An Equilibrium Model for the Cross-Section of Liquidity Premia
AAn Equilibrium Model for the Cross-Section of Liquidity Premia ∗ Johannes Muhle-Karbe † Xiaofei Shi ‡ Chen Yang § November 26, 2020
Abstract
We study a risk-sharing economy where an arbitrary number of heterogenous agents tradesan arbitrary number of risky assets subject to quadratic transaction costs. For linear statedynamics, the forward-backward stochastic differential equations characterizing equilibrium as-set prices and trading strategies in this context reduce to a system of matrix-valued Riccatiequations. We prove the existence of a unique global solution and provide explicit asymptoticexpansions that allow us to approximate the corresponding equilibrium for small transactioncosts. These tractable approximation formulas make it feasible to calibrate the model to timeseries of prices and trading volume, and to study the cross-section of liquidity premia earned byassets with higher and lower trading costs. This is illustrated by an empirical case study.
This paper is dedicated to the memory of our dear colleague Mark H.A. Davis, whose seminalworks [15, 16] ushered in the mathematical analysis of models with transaction costs.
Mathematics Subject Classification: (2010)
JEL Classification:
C68, D52, G11, G12.
Keywords: asset pricing, Radner equilibrium, transaction costs, liquidity premia
In the capital asset pricing model and many of its descendants, agents earn risk premia for holdingassets whose payoffs are uncertain. A number of influential empirical studies [4, 9, 35] suggest that– in reality – agents are also compensated for holding securities that are difficult to trade. To wit,if one sorts assets based on various measures of liquidity, then the returns earned by portfolioscomposed of the more liquid ones are systematically lower than for portfolios of less liquid assets.The theoretical underpinnings of these liqiuidity premia have been studied in an active literaturegoing back to the seminal work of [13]. This paper (and many more recent studies) takes a partialequilibrium approach, where the asset price dynamics are specified endogenously. Liquidity premiathen refer to the amount by which the risky assets’ expected returns have to be increased compared ∗ The authors are grateful to Jean-Philippe Bouchaud, Ibrahim Ekren, Martin Herdegen and Robert Pego forfruitful discussions, and to Steven E. Shreve for pertinent remarks on an earlier version of the manuscript. † Imperial College London, Department of Mathematics, email [email protected] . Research sup-ported by the CFM-Imperial Institute of Quantitative Finance. ‡ Columbia University, Department of Statistics, email [email protected] . § Chinese University of Hong Kong, Department of Systems Engineering and Engineering Management, email [email protected] . a r X i v : . [ q -f i n . GN ] N ov o a hypothetical frictionless version of the asset, in order to offset the utility losses caused by thecosts of trading.Another strand of research derives equilibrium asset prices with transaction costs endogenouslyby matching supply and demand [21, 24, 28, 38, 43, 44, 45]. This allows to study how changes inliquidity feed back into asset prices, e.g., how liquidity premia are affected by the reduction of thefees charged by an exchange or the introduction of a financial transaction tax.Yet, equilibrium models with a single illiquid risky asset still cannot say anything about the cross section of liquidity premia across a spectrum of different assets – that is, the subject ofthe empirical work of [4, 9, 35]. Equilibrium models with several illiquid assets lead to formidablecomputational challenges. These difficulties are of course only exacerbated if one moves beyond two(representative) agents that are typically assumed for tractability. To wit, even the most tractablemodels with linear state dynamics and quadratic transaction costs [19, 24, 38] then lead to coupledsystems of matrix Riccati equations. Whereas general well-posedness results are available for partialequilibrium models [5, 7, 19, 26] or for models with exogenously given constant volatility [8], theonly known results concerning the existence of equilibrium prices require the restrictive assumptionthat the agents’ preferences are sufficiently similar [21], even in the case of only a single illiquidasset and just two agents.In the present study, we establish the existence