Markowitz portfolio selection for multivariate affine and quadratic Volterra models
MMarkowitz portfolio selection for multivariate affine andquadratic Volterra models
Eduardo ABI JABER ∗ Enzo MILLER † Huyˆen PHAM ‡ June 25, 2020
Abstract
This paper concerns portfolio selection with multiple assets under rough covariancematrix. We investigate the continuous-time Markowitz mean-variance problem for amultivariate class of affine and quadratic Volterra models. In this incomplete non-Markovian and non-semimartingale market framework with unbounded random coe-fficients, the optimal portfolio strategy is expressed by means of a Riccati backwardstochastic differential equation (BSDE). In the case of affine Volterra models, we deriveexplicit solutions to this BSDE in terms of multi-dimensional Riccati-Volterra equa-tions. This framework includes multivariate rough Heston models and extends theresults of Han and Wong (2020a). In the quadratic case, we obtain new analytic for-mulae for the the Riccati BSDE and we establish their link with infinite dimensionalRiccati equations. This covers rough Stein-Stein and Wishart type covariance models.Numerical results on a two dimensional rough Stein-Stein model illustrate the impactof rough volatilities and stochastic correlations on the optimal Markowitz strategy. Inparticular for positively correlated assets, we find that the optimal strategy in ourmodel is a ‘buy rough sell smooth’ one.
Keywords:
Mean-variance portfolio theory; rough volatility; correlation matrices; multi-dimensional Volterra process; Riccati equations; non-Markovian Heston, Stein–Stein andWishart models.
MSC Classification: ∗ Universit´e Paris 1 Panth´eon-Sorbonne, Centre d’Economie de la Sorbonne, 106, Boulevard de l’Hˆopital,75013 Paris, eduardo.abi-jaber at univ-paris1.fr. † Universit´e de Paris and Sorbonne Universit´e, Laboratoire de Probabilit´es, Statistique et Mod´elisation(LPSM, UMR CNRS 8001), Building Sophie Germain, Avenue de France, 75013 Paris, enzo.miller at poly-technique.org ‡ Universit´e de Paris and Sorbonne Universit´e, Laboratoire de Probabilit´es, Statistique et Mod´elisation(LPSM, UMR CNRS 8001), Building Sophie Germain, Avenue de France, 75013 Paris, pham at lpsm.paris a r X i v : . [ m a t h . O C ] J un Introduction
The Markowitz (1952) mean-variance portfolio selection problem is the cornerstone of mo-dern portfolio allocation theory. Investment decisions rules are made according to a trade-off between return and risk, and the use of Markowitz efficient portfolio strategies in thefinancial industry has become quite popular mainly due to its natural and intuitive for-mulation. A vast volume of research has been devoted over the last decades to extendMarkowitz problem from static to continuous-time setting, first in Black-Scholes and com-plete markets (Zhou and Li (2000)), and then to consider more general frameworks withrandom coefficients and multiple assets, see e.g. Lim (2004), Chiu and Wong (2014), ormore recently Ismail and Pham (2019) for taking into account model uncertainty on theassets correlation.In the direction of more realistic modeling of asset prices, it is now well-established sincethe seminal paper by Gatheral et al. (2018) that volatility is rough, modeled by fractionalBrownian motion with small Hurst parameter, which captures empirical facts of timesseries of realized volatility and key features of implied volatility surface. Subsequently,an important literature has focused on option pricing and asymptotics in rough volatilitymodels. In comparison, the research on portfolio optimization in fractional and roughmodels is still little developed but has gained an increasing attention with the recentpapers of Fouque and Hu (2018); B¨auerle and Desmettre (2020); Han and Wong (2020b),which consider fractional Ornstein-Uhlenbeck and Heston stochastic volatility models forpower utility function criterion, and the work by Han and Wong (2020a) where the authorsstudy the Markowitz problem in a Volterra Heston model, which covers the rough Hestonmodel of El Euch and Rosenbaum (2018).Most of the developments in rough volatility literature for asset modeling, option pricingor portfolio selection have been carried out in the mono-asset case. However, investment inmulti-assets by taking into account the correlation risk is an importance feature in portfoliochoice in financial markets, see Buraschi et al. (2010). Inspired by the recent papersAbi Jaber (2019b); Abi Jaber et al. (2019); Cuchiero and Teichmann (2019); Rosenbaumand Thomas (2019) that consider multivariate versions of rough Volterra volatility models,the basic goal of this paper is to enrich the literature on portfolio selection:(i) by introducing a class of multivariate Volterra models, which captures stylized factsof financial assets, namely various rough volatility patterns across assets, (possiblyrandom) correlation between stocks, and leverage effects, i.e., correlation between astock and its volatility.(ii) by keeping the model tractable for explicit computations of the optimal Markowitzportfolio strategy, which can be a quite challenging task in multivariate non-Markoviansettings.
Main contributions.
In this paper, we study the continuous-time Markowitz problem ina multivariate setting with a focus on two classes: (i) affine Volterra models as in Abi Jaber2t al. (2019) that include multivariate rough Heston models, (ii) quadratic Volterra models,which are new class of Volterra models, and embrace multivariate rough Stein-Stein models,and rough Wishart type covariance matrix models, in the spirit of Abi Jaber (2019b);Cuchiero and Teichmann (2019). We provide: • A generic verification result for the corresponding mean-variance problem, whichis formulated in an incomplete non-Markovian and non-semimartingale frameworkwith unbounded random coefficients of the volatility and market price of risk, andunder general filtration. This result expresses the solution to the Markowitz problemin terms of a Riccati backward stochastic differential equation (BSDE) by checking inparticular the admissibility condition of the optimal control. We stress that relatedexisting verification results in the literature (see Lim (2004), Jeanblanc et al. (2012),Chiu and Wong (2014), Shen (2015)) cannot be applied directly to our setting, andwe shall discuss more in detail this point in Section 3. • Explicit solutions to the Riccati BSDE in two concrete specifications of multivariateVolterra models exploiting the representation of the solution in terms of a Laplacetransform:(i) the affine case : the optimal Markowitz strategy is expressed in terms of mul-tivariate Riccati-Volterra equations which naturally extends the one obtained inHan and Wong (2020a). We point out that the martingale distorsion argumentsused in Han and Wong (2020a) for the univariate Volterra Heston model, do notapply in higher dimensions, unless the correlation structure is highly degenerate.(ii) the quadratic case: our major result is to derive analytic expressions for theoptimal investment strategy by explicitly solving operator Riccati equations.This gives new explicit formulae for rough Stein-Stein and Wishart type co-variance models. These analytic expressions can be efficiently implemented: theintegral operators can be approximated by closed form expressions involvingfinite dimensional matrices and the underlying processes can be simulated bythe celebrated Cholesky decomposition algorithm. • Numerical simulations of the optimal Markowitz strategy in a two-asset roughStein-Stein model to illustrate our results. We depict the impact of some parametersonto the optimal investment when one asset is rough, and the other smooth (inthe sense of the Hurst index of their volatility), and show in particular that forpositively correlated assets, the optimal strategy is to “buy rough, sell smooth”,which is consistent with the empirical backtesting in Glasserman and He (2020).
Outline of the paper.
The rest of the paper is organized as follows: Section 2 formulatesthe financial market model and the mean-variance problem in a multivariate setting with The code of our implementation can be found at the following link.
Notations.
Given a probability space (Ω , F , P ) and a filtration F = ( F t ) t ≥ satisfying theusual conditions, we denote by L ∞ F ([0 , T ] , R d ) = (cid:110) Y : Ω × [0 , T ] (cid:55)→ R d , F − prog. measurable and bounded a.s. (cid:111) L p F ([0 , T ] , R d ) = (cid:26) Y : Ω × [0 , T ] (cid:55)→ R d , F − prog. measurable s.t. E (cid:104) (cid:90) T | Y s | p ds (cid:105) < ∞ (cid:27) S ∞ F ([0 , T ] , R d ) = (cid:40) Y : Ω × [0 , T ] (cid:55)→ R d , F − prog. measurable s.t. sup t ≤ T | Y t ( w ) | < ∞ a.s. (cid:41) . Here | · | denotes the Euclidian norm on R d . Classically, for p ∈ (cid:74) , ∞ (cid:75) , we define L p,loc F ([0 , T ] , R d ) as the set of progressive processes Y for which there exists a sequence ofincreasing stopping times τ n ↑ ∞ such that the stopped processes Y τ n are in L p F ([0 , T ] , R d )for every n ≥
1, and we recall that it consists of all progressive processes Y s.t. (cid:82) T | Y t | p dt< ∞ , a.s. To unclutter notation, we write L p,loc F ([0 , T ]) instead of L p,loc F ([0 , T ] , R d ) whenthe context is clear. Fix
T > d, N ∈ N . We consider a financial market on [0 , T ] on some filtered probabilityspace (Ω , F , F := ( F t ) t ≥ , P ) with a non–risky asset S dS t = S t r ( t ) dt, with a deterministic short rate r : R + → R , and d risky assets with dynamics dS t = diag( S t ) (cid:2)(cid:0) r ( t ) d + σ t λ t (cid:1) dt + σ t dB t (cid:3) , (2.1)driven by a d -dimensional Brownian motion B , with a d × d -matrix valued stochasticvolatility process σ and a R d -valued continuous stochastic process λ , called market priceof risk . Here d denotes the vector in R d with all components equal to 1. The market is4ypically incomplete, in the sense that the dynamics of the continuous volatility process σ is driven by an N -dimensional process W = ( W , . . . , W N ) (cid:62) defined by: W kt = C (cid:62) k B t + (cid:113) − C (cid:62) k C k B ⊥ ,kt , k = 1 , . . . , N, (2.2)where C k ∈ R d s.t. C (cid:62) k C k ≤
1, and B ⊥ = ( B ⊥ , , . . . , B ⊥ ,N ) (cid:62) is an N –dimensional Brownianmotion independent of B . Note that d (cid:104) W k (cid:105) t = dt but W k and W j can be correlated, hence W is not necessarily a Brownian motion. Observe that processes λ and σ are F -adapted,possibly unbounded, but not necessarily adapted to the filtration generated by W . Wepoint out that F may be strictly larger than the augmented filtration generated by B and B ⊥ as we shall deal with weak solutions to stochastic Volterra equations. Remark 2.1.
In our applications, we will be chiefly interested in the case where λ t is linearin σ t , and where the dynamics of the matrix-valued process σ is governed by a Volterraequation of the form σ t = g ( t ) + (cid:90) t µ ( t, s, ω ) ds + (cid:90) t χ ( t, s, ω ) dW s . (2.3) The class of models that we shall develop in Sections 4 and 5 includes in particular the caseof Volterra Heston model when d = 1 with λ t = θσ t , for some constant θ , as studied inHan and Wong (2020a), and the case of Wishart process for the covariance matrix process V t = σ t σ (cid:62) t , as studied in Chiu and Wong (2014). The class of models that we will developin Sections 4 and 5 includes in particular the case of (i) multivariate Volterra Heston models based on Volterra square-root processes, seeAbi Jaber et al. (2019, Section 6), we refer to Rosenbaum and Thomas (2019) fora microstuctural foundation. When d = 1 , we recover the results of Han and Wong(2020a), which cover the case of the rough Heston model of El Euch and Rosenbaum(2019). (ii) multivariate Volterra Stein-Stein and Wishart type in the sense of Abi Jaber (2019b),where the instantenous covariance is given by squares of Gaussians. Under theMarkovian setting, we recover a similar structure as in the results of Chiu and Wong(2014). Mean-variance optimization problem.
Let π t denote the vector of the amounts in-vested in the risky assets S at time t in a self–financing strategy and set α = σ (cid:62) π . Then,the dynamics of the wealth X α of the portfolio we seek to optimize is given by dX αt = (cid:0) r ( t ) X αt + α (cid:62) t λ t (cid:1) dt + α (cid:62) t dB t , t ≥ , X α = x ∈ R . (2.4)5y a solution to (2.4), we mean an F -adapted continuous process X α satisfying (2.4) on[0 , T ] P -a.s. and such that E (cid:2) sup t ≤ T | X αt | (cid:3) < ∞ . (2.5)The set of admissible investment strategies is naturally defined by A = { α ∈ L ,loc F ([0 , T ] , R d ) such that (2.4) has a solution satisfying (2.5) } . The Markowitz portfolio selection problem in continuous-time consists in solving thefollowing constrained problem V ( m ) := inf α ∈A (cid:8) Var( X T ) : s.t. E [ X T ] = m (cid:9) . (2.6)given some expected return value m ∈ R , where Var( X T ) = E (cid:2)(cid:0) X T − E [ X T ] (cid:1) (cid:3) stands forthe variance. In this section, we establish a generic verification result for the optimization problem (2.6)given the solution of a certain Riccati BSDE. We stress that our mean-variance prob-lem deals with incomplete markets with unbounded random coefficients σ and λ , so thatexisting results cannot be applied directly to our setting: Lim (2004) presents a generalmethodology to solve the MV problem for the wealth process (2.4) in an incomplete marketwithout assuming any particular dynamics on σ nor that the excess return is proportionalto σ . However, a nondegeneracy assumption is made on σσ (cid:62) , see Lim (2004, Assumption(A.1)). The main verification result in Lim (2004, Proposition 3.3), based on a completionof squares argument, states that if a solution to a certain (nonlinear) Riccati BSDE exists,then the MV is solvable. The difficulty resides in proving the existence of solutions to suchnonlinear BSDEs (see also Lim and Zhou (2002) for similar results in complete markets).Here, we assume that the excess return is proportional to σ (instead of the nonde-generacy condition) and state a verification result in terms of solutions of Riccati BSDEs(completion of squares, ie LQ problem with random coefficients). A verification resultdepending on the solution of a Riccati BSDE is also stated in Chiu and Wong (2014),but the admissibility of the optimal candidate control is not proved. We also mention thepaper of Jeanblanc et al. (2012) where the authors adopt a BSDE approach for generalsemimartingales, but focusing on situations in which the existence of an optimal strategyis assumed. In our case, the existence of an admissible optimal control is obtained undera suitable exponential integrability assumption involving the market price of risk and the Z components of the BSDE, which extends the condition in Shen (2015).6ur main result of this section, Theorem 3.1 below, can be seen as unifying frameworkfor the aforementioned results, refer to Table 1. For the sake of presentation, we postponeits proof to Appendix A. Random coef. Unbounded coef. degenerate σ Incomplete marketLim and Zhou (2002) (cid:51) (cid:55) (cid:55) (cid:55)
Lim (2004) (cid:51) (cid:55) (cid:55) (cid:51)
Shen (2015) (cid:51) (cid:51) (cid:55) (cid:55)
Theorem 3.1 (cid:51) (cid:51) (cid:51) (cid:51)
Table 1: Comparison to existing verification results for mean-variance problems.We define C ∈ R N × d by C = ( C , . . . , C N ) (cid:62) , (3.1)where we recall that the vectors C i ∈ R d come from the correlation structure (2.2). Wewill use the matrix norm | A | = tr( A (cid:62) A ) in the subsequent theorem. Theorem 3.1.
