = M (cid:88) m =1 µ m φ m ( x )for parameters µ m such that µ m ≥ m and (cid:80) Mm =1 µ m = 1. Relying on the density andextremity of E laff in ∂ E we replace the infinite dimensional problem (31) by the finite dimensionalproblem min µ (cid:107)∇ ψ µ − (cid:96) (cid:107) H c,p . Notice that for a given convex combination µ we have that (cid:107)∇ g µ − (cid:96) (cid:107) H c,p = E [( ∇ ψ µ − (cid:96) ) (cid:62) c ( ∇ ψ µ − (cid:96) )( Y )]where Y ∼ p . We then approximate this quantity by N (cid:80) Nn =1 ( ∇ ψ µ − (cid:96) ) (cid:62) c ( ∇ ψ µ − (cid:96) )( Y n ) for a largevalue of N and iid samples Y , . . . , Y N ∼ p . A calculation shows that (cid:107)∇ ψ µ − (cid:96) (cid:107) H c,p ≈ µ (cid:62) (cid:32) N N (cid:88) n =1 Q ( X n ) (cid:33) µ − µ (cid:62) (cid:32) N N (cid:88) n =1 r ( X n ) (cid:33) + C (49)where C is some constant independent of µ , while Q : ∆ d − → R M × M and r : ∆ d − → R M aregiven by Q ij ( x ) = 2( ∇ φ i ) (cid:62) c ( ∇ φ j )( x ) r i ( x ) = 2( ∇ φ i ) (cid:62) c(cid:96) ( x ) . As before ∇ φ i refers to a version of the superdifferential of φ i . It follows that Q is positive-semidefinite with v (cid:62) Qv = ( (cid:80) Mm =1 v m ∇ φ m ) (cid:62) c ( (cid:80) Mm =1 v m ∇ φ m ) for every v ∈ R M . Thus, minimizing(49) over µ is a quadratic programming problem that can be numerically solved. Using the optimalsolution ˆ µ we simulate the portfolio generated by exp ψ ˆ µ .34 = 2 Simulation Figure 2: Parameters are a = 3 , σ = 0 . , M = 25 , K = 100 , N = 100We now present simulations under the worst case measure ˜ P , under which the coordinate process X has dynamics given by (7). Figure 2 shows the result in the d = 2 case. We used the aforemen-tioned algorithm to estimate the optimal concave portfolio, represented by the green curve, whichas mentioned is the same as the explicitly known optimal portfolio in feedback form, representedby the red curve. As predicted by the theoretical calculations, the optimal unconstrained portfolioperforms best, closely followed by the optimal long-only portfolio in feedback form ahead of ourestimated optimal concave strategy. In this case we see that the attained growth rates are of thesame order of magnitude though the relationship λ > λ long = λ EC persists. This is in contrastto the results of Figure 3; in this case we set d = 20 and observe that λ EC is an order of mag-nitude lower than λ . It is not surprising to see a larger disparity between the two growth ratesin a higher dimension model. Indeed, as the dimension increases, the region of the simplex wherethe unconstrained strategy takes on a long position grows smaller relative to the size of the entiresimplex. As such, the market weights do not spend as much (if any) time in this region when d = 20 in comparison to when d = 2. The interpretation is that a long-only investor is missingout more heavily on the possible gains from short-selling when the number of assets is large. Thisleads to a relatively smaller growth rate. It is unclear, however, how much of the observed loss ingrowth rate is purely due to the long-only restriction, how much is due to our further restrictionto consider only those portfolios generated by concave functions and how much is additionally lostfrom the aforementioned numerical implementation. It remains an open problem to identify tightupper bounds on the growth rates for these constrained-optimal portfolios.35 = 20 Simulation —— Optimal Unconstrained —— Optimal Concave − − λ Figure 3: Parameters are a = 3 , σ = 5 × − , M = 25 , K = 100 , N = 100 Appendix A Generalized Martingale Problem on ∆ d − d − as an open subset of R d − and identify ∂ ∆ d − with a single absorbingstate Θ. That is, formally we define the domain of our generalized martingale problem to be˜∆ d − := ∆ d − ∪ { Θ } ; the one-point compactification of ∆ d − . Next, since the simplex has a non-smooth boundary we will need a localization procedure to define the martingale problem. To thisend, let { E n } n ∈ N be an increasing sequence of open connected domains such that (i) ¯ E n ⊆ E n +1 forevery n ∈ N , (ii) ∂E n is of class C ,β for every n ∈ N and for some β ∈ (0 ,
1] and (iii) ∆ d − = ∪ n E n .For example one could take E n = (cid:40) x ∈ ∆ d − : d (cid:89) i =1 x i > /n (cid:41) . Define the exit times τ n := inf { t ≥ X t (cid:54)∈ E n } and τ := lim n →∞ τ n . The corresponding sample36pace is ˜Ω := (cid:110) ω ∈ C ([0 , ∞ ); ˜∆ d − ) : ω τ + t = Θ for all t ≥ (cid:111) . We denote by ˜ X the coordinate process on ˜Ω, define ˜ F to be the Borel σ -algebra on ˜Ω and let ˜ F be the right-continuous modification of the filtration generated by ˜ X . Now we are ready to definethe generalized martingale problem. Definition A.1.