of equilibrium prices for an arbitrary numberof illiquid risky assets that are traded by an arbitrary number of agents. These agents have mean-variance preferences as in [18, 19] and trade to share the risk inherent in the fluctuations of theirendowment streams, subject to a deadweight quadratic transaction cost as in [2, 18, 19]. For assetsthat pay exogenous liquidating dividends at a finite terminal time, the “Radner equilibrium” wherethe agents act as price takers then can be characterized by a fully-coupled system of forward-backward stochastic differential equations (FBSDEs). If the terminal dividends and the volatilitiesof the agents’ endowment streams are linear in the driving Brownian motions, then this FBSDEsystem can be reduced to a fully-coupled system of matrix-valued ordinary differential equationsof Riccati form.For the simplest case of a single risky asset traded by two agents, existence for this systemhas been established using Picard iteration by [21]. However, even in this low-dimensional setting,establishing the convergence of the iteration scheme requires the restrictive assumption that theagents’ risk aversions are sufficiently similar. In this paper, we show that this assumption issuperfluous, in that the matrix Riccati system has a unique global solution even for an arbitrarynumber of agents and risky assets.In order to facilitate the calibration of the model to time-series data, we complement thismain result with rigorous asymptotic expansions. In the practically relevant limiting regime ofsmall transaction costs, this leads to explicit formulas for the impact of illiquidity on price levels,volatilities, and the cross section of liquidity premia that are earned by assets with different tradingcosts.To bring these theoretical results to life, we test them using an empirical case study following [1].To wit, we sort the large-cap stocks in the S&P index by Amihud’s “ILLIQ” measure for liquidity [3],leading to three risky portfolios with high, medium, and low liquidity. In the frictionless versionof our model, equilibrium returns solely compensate for risk and turn out to be very similar for allthree portfolios. Using our asymptotic expansions, the calibration of the frictional version of themodel to time series of prices and trading volumes is still feasible. When trading costs are taken intoaccount, the equilibrium returns of the high-liquidity portfolio are indeed decreased in line with the An alternative class of tractable models considers “overlapping generations” of agents that buy the securitieswhen born and then either sell them after a prespecified holding time [1] or gradually (and following a deterministictrajectory) over their lifetime [43].
Notation
Throughout, we fix a filtered probability space (Ω , F , ( F t ) t ∈ [0 ,T ] , P ) with finite timehorizon T >
0, supporting a D -dimensional standard Brownian motion ( W t ) t ∈ [0 ,T ] . For p ≥ L p ( R m ) for the R m -valued random variables X satisfying || X || p := E [ || X || p ] /p < ∞ anddenote by H p ( R m × n ) the R m × n -valued, progressively measurable processes X = ( X t ) t ∈ [0 ,T ] thatsatisfy (cid:107) X (cid:107) H p := (cid:32) E (cid:34)(cid:18)(cid:90) T || X t || dt (cid:19) p/ (cid:35)(cid:33) /p < ∞ . Here, for any vector or matrix, || · || is the Frobenius norm, i.e., the square root of the sum ofsquared entries. For p ≥ S p ( R m ) denotes the R m -valued, progressively measurable processes X = ( X t ) t ∈ [0 ,T ] with continuous paths for which sup ≤ t ≤ T || X t || ∈ L p ( R m ).Finally, we write m for the all-ones vector in R m and I m for the identity matrix in R m × m ; theKronecker product of matrices A ∈ R m × n and B ∈ R m (cid:48) × n (cid:48) is denoted by A ⊗ B := A B · · · A n B ... . . . ... A m B · · · A mn B ∈ R mm (cid:48) × nn (cid:48) , and the Riemannian mean of two symmetric and positive definite matrices A, B ∈ R m × m is denotedby A B := A / ( A − / BA − / ) / A / . We consider N ≥ n = 1 , , . . . , N who receive (cumulative) random endow-ments dζ nt = ( ξ nt ) (cid:62) dW t , where ξ n ∈ H ( R D ) . (2.1)To simplify the analysis below, we follow [28] and assume that the agents’ aggregate endowment iszero ( (cid:80) Nn =1 ξ n = 0). An additional finite-variation drift would not affect the optimizers and hence the equilibrium prices due to themean-variance form of the optimization problems (3.1) and (4.1) below.