Assume that there exists a solution triplet (Γ , Z , Z ) ∈ S ∞ F ([0 , T ] , R ) × L ,loc F ([0 , T ] , R d ) × L ,loc F ([0 , T ] , R N ) to the Riccati BSDE (cid:40) d Γ t = Γ t (cid:104)(cid:0) − r ( t ) + (cid:12)(cid:12) λ t + Z t + CZ t (cid:12)(cid:12) (cid:1) dt + (cid:0) Z t (cid:1) (cid:62) dB t + (cid:0) Z t (cid:1) (cid:62) dW t (cid:105) , Γ T = 1 , (3.2) such that(H1) < Γ < e (cid:82) T r ( s ) ds , and Γ t > , for all t ≤ T ,(H2) E (cid:104) exp (cid:16) a ( p ) (cid:90) T (cid:0) | λ s | + (cid:12)(cid:12) Z s (cid:12)(cid:12) + (cid:12)(cid:12) Z s (cid:12)(cid:12) (cid:1) ds (cid:17)(cid:105) < ∞ , (3.3) for some p > and a constant a ( p ) given by a ( p ) = max (cid:104) p (3 + | C | ) , (8 p − p ) (cid:0) | C | + | C | (cid:1) (cid:105) . (3.4) Then, the optimal investment strategy for the Markowitz problem (2.6) is given by theadmissible control α ∗ t = − (cid:0) λ t + Z t + CZ t (cid:1)(cid:0) X α ∗ t − ξ ∗ e − (cid:82) Tt r ( s ) ds (cid:1) , (3.5)7 here ξ ∗ = m − Γ e − (cid:82) T r ( t ) dt x − Γ e − (cid:82) T r ( t ) dt . (3.6) Furthermore, the value of (2.6) for the optimal wealth process X ∗ = X α ∗ is V ( m ) = Var( X ∗ T ) = Γ (cid:12)(cid:12) x − me − (cid:82) T r ( t ) dt (cid:12)(cid:12) − Γ e − (cid:82) T r ( t ) dt . (3.7) Proof.
We refer to Appendix A.
Remark 3.2.
By setting ˜ Z it = Γ t Z it , i = 1 , , the BSDE (3.2) agrees with the one in Chiuand Wong (2014, Theorem 3.1): d Γ t = Γ t (cid:104)(cid:16) − r ( t ) + (cid:12)(cid:12) λ t + ˜ Z t + C ˜ Z t Γ t (cid:12)(cid:12) (cid:17)(cid:105) dt + (cid:0) ˜ Z t (cid:1) (cid:62) dB t + (cid:0) ˜ Z t (cid:1) (cid:62) dW t , and justifies the terminology Riccati BSDE. In the sequel, we will provide concrete specifications of multivariate stochastic Volterramodels for which the solution to the non-linear Riccati BSDE (3.2) can be computed inclosed and semi-closed forms, while satisfying conditions (H1) and (H2). The key idea isto observe that, first, if such solution exists, then, it admits the following representationas a Laplace transform:Γ t = E (cid:104) exp (cid:16) (cid:90) Tt (cid:0) r ( s ) − (cid:12)(cid:12) λ s + Z s + CZ s (cid:12)(cid:12) (cid:1) ds (cid:17) (cid:12)(cid:12)(cid:12) F t (cid:105) , ≤ t ≤ T. In the special case where λ is deterministic, then the solution to (3.2) trivially exists with Z = Z = 0, and condition (H1) and (H2) are obviously satisfied when λ is nonzeroand bounded. In the general case where λ is an (unbounded) stochastic process, theadmissibility of the optimal control is obtained under finiteness of a certain exponentialmoment of the solution triplet (Γ , Z , Z ) and the risk premium λ as precised in (H2). Suchestimate is crucial to deal with the unbounded random coefficients in (2.4), see for instanceHan and Wong (2020a); Shen et al. (2014); Shen (2015) where similar conditions appear.If the coefficients are bounded, such condition is not needed, see Lim (2004, Lemma 3.1).Our main interest is to find specific dynamics for the volatility σ and for the marketprice of risk λ such that the Laplace transform can be computed in (semi)-explicit form.We shall consider models as mentioned in Remark 2.1, where all the randomness in λ comesfrom the process W driving σ , and for which we naturally expect that Z = 0. We solvemore specifically this problem for two classes of models:8i) Multivariate affine Volterra models of Heston type in Section 4. This extends theresults of Han and Wong (2020a) to the multi dimensional case and provides semi-closed formulas.(ii) Multivariate quadratic Volterra models of Stein-Stein and Wishart type in Section 5for which we derive new closed-form solutions. We let K = diag( K , . . . , K d ) be diagonal with scalar kernels K i ∈ L ([0 , T ] , R ) on thediagonal, ν = diag( ν , . . . , ν d ) and D ∈ R d × d such that D ij ≥ , i (cid:54) = j. We assume that σ in (2.3) is given by σ = (cid:112) diag( V ), where V = ( V , . . . , V d ) (cid:62) is thefollowing R d + –valued Volterra square–root process V t = g ( t ) + (cid:90) t K ( t − s ) DV s ds + (cid:90) t K ( t − s ) ν (cid:112) diag( V s ) dW s . (4.1)Here g : R + → R d + , W is a d -dimensional Brownian motion and the correlation structurewith B is given by W i = ρ i B i + (cid:113) − ρ i B ⊥ ,i , i = 1 , . . . , d, (4.2)for some ( ρ , . . . , ρ d ) ∈ [ − , d . This corresponds to a particular case of the correlationstructure in (2.2) with N = d , and C i = (0 , . . . , ρ i , . . . , (cid:62) . Furthermore, the risk premiumis assumed to be in the form λ = (cid:0) θ √ V , . . . , θ d √ V d (cid:1) (cid:62) , for some θ i ≥
0, so that thedynamics for the stock prices (2.1) reads dS it = S it (cid:0) r ( t ) + θ i V it (cid:1) dt + S it (cid:113) V it dB it , i = 1 , . . . , d. (4.3)We assume that there exists a continuous R d + -valued weak solution ( V, S ) to (4.1)-(4.3)on some filtered probability space (Ω , F , ( F ) t ≥ , P ) such thatsup t ≤ T E [ | V t | p ] < ∞ , p ≥ . (4.4)For instance, weak existence of V such that (4.4) holds is established under suitable assump-tions on the kernel K and specifications g as shown in the following remark. The existenceof S readily follows from that of V . 9 emark 4.1. Assume that, for each i = 1 , . . . , d , K i is completely monotone on (0 , ∞ ) ,andthat there exists γ i ∈ (0 , and k i > such that (cid:90) h K i ( t ) dt + (cid:90) T ( K i ( t + h ) − K i ( t )) dt ≤ k i h γ i , h > . (4.5) This covers, for instance, constant non-negative kernels, fractional kernels of the form t H − / / Γ( H + 1 / with H ∈ (0 , ] , and exponentially decaying kernels e − βt with β > .Moreover, sums and products of completely monotone functions are completely monotone,refer to Abi Jaber et al. (2019) for more details. • If g ( t ) = V + (cid:82) t K ( t − s ) b ds , for some V , b ∈ R d + , then Abi Jaber et al. (2019,Theorem 6.1) ensures the existence of V such that (4.4) holds, • In Abi Jaber and El Euch (2019a), the existence is obtained for more general in-put curves g for the case d = 1 , the extension to the multi-dimensional setting isstraightforward. Exploiting the affine structure of (4.1)-(4.3), see Abi Jaber et al. (2019), we provide anexplicit solution to the Riccati BSDE (3.2) in terms of the Riccati-Volterra equation ψ i ( t ) = (cid:90) t K i ( t − s ) F i ( ψ ( s )) ds, (4.6) F i ( ψ ) = − θ i − θ i ρ i ν i ψ i + ( D (cid:62) ψ ) i + ν i − ρ i )( ψ i ) , i = 1 , . . . , d, (4.7)and the R d -valued process g t ( s ) = g ( s ) + (cid:90) t K ( s − u ) DV u du + (cid:90) t K ( s − u ) ν (cid:112) diag( V u ) dW u , s ≥ t. (4.8)One notes that for each, s ≤ T , ( g t ( s )) t ≤ s is the adjusted forward process g t ( s ) = E (cid:104) V s − (cid:90) st K ( s − u ) DV u du (cid:12)(cid:12)(cid:12) F t (cid:105) . Lemma 4.2.
Assume that there exists a solution ψ ∈ C ([0 , T ] , R d ) to the Riccati-Volterraequation (4.6) - (4.7) . Let (cid:0) Γ , Z , Z (cid:1) be defined as Γ t = exp (cid:16) (cid:82) Tt r ( s ) ds + (cid:80) di =1 (cid:82) Tt F i ( ψ ( T − s )) g it ( s ) ds (cid:17) ,Z t = 0 ,Z ,it = ψ i ( T − t ) ν i (cid:112) V it , i = 1 , . . . , d, ≤ t ≤ T, (4.9) A function f is completely monotone on (0 , ∞ ) if it is infinitely differentiable on (0 , ∞ ) such that( − n f n ( t ) ≥
0, for all n ≥ t > here g = ( g , . . . , g d ) (cid:62) is given by (4.8) . Then, (cid:0) Γ , Z , Z (cid:1) is a S ∞ F ([0 , T ] , R ) × L F ([0 , T ] , R d ) × L F ([0 , T ] , R d ) -valued solution to (3.2) .Proof. We first observe that the correlation structure (4.2) implies that C in (3.1) is givenby C = diag( ρ , . . . , ρ d ). Set G t = 2 (cid:90) Tt r ( s ) ds + d (cid:88) i =1 (cid:90) Tt F i ( ψ ( T − s )) g it ( s ) ds, t ≤ T. Then, Γ = exp( G ) and d Γ t = Γ t (cid:16) dG t + 12 d (cid:104) G (cid:105) t (cid:17) . (4.10)Using (4.8), and by stochastic Fubini’s theorem, see Veraar (2012, Theorem 2.2), thedynamics of G reads as dG t = (cid:16) − r ( t ) − d (cid:88) i =1 F i ( ψ ( T − t )) V it + d (cid:88) j =1 (cid:90) Tt F j ( ψ ( T − s )) K j ( s − t ) ds d (cid:88) i =1 D ji V it (cid:17) dt + d (cid:88) i =1 (cid:90) Tt F i ( ψ ( T − s )) K i ( s − t ) dsν i (cid:113) V it dW it = (cid:16) − r ( t ) − d (cid:88) i =1 F i ( ψ ( T − t )) V it + d (cid:88) j =1 ψ j ( T − t ) d (cid:88) i =1 D ji V it (cid:17) dt + d (cid:88) i =1 ψ i ( T − t ) ν i (cid:113) V it dW it , where we changed variables and used the Riccati–Volterra equation (4.6) for ψ for the lastequality. This yields that the dynamics of Γ in (4.10) is given by d Γ t = Γ t (cid:16) − r ( t ) + d (cid:88) i =1 V it (cid:0) − F i ( ψ ( T − t )) + d (cid:88) j =1 D ji ψ j ( T − t ) + ν i ψ i ( T − t )) (cid:1)(cid:17) dt + Γ t d (cid:88) i =1 ψ i ( T − t ) ν i (cid:113) V it dW it = Γ t (cid:104)(cid:0) − r ( t ) + d (cid:88) i =1 V it ( θ i + ρ i ν i ψ i ( T − t )) (cid:1) dt + ( Z t ) (cid:62) dW t (cid:105) , (4.11)where we used (4.7) for the last identity. Finally, observing that (cid:12)(cid:12) λ t + Z t + CZ t (cid:12)(cid:12) = d (cid:88) i =1 (cid:0) θ i + ρ i ν i ψ i ( T − t ) (cid:1) V it , T = 1, we get that (Γ , Z , Z ) as defined in (4.9) solves the BSDE (3.2).It remains to show that (cid:0) Γ , Z , Z (cid:1) ∈ S ∞ F ([0 , T ] , R ) × L F ([0 , T ] , R d ) × L F ([0 , T ] , R d ). Forthis, define the process M t = Γ t exp (cid:16) (cid:90) Tt (cid:0) − r ( s ) + d (cid:88) i =1 V is ( θ i + ρ i ν i ψ i ( T − s )) (cid:1) ds (cid:17) , t ≤ T. An application of Itˆo’s formula combined with the dynamics (4.11) shows that dM t = M t ( Z t ) (cid:62) dW t , and so M is a local martingale of the form M t = E (cid:16) (cid:90) Tt d (cid:88) i =1 ψ i ( T − s ) ν i (cid:113) V is dW is (cid:17) . Since ψ is continuous, it is bounded so that a straightforward adaptation of Abi Jaberet al. (2019, Lemma 7.3) to the multi-dimensional setting, recall (4.4), yields that M is atrue martingale. Since M T = 1, writing E [ M T |F t ] = M t , we obtainΓ t = E (cid:104) exp (cid:16) (cid:90) Tt (cid:0) r ( s ) − d (cid:88) i =1 V is ( θ i + ρ i ν i ψ i ( T − s )) (cid:1) ds (cid:17) | F t (cid:105) , t ≤ T, (4.12)which ensures that 0 < Γ t ≤ e (cid:82) Tt r ( s ) ds , P − a.s. , since V ∈ R d + . As for Z , it is clearthat it belongs to L F ([0 , T ] , R d ) since Γ and ψ are bounded and E (cid:104) (cid:82) T (cid:80) di =1 V is ds (cid:105) < ∞ by(4.4).The following remark makes precise the existence of a continuous solution to the Riccati-Volterra equation (4.6)-(4.7). Remark 4.3.
