Asymptotic Optimal Portfolio in Fast Mean-reverting Stochastic Environments
aa r X i v : . [ q -f i n . M F ] J a n Asymptotic Optimal Portfolio in Fast Mean-reverting StochasticEnvironments
Ruimeng Hu
Abstract — This paper studies the portfolio optimization prob-lem when the investor’s utility is general and the return andvolatility of the risky asset are fast mean-reverting, which areimportant to capture the fast-time scale in the modeling ofstock price volatility. Motivated by the heuristic derivation in[J.-P. Fouque, R. Sircar and T. Zariphopoulou,
MathematicalFinance , 2016], we propose a zeroth order strategy, and showits asymptotic optimality within a specific (smaller) family ofadmissible strategies under proper assumptions. This optimalityresult is achieved by establishing a first order approximationof the problem value associated to this proposed strategyusing singular perturbation method, and estimating the risk-tolerance functions. The results are natural extensions of ourprevious work on portfolio optimization in a slowly varyingstochastic environment [J.-P. Fouque and R. Hu,
SIAM Journalon Control and Optimization , 2017], and together they form awhole picture of analyzing portfolio optimization in both fastand slow environments.
Index Terms — Stochastic optimal control, asset allocation,stochastic volatility, singular perturbation, asymptotic optimal-ity.
I. I
NTRODUCTION
The portfolio optimization problem in continuous time,also known as the Merton problem, was firstly studied in[17], [18]. In his original work, explicit solutions on howto allocate money between risky and risk-less assets and/orhow to consume wealth are provided so that the investor’sexpected utility is maximized, when the risky assets followsthe Black–Scholes (BS) model and the utility is of ConstantRelative Risk Aversion (CRRA) type. Since these seminalworks, lots of research has been done to relax the originalmodel assumptions, for example, to allow transaction cost[16], [10], drawdown constraints [9], [4], [5], price impact[3], and stochastic volatility [20], [2], [8] and [15].Our work extends Merton’s model by allowing moregeneral utility, and by modeling the return and volatility ofthe risky asset S t by a fast mean-reverting process Y t : d S t = µ ( Y t ) S t d t + σ ( Y t ) S t d W t , (1) d Y t = 1 ǫ b ( Y t ) d t + 1 √ ǫ a ( Y t ) d W Yt . (2)The two standard Brownian motion (Bm) are imperfectlycorrelated: d (cid:10) W, W Y (cid:11) = ρ d t, ρ ∈ ( − , . We areinterested in the terminal utility maximization problem V ǫ ( t, x, y ) ≡ sup π ∈A ǫ E [ U ( X πT ) | X πt = x, Y t = y ] , (3) Ruimeng Hu is with the Department of Statistics, Columbia University,New York, NY 10027, USA [email protected] . The workwas mainly done when RH was a graduate student at the University ofCalifornia, Santa Barbara, and was partially supported by the NationalScience Foundation under DMS-1409434. where X πt is the wealth associated to self-financing π : d X πt = π ( t, X πt , Y t ) µ ( Y t ) d t + π ( t, X πt , Y t ) σ ( Y t ) d W t , (4)(assume the risk-free interest rate varnishes r = 0 ) and A ǫ isthe set of strategies that X πt stays nonnegative. Using singu-lar perturbation technique, our work provide an asymptoticoptimal strategy π (0) within a specific class of admissiblestrategies A ǫ that satisfies certain assumptions: A ǫ (cid:2)e π , e π , α (cid:3) = (cid:8)e π + ǫ α e π (cid:9) ≤ ǫ ≤ . (5) Motivation and Related Literature.
The reason to studythe proposed problem is threefold. Firstly, in the directionof asset modeling (1)-(2), the well-known implied volatilitysmile/smirk phenomenon leads us to employ a BS-likestochastic volatility model. Empirical studies have identifiedscales in stock price volatility: both fast-time scale on theorder of days and slow-scale on the order of months [7].This results in putting a parameter ǫ in (2). The slow-scalecase (corresponding to large ǫ in (2)), which is particularlyimportant in long-term investments, has been studied inour previous work [6]. An asymptotic optimality strategy isproposed therein using regular perturbation techniques. Thismakes it natural to extend the study to fast-varying regime,where one needs to use singular perturbation techniques.Secondly, in the direction of utility modeling, apparently noteveryone’s utility is of CRRA type [1], therefore it is impor-tant to consider to work under more general utility functions.Thirdly, although it is natural to consider multiscale factormodels for risky assets, with a slow factor and a fast factoras in [8], more involved technical calculation and proof arerequired in combining them, and thus, we leave it to anotherpaper in preparation [11].Our proposed strategy π (0) is motivated by the heuristicderivation in [8], where a singular perturbation is performedto the PDE satisfied by V ǫ . This gave a formal approximation V ǫ = v (0) + √ ǫv (1) + ǫv (2) + · · · . They then conjectured thatthe zeroth order strategy π (0) ( t, x, y ) = − λ ( y ) σ ( y ) v (0) x ( t, x, y ) v (0) xx ( t, x, y ) , λ ( y ) = µ ( y ) σ ( y ) (6)reproduces the optimal value up to the first order v (0) + √ ǫv (1) , with v (0) and v (1) given by (11) and (13). Main Theorem.