3o hedge against the fluctuations of their endowment streams driven by the D -dimensionalBrownian motion, the agents trade a safe and K ≤ D risky assets. The price of the safe asset isexogenous and normalized to one. The prices of the risky assets have dynamics dS t = µ t dt + σ t dW t , S T = S . (2.2)Here, the liquidating dividend S ∈ L ( R K ) is given exogenously. In contrast, the expected returnsprocess µ ∈ H ( R K ) and the volatility process σ ∈ H ( R K × D ) are to be determined endogenouslyby matching the agents’ demand to the fixed supply s ∈ R K of the risky assets. As a benchmark, we first consider the frictionless version of the model. Starting from fixed initialpositions ϕ n − ∈ R K , n = 1 , . . . , N that clear the market ( (cid:80) Nn =1 ϕ n − = s ), the agents choose theirpositions ( ϕ t ) t ∈ [0 ,T ] in the risky assets to maximize one-period expected returns penalized for thecorresponding variances as in [17, 18, 19, 25, 30, 31]. Without transaction costs, the continuous-timeversion of this criterion is¯ J nT ( ϕ ) = E (cid:20)(cid:90) T ( ϕ (cid:62) t dS t + dζ nt ) − γ n d (cid:104) (cid:82) · ϕ (cid:62) u dS u + ζ n (cid:105) t (cid:21) = E (cid:20)(cid:90) T (cid:16) ϕ (cid:62) t µ t − γ n (cid:107) σ (cid:62) t ϕ t + ξ nt (cid:107) (cid:17) dt (cid:21) . (3.1)Here, γ n > n ; we assume without loss of generality that γ N = max { γ , . . . , γ N } . (3.2)To ensure that the goal functional (3.1) is well defined for any price dynamics (2.2) with µ ∈ H ( R K ), σ ∈ H ( R K × D ), we focus on admissible strategies ϕ ∈ H ( R K ). Given that the covariance matrix σ t σ (cid:62) t ∈ R K × K is invertible for every t ∈ [0 , T ] , each agent’s optimal strategy for the frictionlessproblem (3.1) is readily determined by pointwise optimization as ϕ nt = (cid:0) σ t σ (cid:62) t (cid:1) − µ t γ n − (cid:0) σ t σ (cid:62) t (cid:1) − σ t ξ nt , t ∈ [0 , T ] . (3.3)We are interested in “competitive” Radner equilibria [36], where each (small) agent takes theprice dynamics of the risky assets as given in their individual optimization problem (3.1): Definition 3.1.
A price process (2.2) for the risky assets is called a (Radner) equilibrium if:i) (
Individual Optimality ) the corresponding individual optimization problem (3.1) has a solu-tion ϕ n for each agent n = 1 , . . . , N ;ii) ( Market Clearing ) the agents’ total demand matches the supply of the risky assets at all times,in that (cid:80) Nn =1 ϕ nt = s for all t ∈ [0 , T ] . The precise notion of admissibility is not crucial. We just need to ensure that the local martingale part of thewealth process (cid:82) · ϕ t dS t is a true martingale. This will be inherited from the terminal condition S in the equilibrium we construct below. S with dynamics (2.2), matching the sum of the agents’ correspondingdemands (3.3) to the supply s requires the following relation between the equilibrium expectedreturns ¯ µ t and volatility matrix ¯ σ t :¯ µ t = ¯ γ ¯ σ t ¯ σ (cid:62) t s, t ∈ [0 , T ] , where ¯ γ = (cid:32) N (cid:88) n =1 γ n (cid:33) − . (3.4)Together with the terminal condition from (2.2), it follows that equilibrium prices correspond tosolutions of the following system of quadratic backward stochastic differential equations (BSDEs): d ¯ S t = (cid:0) ¯ γ ¯ σ t ¯ σ (cid:62) t s (cid:1) dt + ¯ σ t dW t , S T = S . (3.5)If the terminal condition S is linear in the driving Brownian motion, then the BSDE (3.5) can besolved explicitly, leading to an equilibrium price with Bachelier dynamics. Assumption 3.2.
The terminal dividend is of the linear form S = αW T + βT, for β ∈ R D and α ∈ R K × D with rank( α ) = rank( αα (cid:62) ) = K . Proposition 3.3.
Under Assumption 3.2, a solution of the BSDE system (3.5) and in turn africtionless equilibrium price is given by d ¯ S t = (cid:0) ¯ γαα (cid:62) s (cid:1) dt + αdW t , ¯ S = (cid:0) β − ¯ γαα (cid:62) s (cid:1) T. (3.6) This equilibrium is unique among price dynamics with uniformly bounded volatility.