Assume that K satisfies the assumptions of Remark 4.1. • If − ρ i ≥ , then Abi Jaber et al. (2019, Lemma 6.3) provides the existence ofa unique solution ψ ∈ L ([0 , T ] , R d − ) . Continuity of such solution can then be easilyestablished, since as opposed to Abi Jaber et al. (2019, Lemma 6.3), (4.6) starts from . • If d = 1 and − ρ < , Han and Wong (2020a, Lemma A.4) establishes theexistence of a continuous solution ψ . Using Theorem 3.1, we can now explicitly solve the Markowitz problem (2.6) in themultivariate Volterra Heston model (4.1)-(4.2)-(4.3). The next theorem extends (Han andWong, 2020a, Theorem 4.2) to the multivariate case. Notice that the martingale distorsionargument in this cited paper is specific to the dimension d = 1, and here, instead, we relyon the generic verification result in Theorem 3.1.12 heorem 4.4. Assume that there exists a solution ψ ∈ C ([0 , T ] , R d ) to the Riccati-Volterraequation (4.6) - (4.7) such that max ≤ i ≤ d max t ∈ [0 ,T ] (cid:0) θ i + ν i ψ i ( t ) (cid:1) ≤ aa ( p ) , for some p > , (4.13) where a ( p ) is given by (3.4) and the constant a > is such that E (cid:104) exp (cid:0) a (cid:82) T (cid:80) di =1 V is ds (cid:1)(cid:105) < ∞ . Assume that g i (0) > for some i ≤ d . Then, the optimal investment strategy for themaximization problem (2.6) in the multivariate Volterra Heston model (4.1) - (4.2) - (4.3) isgiven by the admissible control α ∗ it = − (cid:0) θ i + ρ i ν i ψ i ( T − t ) (cid:1) (cid:113) V it (cid:16) X α ∗ t − ξ ∗ e − (cid:82) Tt r ( s ) ds (cid:17) , ≤ i ≤ d, (4.14) where ξ ∗ is defined as in (3.6) , the wealth process X ∗ = X α ∗ by (2.4) with λ = (cid:0) θ √ V ,. . . , θ d √ V d (cid:1) (cid:62) , and the optimal value is given by (3.7) with Γ as in (4.12) .Proof. First note that under the specification (4.9), the candidate for the optimal feedbackcontrol defined in (3.5) takes the form α ∗ t = − (cid:0) λ t + Z t + CZ t (cid:1)(cid:0) X ∗ t − ξ ∗ e − (cid:82) Tt r ( s ) ds (cid:1) = (cid:16) − (cid:0) θ i + ρ i ν i ψ i ( T − t ) (cid:1)(cid:113) V it (cid:0) X ∗ t − ξ ∗ e − (cid:82) Tt r ( s ) ds (cid:1)(cid:17) ≤ i ≤ d . It then suffices to check that the assumptions of Theorem 3.1 are verified to ensure thatsuch α ∗ is optimal and to get that (3.7) is the optimal value. The existence of a solutiontriplet (Γ , Z , Z ) ∈ S ∞ F ([0 , T ] , R ) × L F ([0 , T ] , R d ) × L F ([0 , T ] , R N ) to the stochastic back-ward Riccati equation (3.2) is ensured by Lemma 4.2. In addition, (4.12) implies thatΓ < e (cid:82) T r ( s ) ds since g i (0) > i ≤ d by assumption and V i is continuous. Thuscondition (H1) of Theorem 3.1 is verified. As for condition (H2) of Theorem 3.1, note that a ( p ) (cid:16) | λ s | + (cid:12)(cid:12) Z s (cid:12)(cid:12) + (cid:12)(cid:12) Z s (cid:12)(cid:12) (cid:17) = a ( p ) d (cid:88) i =1 V is (cid:0) θ i + ν i ψ i ( t ) (cid:1) ≤ a d (cid:88) i =1 V is , which implies that E (cid:104) exp (cid:16) a ( p ) (cid:82) T (cid:16) | λ s | + (cid:12)(cid:12) Z s (cid:12)(cid:12) + (cid:12)(cid:12) Z s (cid:12)(cid:12) (cid:17) ds (cid:17)(cid:105) < ∞ and ends the proof. Remark 4.5.
Condition (4.13) concerns the risk premium constants ( θ , . . . , θ d ) . For a > , a sufficient condition ensuring E (cid:2) exp (cid:0) a (cid:82) T (cid:80) di =1 V is ds (cid:1)(cid:3) < ∞ is the existence of acontinuous solution ˜ ψ to the Riccati–Volterra ˜ ψ i ( t ) = (cid:90) t K i ( t − s ) (cid:16) a + (cid:0) D ˜ ψ ( s ) (cid:1) i + ν i ψ i ( s ) (cid:17) ds, ee Abi Jaber et al. (2019, Theorem 4.3). In the one dimensional case d = 1 , such existenceis established in Han and Wong (2020a, Lemma A.2) for the case where g ( t ) = V + κ (cid:82) t K ( t − s ) φds , φ ≥ , D = − κ and a < κ ν . Remark 4.6.
Note that in the one dimensional case the condition (4.13) can be mademore explicit by bounding ψ with respect to θ . Indeed since − θ < we get from Abi Jaberand El Euch (2019b, Theorem C.1) that ψ is non-positive. Furthermore, the fact that ψ issolution to the following linear Volterra equation χ ( t ) = (cid:90) t K ( t − s ) (cid:16) − θ + (cid:0) ( D − θρν ) + ν − ρ ) ψ ( s ) (cid:1) χ ( s ) (cid:17) ds, leads to, see Abi Jaber and El Euch (2019b, Corollary C.4), sup t ∈ [0 ,T ] | ψ t | ≤ | θ | (cid:90) T R D ( s ) ds, where R D is the resolvent of KD . Consequently, a sufficient condition on θ to ensure (4.13) would be θ (cid:18) θν ) (cid:90) T R D ( s ) ds (cid:19) ≤ aa ( p ) . Remark 4.7.
In order to numerically implement the optimal strategy (4.14) , one needsto simulate the possibly non-Markovian process V and to discretize the Riccati-Volterraequation for ψ . Abi Jaber (2019a); Abi Jaber and El Euch (2019b) develop a taylor-made approximating procedure for the stochastic Volterra equation (4.1) (resp. the Riccati-Volterra equation (4.6) ), using finite-dimensional Markovian semimartingales (resp. finite-dimensional Riccati ODE’s). An illustration of such procedure on the mean-variance pro-blem in the univariate Volterra Heston model for the fractional kernel is given in Han andWong (2020a, Section 5). Before we introduce the class of multivariate quadratic Volterra models, we need to defineand recall some properties on integral operators.
Fix
T >
0. We denote by (cid:104)· , ·(cid:105) L the inner product on L (cid:0) [0 , T ] , R N (cid:1) that is (cid:104) f, g (cid:105) L = (cid:90) T f ( s ) (cid:62) g ( s ) ds, f, g ∈ L (cid:0) [0 , T ] , R N (cid:1) .
14e define L (cid:0) [0 , T ] , R N × N (cid:1) to be the space of measurable kernels K : [0 , T ] → R N × N such that (cid:90) T (cid:90) T | K ( t, s ) | dtds < ∞ . For any
K, L ∈ L (cid:0) [0 , T ] , R N × N (cid:1) we define the (cid:63) -product by( K (cid:63) L )( s, u ) = (cid:90) T K ( s, z ) L ( z, u ) du, ( s, u ) ∈ [0 , T ] , which is well-defined in L (cid:0) [0 , T ] , R N × N (cid:1) due to the Cauchy-Schwarz inequality. For anykernel K ∈ L (cid:0) [0 , T ] , R N × N (cid:1) , we denote by K the integral operator induced by the kernel K that is ( K g )( s ) = (cid:90) T K ( s, u ) g ( u ) du, g ∈ L (cid:0) [0 , T ] , R N (cid:1) . K is a linear bounded operator from L (cid:0) [0 , T ] , R N (cid:1) into itself. If K and L are two integraloperators induced by the kernels K and L in L (cid:0) [0 , T ] , R N × N (cid:1) , then KL is the integraloperator induced by the kernel K (cid:63) L .We denote by K ∗ the adjoint kernel of K for (cid:104)· , ·(cid:105) L , that is K ∗ ( s, u ) = K ( u, s ) (cid:62) , ( s, u ) ∈ [0 , T ] , and by K ∗ the corresponding adjoint integral operator. Definition 5.1.
A kernel K ∈ L (cid:0) [0 , T ] , R N × N (cid:1) is symmetric nonnegative if K = K ∗ and (cid:90) T (cid:90) T f ( s ) (cid:62) K ( s, u ) f ( u ) duds ≥ , ∀ f ∈ L (cid:0) [0 , T ] , R N (cid:1) . In this case, the integral operator K is said to be symmetric nonnegative and K = K ∗ and (cid:104) f, K f (cid:105) L ≥ . K is said to be symmetric nonpositive, if ( − K ) is symmetric nonnegative. We recall the definition of Volterra kernels of continuous and bounded type in theterminology of Gripenberg et al. (1990, Definitions 9.2.1, 9.5.1 and 9.5.2).
Definition 5.2.
A kernel K : R → R N × N is a Volterra kernel of continuous and boundedtype in L if K ( t, s ) = 0 whenever s > t and sup t ∈ [0 ,T ] (cid:90) T | K ( t, s ) | ds < ∞ , and lim h → (cid:90) T | K ( u + h, s ) − K ( u, s ) | ds = 0 , u ≤ T. (5.1)15ny convolution kernel of the form K ( t, s ) = k ( t − s ) s ≤ t with k ∈ L (cid:0) [0 , T ] , R N × N (cid:1) satisfies (5.1), we refer to Abi Jaber (2019b, Example 3.1) for additional examples. Notethat ( s, t ) (cid:55)→ K ( s, t ) is not necessarily continuous nor bounded. Lemma 5.3.
Let K satisfy (5.1) and L ∈ L ([0 , T ] , R N × N ) . Then, K (cid:63) L satisfies (5.1) .Furthemore, if L satisfies (5.1) , then, ( s, u ) (cid:55)→ ( K (cid:63) L ∗ )( s, u ) is continuous.Proof. An application of the Cauchy-Schwarz inequality yields the first part. The secondpart follows along the same lines as in the proof of Abi Jaber (2019b, Lemma 3.2).For a kernel K ∈ L ([0 , T ] , R N × N ), we define its resolvent R T ∈ L ([0 , T ] , R N × N ) bythe unique solution to R T = K + K (cid:63) R T , K (cid:63) R T = R T (cid:63) K. (5.2)In terms of integral operators, this translates into R T = K + KR T , KR T = R T K . In particular, if K admits a resolvent, (Id − K ) is invertible and(Id − K ) − = Id + R T , (5.3)where Id denotes the identity operator, i.e. (Id f ) = f for all f ∈ L (cid:0) [0 , T ] , R N (cid:1) .The following lemma establishes the existence of resolvents for the two classes of kernelsintroduced above. Lemma 5.4.