Let V π (0) ,ǫ (resp. e V ǫ ) be the expectedutility of terminal wealth associated to π (0) (resp. π ∈ A ǫ ): V π (0) ,ǫ := E [ U ( X π (0) t ) | X π (0) t = x, Y t = y ] , and X π (0) t bethe wealth process given by (4) with π = π (0) (resp. π in A ǫ ). By comparing V π (0) ,ǫ and e V ǫ , we claim that π (0) erforms asymptotically better up to order √ ǫ than the family (cid:8)e π + ǫ α e π (cid:9) . Mathematically, this is formulated as: Theorem 1.1:
Under assumptions detailed in Sections IIand IV, for any family of trading strategies A ǫ (cid:2)e π , e π , α (cid:3) = { e π + ǫ α e π } ≤ ǫ ≤ , the following limit exists and satisfies ℓ := lim ǫ → (cid:0) e V ǫ ( t, x, y ) − V π (0) ,ǫ ( t, x, y )) / √ ǫ ≤ . Proof will be given in Section IV as well as the interpreta-tions of this inequality according to different α ’s. Our maintheorem gives some insights on how to construct expansionof the optimal π , which, however, is still an open question.We remark that, in a related work [19], expansion results for π ∗ exist under a discrete-time filtering setting.The rest of the paper is organized as follows. Section IIintroduces some preliminaries of the Merton problem andstanding assumptions in this paper. Section III gives V π (0) ,ǫ ’sfirst order approximation v (0) + √ ǫv (1) . Section IV is ded-icated to the proof of Theorem 1.1. The expansion of e V ǫ is analyzed first, with precise derivations, while the detailedtechnical assumption is referred to our recent work [6].II. P RELIMINARIES AND A SSUMPTIONS
In this section, we firstly review the classical Merton prob-lem, and the notation of risk tolerance function R ( t, x ; λ ) .Then heuristic expansion results of V ǫ in [8] are summarized.Standing assumptions of this paper are listed, as well as someestimations regarding R ( t, x ; λ ) and v (0) . We refer to ourrecent work [6, Section 2, 3] for proofs of all these results.We shall first consider the case of constant µ and σ in(1). This is the classical Merton problem, which plays acrucial role in interpreting the leading order term v (0) andanalyzing the singular perturbation. This problem has beenstudied intensively, for instance, in [13]. Let X t be the wealthprocess in this case. Using the notation in [8], we denoteby M ( t, x ; λ ) the problem value. In Merton’s original work,closed-form M ( t, x ; λ ) was obtained when the utility U ( · ) isof power type. In general, one has the following results, withproofs given in [6, Section 2.1] or the references therein. Proposition 2.1:
Assume that the utility function U ( x ) is C (0 , ∞ ) , strictly increasing, strictly concave, such that U (0+) is finite, and satisfies the Inada and AsymptoticElasticity conditions: U ′ (0+) = ∞ , U ′ ( ∞ ) = 0 , AE [ U ] :=lim x →∞ x U ′ ( x ) U ( x ) < , then, the Merton value function M ( t, x ; λ ) is strictly increasing, strictly concave in thewealth variable x , and decreasing in the time variable t . It is C , ([0 , T ] × R + ) and is the unique solution to the Hamilton-Jacobi-Bellman(HJB) equation, with M ( T, x ; λ ) = U ( x ) , M t + sup π (cid:26) σ π M xx + µπM x (cid:27) = 0 , (7)where λ = µσ is the Sharpe ratio. It is C w.r.t λ , and theoptimal strategy is given by π ⋆ ( t, x ; λ ) = − λσ M x ( t,x ; λ ) M xx ( t,x ; λ ) . We next define the risk-tolerance function R ( t, x ; λ ) = − M x ( t,x ; λ ) M xx ( t,x, ; λ ) , and operators following the notations in [8], D k = R ( t, x ; λ ) k ∂ kx , L t,x ( λ ) = ∂ t + 12 λ D + λ D . (8) By the concavity of M ( t, x ; λ ) , R ( t, x ; λ ) is continuous andstrictly positive. Using the relation D M = − D M , the non-linear Merton PDE (7) can be re-written in a “linear” way: L t,x ( λ ) M ( t, x ; λ ) = 0 . We now mention a uniqueness resultto this PDE, which will be used repeatedly in Sections III. Proposition 2.2:
Let L t,x ( λ ) be the operator defined in(8), and assume that the utility function U ( x ) satisfies theconditions in Proposition 2.1, then L t,x ( λ ) u ( t, x ; λ ) = 0 , u ( T, x ; λ ) = U ( x ) , (9)has a unique nonnegative solution.Next, we review the formal expansion results of V ǫ derived in [8]. To apply singular perturbation technique, weassume that the process Y (1) t D = Y tǫ is ergodic and equippedwith a unique invariant distribution Φ . We use the notation h·i for averaging w.r.t. Φ , namely, h f i = R f dΦ . Let L bethe infinitesimal generator of Y (1) : L = a ( y ) ∂ y + b ( y ) ∂ y . Then, by dynamic programming principle, the value function V ǫ solves the HJB equation in the viscosity sense: V ǫt + 1 ǫ L V ǫ + max π ∈A ǫ (cid:16) σ ( y ) π V ǫxx / π (cid:0) µ ( y ) V ǫx + ρa ( y ) σ ( y ) V ǫxy / √ ǫ (cid:1)(cid:17) = 0 . (10)and its regularity is not clear. In [8], a unique classicalsolution is assumed in order to perform heuristic derivations.Moreover, the optimizer in (10) is well-defined: π ∗ = − λ ( y ) V ǫx σ ( y ) V ǫxx − ρa ( y ) V ǫxy √ ǫσ ( y ) V ǫxx , and the simplified HJB equationreads: V ǫt + ǫ L V ǫ − (cid:16) λ ( y ) V ǫx + √ ǫ ρa ( y ) V ǫxy (cid:17) / (2 V ǫxx ) =0 , for ( t, x, y ) ∈ [0 , T ] × R + × R . We remark that, to obtainour Theorem (1.1), the smooth condition is not needed, as wefocus on the quantity V π (0) ,ǫ defined in (16). It correspondsto a linear PDE, for which classical solutions exist.The equation (10) is fully nonlinear and is only explicitlysolvable in some cases; see [2] for instance. The heuristic ex-pansions overcome this by providing approximations to V ǫ .This is done by the so-called singular perturbation method,as often seen in homogenization theory. To be specific, onesubstitutes the expansion V ǫ = v (0) + √ ǫv (1) + ǫv (2) + · · · into the above equation, establishes equations about v ( k ) bycollecting terms of different orders. In [8, Section 2], this isperformed for k = 0 , and we list their results as follows:(i) The leading order term v (0) ( t, x ) is defined as the so-lution to the Merton PDE associated with the averagedSharpe ratio λ = p h λ i : v (0) t − λ (cid:16) v (0) x (cid:17) v (0) xx = 0 , v (0) ( T, x ) = U ( x ) , (11)and by Proposition 2.1 v (0) is identified as: v (0) ( t, x ) = M (cid:0) t, x ; λ (cid:1) . (12)(ii) The first order correction v (1) is identified as thesolution to the linear PDE: v (1) t + λ v (0) x v (0) xx ) v (1) xx − λ v (0) x v (0) xx v (1) x = ρ BD v (0) , (13)ith v (1) ( T, x ) = 0 . The constant B = h λaθ ′ i , and θ ( y ) solves L θ ( y ) = λ ( y ) − λ . Rewrite equation(13) in terms of the operators in (8), v (1) solves thefollowing PDE which admits a unique solution: L t,x ( λ ) v (1) = 12 ρBD v (0) , v (1) ( T, x ) = 0 . (14)(iii) v (1) is explicitly given in term of v (0) by v (1) ( t, x ) = − ( T − t ) ρBD v (0) ( t, x ) . Now we introduce the assumptions on the utility U ( · ) andthe state processes ( S t , X π (0) t , Y t ) , and refer to [6, Section 2]for further discussions and remarks. Assumption 2.3:
Throughout the paper, we make the fol-lowing assumptions on the utility U ( x ) :(i) U ( x ) is C (0 , ∞ ) , strictly increasing, strictly concaveand satisfying the following conditions: U ′ (0+) = ∞ , U ′ ( ∞ ) = 0 , AE [ U ] := lim x →∞ x U ′ ( x ) U ( x ) < . (ii) U (0+) is finite. Without loss of generality, U (0+) = 0 .(iii) Denote by R ( x ) the risk tolerance, R ( x ) := − U ′ ( x ) U ′′ ( x ) . Assume that R (0) = 0 , R ( x ) is strictly increasing and R ′ ( x ) < ∞ on [0 , ∞ ) , and there exists K ∈ R + , suchthat for x ≥ , and ≤ i ≤ , (cid:12)(cid:12) ∂ ix R i ( x ) (cid:12)(cid:12) ≤ K. (15)(iv) Define the inverse function of the marginal utility U ′ ( x ) as I : R + → R + , I ( y ) = U ′ ( − ( y ) , and assumethat, for some positive α , I ( y ) satisfies the polynomialgrowth condition: I ( y ) ≤ α + κy − α . Assumption 2.3(ii) is a sufficient condition, and rules outthe cases U ( x ) = x γ γ , γ < , and U ( x ) = log( x ) . However,all theorems in the paper still hold, as it is to ensure that termsin (18) are of the form (19), which is automatically satisfiedfor aforementioned cases. Next are the model assumptions. Assumption 2.4:
We make the following assumptions onthe state processes ( S t , X π (0) t , Y t ) :(i) For any starting points ( s, y ) and fixed ǫ , the systemof SDEs (1)–(2) has a unique strong solution ( S t , Y t ) .The functions λ ( y ) and a ( y ) have polynomial growth.(ii) The process Y (1) with infinitesimal generator L isergodic with a unique invariant distribution, and ad-mits moments of any order uniformly in t ≤ T : sup t ≤ T (cid:26) E (cid:12)(cid:12)(cid:12) Y (1) t (cid:12)(cid:12)(cid:12) k (cid:27) ≤ C ( T, k ) . The solution φ ( y ) of the Poisson equation L φ = g is assumed to bepolynomial for polynomial functions g .