Now suppose as in [2, 18, 19] that trading incurs quadratic costs on the turnover rate ˙ ϕ t = dϕ t /dt .The frictional analogue of the mean-variance goal functional (3.1) then is J nT ( ˙ ϕ ) = E (cid:20)(cid:90) T (cid:16) ϕ (cid:62) t µ t − γ n (cid:107) σ (cid:62) t ϕ t + ξ nt (cid:107) −
12 ˙ ϕ (cid:62) t Λ ˙ ϕ t (cid:17) dt (cid:21) . (4.1)Here, the transaction cost matrix Λ is symmetric and positive definite, and we focus on admissible trading strategies that are absolutely continuous with rate ˙ ϕ ∈ H ( R K ). Remark 4.1.
As in [19, Section 3.2], the deadweight transaction costs can be seen as a compensa-tion paid to liquidity providers who intermediate between the agents we model in the present paper.Non-trivial off-diagonal elements of Λ then correspond to cross price impact due to each assets’contribution to the intermediaries’ portfolio. Alternatively, if the quadratic costs are interpreted asmore tractable proxies for linear costs such as bid-ask spreads or a transaction tax, then a diagonalmatrix is the natural specification for Λ. As pointed out by [18], symmetry of Λ can be assumed without loss of generality because otherwise the sym-metrized version (Λ + Λ (cid:62) ) / / for the unique symmetric and positive definite square root of Λ, and note that Λ andΛ / both are invertible. The corresponding positions then also automatically belong to H ( R K ) as in the frictionless case, so that thefrictional goal functional is well defined for expected returns process µ ∈ H ( R K ) and volatility matrix σ ∈ H ( R K × D ). ϕ nt = E t (cid:104) (cid:90) Tt (cid:0) µ u − γ n σ u ( σ (cid:62) u ϕ nu + ξ nu ) (cid:1) du (cid:105) = E t (cid:104) (cid:90) T (cid:0) µ u − γ n σ s ( σ (cid:62) s ϕ nu + ξ nu ) (cid:1) du (cid:105) + (cid:90) t (cid:0) γ n σ u ( σ (cid:62) u ϕ nu + ξ nu ) − µ u (cid:1) du. (4.2)To clear the market, the sum of all agents’ trading rates has to vanish at all times. Therefore, aftersumming the agents’ first-order conditions (4.2), both the martingale and the drift terms need tovanish for all t ∈ [0 , T ]. The frictional equilibrium return in turn has to satisfy0 = N (cid:88) n =1 (cid:16) µ t − γ n σ t ( σ (cid:62) t ϕ nt + ξ nt ) (cid:17) . Taking into account the market clearing condition (cid:80) Nn =1 ϕ nt = s and recalling that the aggregateendowment is zero ( (cid:80) Nn =1 ξ nt = 0), the price dynamics (2.2) therefore again lead to a BSDE systemfor the equilibrium asset price: dS t = (cid:32) γ N N σ t σ (cid:62) t s + σ t σ (cid:62) t N N − (cid:88) n =1 ( γ n − γ N ) (cid:16) σ (cid:62) t ϕ nt + ξ nt (cid:17)(cid:33) dt + σ t dW t , S T = S . (4.3)However, these equations are now no longer autonomous but coupled to the forward equations forthe optimal positions, dϕ nt = ˙ ϕ nt dt, ϕ n = ϕ n − , n = 1 , . . . , N − , (4.4)as well as the backward equations for the corresponding optimal trading rates ˙ ϕ nt implied by thefirst-order conditions (4.2): d ˙ ϕ nt = Λ − (cid:0) γ n σ t ( σ (cid:62) t ϕ nt + ξ nt ) − µ t (cid:1) dt + ˙ Z nt dW t ˙ ϕ nT = 0 , n = 1 , . . . , N − , (4.5)= Λ − σ t (cid:0) σ (cid:62) t ( γ n ϕ nt − N (cid:80) N − m =1 ( γ m − γ N ) ϕ mt ) + ( γ n ξ nt − N (cid:80) N − m =1 ( γ m − γ N ) ξ mt ) − γ N N σ (cid:62) t s (cid:1) dt + ˙ Z nt dW t . (The position and trading rate of agent N are in turn pinned down by market clearing.) To expressthis forward-backward system more compactly in matrix-vector notation, we write ϕ t := ϕ t ... ϕ N − t , ˙ ϕ t := ˙ ϕ t ...˙ ϕ N − t , ˙ Z t := ˙ Z t ...