Let K ∈ L (cid:0) [0 , T ] , R N × N (cid:1) . K admits a resolvent if either one of thefollowing conditions hold: (i) K is a Volterra kernel of continuous and bounded type in L in the sense of Defi-nition 5.2. In this case, the resolvent is again a Volterra kernel of continuous andbounded type. (ii) K is symmetric nonpositive in the sense of Definition 5.1 and ( s, u ) (cid:55)→ K ( s, u ) iscontinuous.Proof. (i) follows from Gripenberg et al. (1990, Lemma 9.3.3, Theorem 9.5.5(i)). (ii) followsfrom an application of Mercer’s theorem, see Abi Jaber (2019b, Section 2.1).16 .2 The model In this section, we assume that the components of the stochastic volatility matrix σ in(2.1) are given by σ ij = γ (cid:62) ij Y , where γ ij ∈ R N and Y = ( Y , . . . , Y N ) (cid:62) is the following N -dimensional Volterra Ornstein–Uhlenbeck process Y t = g ( t ) + (cid:90) t K ( t, s ) DY s ds + (cid:90) t K ( t, s ) ηdW s , (5.4)where D, η ∈ R N × N , g : R + → R N is locally bounded, W is a N -dimensional process asin (2.2), i.e., W kt = C (cid:62) k B t + (cid:113) − C (cid:62) k C k B ⊥ ,kt , (5.5)where C k ∈ R d , such that C (cid:62) k C k ≤ k = 1 , . . . , N , and K : [0 , T ] → R N × N is a Volterrakernel of continuous and bounded type in L as in Definition 5.2. We stress that the process W is not necessarily a N -dimensional Brownian motion due to the possible correlations.Furthermore, the risk premium is assumed to be in the form λ t = Θ Y t , t ≤ T, for some Θ ∈ R d × N , so that the dynamics for the stock prices (2.1) reads as dS it = S it (cid:16) r ( t ) + N (cid:88) k,(cid:96) =1 d (cid:88) j =1 γ (cid:96)ij Θ jk Y (cid:96)t Y kt (cid:17) dt + S it d (cid:88) j =1 γ (cid:62) ij Y t dB jt , i = 1 , . . . , d. (5.6)The appellation quadratic reflects the quadratic dependence of the drift and the covari-ance matrix of log S in Y . Such models nest as special cases the Volterra extensions of thecelebrated Stein and Stein (1991) or Sch¨obel and Zhu (1999) model and certain Wishartmodels of Bru (1991) as shown in the following example. Example 5.5. (i)
The multivariate Volterra Stein-Stein model:
For N = d , K = diag( K , . . . , K d ) and γ ij = β ij e i with β ij ∈ R such that (cid:80) dj =1 β ij = 1and ( e , . . . , e d ) the canonical basis of R d , we recover the multivariate Volterra Stein-Steinmodel defined by (cid:40) dS it = S it (cid:16) r ( t ) + (cid:80) dj,k =1 β ij Θ jk Y it Y kt (cid:17) dt + S it Y it (cid:80) dj =1 β ij dB jt ,Y it = g i ( t ) + (cid:82) t K i ( t, s ) (cid:80) dj =1 D ij Y js ds + (cid:82) t K i ( t, s ) η i dW is , i = 1 , . . . , d, and C i = ρ i ( β i , . . . , β id ) (cid:62) to take into account the leverage effect. Recall that W is possiblycorrelated and is not necessarily a Brownian motion.(ii) The Volterra Wishart covariance model: N = d : (cid:40) dS t = diag( S t ) (cid:2) r ( t ) d dt + ˜ Y t dB t (cid:3) , S ∈ R d + , ˜ Y t = ˜ g ( t ) + (cid:82) t ˜ K ( t, s ) DY s ds + (cid:82) t ˜ K ( t, s ) ηdW s , with ˜ g : [0 , T ] → R d × d , a suitable measurable kernel ˜ K : [0 , T ] → R d × d , a d × d Brownianmotion W and W ij = ρ (cid:62) ij B + (cid:113) − ρ (cid:62) ij ρ ij B ⊥ ,ij , i, j = 1 , . . . , d, for some ρ ij ∈ R d × d such that ρ (cid:62) ij ρ ij ≤
1, for i, j = 1 , . . . , d , where B ⊥ is a d × d –dimensionalBrownian motion independent of B . Here the process ˜ Y is d × d -matrix valued. Remark 5.6.
Note that with (5.5) , there are no restrictions on the correlations between Y i and the stocks S i in (5.4) and (5.6) , in contrast with the correlation structure (4.1) ofthe multivariate Volterra Heston model. Moreover, the models in Example 5.5 allow us todeal with correlated stocks in contrast with the multivariate Heston model in (4.3) whereno correlation between the driving Brownian motion of the assets S i and S j is allowed inorder to keep the affine structure. Since K is a Volterra kernel of continuous and bounded type in L , there exists aprogressively measurable R N × R d + -valued strong solution ( Y, S ) to (5.4) and (5.6) suchthat sup t ≤ T E [ | Y t | p ] < ∞ , p ≥ . Indeed, the solution for (5.4) is given in the following closed form Y t = g ( t ) + (cid:90) t R D ( t, s ) g ( s ) ds + (cid:90) t ( K ( t, s ) + R D ( t, s )) ηdW s , (5.7)where R D is the resolvent of KD , whose existence is ensured by Lemma 5.4-(i). Theexistence of S readily follows from that of Y and is given as a stochastic exponential. Inthe sequel, we will assume that the solution Y is continuous. Additional conditions on K ,in the spirit of (4.5), are needed to ensure the existence of continuous modification, by anapplication of the Kolmogorov-Chentsov continuity criterion, for instance, as shown in thefollowing remark. 18 emark 5.7. For s ≤ t and p ≥ , an application of Jensen and Burkholder-Davis-Gundy’s inequalities yield E [ | ( Y t − g ( t )) − ( Y s − g ( s )) | p ] ≤ c (cid:16) r ≤ T E (cid:2) | Y s | p (cid:3)(cid:17) × (cid:18)(cid:90) ts | K ( t, r ) | dr + (cid:90) T | K ( t, r ) − K ( s, r ) | dr (cid:19) p/ . This shows that ( Y − g ) admits a continuous modification, by the Kolmogorov-Chentsovcontinuity criterion, provided that (cid:90) ts | K ( t, r ) | dr + (cid:90) T | K ( t, r ) − K ( s, r ) | dr ≤ c | t − s | γ , for some γ > . In this section, we provide an explicit solution for the Markowitz problem for quadraticVolterra models, and our main result is stated in Theorem 5.11 below.Exploiting the quadratic structure of (5.4)-(5.6), see Abi Jaber (2019b), we provide anexplicit solution to the Riccati BSDE in Lemma 5.9 below, in terms of the following familyof linear operators ( Ψ t ) ≤ t ≤ T acting on L (cid:0) [0 , T ] , R N (cid:1) : Ψ t = − (cid:16) Id − ˆ K (cid:17) −∗ Θ (cid:62) (cid:16) Id + 2Θ ˜ Σ t Θ (cid:62) (cid:17) − Θ (cid:16) Id − ˆ K (cid:17) − , ≤ t ≤ T, (5.8)where F −∗ := ( F − ) ∗ , and ˆ K is the integral operator induced by the kernel ˆ K = K ( D − ηC (cid:62) Θ) and ˜ Σ t the integral operator defined by ˜Σ t = (Id − ˆ K ) − Σ t (Id − ˆ K ) −∗ , t ∈ [0 , T ] , (5.9)with Σ t defined as the integral operator associated to the kernelΣ t ( s, u ) = (cid:90) s ∧ ut K ( s, z ) η (cid:0) U − C (cid:62) C (cid:1) η (cid:62) K ( u, z ) (cid:62) dz, t ∈ [0 , T ] , (5.10)where U = d (cid:104) W (cid:105) t dt = (cid:0) i = j + 1 i (cid:54) = j ( C i ) (cid:62) C j (cid:1) ≤ i,j ≤ N .We start by deriving some first properties of t (cid:55)→ Ψ t , namely that it is well-defined,strongly differentiable and satisfies an operator Riccati equation under the following addi-tional assumption on the kernel:sup t ≤ T (cid:90) T | K ( s, t ) | ds < ∞ . (5.11)19e recall that t (cid:55)→ Ψ t is said to be strongly differentiable at time t ≥
0, if there exists abounded linear operator ˙ Ψ t from L (cid:0) [0 , T ] , R N (cid:1) into itself such thatlim h → h (cid:107) Ψ t + h − Ψ t − h ˙ Ψ t (cid:107) op = 0 , where (cid:107) G (cid:107) op = sup f ∈ L ([0 ,T ] , R N ) (cid:107) G f (cid:107) L (cid:107) f (cid:107) L . Lemma 5.8.
Fix a kernel K as in Definition 5.2 satisfying (5.11) . Assume that ( U − C (cid:62) C ) ∈ S N + . Then, for each t ≤ T , Ψ t given by (5.8) is well-defined and is a boundedlinear operator from L (cid:0) [0 , T ] , R N (cid:1) into itself. Furthermore, (i) (Θ (cid:62) ΘId + Ψ t ) is an integral operator induced by a kernel ψ t ( s, u ) such that sup t ≤ T (cid:90) [0 ,T ] | ψ t ( s, u ) | dsdu < ∞ . (5.12)(ii) For any f ∈ L (cid:0) [0 , T ] , R N (cid:1) , ( Ψ t f t )( t ) =( − Θ (cid:62) ΘId + ˆ K ∗ Ψ t )( f )( t ) , where t : s (cid:55)→ t ≤ s . (iii) t (cid:55)→ Ψ t is strongly differentiable and satisfies the operator Riccati equation ˙ Ψ t = 2 Ψ t ˙ Σ t Ψ t , t ∈ [0 , T ] Ψ T = − (cid:16) Id − ˆ K (cid:17) −∗ Θ (cid:62) Θ (cid:16) Id − ˆ K (cid:17) − (5.13) where ˙ Σ t is the strong derivative of t (cid:55)→ Σ t induced by the kernel ˙Σ t ( s, u ) = − K ( s, t ) η (cid:0) U − C (cid:62) C (cid:1) η (cid:62) K ( u, t ) (cid:62) , a.e. (5.14) Proof.