(iii) The wealth process X π (0) · is in L ([0 , T ] × Ω) uniformlyin ǫ , i.e., E (0 ,x,y ) (cid:20)R T (cid:16) X π (0) s (cid:17) d s (cid:21) ≤ C ( T, x, y ) ,where C ( T, x, y ) is independent of ǫ and E (0 ,x,y ) [ · ] = E [ ·| X = x, Y = y ] .Here we provide several estimations of the risk tolerancefunction R ( t, x ; λ ) and the zeroth order value function v (0) ,which are crucial in the proof of Theorem 3.1.By Proposition 2.1 and the relation (12), v (0) is concavein the wealth variable x , and decreasing in the time variable t , therefore has a linear upper bound, for ( t, x ) ∈ [0 , T ] × R + : v (0) ( t, x ) ≤ v (0) (0 , x ) ≤ c + x , for some constant c .Combining it with Assumption 2.4(iii), we deduce: Lemma 2.5:
Under Assumption 2.3 and 2.4, the process v (0) ( · , X π (0) · ) is in L ([0 , T ] × Ω) uniformly in ǫ , i.e. ∀ ( t, x ) ∈ [0 , T ] × R + : E ( t,x ) (cid:20)R Tt (cid:16) v (0) ( s, X π (0) s ) (cid:17) d s (cid:21) ≤ C ( T, x ) , where v (0) ( t, x ) satisfies equation (11). Proposition 2.6:
Suppose the risk tolerance R ( x ) = − U ′ ( x ) U ′′ ( x ) is strictly increasing for all x in [0 , ∞ ) (this is partof Assumption 2.3 (iii)), then, for each t ∈ [0 , T ) , R ( t, x ; λ ) is strictly increasing in the wealth variable x . Proposition 2.7:
Under Assumption 2.3, the risk tolerancefunction R ( t, x ; λ ) satisfies: ∀ ≤ j ≤ , ∃ K j > , such that ∀ ( t, x ) ∈ [0 , T ) × R + , (cid:12)(cid:12) R j ( t, x ; λ ) (cid:0) ∂ j +1 x R ( t, x ; λ ) (cid:1)(cid:12)(cid:12) ≤ K j . Or equivalently, ∀ ≤ j ≤ , there exists e K j > , such that (cid:12)(cid:12) ∂ jx R j ( t, x ; λ ) (cid:12)(cid:12) ≤ e K j . Moreover, one has R ( t, x ; λ ) ≤ K x. III. P
ORTFOLIO PERFORMANCE OF A GIVEN STRATEGY
Recall the strategy π (0) defined in (6), and assume π (0) is admissible. In this section, we are interested in studyingits performance. That is, to give approximation results of thevalue function associated to π (0) , denote by V π (0) ,ǫ : V π (0) ,ǫ ( t, x, y ) = E n U ( X π (0) T ) | X π (0) t = x, Y t = y o , (16)where U ( · ) is a general utility function satisfying Assump-tion 2.3, X π (0) t is the wealth process associated to the strategy π (0) and Y t is the fast factor. Our main result of this sectionis the following, with the proof delayed in Section III-B. Theorem 3.1:
Under assumptions 2.3 and 2.4, the residualfunction E ( t, x, y ) defined by E ( t, x, y ) := V π (0) ,ǫ ( t, x, y ) − v (0) ( t, x ) − √ ǫv (1) ( t, x ) , is of order ǫ . In other words, ∀ ( t, x, y ) ∈ [0 , T ] × R + × R , E ( t, x, y ) ≤ Cǫ , for someconstant C depending on ( t, x, y ) but not on ǫ . Corollary 3.2:
In the case of power utility U ( x ) = x γ γ , π (0) is asymptotically optimal in A ǫ ( t, x, y ) up to order √ ǫ . Proof:
This is obtained by comparing expansions of V ǫ given in [8, Corollary 6.8], and of V π (0) ,ǫ from theabove Theorem. Since both quantities have the approxima-tion v (0) + √ ǫv (1) at order √ ǫ , we have the desired result. A. Formal expansion of V π (0) ,ǫ In the following derivation, to condense the notation, weuse R for R ( t, x ; λ ) , and π (0) for π (0) ( t, x, y ) given in (6).By the martingale property, V π (0) ,ǫ solves the linearPDE: V π (0) ,ǫt + π (0) µ ( y ) V π (0) ,ǫx + π (0) √ ǫ ρa ( y ) σ ( y ) V π (0) ,ǫxy + ǫ L V π (0) ,ǫ + σ ( y ) (cid:0) π (0) (cid:1) V π (0) ,ǫxx = 0 . Define twooperators L and L by L = ρa ( y ) σ ( y ) π (0) ∂ xy = ρa ( y ) λ ( y ) R ( t, x ; λ ) ∂ xy , and L = ∂ t + σ ( y ) (cid:0) π (0) (cid:1) ∂ x + µ ( y ) π (0) ∂ x = ∂ t + λ ( y ) D + λ ( y ) D respectively, thenthis linear PDE can be rewritten as: (cid:0) L + L / √ ǫ + L /ǫ (cid:1) V π (0) ,ǫ = 0 . (17)We look for an expansion of V π (0) ,ǫ of the form V π (0) ,ǫ = v π (0) , (0) + √ ǫv π (0) , (1) + ǫv π (0) , (2) + · · · , with π (0) , (0) ( T, x, y ) = U ( x ) and v π (0) , ( k ) ( T, x, y ) = 0 , for k ≥ . Inserting the above expansion of V π (0) ,ǫ into (17),and collecting terms of O ( ǫ ) and O ( √ ǫ ) give: L v π (0) , (0) =0 , L v π (0) , (1) + L v π (0) , (0) = 0 . Since L and L areoperators taking derivatives in y , we make the choice that v π (0) , (0) and v π (0) , (1) are independent of y . Next, collectingterms of O (1) yields L v π (0) , (2) + L v π (0) , (0) = 0 , whosesolvability condition requires that D L v π (0) , (0) E = 0 . Thisleads to a PDE for v π (0) , (0) : v π (0) , (0) t + λ ( R ) v π (0) , (0) xx + λ Rv π (0) , (0) x = 0 , v π (0) , (0) ( T, x ) = U ( x ) , which has aunique solution (c.f. Proposition 2.2). Since v (0) also solvesthis equation, we deduce that v π (0) , (0) ( t, x ) ≡ v (0) ( t, x ) = M ( t, x ; λ ) , and v π (0) , (2) admits a solution v π (0) , (2) ( t, x, y ) = − θ ( y ) D v (0) + C ( t, x ) , with θ ( y ) given by L θ ( y ) = λ ( y ) − λ and D k in (8).Then, collecting terms of order √ ǫ yields L v π (0) , (1) + L v π (0) , (2) + L v π (0) , (3) = 0 , and the solvability conditionreads D L v π (0) , (1) + L v π (0) , (2) E = 0 . This gives an equa-tion satisfied by v π (0) , (1) : v π (0) , (1) t + λ ( R ) v π (0) , (1) xx + λ Rv π (0) , (1) x − ρBD v (0) = 0 , v π (0) , (1) ( T, x ) = 0 , whichis exactly equation (13). This equation is uniquely solvedby v (1) (see (14)). Thus, we obtain v π (0) , (1) ≡ v (1) = − ( T − t ) ρBD v (0) .Using the solution of v π (0) , (1) and v π (0) , (2) we just iden-tified, one deduces an expression for v π (0) , (3) : v π (0) , (3) = ( T − t ) θ ( y ) ρB (cid:0) D + D (cid:1) D v (0) + ρθ ( y ) D v (0) + C ( t, x ) , where θ ( y ) is the solution to the Poisson equation: L θ ( y ) = a ( y ) λ ( y ) θ ′ ( y ) − h aλθ ′ i . B. First order accuracy: proof of Theorem 3.1
This section completes the proof of Theorem 3.1, whichshows the residual function E ( t, x, y ) is of order ǫ . To thisend, we define the auxiliary residual function e E ( t, x, y ) by e E = V π (0) ,ǫ − ( v (0) + ǫ / v (1) + ǫv π (0) , (2) + ǫ / v π (0) , (3) ) , where we choose C ( t, x ) = C ( t, x ) ≡ in the expressionof v π (0) , (2) and v π (0) , (3) . Then, it remains to show e E ∼ ǫ .According to the derivation in Section III-A, the auxil-iary residual function e E solves (cid:16) ǫ L + √ ǫ L + L (cid:17) e E + ǫ ( L v π (0) , (3) + L v π (0) , (2) ) + ǫ / L v π (0) , (3) = 0 , witha terminal condition e E ( T, x, y ) = − ǫv π (0) , (2) ( T, x, y ) − ǫ / v π (0) , (3) ( T, x, y ) . Note that ǫ L + √ ǫ L + L is theinfinitesimal generator of the processes (cid:16) X π (0) t , Y t (cid:17) , oneapplies Feynman-Kac formula and deduces: e E ( t, x, y ) = ǫ E ( t,x,y ) (cid:20)Z Tt L v π (0) , (3) ( s, X π (0) s , Y s ) d s (cid:21) + ǫ E ( t,x,y ) (cid:20)Z Tt L v π (0) , (2) ( s, X π (0) s , Y s ) d s (cid:21) + ǫ / E ( t,x,y ) (cid:20)Z Tt L v π (0) , (3) ( s, X π (0) s , Y s ) d s (cid:21) − ǫ E ( t,x,y ) h v π (0) , (2) ( T, X π (0) T , Y T ) i − ǫ / E ( t,x,y ) h v π (0) , (3) ( T, X π (0) T , Y T ) i . (18) The first three expectations come from the source termswhile the last two come from the terminal condition. Weshall prove that each expectation above is uniformly boundedin ǫ . The idea is to relate them to the leading order term v (0) and the risk-tolerance function R ( t, x ; λ ) , where some niceproperties and estimates are already established in Section II.For the source terms, straightforward but tedious compu-tations give: L vπ (0) , (2) = − θ ( y ) (cid:16) λ y ) − λ (cid:17) D v (0) , L vπ (0) , (3) = 12 ρ a ( y ) λ ( y ) θ ′ y ) D v (0)+12 ( T − t ) ρ Ba ( y ) λ ( y ) θ ′ ( y ) D (cid:20) D D (cid:21) D v (0) , L vπ (0) , (3) = 14 ρθ y ) (cid:16) λ y ) − λ (cid:17) D v (0)+ 12 θ ( y ) ρB (cid:26) − (cid:20) D D (cid:21) D v (0) + 12 ( T − t ) (cid:16) λ y ) − λ (cid:17) D v (0) (cid:27) + 14 θ ( y ) ρB ( T − t ) × (cid:20) (cid:16) λ y ) − λ (cid:17) D D v (0) − λ y ) RRxx ( D D D v (0) (cid:21) , where in the computation of L v π (0) , (3) , we use the commu-tator between operators D and L : [ L , D ] w = L D w − D L w = − λ ( y ) R R xx ( Rw xx + w x ) . At terminal time t = T , they become v π (0) , (2) ( T, x, y ) = − θ ( y ) D v (0) ( T, x ) and v π (0) , (3) ( T, x, y ) = ρθ ( y ) BD v (0) ( T, x ) .Note that the quantity RR xx ( t, x ; λ ) is bounded by aconstant K . This is proved for ( t, x ; λ ) ∈ [0 , T ) × R + × R inProposition 2.7, and guaranteed by Assumption 2.3(iii) for t = T , since by definition R ( T, x ; λ ) = R ( x ) . Therefore,the expectations related to the source terms in (18) are sumof terms of the following form: E ( t,x,y ) "Z Tt h ( Y s ) D v (0) ( s, X π (0) s ) d s , (19)where h ( y ) is at most polynomially growing, and D v (0) is one of the following: D v (0) , D v (0) , D v (0) , D D D v (0) , D D v (0) , D D v (0) .Applying Cauchy-Schwartz inequality, it becomes E / t,y ) hR