˙ Z N − t , ξ t := ξ t ... ξ N − t , (4.6)and define the risk-aversion matrixΓ := diag { γ , · · · , γ N − } − N N − (cid:62) N − diag { γ − γ N , · · · , γ N − − γ N } ∈ R ( N − × ( N − . (4.7)The above discussion then can be summarized as follows:6 emma 4.2. Suppose there exists a solution ( ϕ, ˙ ϕ, ˙ Z, S, σ ) ∈ H ( R K ( N − ) × H ( R K ( N − ) × H ( R K ( N − × D ) × S ( R K ) × H ( R K × D ) of the following FBSDE system: dϕ t = ˙ ϕ t dt, ϕ = ϕ − ,d ˙ ϕ t = (cid:18) (Γ ⊗ Λ − σ t σ (cid:62) t ) ϕ t + (Γ ⊗ Λ − σ t ) ξ t − γ N N N − ⊗ Λ − σ t σ (cid:62) t s (cid:19) dt + ˙ Z t dW t , ˙ ϕ T = 0 ,dS t = (cid:32) γ N N σ t σ (cid:62) t s + σ t N N − (cid:88) n =1 ( γ n − γ N ) (cid:16) σ (cid:62) t ϕ nt + ξ nt (cid:17)(cid:33) dt + σ t dW t , S T = S . Then, S is a Radner equilibrium with transaction costs, in that the trading rates ˙ ϕ , . . . , ˙ ϕ N − and ˙ ϕ N = − (cid:80) N − n =1 ˙ ϕ n are optimal for the frictional optimization problems (4.1) of agents n = 1 , . . . , N ,and clear the market. For the simplest case of a single risky asset and two agents, the FBSDE system from Lemma 4.2has been studied by [21]. More specifically, local existence is established there under the restrictivecondition that the agents’ risk aversion coefficients are sufficiently similar. If the terminal condition S is linear in the driving Brownian motion as in Assumption 3.2 and the volatilities ξ nt of theagents’ endowments are of the same linear form, then the FBSDE system can be reduced to asystem of Riccati equations by an appropriate ansatz. However, the system consists of four fullycoupled equations even for a single risky asset and two agents, so that existence (established viaPicard iteration) is again only known if the agents’ preferences are sufficiently homogenous [21,Theorem 5.2]. These difficulties are of course only exacerbated for multiple assets and agents,because each of the Riccati equations becomes matrix valued in this case.In the present paper, we overcome these difficulties and establish global existence for the FBSDEsystem from Lemma 4.2 for linear terminal conditions and endowment volatilities:
Assumption 4.3.
The volatilities of the agents’ endowment streams (2.1) are of the form ξ nt = ξ n W t , for ξ n ∈ R D × D . With a slight abuse of notation, we set ξ = [ ξ , . . . , ξ N − ] (cid:62) ∈ R ( N − D × D .Like [21, Theorem 5.2], our existence result in Theorem 4.5 exploits the link between the FBSDEsystem and a system of Riccati ODEs. However, to make the latter more amenable to analyticalestimates, we perform a number of changes of variables that allow to reduce the number of coupled(matrix) equations from four to two. Standard comparison arguments still do not apply to thismultidimensional system, in particular, when the equations are matrix-valued for many risky assetsand agents. However, another reparametrization finally leads to a system where the right-hand sideof one equation is linear in this component. A matrix version of the variation of constants formulain turn allows to derive bounds on the unique local solution of this equation. This in turn finally If one penalizes squared inventories rather than the corresponding fluctuations (which depend on the endogenous volatility), then the FBSDE system becomes linear and can be analyzed in very general settings, in particular, forarbitrary numbers of agents, compare [6, 33]. If all agents have the same risk aversion coefficient, then the BSDE for the frictional equilibrium price decouplesfrom the other components of the FBSDE system in Lemma 4.2 and reduces to its frictionless counterpart similarlyas in [21]. Similar risk aversions in turn lead to frictional equilibrium prices in the vicinity of their frictionlesscounterparts, so that existence can be established using a Picard iteration under smallness conditions inspired by [41]. Equilibria in linear-quadratic models are also linked to systems of nonlinear equations in [24, 38], but the existenceof a unique solution is left open in these studies. scalar function – thenorm of the local solution of the other equation on the cone of positive semidefinite matrices.To formulate these results, we first state our global wellposedness result for the reduced ODEsystem. (The proof is deferred to Section 7.2 for better readability.)
Lemma 4.4.