Fix t ≤ T . We start by proving that Ψ t is well defined and is a bounded linearoperator from L (cid:0) [0 , T ] , R N (cid:1) to L (cid:0) [0 , T ] , R N (cid:1) . First, since K is a Volterra kernel ofcontinuous and bounded type in L , so is ˆ K , and Lemma 5.4-(i) yields the existence of itsresolvent ˆ R such thatsup s ≤ T (cid:90) T | ˆ R ( s, u ) | ds < ∞ , sup u ≤ T (cid:90) T | ˆ R ( s, u ) | du < ∞ . (5.15)In particular, denoting by ˆ R the integral operator induced by ˆ R , we obtain that (Id − ˆ K )is invertible with an inverse given by (Id − ˆ K ) − = Id + ˆ R , recall (5.3). Next, we provethat (cid:0) Id + 2Θ ˜ Σ t Θ (cid:62) (cid:1) is invertible. It follows from (5.9) that ˜Σ t = (Id + ˆ R ) Σ t (Id + ˆ R ) ∗ = Σ t + Σ t ˆ R ∗ + ˆ R Σ t + ˆ R Σ t ˆ R ∗ . ˜Σ t is an integral operator generated by the kernel˜Σ t = Σ t + Σ t (cid:63) ˆ R ∗ + ˆ R (cid:63) Σ t + ˆ R (cid:63) Σ t (cid:63) ˆ R ∗ . (5.16)Since K satisfies (5.1) and ( U − C (cid:62) C ) ∈ S N + , Σ t defined in (5.10) is clearly a symmetricnonnegative kernel. Combined with (5.16), we get that ˜Σ t is symmetric nonnegative.Succesive applications of Lemma 5.3 yield that ( s, u ) (cid:55)→ ˜Σ t ( s, u ) is continuous. Therefore,( − ˜Σ t Θ (cid:62) ) is symmetric nonpositive and continuous so that an application of Lemma 5.4-(ii) yields the existence of its resolvent R Θ t . In particular, (cid:0) Id + 2Θ ˜ Σ t Θ (cid:62) (cid:1) is invertible withan inverse given by (Id + R Θ t ), recall (5.3). Combining the above, we get that Ψ t iswell-defined, and satisfies Ψ t = − (Id + ˆ R ) ∗ Θ (cid:62) (Id + R Θ t )Θ(Id + ˆ R )= − Θ (cid:62) ΘId − ˆ R ∗ Θ (cid:62) Θ − Θ (cid:62) Θ ˆ R − ˆ R ∗ Θ (cid:62) R Θ t Θ − Θ (cid:62) R Θ t Θ ˆ R − ˆ R ∗ Θ (cid:62) R Θ t Θ ˆ R − ˆ R ∗ Θ (cid:62) Θ ˆ R − Θ (cid:62) R Θ t Θ , (5.17)showing that Ψ t is a bounded operator.(i): From (5.17), we see that (Θ (cid:62) ΘId + Ψ t ) is an integral operator whose kernel is of theform ψ t = − ˆ R ∗ Θ (cid:62) Θ − Θ (cid:62) Θ ˆ R − ˆ R ∗ Θ (cid:62) (cid:63) R Θ t Θ − Θ (cid:62) R Θ t (cid:63) Θ ˆ R − ˆ R ∗ Θ (cid:62) (cid:63) R Θ t (cid:63) Θ ˆ R − ˆ R ∗ Θ (cid:62) (cid:63) Θ ˆ R + Θ (cid:62) R Θ t Θ . Then, from Abi Jaber (2019b, Lemma C.1) we get thatsup t ≤ T (cid:90) [0 ,T ] | R Θ t ( s, u ) | dsdu < ∞ , which, combined with (5.15) ensures (5.12).(ii): Fix f ∈ L (cid:0) [0 , T ] , R N (cid:1) and t ≤ T . We first argue that R Θ t ( t, . ) = 0 and ˆ R ( s, u ) = 0 , for any s < u. (5.18)Indeed, since ˆ K is a Volterra kernel, its resolvent ˆ R is also a Volterra kernel so that R ( s, u ) = 0 whenever s < u . This, combined with the fact that Σ t ( t, · ) = 0 and (5.16),yields that ˜Σ t ( t, · ) = 0, so that R Θ t ( t, · ) = 0 by virtue of the resolvent equation (5.2). Usingthe relations (5.18), we compute (cid:16) ΘΘ (cid:62) ˆ R (cid:17) ( f t )( t ) = ΘΘ (cid:62) (cid:90) T ˆ R ( t, s ) f ( s )1 t ( s ) ds = 0 , (cid:16) Θ (cid:62) R Θ t Θ (cid:17) ( f t )( t ) = Θ (cid:62) (cid:90) T R Θ t ( t, s )Θ f ( s )1 t ( s ) ds = 0 , (cid:16) Θ (cid:62) R Θ t Θ ˆ R (cid:17) ( f t )( t ) = Θ (cid:62) (cid:90) T (cid:90) T R Θ t ( t, u )Θ ˆ R ( u, s ) f ( s )1 t ( s ) duds = 0 . (5.19)21hus, (5.19) combined with (5.17) and the resolvent’s relations ˆ R = ˆ K + ˆ K ˆ R and ˆ R ∗ = ˆ K ∗ + ˆ K ∗ ˆ R ∗ yield − (Θ (cid:62) ΘId + Ψ t )( f t )( t ) =( ˆ R ∗ Θ (cid:62) Θ + ˆ R ∗ Θ (cid:62) R Θ Θ + ˆ R ∗ Θ (cid:62) R Θ t Θ ˆ R + ˆ R ∗ ΘΘ (cid:62) ˆ R )( f t )( t )= − ( K ∗ Ψ t )( f t )( t )which proves the second claim (ii).(iii): Under (5.11), Abi Jaber (2019b, Lemma 3.2) yields that t (cid:55)→ Σ t is strongly differen-tiable on [0 , T ] with a derivative given by t (cid:55)→ ˙ Σ t induced by the kernel (5.14). Whence, itfollows from (5.9), that t (cid:55)→ ˜Σ t is also differentiable such that ˙ ˜Σ t = (Id − ˆ K ) − ˙ Σ t (Id − ˆ K ) −∗ .Thus, (5.8) yields that t (cid:55)→ Ψ t is strongly differentiable with a derivative given by˙ Ψ t = 2(Id − ˆ K ) −∗ Θ (cid:62) (Id −
2Θ ˜ Σ t Θ (cid:62) ) − Θ ˙ ˜Σ t Θ (cid:62) (Id −
2Θ ˜ Σ t Θ (cid:62) ) − Θ(Id − ˆ K ) − = 2 Ψ t ˙ Σ t Ψ t . Finally, evaluating (5.10) at t = T , yields that Σ T ( s, u ) = 0 for all s, u ≤ T , leading to Σ T = so that Ψ T = − (cid:16) Id − ˆ K (cid:17) −∗ Θ (cid:62) Θ (cid:16) Id − ˆ K (cid:17) − . This proves (5.13).We are now ready to provide a solution for the Riccati-BSDE (3.2). For this, denoteby g the process g t ( s ) = t ≤ s (cid:16) g ( s ) + (cid:90) t K ( s, u ) DY u du + (cid:90) t K ( s, u ) ηdW u (cid:17) . (5.20)One notes that for each, s ≤ T , ( g t ( s )) t ≤ s is the adjusted forward process g t ( s ) = E (cid:104) Y s − (cid:90) st K ( s, u ) DY u du | F t (cid:105) , s ≥ t. We also denote the trace of an integral operator F by Tr( F ) = (cid:82) T tr( F ( s, s )) ds , where tris the usual trace of a matrix, and we define the function φ by ˙ φ t = Tr (cid:0) Ψ t ˙ Λ t (cid:1) − r ( t )= (cid:82) ( t,T ] tr (cid:0) Θ (cid:62) Θ K ( s, t ) ηU η (cid:62) K ( s, t ) (cid:62) (cid:1) ds − (cid:82) ( t,T ] tr (cid:0) ψ t ( s, u ) K ( u, t ) ηU η (cid:62) K ( s, t ) (cid:62) (cid:1) dsdu − r ( t ) ,φ T = 0 , (5.21)where ˙ Λ t is the integral operator induced by the kernel given by˙Λ t ( s, u ) = − K ( s, t ) ηU η (cid:62) K ( u, t ) (cid:62) , u, s ≤ T. emma 5.9. Fix a kernel K as in Definition 5.2 satisfying (5.11) . Assume that ( U − C (cid:62) C ) ∈ S N + . Let Ψ be the operator defined in (5.8) . Then, the process (cid:0) Γ , Z , Z (cid:1) definedby Γ t = exp ( φ t + (cid:104) g t , Ψ t g t (cid:105) L ) ,Z t = 0 ,Z t = 2 (cid:0) ( Ψ t K η ) ∗ g t (cid:1) ( t ) , (5.22) where g and φ are respectively given by (5.20) and (5.21) , is a S ∞ F ([0 , T ] , R ) × L F ([0 , T ] , R d ) × L F ([0 , T ] , R N ) -valued solution to the Riccati-BSDE (3.2) .Proof. Set G t = φ t + (cid:104) g t , Ψ t g t (cid:105) L , so that Γ t = exp( G t ) and d Γ t = Γ t (cid:0) dG t + 12 d (cid:104) G (cid:105) t (cid:1) . (5.23)To obtain the dynamics of G it suffices to determine the dynamics of the process t (cid:55)→(cid:104) g t , Ψ t g t (cid:105) L . Step 1.
In this step we prove that the dynamics of t (cid:55)→ (cid:104) g t , Ψ t g t (cid:105) L is given by d (cid:104) g t , Ψ t g t (cid:105) L = (cid:16) (cid:104) g t , ˙ Ψ t g t (cid:105) L + λ (cid:62) t λ t + 2 λ (cid:62) t CZ t + Tr (cid:0) Ψ t ˙ Λ t (cid:1)(cid:17) dt + ( Z t ) (cid:62) dW t . (5.24)We first note that (cid:104) g t , Ψ t g t (cid:105) L = (cid:90) T g t ( s ) (cid:62) ( Ψ t g t )( s ) ds, and compute the dynamics of t (cid:55)→ g t ( s ) (cid:62) ( Ψ t g t )( s ). For fixed s ≤ T , it follows from (5.20)and the fact that Y t = g t ( t ), that dg t ( s ) = − δ t = s g t ( t ) dt + K ( s, t ) Dg t ( t ) dt + K ( s, t ) ηdW t . Together with Lemma 5.8-(iii), we deduce that t (cid:55)→ ( Ψ t g t )( s ) is a semimartingale with thefollowing dynamics d ( Ψ t g t )( s ) = ( ˙ Ψ t g t )( s ) dt + ( Ψ t dg t )( s )= ( ˙ Ψ t g t )( s ) dt − ψ t ( s, t ) g t ( t ) dt + ( Ψ t K ( · , t ) Dg t ( t ))( s ) dt + ( Ψ t K ( · , t ) ηdW t )( s ) . Here, we used the fact that Id δ t = 0: indeed, for every f ∈ L ([0 , T ] , R d ) we have(Id δ t )( f ) = ( f ( · ) t = · ) = 0 L . Moreover, d (cid:104) g · ( s ) , ( Ψ · g · )( s ) (cid:105) t = − tr (cid:0) Θ (cid:62) Θ K ( s, t ) ηU η (cid:62) K ( s, t ) (cid:62) (cid:1) dt + (cid:90) Tt tr (cid:0) ψ t ( s, u ) K ( u, t ) ηU η (cid:62) K ( s, t ) (cid:62) (cid:1) dudt = − tr (cid:0) Θ (cid:62) Θ ˙Λ t ( s, s ) (cid:1) dt + (cid:90) Tt tr (cid:0) ψ t ( s, u ) ˙Λ t ( u, s ) (cid:1) dudt = tr (cid:16)(cid:0) Ψ t ˙Λ( · , s ) (cid:1) ( s ) (cid:17) . d (cid:16) g t ( s ) (cid:62) ( Ψ t g t )( s ) (cid:17) = dg t ( s ) (cid:62) ( Ψ t g t )( s ) + g t ( s ) (cid:62) d ( Ψ t g t )( s ) + d (cid:104) g · ( s ) , ( Ψ · g · )( s ) (cid:105) t = − δ t = s g t ( t ) (cid:62) ( Ψ t g t )( s ) dt + g t ( t ) (cid:62) D (cid:62) K ( s, t ) (cid:62) ( Ψ t g t )( s ) dt + g t ( s ) (cid:62) ( ˙ Ψ t g t )( s ) dt − g t ( s ) (cid:62) ψ ( s, t ) g t ( t ) dt + g t ( s ) (cid:62) ( Ψ t K ( · , t ) Dg t ( t ))( s ) dt + tr (cid:16)(cid:0) Ψ t ˙Λ( · , s ) (cid:1) ( s ) (cid:17) + dW (cid:62) t η (cid:62) K ( s, t ) (cid:62) ( Ψ t g t )( s ) + g t ( s ) (cid:62) ( Ψ t K ( · , t ) ηdW t )( s )= (cid:104) I ( s ) + II ( s ) + III ( s ) + IV ( s ) + V ( s ) + VI ( s ) (cid:105) dt + VII ( s ) + VIII ( s ) . We now integrate in s . First, using Lemma 5.8-(i) we get that (cid:90) T (cid:2) I ( s ) + IV ( s ) (cid:3) ds = − g t ( t ) (cid:62) ( Ψ t g t )( t ) = λ (cid:62) t λ t − g t ( t ) (cid:62) (cid:90) Tt ψ t ( t, u ) g t ( u ) du = λ (cid:62) t λ t − g t ( t ) (cid:62) (( Ψ t + Θ (cid:62) ΘId) g t )( t ) . On the other hand, since, Ψ ∗ = Ψ , we have (cid:90) T (cid:2) II ( s ) + V ( s ) (cid:3) ds = 2 g t ( t ) (cid:62) (cid:16)(cid:0) ( K D ) ∗ Ψ t (cid:1) g t (cid:17) ( t ) . Therefore, summing the above, using Lemma 5.8-(ii), and the definition of ˆ K , we get (cid:90) T (cid:2) I ( s ) + IV ( s ) + II ( s ) + V ( s ) (cid:3) ds = λ (cid:62) t λ t − g t ( t ) (cid:62) (cid:16)(cid:0) Ψ t + Θ (cid:62) ΘId − (( K D ) ∗ ) Ψ t (cid:1) g t (cid:17) ( t )= λ (cid:62) t λ t + 4 g t ( t ) (cid:62) (( K ηC (cid:62) Θ) ∗ Ψ t ) g t )( t )= λ (cid:62) t λ t + 2 λ (cid:62) t CZ t . Finally, observing that (cid:90) T III ( s ) ds = (cid:104) g t , ˙ Ψ t g t (cid:105) L , (cid:90) T VI ( s ) ds = Tr (cid:16) Ψ t ˙ Λ t (cid:17) , (cid:90) T (cid:2) VI ( s ) + VIII ( s ) (cid:3) ds = (cid:0) Z t (cid:1) (cid:62) dW t , we obtain the claimed dynamics (5.24). Step 2.
Plugging the dynamics (5.24) in (5.23) yields d Γ t Γ t = (cid:104) ˙ φ t,T + Tr (cid:16) Ψ t ˙ Λ t (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) + (cid:104) g t , ˙ Ψ t g t (cid:105) L + ( Z t ) (cid:62) U Z t (cid:124) (cid:123)(cid:122) (cid:125) + λ (cid:62) t λ t + 2 λ (cid:62) t CZ t (cid:124) (cid:123)(cid:122) (cid:125) (cid:105) dt + (cid:0) Z t (cid:1) (cid:62) dW t .
24y (5.21), we have: = − r ( t ). From the definition of Z , we have( Z t ) (cid:62) U Z t (cid:104)(cid:16)(cid:0) Ψ t K η (cid:1) ∗ g t (cid:17) ( t ) (cid:105) (cid:62) U (cid:16)(cid:0) Ψ t K η (cid:1) ∗ g t (cid:17) ( t )= − (cid:104) g t , ( Ψ t ˙ Λ t Ψ t ) g t (cid:105) L . Thus, using the Riccati relation (5.13), we get = (cid:104) g t , ( ˙ Ψ t − Ψ t ˙ Λ t Ψ t ) g t (cid:105) L = 4 (cid:104)(cid:16)(cid:0) Ψ t K η (cid:1) ∗ g t (cid:17) ( t ) (cid:105) (cid:62) C (cid:62) C (cid:16)(cid:0) Ψ t K η (cid:1) ∗ g t (cid:17) ( t )= ( Z t ) (cid:62) CC (cid:62) Z t . Combining , and yields d Γ t Γ t = (cid:16) − r ( t ) + (cid:12)(cid:12) λ t + Z t + CZ t (cid:12)(cid:12) (cid:17) dt + ( Z t ) (cid:62) dW t . This shows that (Γ , Z , Z ) solves (3.2). Step 3.