Tt h ( Y s ) d s i E / t,x,y ) (cid:20)R Tt (cid:16) D v (0) ( s, X π (0) s ) (cid:17) d s (cid:21) . The first part is uniformly bounded in ǫ since Y t admitsbounded moments at any order (cf. Assumption 2.4(ii)).It remains to show the second part is also uniformlybounded in ǫ . The proof consists a repeated use of theconcavity of v (0) and the results in Proposition 2.7 andLemma 2.5. For the sake of simplicity, we shall onlydetail the proof when D v (0) = D v (0) and omit the rest.Since (cid:12)(cid:12) D v (0) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) RR x v (0) x − Rv (0) x (cid:12)(cid:12)(cid:12) ≤ ( K + 1) Rv (0) x ≤ ( K + 1) K xv (0) x ≤ K ( K + 1) v (0) , we conclude E ( t,x,y ) (cid:20)Z Tt (cid:16) D v (0) ( s, X π (0) s , Y s ) (cid:17) d s (cid:21) ≤ K ( K + 1) E ( t,x,y ) (cid:20)Z Tt (cid:16) v (0) ( s, X π (0) s ) (cid:17) d s (cid:21) is uniformly bounded in ǫ by Lemma 2.5. Straightforwardbut tedious computations show that the rest terms in (19) arealso bounded by multiples of Rv (0) , then the boundednessis again ensured by the relation R ( t, x ; λ ) ≤ K x , theconcavity of v (0) , and Lemma 2.5.he last two expectations in (18) are treated similarly byusing Assumption 2.3 (15) and the concavity of U ( x ) . There-fore we have shown that (cid:12)(cid:12)(cid:12) e E ( t, x, y ) (cid:12)(cid:12)(cid:12) ≤ e Cǫ . By the inequality | E ( t, x, y ) | ≤ e Cǫ + ǫv π (0) , (2) ( t, x, y )+ ǫ / v π (0) , (3) ( t, x, y ) ≤ Cǫ, we obtain the desired result.IV. T HE A SYMPTOTIC O PTIMALITY OF π (0) We now show that the strategy π (0) defined in (6)asymptotically outperforms every family A ǫ (cid:2)e π , e π , α (cid:3) asprecisely stated in our main Theorem 1.1 in Section I.For a fixed choice of ( e π , e π ) and positive α , recallthe definition of A ǫ (cid:2)e π , e π , α (cid:3) in (5). Working with A ǫ ismotivated by the following. The optimal control to problem(3), whose existence is ensured by [14], clearly depends on ǫ . It is not known whether π ∗ will converge as ǫ goes to zero.But if ǫ had a limit, say e π , it is then natural to consider afamily of controls of the form e π + ǫ α e π as the perturbationof the limit e π . We think the subset A ǫ is not so smallcomparing to the full one A ǫ , as we only restrict α > ,which allows for correction of any order in ǫ . Assumption 4.1:
For the triplet ( e π , e π , α ) , we require:(i) The whole family of strategies { e π + ǫ α e π } ǫ ≤ ∈ A ǫ ;(ii) Let ( e X t,xs ) t ≤ s ≤ T be the solution to: d e X s = D µ ( · ) e π ( s, e X s , · ) E d s + rD σ ( · ) e π ( s, e X s , · ) E d W s , start-ing at x at time t . By (i), e X t,xs ≥ . We further assumethat it has full support R + for any t < s ≤ T . Remark 4.2:
Part (ii) is motivated as follows. Con-sider d b X s = h µπ (0) i d s + h σ π (0)2 i d W s . Noticing that (cid:10) µ ( · ) π (0) ( t, x, · ) (cid:11) = λ R ( t, x ; λ ) , q(cid:10) σ ( · ) π (0) ( t, x, · ) (cid:11) = λR ( t, x ; λ ) , then b X s can be interpreted as the optimal wealthprocess of the classical Merton problem with averagedSharpe-ratio λ . From [12, Proposition 7], one has b X t,xs = H (cid:16) H − ( x, t, λ ) + λ ( s − t ) + λ ( W s − W t ) , s, λ (cid:17) , where H : R × [0 , T ] × R → R + solves the heat equation H t + λ H xx = 0 , and is of full range in x . Consequently, b X t,xs has full support R + , and thus, it is natural to requirethat e X t,xs has full support R + .Denote by e V ǫ the value function associated to the tradingstrategy π := e π + ǫ α e π ∈ A ǫ (cid:2)e π , e π , α (cid:3) : e V ǫ ( t, x, y ) = E [ U ( X πT ) | X πt = x, Y t = y ] , (20)where X πt is the wealth process following the strategy π ∈A ǫ , and Y t is fast mean-reverting with the same ǫ . The ideais to compare e V ǫ with V π (0) ,ǫ defined in (16), for which arigorous first order approximation v (0) + √ ǫv (1) has beenestablished in Theorem 3.1. After finding the expansion of e V ǫ , the comparison is done asymptotically in ǫ up to O ( √ ǫ ) . Approximations of the Value Function e V ǫ . Denote by L theinfinitesimal generator of the state processes ( X πt , Y t ) : L := ǫ L + σ ( y ) (cid:0)e π + ǫ α e π (cid:1) ∂ xx + (cid:0)e π + ǫ α e π (cid:1) µ ( y ) ∂ x + √ ǫ ρa ( y ) σ ( y ) (cid:0)e π + ǫ α e π (cid:1) ∂ xy , then by the martingale prop-erty, the value function e V ǫ defined in (20) satisfies ∂ t e V ǫ + L e V ǫ = 0 , e V ǫ ( T, x, y ) = U ( x ) . (21) Motivate by the fact that the first order in the operator L is ǫ α , we propose the following expansion form for e V ǫ e V ǫ = e v (0) + ǫ α e v α + ǫ α e v α + · · · + ǫ nα e v nα + √ ǫ e v (1) + · · · , where n is the largest integer such that nα < / , and for the case α > / , n is simply zero. In the derivation, we aim atidentifying the zeroth order term e v (0) and the first non-zeroterm up to O ( √ ǫ ) . Apparently, the term following e v (0) willdepend on the value of α .To further simplify the notation, we decompose ∂ t + L according to different powers of ǫ as follows: ∂ t + L = ǫ L + √ ǫ e L + e L + ǫ α e L + ǫ α e L + ǫ α − / e L , wherethe operators e L i are defined by: e L = e π ρ a ( y ) σ ( y ) ∂ xy , e L = ∂ t + σ ( y ) (cid:0)e π (cid:1) ∂ xx + e π µ ( y ) ∂ x , e L = σ ( y ) e π e π ∂ xx + e π µ ( y ) ∂ x , e L = σ ( y ) (cid:0)e π (cid:1) ∂ xx and e L = e π ρ a ( y ) σ ( y ) ∂ xy .In all cases, we first collect terms of O ( ǫ β ) in (21) with β ∈ [ − , . Noticing that L and e L (also e L when α < / ) take derivatives in y , we are able to make the choicethat the approximation of e V ǫ up to O ( ǫ β ′ ) is independent of y , for β ′ < . In the following derivation, this choice is madefor every case, and consequently, we will not mention thisagain and will start the argument by collecting terms of O (1) .Different order of approximations are obtained depending on e π being identical to π (0) or not.
1) Case e π ≡ π (0) : We first analyze the case e π ≡ π (0) ,in which e L and e L coincide with L and L , and e L v (0) =0 . The terms of O (1) form a Poisson equation for e v (2) L e v (2) + L e v (0) = 0 , e v (0) ( T, x ) = U ( x ) . For different values of α , there might be extra terms whichare eventually zero, thus are not included in the aboveequation: L e v (1) (all cases), e L e v (0) when α = 1 / , and e L e v kα when ( k + 1) α = 1 / . By the solvability condition, e v (0) solves (9), which possesses a unique solution v (0) .Therefore, we deduce e v (0) ≡ v (0) , and e v (2) ≡ v π (0) , (2) . (i) α = 1 / . We then collect terms of O ( ǫ / ) : L e v (3) + L e v (1) + L e v (2) + e L e v (0) + e L e v (1) = 0 . This is a Poisson equation for e v (3) , for which thesolvability condition gives: e v (1) satisfies (14). Here wehave used e L e v (0) = e L v (0) = 0 , e v (2) = v π (0) , (2) and e L e v (1) = 0 . This equation is uniquely solved, onededuces e v (1) = v (1) , and e v (3) ≡ v π (0) , (3) . (ii) α > / . Collecting terms of O ( √ ǫ ) yields a Poissonequation for e v (3) , L e v (3) + L e v (1) + L e v (2) + e L e v (0) =0 , where the term e L e v (0) only exists when α = 1 (but anyway L e v (1) and e L e v (0) disappear due to theirindependence of y). Arguments similar to the case α = 1 / give that e v (1) = v (1) , and e v (3) = v π (0) , (3) . (iii) α < / . The next order is ǫ α , L e v α +1 + L e v α + e L e v (0) + L e v α +1 / + e L e v (1) = 0 . Again the last three terms disappear due to the fact e L e v (0) = e L v (0) = 0 , and e v α +1 / and e v (1) ’s indepen-dence of y . Then using solvability condition, e v α solves t,x ( λ ) e v α ( t, x ; λ ) = 0 , e v α ( T, x ) = 0 , which onlyhas the trivial solution e v α ≡ . Consequently, we needto identify the next non-zero term. / < α < / / < α < / / < α < / . The next order is √ ǫ , which gives L e v (3) + L e v (1) + L e v (2) = 0 . It coincides with (14)after using the solvability condition, and we deduce e v (1) ≡ v (1) and e v (3) ≡ v π (0) , (3) . α = 1 / α = 1 / α = 1 / . The next order is √ ǫ , and the Pois-son equation for e v (3) becomes L e v (3) + L e v (1) + L e v (2) + e L e v (0) = 0 . The solvability condition reads L t,x ( λ ) e v (1) − ρBD v (0) − λ D v (0) = 0 . Compar-ing this equation with (14) and using the concavity of v (0) , one deduces e v (1) ≤ v (1) . α < / α < / α < / . The next order is ǫ α since α < / , and L e v α +1 + L e v α + L e v α +1 / + e L e v α + e L e v (0) + e L e v α +1 / = 0 , e v α ( T, x ) = 0 . The third, fourthand sixth terms varnish since e v α ≡ , and e v α +1 / and e v α +1 / are independent of y . One has e v αt + λ R e v αxx + λR e v αx + D σ ( · ) (cid:0)e π ( t, x, · ) (cid:1) E v (0) xx = 0 ,by the solvablility condition. Assuming that e π is notidentically zero, we claim e v α < .