Define c := (cid:2) c · · · c N − (cid:3) (cid:62) , where c n := ¯ γ (cid:18) γ n − γ N (cid:19) > . There exists a unique global solution ( F, H ) on [0 , T ] of the following initial value problem: F (cid:48) = Γ ⊗ (cid:16) α + ( c ⊗ I K ) (cid:62) H (cid:17) (cid:16) α + ( c ⊗ I K ) (cid:62) H (cid:17) (cid:62) − F (cid:0) I N − ⊗ Λ − (cid:1) F, F (0) = 0 ,H (cid:48) = (cid:16) Γ ⊗ (cid:16) α + ( c ⊗ I K ) (cid:62) H (cid:17)(cid:17) ξ − F (cid:0) I N − ⊗ Λ − (cid:1) H, H (0) = 0 . (4.8) Moreover, F takes values in the positive semidefinite matrices. With the solution of the matrix Riccati equations (4.8) at hand, we can then construct asolution of the FBSDE system from Lemma 4.2. The latter in turn leads to a Radner equilibriumwith transaction costs. (The proof is again delegated to Section 7.2 for better readability.)
Theorem 4.5.
With the functions F , H from Lemma 4.4, let Φ( τ ) be the solution of the linearmatrix ODE Φ (cid:48) ( t ) = (cid:16) I N − ⊗ Λ − / (cid:17) F (cid:62) ( T − t ) (cid:16) I N − ⊗ Λ − / (cid:17) Φ( t ) , Φ(0) = I K ( N − , (4.9) and define Ψ( r ; t ) := (cid:16) I N − ⊗ Λ / (cid:17) Φ( r )Φ − ( t ) (cid:16) I N − ⊗ Λ − / (cid:17) , for r, t ∈ [0 , T ] . (4.10) Suppose Assumptions 3.2 and 4.3 are satisfied. With the frictionless equilibrium price and volatility ( ¯ S, ¯ σ ) from Proposition 3.3, a solution ( ϕ, ˙ ϕ, ˙ Z, ¯ S + Y − ( c ⊗ Λ) (cid:62) ˙ ϕ, ¯ σ − ( c ⊗ Λ) (cid:62) ˙ Z ) of the FBSDEsystem from Lemma 4.2 is then given by ϕ t = ¯ ϕ + Ψ (cid:62) (0; t ) ( ϕ − − ¯ ϕ ) − (cid:90) t Ψ (cid:62) ( r ; t ) (cid:0) I N − ⊗ Λ − (cid:1) H ( T − r ) W r dr, (4.11)˙ ϕ t = − (cid:0) I N − ⊗ Λ − (cid:1) [ F ( T − t ) ( ϕ t − ¯ ϕ ) + H ( T − t ) W t ] , (4.12) Y t = − ¯ γ (cid:18)(cid:90) T − t (cid:16) ( c ⊗ I K ) (cid:62) Hα (cid:62) + αH (cid:62) ( c ⊗ I K ) + ( c ⊗ I K ) (cid:62) HH (cid:62) ( c ⊗ I K ) (cid:17) ( r ) dr (cid:19) s, (4.13)˙ Z t = − (cid:0) I N − ⊗ Λ − (cid:1) H ( T − t ) . (4.14) In particular, S = ¯ S + Y t − ( c ⊗ Λ) (cid:62) ˙ ϕ t is a Radner equilibrium with transaction costs. In much of the literature, positive definite matrices are additionally required to be symmetric. This does notgenerally hold for F , however, so that the arguments below need to be developed without this convenient property. This is the exponential of (cid:82) · ( I N − ⊗ Λ − / ) F (cid:62) ( r )( I N − ⊗ Λ − / ) dr in the scalar case or if the matrices involvedcommute. Small-costs Asymptotics
The Riccati system (4.8) can be solved numerically using standard ODE solvers by vectorizingthe matrix equations. In order to glean qualitative insights into the structure of the solution andfacilitate the calibration of the model parameter to time series data, it is nevertheless instructive toexpand the solution in the practically relevant limiting regime of small transaction costs. (Again,the proof of Theorem 5.1 is deferred to Section 7.4 for better readability.)
Theorem 5.1.