It remains to check that (cid:0) Γ , Z , Z (cid:1) ∈ S ∞ F ([0 , T ] , R ) × L F ([0 , T ] , R d ) × L F ([0 , T ] , R N ).For this, observe that since Ψ is a nonpositive operator over [0 , T ], we have the bound0 < Γ t ≤ e (cid:82) Tt r ( s ) ds . Finally, to show that Z ∈ L F ([0 , T ] , R d ), it is enough to show that E (cid:104) (cid:90) T (cid:12)(cid:12)(cid:12) (cid:90) Tt K ( s, t ) (cid:62) g t ( s ) ds (cid:12)(cid:12)(cid:12) dt (cid:105) < ∞ , and E (cid:104) (cid:90) T (cid:12)(cid:12)(cid:12) (cid:90) ( t,T ] K ( v, t ) (cid:62) ψ t ( v, s ) g t ( s ) dvds (cid:12)(cid:12)(cid:12) dt (cid:105) < ∞ . This follows from the fact that K and ψ satisfy (5.1)-(5.12) respectively, andsup ≤ t ≤ s ≤ T E (cid:2) | g t ( s ) | (cid:3) ≤ sup s ≤ T | g ( s ) | (cid:16) s ≤ T (cid:90) T | R D ( s, u ) | du (cid:17) < ∞ , where R D is the resolvent of KD .From Theorem 3.1, we can now explicitly solve the Markowitz problem (2.6) in thequadratic Volterra model (5.4), (5.5) and (5.6), see Theorem 5.11 below. In order to verifycondition (H2) of Theorem 3.1, we will first need the following lemma whose proof ispostponed to Appendix B.1. Lemma 5.10.
Let the assumptions of Lemma 5.9 be in force. Then, there exists a constant κ > such that | λ s | + (cid:12)(cid:12) Z s (cid:12)(cid:12) + (cid:12)(cid:12) Z s (cid:12)(cid:12) ≤ | Θ | | g s ( s ) | + κ | Θ | (cid:90) T | g s ( u ) | du, s ≤ T, Θ ∈ R d × N .
25e now arrive to the main result of this section.
Theorem 5.11.
Fix a kernel K as in Definition 5.2 satisfying (5.11) and assume that ( U − C (cid:62) C ) ∈ S N + . Let a ( p ) be as in (3.4) and assume that there exists Θ ∈ R d × N suchthat E (cid:104) exp (cid:16) a ( p ) (cid:90) T (cid:0) | Θ | | g s ( s ) | + κ | Θ | (cid:90) T | g s ( u ) | du (cid:1) ds (cid:17)(cid:105) < ∞ , (5.25) for some p > , where κ is the constant appearing in Lemma 5.10. Assume that g i (0) > for some i ≤ d . Then, the optimal investment strategy for the Markowitz problem (2.6) isgiven by the admissible control α ∗ t = − (cid:16)(cid:0) Θ + 2 C [ Ψ t K η ] ∗ (cid:1) g t (cid:17) ( t ) (cid:16) X α ∗ t − ξ ∗ e − (cid:82) Tt r ( s ) ds (cid:17) , (5.26) where ξ ∗ is defined in (3.6) , and the optimal value is given by (3.7) with Γ as in (5.22) .Proof. First note that under the specification (5.22), and λ t = Θ Y t = Θ g t ( t ), the candidatefor the optimal feedback control defined in (3.5) takes the form α ∗ t = − (cid:0) λ t + Z t + CZ t (cid:1) (cid:0) X α ∗ t − ξe − (cid:82) Tt r ( s ) ds (cid:1) = (cid:16)(cid:0) Θ + 2 C [ Ψ t K η ] ∗ (cid:1) g t (cid:17) ( t ) (cid:0) X α ∗ t − ξe − (cid:82) Tt r ( s ) ds (cid:1) . It thus suffices to check that the assumptions of Theorem 3.1 are verified to ensure that α ∗ ( ξ ∗ ) is optimal and to get that (3.7) is the optimal value. The existence of a solu-tion triplet (Γ , Z , Z ) ∈ S ∞ F ([0 , T ] , R ) × L F ([0 , T ] , R d ) × L F ([0 , T ] , R N ) to the stochasticbackward Riccati equation (3.2) is ensured by Lemma 5.9. In addition, we haveΓ = E (cid:104) e (cid:82) T (cid:0) r ( s ) − | λ s + Z s + CZ s | (cid:1) ds (cid:105) = E (cid:104) e (cid:82) T (cid:2) r ( s ) − (cid:12)(cid:12)(cid:0)(cid:0) Θ+2 C [ Ψ s K η ] ∗ (cid:1) g s (cid:1) ( s ) (cid:12)(cid:12) (cid:3) ds (cid:105) , which implies that Γ < e (cid:82) T r ( s ) ds since g i (0) > i ≤ d by assumption. Thuscondition (H1) of Theorem 3.1 is verified. Condition (H2) follows directly from Lemma 5.10and (5.25). The proof is complete.The following lemma provides a general sufficient condition for the existence of Θsatisfying (5.25). Without loss of generality, we assume that D = 0 in (5.4). Under (5.1), K is a compact linear operator from L ([0 , T ] , R N ) into itself, see Brunner (2017, ExampleA.2.13). An application of the singular value decomposition in Hilbert spaces (Reed andSimon, 2012, Theorem IV.17) yields the existence of two orthonormal basis ( e n ) n ≥ and If D (cid:54) = 0, then making use of the resolvent kernel R D of KD , we reduce to the case D = 0 as illustratedon (5.7) by working on the kernel ( K + R D ) instead of K . f n ) n ≥ in L ([0 , T ] , R N ) and a sequence of real numbers λ ≥ λ ≥ . . . ≥ λ n → n → ∞ , such that K ( t, s ) = (cid:88) n ≥ √ λ n e n ( t ) f n ( s ) (cid:62) . (5.27)In addition, we observe by virtue of (5.1) that (cid:88) n ≥ λ n = (cid:90) T (cid:90) T | K ( t, s ) | dtds < ∞ . (5.28) Lemma 5.12.
Set D = 0 . Let a, b > be such that a + bT ) < λ . Then, E (cid:104) exp (cid:16) (cid:90) T (cid:0) a | g s ( s ) | + b (cid:90) T | g s ( u ) | du (cid:1) ds (cid:17)(cid:105) < ∞ . In particular, (5.25) holds if a ( p )( | Θ | + | Θ | κT ) < λ for some p > .Proof. We refer to Appendix B.2.
Remark 5.13.
In practice, it follows from Lemma 5.12 and (5.28) , that a sufficient con-dition for the existence of Θ satisfying (5.25) would be a ( p )( | Θ | + | Θ | κT ) < (cid:82) T (cid:82) T | K ( t, s ) | dsdt . For instance, for the fractional convolution kernel K ( t, s ) = s ≤ t ( t − s ) H − / , we have (cid:82) T (cid:82) T | K ( t, s ) | dsdt = T H +1 . The following corollary treats the standard Markovian and semimartingale case for K = I N and shows how to recover the well-known formulae in the spirit of Chiu and Wong(2014). Corollary 5.14.
Set K ( t, s ) = I N s ≤ t . Then, the solution to the Riccati BSDE can bere-written in the form Γ t = exp (cid:0) φ t + Y (cid:62) t P t Y t (cid:1) , and Z t = 2( Dη ) (cid:62) P t Y t , (5.29) where P : [0 , T ] (cid:55)→ R N × N and φ solve the conventional system of N × N -matrix Riccatiequations ˙ P t = Θ (cid:62) Θ + P t (2 ηC (cid:62) Θ − D ) + (2 ηC (cid:62) Θ − D ) (cid:62) P t + 2 P t ( η ( U − C (cid:62) C ) η (cid:62) ) P t ,P T = 0 , ˙ φ t = − r ( t ) − tr( P t ηU η (cid:62) ) , t ∈ [0 , T ] ,φ T = 0 . urthermore, the optimal control reads α ∗ t = − (cid:16) Θ + 2 C ( Dη ) (cid:62) P t Y t (cid:17)(cid:16) X α ∗ t − ξ ∗ e − (cid:82) Tt r ( s ) ds (cid:17) . (5.30) Proof.
For K ( t, s ) = I N s ≤ t , Y s = Y t + (cid:90) st DY u du + (cid:90) st ηdW u , s ≥ t, so that the adjusted forward process reads g t ( s ) = E (cid:104) Y s − (cid:90) st DY u du | F t (cid:105) = t ≤ s Y t , and the solution to the Riccati BSDE can be re-written in the formΓ t = exp (cid:0) φ t + (cid:104) g t , Ψ t g t (cid:105) L (cid:1) = exp (cid:0) φ t + Y (cid:62) t P t Y t (cid:1) , where P t = (cid:82) Tt ( Ψ t t )( s ) ds with the R N -valued indicator function t : ( s ) (cid:55)→ (1 t ≤ s , . . . , t ≤ s ) (cid:62) .We now derive the equations satisfied by P and φ . First we have K T = 0 and˙ P t = − ( Ψ t t )( t ) + (cid:90) Tt d ( Ψ t t )( s ) dt ds = − ( Ψ t t )( t ) + (cid:90) Tt ( ˙ Ψ t t )( s ) ds − (cid:90) Tt ψ t ( s, t ) ds = + + . Using Lemma 5.8–(ii) and the expression ˆ K ( s, u ) = 1 u ≤ s ( D − ηC (cid:62) Θ) we get = ( − Θ (cid:62) ΘId + ˆ K ∗ Ψ t )(1)( t ) = − Θ (cid:62) Θ + ( D − ηC (cid:62) Θ) (cid:62) P t . Furthermore, Lemma 5.8–(iii) and ˙Σ t ( s, u ) = 1 t ≤ s ∧ u η ( U − C (cid:62) C ) η (cid:62) yield = (cid:90) Tt ( ˙ Ψ t t )( s ) ds = (cid:90) Tt ( Ψ t ˙ Σ t Ψ t t )( s ) ds = (cid:18)(cid:90) Tt ( Ψ t t )( s ) ds (cid:19) η ( U − C (cid:62) C ) η (cid:62) (cid:18)(cid:90) Tt ( Ψ t t )( s ) ds (cid:19) = P t ( η ( U − C (cid:62) C ) η (cid:62) ) P t . Moreover, by using Lemma 5.8–(i)-(ii), we obtain = − (cid:90) Tt ψ t ( s, t ) ds = − ( Ψ t + Θ (cid:62) Θ id ) ∗ (1 t ) = − ( ˆ K ∗ Ψ t ) ∗ (1)( t ) = − P t ( D − ηC (cid:62) Θ) . This proves the equation for P , and that of φ is immediate. Finally to prove the formulaof Z in (5.29) and α ∗ in (5.30) it suffices to observe the following identity (cid:0) ( Ψ t K η ) ∗ g t (cid:1) ( t ) = ( Dη ) (cid:62) P t Y t . Numerical experiment: rough Stein-Stein for two assets
We illustrate the results of Section 5 on a special case of the two dimensional rough Stein-Stein model as described in Example 5.5. We consider a four dimensional Brownian motion( B , B , B , ⊥ , B , ⊥ ), and define˜ B = B , ˜ B = ρB + (cid:112) − ρ B , W i = c i ˜ B i + (cid:113) − c i ˜ B i, ⊥ , for some ρ ∈ [ − , c i ∈ [ − , i = 1 , r ≡
0, and consider two stocks of price process S and S withthe following dynamics (cid:40) dS it = S it θ i ( Y it ) dt + S it Y it d ˜ B it ,Y it = Y i + H i +1 / (cid:82) t ( t − s ) H i − / η i dW is , i = 1 , , with H i > η i , θ i ≥ Y i ∈ R .Although the framework of Section 5 allows for a more general correlation structure forthe Brownian motion, the model is already rich enough to capture the following stylizedfacts: • the two stocks S i , i = 1 ,
2, are correlated through ρ , • each stock S i has a stochastic rough volatility | Y i | with possibly different Hurstindices H i , • each stock S i is correlated with its own volatility process through c i to take intoaccount the leverage effect.Our main motivation for considering the multivariate rough Stein-Stein model is tostudy the ‘buy rough sell smooth’ strategy of Glasserman and He (2020) that was back-tested empirically: this strategy consisting in buying the roughest assets while shorting onthe smoothest ones was shown to be profitable. We point out that the numerical simula-tions for the one dimensional rough Heston model carried in Han and Wong (2020a) byvarying the Hurst index H could not provide much insight on such strategy, apart fromsuggesting that the vol-of-vol has a possible impact on the ‘buy rough sell smooth strategy’.Our quadratic multivariate framework allows for more flexible simulations, with a richercorrelation structure compared to multivariate extensions of the rough Heston model, recallRemark 5.6. Our results below provide new insights on the strategy by showing that thecorrelation between stocks plays a key role.Our present goal is to illustrate the influence of some parameters, namely the horizon T , the vol-of-vol η and the correlation ρ between the stocks, onto the optimal investment This corresponds to Example 5.5-(i) with ( β , β , β , β ) = (1 , , ρ, (cid:112) − ρ ) and Θ = β − diag ( θ , θ ). H < H , are at stakes. To easecomparison, we set c = c = − . Y = Y and we normalizethe vol-of-vols by setting η = η . We consider the evolution of optimal vector of amountinvested into each stock, i.e., t (cid:55)→ π ∗ t (recall that α ∗ t = σ (cid:62) t π ∗ t with σ = diag( Y , Y ) β and α ∗ is given by (5.26)). π being a stochastic process, we also consider the deterministicfunction t (cid:55)→ ((Θ + 2 C [ Ψ t K η ] ∗ ) Y )( t )( ξ ∗ ), where ξ ∗ is defined in (3.6), to help us in ouranalysis.For our implementation of α ∗ given by (5.26), we discretize in time the operators actingon L , so that the kernel of the operator Ψ in (5.8) is approximated by a finite dimensionalmatrix (see for instance Abi Jaber (2019b, Section 2.3) for a similar procedure) and theGaussian process ( g t ( s )) t ≤ s ≤ T defined in (5.20) is simulated by Cholesky’s decompositionalgorithm. We refer to the following url for the full code and additional simulations.Our observations from the simulations are the following.