2) Case e π π (0) : In this case, after collect-ing terms of O (1) , and using the solvability condi-tion, one has the following PDE for e v (0) : e v (0) t + (cid:10) σ ( · ) e π ( t, x, · ) (cid:11) e v (0) xx + (cid:10)e π ( t, x, · ) µ ( · ) (cid:11) e v (0) x = 0 . Tocompare e v (0) to v (0) , we rewrite (11) in the same pat-tern: v (0) t + (cid:10) σ ( · ) e π ( t, x, · ) (cid:11) v (0) xx + (cid:10)e π ( t, x, · ) µ ( · ) (cid:11) v (0) x − D σ ( · ) (cid:0)e π − π (0) (cid:1) ( t, x, · ) E v (0) xx = 0 , via the relation − (cid:10) σ ( y )( e π − π (0) ) π (0) (cid:11) v (0) xx = (cid:10) ( e π − π (0) ) µ ( y ) (cid:11) v (0) x . Again by the strict concavity of v (0) and Feynman–Kacformula, we obtain e v (0) < v (0) .To fully justify the above expansions, additional assump-tions similar to [6, Appendix C] are needed. They aretechnical uniform (in ǫ ) integrability conditions on the strate-gies A ǫ [ e π , e π , α ] . For the sake of simplicity, we omit theconditions here and refer to [6, Appendix C] for furtherdetails. Now we summarize the above derivation as follows. Proposition 4.3:
Summary of the accuracy results:
TABLE IA
CCURAY OF APPROXIMATIONS OF e V ǫ .Case Value of α Approximation Accuracy α ≥ / v (0) + √ ǫv (1) O ( ǫ ) e π ≡ π (0) / < α < / O ( ǫ α ) α = 1 / v (0) + √ ǫ e v (1) O ( ǫ / ) α < / v (0) + ǫ α e v α O ( ǫ α ∧ (1 / ) e π π (0) all e v (0) O ( ǫ α ∧ (1 / ) where the accuracy column gives the order of the differencebetween e V ǫ and its approximation. Moreover, when e π ≡ π (0) , we have the relation e v (1) ≤ v (1) if α = 1 / , and e v α < if α < / ; while if e π π (0) , then e v (0) < v (0) . Asymptotic Optimality: Proof of Theorem 1.1.
We nowgive the proof of Theorem 1.1, via comparing the firstorder approximation v (0) + √ ǫv (1) of V π (0) ,ǫ obtained inTheorem 3.1, and the one of e V ǫ summarized in Tab. I. In the case that the approximation of e V ǫ is v (0) + √ ǫv (1) ,the limit is easily verified to be zero. When the approxi-mation of e V ǫ is v (0) + √ ǫ e v (1) , the limit ℓ is non-positivebut stay finite, by the fact e v (1) ≤ v (1) . If e π ≡ π (0) and α < / , the limit ℓ is computed as ℓ = lim ǫ → (cid:0) ǫ α e v α −√ ǫv (1) + O ( ǫ α ∧ / ) (cid:1) / √ ǫ = −∞ , since e v α < . Thesimilar arguments also apply to the case e π π (0) , and leadto ℓ = −∞ . Thus we complete the proof. In fact, this limitcan be understood according to the following four cases:(i) e π ≡ π (0) and ℓ = 0 : e V ǫ = V π (0) ,ǫ + o ( √ ǫ ) ;(ii) e π ≡ π (0) and −∞ < ℓ < : e V ǫ = V π (0) ,ǫ + O ( √ ǫ ) with O ( √ ǫ ) < ;(iii) e π ≡ π (0) and ℓ = −∞ : e V ǫ = V π (0) ,ǫ + O ( ǫ α ) with O ( ǫ α ) < and α < / ;(iv) e π π (0) : lim ǫ → e V ǫ ( t, x, z ) < lim ǫ → V π (0) ,ǫ ( t, x, z ) . R EFERENCES[1] M. K. Brunnermeier and S. Nagel. Do wealth fluctuations generatetime-varying risk aversion? micro-evidence on individuals asset allo-cation.
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