Fix a positive definite matrix ¯Λ and set M := (cid:18) c (cid:62) Γ / ⊗ ¯Λ (cid:16) ¯Λ αα (cid:62) (cid:17) − α (cid:19) . For small transaction costs
Λ = λ ¯Λ with λ → , the difference between the frictional equilibriumvolatility from Theorem 4.5 and its frictionless counterpart ¯ σ = α from Proposition 3.3 has thefollowing leading-order expansion: (cid:90) T (cid:107) σ t − ¯ σ − λ / M ξ (cid:107) op dt = O ( λ ) . (5.1) For ϕ − = ¯ ϕ , the leading-order adjustment of the initial price level can be approximated as S − ¯ S = − λ / ¯ γ (cid:16) M ξα (cid:62) + αξ (cid:62) M (cid:62) (cid:17) sT + O ( λ ) . (5.2) Finally, the equilibrium expected returns satisfy (cid:13)(cid:13)(cid:13) µ − (cid:16) ¯ µ + ∆¯ µ + λ / (cid:16) c (cid:62) Γ / ⊗ (cid:16) ¯Λ αα (cid:62) (cid:17)(cid:17) ˙¯ ϕ (cid:17)(cid:13)(cid:13)(cid:13) H p = O ( λ ) . Here, the average adjustment compared to the frictionless case are given by ∆¯ µ := λ / ¯ γ (cid:16) M ξα (cid:62) + αξ (cid:62) M (cid:62) (cid:17) s = O ( λ / ) . The process ˙¯ ϕ , that describes the mean-zero fluctuations around this constant value, follows an K ( N − -dimensional Ornstein-Uhlenbeck process: d ˙¯ ϕ t = − λ − / (cid:16) Γ / ⊗ ¯Λ − (cid:16) ¯Λ αα (cid:62) (cid:17)(cid:17) (cid:18) ˙¯ ϕ t dt + (cid:18) I N − ⊗ (cid:16) αα (cid:62) (cid:17) − α (cid:19) ξdW t (cid:19) . This process also provides a leading-order approximation of the equilibrium (signed) trading volume,in that (cid:107) ˙ ϕ − ˙¯ ϕ (cid:107) H p = O (1) for every p > . These formulas simplify considerably in the case of two agents ( N = 2). To wit, the risk-aversionmatrix Γ and the risk-aversion vector c then collapse to the scalarsΓ = γ + γ , c = ¯ γ γ − γ γ γ = γ − γ γ + γ . Note that the square root of the risk-aversion matrix Γ is well defined by [22, Theorem 1.29], even though thismatrix is generally only positive semidefinite but not symmetric. As in [32], the same expansion remains valid if the initial condition is close enough to the frictionless allocation,which is a natural assumption for a market with small trading costs.
9s a result, the average adjustments of the expected returns compared to the frictionless casesimplify to λ / ¯ γ (cid:16) M ξα (cid:62) + αξ (cid:62) M (cid:62) (cid:17) s, where M = γ − γ (cid:112) γ + γ ) ¯Λ (cid:16) ¯Λ αα (cid:62) (cid:17) − α. (5.3)The corresponding leading-order approximation of the (signed) trading volume is d ˙¯ ϕ t = − λ − / (cid:114) γ + γ − (cid:16) ¯Λ αα (cid:62) (cid:17) (cid:18) ˙¯ ϕ t dt + (cid:16) αα (cid:62) (cid:17) − αξdW t (cid:19) . (5.4)These explicit formulas clearly separate the impact of risk, (heterogeneity of) risk aversions,trading costs, and individual trading motives. This is makes it feasible to calibrate the model totime series of prices and trading volume, as we discuss now. Following empirical research of [3, 1] and industry practice as documented in [42], we study liquiditypremia for US equities by constructing portfolios corresponding to different levels of liquidity.To wit, we build portfolios H, M and L, which correspond to High, Medium and Low liquidity,respectively, from 1991 to 2016. The portfolios are constructed in a tradable manner: for eachportfolio, the number of shares in each constituent stock in year T is computed using only the datain year T − T . We choose this 26year investment period to match the estimation in [42] on Russell indices. We obtain the S&P500constituents from 1990 to 2016 from Compustat, match them to the CRSP daily stock file basedon the
CUSIP identifier, and then obtain the daily adjusted closing prices, trading volumes, andshares outstanding.The constituent stocks for each portfolio in year T are selected as follows. First, we carryout a prescreening using the data in year T − T −
1, (2) have more than 200 trading days with available pricedata and positive volume in year T −
1, and (3) have available prices on the first trading day ofyear T . Second, among these prescreened stocks, we pick the 200 stocks with the highest averagedaily market capitalization, the same number of stocks as the in large-cap portfolio considered in[42]. These 200 stocks are then sorted by their transaction costs proxied by ILLIQ in year T − for each group, weform a portfolio that is equal-weighted in year T − Also note that an even longer period would be increasingly at odds with our arithmetic model and the Bachelier-type price dynamics it implies. See . [1] observed liquidity premia for the equal-weighted returns of various portfolios. However, to achieve such returnsin practice, these portfolios need to be rebalanced daily. We rebalance the portfolio at the beginning of each year tostay close to a buy-and-hold strategy, which seems natural given that the portfolios are interpreted as assets that canbe bought and hold in our model. T −