1. Horizon T : With the goal of understanding the effect of the horizon T on theinvestment strategy, we fix all parameters but T with ρ = 0. The results are illustrated onFigures 1a-1b-1c and 2a-2b-2c. We can distinguish 3 regimes: • T (cid:28) • T ≈ • T (cid:29)
1: The smooth asset is overweighted all along the experiment, letting its firstposition only when the maturity is close, suggesting that the transition point becomescloser to T as T grows.One possible interpretation of this transition is the following. Rough processes are morevolatile than smooth processes in the short term but less volatile in the long term, sincetheir variances evolve approximately as t H . Thus, when there is not much time left, itseems natural to look for rough processes to obtain some performance. Conversely, themore time we have, the more we favor the smooth asset.
2. Vol-of-vol η : The volatility of volatility seems to have the opposite effect of thehorizon T over the investment strategy as shown on Figures 3a-3b. • η (cid:28) • η (cid:29) .0 0.1 0.2 0.3 0.4 0.5t0.100.110.120.130.140.15 t t H H t (( + 2 C [ t K ] * ) Y )( t )( ) H H (a) T = 0 . t t H H t (( + 2 C [ t K ] * ) Y )( t )( ) H H (b) T = 1 . t t H H t (( + 2 C [ t K ] * ) Y )( t )( ) H H (c) T = 2 . Figure 1: Effect of the horizon T on the optimal allocation strategy. When the horizon T approaches, the rough stock in blue is preferred. When T is big enough and the horizon farenough the smooth stock in green is preferred. (The parameters are: H = 0 . H = 0 . ρ = 0 , η = η = 1 , c i = − . .0 0.1 0.2 0.3 0.4 0.5 m X V ( X T ) Efficient frontier V ( m )= ( m X ) , T =0.5 H i =0.1 H i =0.25 H i =0.35 H i =0.5 (a) T = 0 . m X V ( X T ) Efficient frontier V ( m )= ( m X ) , T =1.9 H =0.1 H =0.25 H =0.35 H =0.5 (b) T = 1 . m X V ( X T ) Efficient frontier V ( m )= ( m X ) , T =2.8 H =0.1 H =0.25 H =0.35 H =0.5 (c) T = 2 . Figure 2: The efficient frontier in the case where both assets have the same roughness H = H = H . When the horizon T is small, the rough stocks allows for lower variance.When T increases we observe a transition and an inversion of the relation order. Indeed,when T increases, it is the smoothest stocks that allow for a lower variance.the horizon T , since increasing the vol-of-vol is similar to accelerating the time scale at acertain rate depending on H (think of the self-similarity property of fractional Brownianmotion). t (( + 2 C [ t K ] * ) Y )( t )( ) H H (a) η = 0 . t (( + 2 C [ t K ] * ) Y )( t )( ) H H (b) η = 1 . Figure 3: As the vol-of-vol η increases, it is as if the horizon T was decreasing and therough stock in blue begins to be preferred. H = 0 . H = 0 . T = 2 . , ρ = 0 , c i = − .
3. Correlation ρ : • ρ < ρ < ρ = 0 except that the transitionfrom T (cid:28) T (cid:29) T . This is what we observe on32igures 5a-5b-5c. We interpret this evolution towards the equally weighted portfolioas the possibility to be protected from volatility by holding both assets. • ρ > ρ >
0, there is nominimization of variance through diversification by going long in both assets. Thusin the case a positively correlated assets, it is natural to expect the emergence of astarker choice between the assets. In the ρ > buy rough sell smooth strategy as the one empirically found in Glasserman and He(2020). t t H H t (( + 2 C [ t K ] * ) Y )( t )( ) H H Figure 4: ρ = 0 .
7, when the two assets are positively correlated we recover the buy roughsell smooth strategy as it is described in Glasserman and He (2020). (the parameters are: H = 0 . H = 0 . T = 2 . η = η = 1, c i = − . • A theoretical study of influence of the parameters onto the investments strategies. • An empirical study testing the different conjectures made about the influence of someparameters such as
T, η, ρ, H , etc.
A Proof of the verification result
In this section, we provide a detailed proof of Theorem 3.1. It is well-known that Markowitzproblem (2.6) is equivalent to the following max-min problem, see e.g. (Pham, 2009,Proposition 6.6.5): V ( m ) = max η ∈ R min α ∈A (cid:110) E (cid:104)(cid:12)(cid:12) X T − ( m − η ) (cid:12)(cid:12) (cid:105) − η (cid:111) . (A.1)Thus, solving problem (2.6) involves two steps. First, the internal minimization problemin term of the Lagrange multiplier η has to be solved. Second, the optimal value of η for33 .0 0.1 0.2 0.3 0.4 0.5t0.100.120.140.160.180.20 t t H H t ( ) (( + 2 C [ t K ] * )1)( t )( ) H H (a) T = 0 . t t H H t ( ) (( + 2 C [ t K ] * )1)( t )( ) H H (b) T = 1 . t t H H t ( ) (( + 2 C [ t K ] * )1)( t )( ) H H (c) T = 2 . Figure 5: Effect of the horizon T on the optimal allocation strategy when the two assets arenegatively correlated ( ρ = − . H = 0 . H = 0 .
4. As T increases the smooth stock ingreen is more and more weighted in comparison to the rough one in blue. But the transitiontakes more time compared to the case ρ = 0, see Figures 1a-1c. η = η = 1 , c i = − . T reaches T = 2 .
4, as it could be forseen by thecondition of Lemma 5.12. 34he external maximization problem has to be determined. Let us then introduce the inneroptimization problem: ˜ V ( ξ ) := min α ∈A E (cid:104)(cid:12)(cid:12) X αT − ξ (cid:12)(cid:12) (cid:105) , ξ ∈ R . (A.2)First, we provide a verification result for the inner optimization problem (A.2) via thestandard completion of squares technique, see for instance Lim and Zhou (2002, Proposition3.1), Lim (2004, Proposition 3.3) and Chiu and Wong (2014, Theorem 3.1). Lemma A.1.
Assume there exists a solution triplet (Γ , Z , Z ) ∈ S ∞ F ([0 , T ] , R ) × L ,loc F ([0 , T ] , R d ) × L ,loc F ([0 , T ] , R N ) to the Riccati BSDE (3.2) such that Γ t > , for all t ≤ T . Fix ξ ∈ R , and assume that there exists an admissible control α ∗ ( ξ ) satisfying α ∗ t ( ξ ) = − (cid:0) λ t + Z t + CZ t (cid:1) (cid:16) X α ∗ ( ξ ) t − ξe − (cid:82) Tt r ( s ) ds (cid:17) , ≤ t ≤ T. (A.3) Then, the inner minimization problem (A.2) admits α ∗ ( ξ ) as an optimal feedback controland the optimal value is ˜ V ( ξ ) = Γ (cid:12)(cid:12)(cid:12) x − ξe − (cid:82) T r ( s ) ds (cid:12)(cid:12)(cid:12) . (A.4) Proof.
Let us first define ˜ X αt = X αt − ξe − (cid:82) Tt r ( s ) ds , for any α ∈ A . Then, by Itˆo’s lemmawe have d ˜ X αt = (cid:0) r ( t ) ˜ X αt + α (cid:62) t λ t (cid:1) dt + α (cid:62) t dB t , ≤ t ≤ T, ˜ X α = x − ξe − (cid:82) T r ( s ) ds . As a result, ˜ X α and X α have the same dynamics and ˜ X αT = X αT − ξ so that problem (A.2)can be alternatively written as min α ∈A E (cid:104)(cid:12)(cid:12) ˜ X αT (cid:12)(cid:12) (cid:105) . To ease notations, we set h t = λ t + Z t + CZ t . For any α ∈ A , Itˆo’s lemma combined with(3.2) and a completion of squares in α yield d (cid:16) Γ t (cid:12)(cid:12) ˜ X αt (cid:12)(cid:12) (cid:17) = (cid:12)(cid:12) ˜ X αt (cid:12)(cid:12) Γ t (cid:0) − r ( t ) + h (cid:62) t h t (cid:1) dt + Γ t (cid:12)(cid:12) ˜ X αt (cid:12)(cid:12) (cid:16)(cid:0) Z t (cid:1) (cid:62) dB t + (cid:0) Z t (cid:1) (cid:62) dW t (cid:17) + Γ t (cid:16) X αt (cid:0) r ( t ) ˜ X αt + α (cid:62) t λ t (cid:1) + α (cid:62) t α t (cid:17) dt + 2Γ t ˜ X αt α (cid:62) t dB t + 2 α (cid:62) t (cid:0) Z t + CZ t (cid:1) ˜ X αt dt = (cid:0) α t + h t ˜ X αt (cid:1) (cid:62) Γ t (cid:0) α t + h t ˜ X αt (cid:1) dt + 2Γ t ˜ X αt α (cid:62) t dB t + Γ t (cid:12)(cid:12) ˜ X αt (cid:12)(cid:12) (cid:16)(cid:0) Z t (cid:1) (cid:62) dB t + (cid:0) Z t (cid:1) (cid:62) dW t (cid:17) .
35s a consequence, using Γ T = 1, we get (cid:12)(cid:12) ˜ X αT (cid:12)(cid:12) =Γ (cid:12)(cid:12) ˜ X α (cid:12)(cid:12) + (cid:90) T (cid:0) α s + h s ˜ X αs (cid:1) (cid:62) Γ s (cid:0) α s + h s ˜ X αs (cid:1) ds + (cid:90) T s ˜ X αs α (cid:62) s dB s + (cid:90) T s (cid:12)(cid:12) ˜ X αs (cid:12)(cid:12) (cid:16)(cid:0) Z s (cid:1) (cid:62) dB s + (cid:0) Z s (cid:1) (cid:62) dW s (cid:17) . Note that the stochastic integrals (cid:90) . s ˜ X αs α (cid:62) s dB s , (cid:90) . Γ s (cid:12)(cid:12) ˜ X αs (cid:12)(cid:12) (cid:0) Z s (cid:1) (cid:62) dB s , (cid:90) . Γ t (cid:16) ˜ X αs (cid:17) (cid:0) Z s (cid:1) (cid:62) dW s , are well-defined since X α is continuous, ( α, Z , Z ) are in L ,loc F ([0 , T ]) and Γ in S ∞ F ([0 , T ] , R ).Furthermore, they are local martingales. Let { τ k } k ≥ be a common localizing increasingsequence of stopping times converging to T . Then, E (cid:104)(cid:12)(cid:12) ˜ X αT ∧ τ k (cid:12)(cid:12) (cid:105) = Γ (cid:12)(cid:12) ˜ X α (cid:12)(cid:12) + E (cid:104) (cid:90) T ∧ τ k (cid:0) α s + h s ˜ X αs (cid:1) (cid:62) Γ s (cid:0) α s + h s ˜ X αs (cid:1) ds (cid:105) . Since α ∈ A , X α satisfies (2.5), and so E (cid:104) sup t ≤ T | ˜ X αt | (cid:105) < ∞ . An application of thedominated convergence theorem on the left term combined with the monotone convergencetheorem on the right term, recall that Γ is S d + -valued, yields, as k → ∞ , E (cid:104)(cid:12)(cid:12) ˜ X αT (cid:12)(cid:12) (cid:105) = Γ (cid:12)(cid:12) ˜ X α (cid:12)(cid:12) + E (cid:104) (cid:90) T (cid:0) α s + h s ˜ X αs (cid:1) (cid:62) Γ s (cid:0) α s + h s ˜ X αs (cid:1) ds (cid:105) . Since Γ s is positive definite for any s ≤ T , we obtain that the optimal strategy α ∗ ( ξ ) isgiven by (A.3) and the optimal value of (A.2) is equal to˜ V ( ξ ) = Γ (cid:12)(cid:12) ˜ X α ∗ ( ξ )0 (cid:12)(cid:12) = Γ (cid:12)(cid:12) X − ξe − (cid:82) T r ( s ) ds (cid:12)(cid:12) , which gives (A.4).We next address the admissibility of the candidate for the optimal control. Lemma A.2.