The generalized martingale problem on ∆ d − associated to an operator ˜ L is forevery x ∈ ˜∆ d − to find a probability measure ˜ P x on ( ˜Ω , ˜ F ) such that(a) ˜ P x ( ˜ X = x ) = 1 ,(b) f ( ˜ X T ∧ τ n ) − (cid:82) T ∧ τ n ˜ Lf ( ˜ X t ) dt is a martingale on ( ˜Ω , ˜ F , ˜ F , ˜ P x ) for every n ∈ N and f ∈ C (∆ d − ) .We say that the solution is nonexplosive if ˜ P x ( τ < ∞ ) = 0 for every x ∈ ∆ d − . We note that a nonexplosive solution to the martingale problem does not charge the set of pathsthat reach Θ and as such a nonexplosive solution ˜ P x can be viewed as a measure on (Ω , F ). Wedefine the function R = p / (cid:81) di =1 f / i and using the notation of Assumption 2.4 (iii) define theoperator L R via L R g := Lg − ∇ log R (cid:62) c ∇ g for every g ∈ C (∆ d − ). The following theorem from[18] establishes existence and uniqueness for the generalized martingale problem associated to thisoperator: Theorem A.2 (Theorem 1.13.1 in [18]) . Under Assumption 2.2(i) on the instantaneous covariationmatrix c there exits a unique solution to the generalized martingale corresponding to the operator L R . In general the unique solution guaranteed by this theorem may explode in finite time. Thus,to verify Assumption 2.4 (iii), one must ensure that the solution is nonexplosive. We will establishsufficient conditions on the inputs p and f i for i = 1 , . . . , d that guarantee this is the case. Themain tool to establish this is the following test-function method: Proposition A.3 (Theorem 6.7.1 (i) in [18]) . Let ˜ L be an operator for which there exists a uniquesolution to the generalized martingale problem given in Definition A.1. Assume there exists aconstant λ > and a function u ∈ C (∆ d − ) such that ˜ Lu ( x ) ≤ λu ( x ) for every x ∈ ∆ d − and lim x → ∂ ∆ d − u ( x ) = ∞ . Then the solution to the generalized martingale problem corresponding to ˜ L does not explode. With these preliminary results established we are ready to prove Theorem 3.5.
Proof of Proposition 3.5 .
We note that (cid:96) = ∇ log R and a direct calculation shows that12 (cid:96) (cid:62) c(cid:96) = LRR − L log R so that Assumption 2.4 ( i ) holds by the integrability assumptions on LRR and L log R . Next wecompute that div pc(cid:96) = d (cid:88) i =1 ∂ i d (cid:88) j =1 c ij (cid:96) j p = d (cid:88) i,j =1 ∂ i c ij (cid:96) j p + c ij ∂ i (cid:96) j p + c ij (cid:96) j ∂ i p p div c (cid:62) (cid:96) + 2 pL log R + ∇ pc(cid:96). Recalling that (cid:96) = ( c − div c + ∇ log p ) we obtaindiv pc(cid:96) = 2 (cid:96) (cid:62) c(cid:96)p + 2 pL log R. Thus we again see that Assumption 2.4 (ii) holds by the integrability assumptions.Next we note by Theorem A.2 and Proposition A.3 it is enough to find a function u ∈ C (∆ d − )such that lim x → ∂ ∆ d − u ( x ) = ∞ , u > d − and L R u ≤ λu for some λ > L R u = L ( uR ) R − LRR u for every u ∈ C (∆ d − ). Choosing u = 1 /R we see that L R u ≤ λu ⇐⇒ − LRR ≤ λ and the right hand side holds for some λ > LR/R is bounded from below.Moreover, since R − > d − and R ( x ) → x → ∂ ∆ d − we see that this test functionpossesses all the required properties. References [1] Gholamreza Alirezaei and Rudolf Mathar. On exponentially concave functions and their impactin information theory. In , pages1–10. IEEE, 2018.[2] Adrian D Banner and Daniel Fernholz. Short-term relative arbitrage in volatility-stabilizedmarkets.
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