Assume that there exists a solution triplet (Γ , Z , Z ) ∈ S ∞ F ([0 , T ] , R ) × L ,loc F ([0 , T ] , R d ) × L ,loc F ([0 , T ] , R N ) to the Riccati BSDE (3.2) such that (3.3) holds forsome p > and a constant a ( p ) given by (3.4) . Then, for any ξ ∈ R , there exists anadmissible control process α ∗ ( ξ ) satisfying (A.3) .Proof. Fix ξ ∈ R . We first prove that there exists a control α ∗ ( ξ ) satisfying (A.3). Forthis, we prove that the corresponding wealth equation (2.4) admits a solution. As in theproof of Lemma A.1, it is enough to consider the modified equation d ˜ X ∗ t = (cid:0) r ( t ) ˜ X ∗ t + λ (cid:62) t A t ˜ X ∗ t (cid:1) dt + (cid:0) A t ˜ X ∗ t (cid:1) (cid:62) dB t , ˜ X ∗ = x − ξe − (cid:82) T r ( s ) ds , A t = − (cid:0) λ t + Z t + CZ t (cid:1) , and then set X ∗ t = ˜ X ∗ t + ξe − (cid:82) Tt r ( s ) ds . By virtue of Itˆo’slemma the unique continuous solution is given by˜ X ∗ t = ˜ X ∗ exp (cid:16) (cid:90) t (cid:0) r ( s ) + λ (cid:62) s A s − A (cid:62) s A s (cid:1) ds + (cid:90) t A (cid:62) s dB s (cid:17) . Setting α ∗ t ( ξ ) := A t ˜ X ∗ t , we obtain that α ∗ ( ξ ) satisfies (A.3) with the controlled wealth X α ( ξ ) ∗ = X ∗ . The crucial step is now to obtain the admissibility condition (2.5). Forthat purpose, observe by virtue of (3.3), that the Dol´eans-Dade exponential E (cid:0)(cid:82) · A (cid:62) s dB s (cid:1) satisfies Novikov’s condition, and is therefore a true martingale. Whence, successive appli-cations of the inequality ab ≤ ( a + b ) / K > E (cid:104) sup t ∈ [0 ,T ] | ˜ X ∗ t | p (cid:105) ≤ K E (cid:104) sup t ∈ [0 ,T ] (cid:12)(cid:12) e (cid:82) t ( r ( s )+ λ (cid:62) s A s ) ds (cid:12)(cid:12) p (cid:105) + K E (cid:104) sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) e − (cid:82) t A (cid:62) s As ds + (cid:82) t A (cid:62) s dB s (cid:12)(cid:12)(cid:12) p (cid:105) ≤ K E (cid:104) e (cid:82) T p (cid:12)(cid:12) λ (cid:62) s A s (cid:12)(cid:12) ds (cid:105) + K E (cid:104) e − p (cid:82) T A (cid:62) s A s ds +2 p (cid:82) T A (cid:62) s dB s (cid:105) = K ( + ) , which is finite since ≤ E (cid:20) exp (cid:18) a ( p ) (cid:90) T (cid:0) | λ s | + | Z s | + | Z s | (cid:1) ds (cid:19)(cid:21) < ∞ , and, by virtue of the Cauchy-Schwarz inequality, ≤ (cid:16) E (cid:104) e (8 p − p ) (cid:82) T A (cid:62) s A s ds (cid:105)(cid:17) / (cid:16) E (cid:104) e − p (cid:82) T A (cid:62) s A s ds +4 p (cid:82) T A (cid:62) s dB s (cid:105)(cid:17) / ≤ (cid:16) E (cid:104) e a ( p ) (cid:82) T ( | λ s | + | Z s | + | Z s | ) ds (cid:105)(cid:17) / × < ∞ , where we used assumption (H2) combined with Novikov’s condition to the Dol´eans-Dadeexponential E (4 p (cid:82) · A (cid:62) s dB s ). Finally, to get that α ∗ ( ξ ) is admissible, we are left to prove37hat α ∗ ( ξ ) ∈ L F ([0 , T ] , R d ). Let 2 /p + 1 / ˆ q = 1, by H¨older’s inequality we obtain E (cid:20)(cid:90) T | α ∗ s ( ξ ) | ds (cid:21) = E (cid:20)(cid:90) T | A s ˜ X ∗ s | ds (cid:21) ≤ E (cid:34) sup t ∈ [0 ,T ] | ˜ X ∗ t | (cid:90) T | A s | ds (cid:35) ≤ (cid:32) E (cid:34) sup t ∈ [0 ,T ] | ˜ X ∗ t | p (cid:35)(cid:33) /p (cid:32) E (cid:34)(cid:18)(cid:90) T | A s | ds (cid:19) ˆ q (cid:35)(cid:33) / ˆ q ≤ C (cid:32) E (cid:34) sup t ∈ [0 ,T ] | ˜ X ∗ t | p (cid:35)(cid:33) /p (cid:32) E (cid:34)(cid:18)(cid:90) T (cid:0) | λ s | + | Z s | + | Z s | (cid:1) ds (cid:19) ˆ q (cid:35)(cid:33) / ˆ q < ∞ , where the last term is finite due to condition (3.3) and the inequality | z | q ≤ c q e | z | . Theproof is complete.Finally, combining the above, we deduce the solution for the outer optimization problem(2.6) under a non-degeneracy condition on the solution Γ to the Riccati BSDE, yieldingTheorem 3.1. Proof of Theorem 3.1.
From Lemmas A.1 and A.2, we have that the max-min problem(A.1) (which is equivalent to the Markowitz problem (2.6)) is equivalent tomax η ∈ R J ( η ) , with J ( η ) = Γ (cid:12)(cid:12) X − ( m − η ) e − (cid:82) T r ( s ) ds (cid:12)(cid:12) − η . Furthermore, condition (H1): Γ < e (cid:82) T r ( s ) ds , ensures that the quadratic function J isstrictly concave. This yields that the maximum is achieved from the first-order condition J (cid:48) ( η ∗ ) = 0, which gives η ∗ = Γ e − (cid:82) T r ( s ) ds (cid:16) x − me − (cid:82) T r ( s ) ds (cid:17) − Γ e − (cid:82) T r ( s ) ds , and thus ξ ∗ = m − η ∗ is given by (3.6). We conclude that the optimal control is equalto α ∗ = α ∗ ( ξ ∗ ) as in (3.5), and by (A.1), the optimal value of (2.6) is equal to V ( m ) =˜ V ( ξ ∗ ) − ( η ∗ ) , given by (3.7). 38 Proofs of some technical lemmas
B.1 Proof of Lemma 5.10
Proof.
Fix s ≤ T and Θ ∈ R d × N . We first note that | λ s | + (cid:12)(cid:12) Z s (cid:12)(cid:12) + (cid:12)(cid:12) Z s (cid:12)(cid:12) = | Θ g s ( s ) | + 4 | (( Ψ s K η ) ∗ g s ) ( s ) | . Using 5.8-(i), and denoting by ψ op s the operator induced by the kernel ψ s there, we write | (( Ψ s K η ) ∗ g s ) ( s ) | = | − ((Θ (cid:62) Θ K η ) ∗ g s )( s ) + (( ψ op s K η ) ∗ g s )( s ) | = | + | ≤ | | + | | ) . An application of the Cauchy-Schwarz inequality combined with (5.11) leads to | | = (cid:12)(cid:12)(cid:12)(cid:12) − (cid:90) T η T K ( z, s ) (cid:62) Θ (cid:62) Θ g s ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ | η | | ΘΘ (cid:62) | sup u (cid:48) ≤ T (cid:90) T | K ( z, u (cid:48) ) | du (cid:48) (cid:90) T | g s ( u ) | du. Similarly, = (cid:18)(cid:90) T η (cid:62) (cid:18)(cid:90) T K ( r, s ) (cid:62) ψ s ( r, z ) dr (cid:19) g s ( z ) dz (cid:19) ≤| η | (cid:18)(cid:90) T (cid:90) T | K ( r, s ) | | ψ s ( r, z ) | drdz (cid:19) (cid:18)(cid:90) T | g s ( z ) | dz (cid:19) ≤| η | sup u (cid:48) ≤ T (cid:90) T | K ( r, u (cid:48) ) | dr (cid:18)(cid:90) T (cid:90) T | ψ s ( r, z ) | drdz (cid:19) (cid:18)(cid:90) T | g s ( z ) | dz (cid:19) , where we stress that ψ s is the only term on the right hand side depending on Θ. Let usnow show that there exists a scalar κ independent of Θ such thatsup s ∈ [0 ,T ] (cid:90) T (cid:90) T | ψ s ( r, z ) | drdz ≤ κ | Θ | . (B.1)Recall from (5.17) that we have ψ t = − ˆ R ∗ Θ (cid:62) Θ − Θ (cid:62) Θ ˆ R − ˆ R ∗ Θ (cid:62) (cid:63) R Θ t Θ − Θ (cid:62) R Θ t (cid:63) Θ ˆ R − ˆ R ∗ Θ (cid:62) (cid:63) R Θ t (cid:63) Θ ˆ R − ˆ R ∗ Θ (cid:62) (cid:63) Θ ˆ R + Θ (cid:62) R Θ t Θ . Thus, recalling (5.15), to obtain (B.1) it is enough to show thatsup t ∈ [0 ,T ] (cid:90) T (cid:90) T | R Θ t ( s, u ) | dsdu ≤ c | Θ | , (B.2)39here c does not depend on Θ. For this recall that R Θ t is the resolvent of − ˜Σ t Θ (cid:62) whichimplies that R Θ t = (Id + 2Θ ˜Σ t Θ T ) − − Id. Since, for each t ≤ T , Θ ˜Σ t Θ (cid:62) is a positivesymmetric operator on L ([0 , T ] , R d ) induced by a continuous kernel, an application ofMercer’s theorem, see Shorack and Wellner (2009, Theorem 1, p.208), yields the existenceof a countable orthonormal basis ( e nt, Θ ) n ≥ of L ([0 , T ] , R d ) such that2Θ ˜Σ t ( s, u )Θ = (cid:88) n ≥ λ nt, Θ e nt, Θ ( s ) e nt, Θ ( u ) (cid:62) , where λ nt, Θ ≥
0, for all n ≥
1. Consequently R Θ t ( s, u ) = (cid:88) n ≥ − λ nt, Θ λ nt, Θ e nt, Θ ( s ) e nt, Θ ( u ) (cid:62) , which yields (cid:90) T (cid:90) T | R Θ t ( s, u ) | dsdu = (cid:88) n ≥ ( λ nt, Θ ) (1 + λ nt, Θ ) ≤ (cid:88) n ≥ ( λ nt, Θ ) = (cid:90) T (cid:90) T |
2Θ ˜Σ t ( s, u )Θ (cid:62) | dsdu ≤ | Θ | sup t ≤ T (cid:90) T (cid:90) T | ˜Σ t ( s, u ) | dsdu. This gives (B.2), so that (B.1) holds and the claimed estimate follows.
B.2 Proof of Lemma 5.12
Proof.
We follow the same ideas as in Abi Jaber (2019b, Section 2.2). Fix s ≤ T , we firstdecompose g on ( e n ) n ≥ in the form g ( s ) = (cid:88) n ≥ γ n e n ( s ) , with γ n = (cid:82) T g ( s ) (cid:62) e n ( s ) ds . By the orhonormality of ( e n ) n ≥ and ( f n ) n ≥ , plugging (5.27)in the dynamics of g s in (5.20) leads to the decomposition g s ( u ) = u ≥ s (cid:88) n ≥ (cid:16) γ n + √ λ n ξ n (cid:17) e n ( u ) , where ξ n = (cid:82) T ( f n ( r )) (cid:62) dW r are independent standard Gaussians. Whence, due to theorthonormality of ( e n ) n ≥ , it follows that (cid:90) T (cid:18) a | g s ( s ) | + b (cid:90) T | g s ( u ) | du (cid:19) ds ≤ ( a + bT ) (cid:88) n ≥ (cid:16) γ n + √ λ n ξ n (cid:17) . ξ n ) n ≥ , we get that E (cid:104) exp (cid:16) (cid:90) T (cid:0) a | g s ( s ) | + b (cid:90) T | g s ( u ) | du (cid:1) ds (cid:17)(cid:105) , is bounded by E (cid:104) exp (cid:16) (cid:88) n ≥ ( a + bT ) (cid:0) γ n + √ λ n ξ n (cid:1) (cid:17)(cid:105) = (cid:89) n ≥ E (cid:104) exp (cid:16) ( a + bT ) (cid:0) γ n + √ λ n ξ n (cid:1) (cid:17)(cid:105) = (cid:89) n ≥ exp (cid:16) ( a + bT ) ( γ n ) − a + bT ) λ n (cid:17) − a + bT ) λ n , where the last equality follows from the standard chi-square distribution since 2( a + bT ) λ n < n ≥ − a + bT ) (cid:88) n ≥ λ n ≤ (cid:89) n ≥ (1 − a + bT ) λ n ) ≤ exp a + bT ) (cid:88) n ≥ λ n , and (5.28), we obtain that (cid:81) n ≥ (1 − a + bT ) λ n ) < ∞ . For the numerator, since λ n →
0, as n → ∞ , (cid:16) − a + bT ) λ n (cid:17) n ≥ is uniformly bounded by a constant c > (cid:89) n ≥ exp (cid:18) ( a + bT ) ( γ n ) − a + bT ) λ n (cid:19) ≤ exp (cid:0) c ( a + bT ) (cid:107) g (cid:107) L (0 ,T ) (cid:1) < ∞ . The proof is complete.
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