1. In summary, this leads to threeportfolios H, M and L with the lowest, medium, and highest transaction costs, respectively.We view these three portfolios as three risky assets with different liquidity. The trading volumesand outstanding shares for each portfolio are calculated as the aggregated values for all constituentstocks. On the first trading day, we set the price of each portfolio to be the average prices ofconstituents weighted by their shares outstanding, so that the price multiplied by the shares out-standing equals the total market capitalization for the constituents of each portfolio. The portfoliois then rebalanced at the beginning of each subsequent year. To determine the transaction costassociated with each portfolio, we first calculate the daily values as the equal-weighted average ofthe transaction cost of all constituent stocks on each day, and then calculate the average of thesedaily values during the whole sample period.For our 26 years of data the average historical shares outstanding are s = (1 . , . , . (cid:62) × ; the average prices (in dollars) are (45 . , . , . (cid:62) . The annualized arithmetic returnis ˆ µ = (2 . , . , . (cid:62) ; dividing by the average prices, this corresponds to a (relative) Black-Scholes return of (6 . , . , . (cid:62) . In particular, the liquidity premium of the low-liquidityportfolio L compared to the high-liquidity portfolio H (i.e., the difference between the respectiveBlack-Scholes returns) is 2.69%, in line with the 2.4% reported for Russell data in [42]. Thecorresponding estimate for the annualized arithmetic variance isˆΣ = .
00 71 .
49 54 . .
49 85 .
42 65 . .
80 65 .
86 56 . . We first consider the frictionless version of the model and check whether the liquidity premiumis in fact just a risk premium that compensates for higher volatilities of less liquid stocks. ByProposition 3.3, the frictionless equilibrium expected return is¯ µ = ¯ γαα (cid:62) s, where αα (cid:62) is the frictionless equilibrium variance. We proxy µ and αα (cid:62) by the empirical esti-mates ˆ µ and ˆΣ reported above. The aggregate risk aversion ¯ γ is in turn estimated via a linearregression model without intercept as ¯ γ = 2 . × − . Using the empirical covariance matrixand this calibrated value for the aggregate risk aversion ¯ γ , the frictionless Black-Scholes returnare (7 . , . , . (cid:62) . To wit, the (co-)variances of the high-, medium-, and low-liquidityportfolios observed empirically suggest nearly identical risk premia for all of them. This in starkcontrast to the empirical data, where the low-liquidity portfolio has a substantially higher returnthan the portfolio composed of the highly liquid assets. We now discuss how the above calibration results change when trading costs (again proxied byILLIQ) are taken into account. To ease the computational burden, we assume that the dividendvolatility αα (cid:62) and in turn the leading-order equilibrium price volatilities are the same as in thefrictionless version of the model. Likewise, we use the same value for the aggregate risk aversion¯ γ . Unlike in the frictionless version of the model, not just this aggregate risk aversion, but also the11eterogeneity between the individual agents now play a crucial role. For tractability, we focus onthe simplest model with two agents and write γ = kγ , where k ≥ k = 2 to illustratethe following calibration process; then, γ = 4 . × − and γ = 8 . × − . However, byvirtue of our explicit asymptotic formulas, different values of k will just lead to a rescaling of theleading-order equilibrium returns implied by the model, which we outline at the end of this section.For simplicity, we assume that the transaction costs matrix Λ is diagonal, which is reasonable ifthe quadratic trading costs are seen as a more tractable proxy for proportional costs. The diagonalelements of Λ are the transaction costs for three portfolios proxied by ILLIQ as described above,multiplied by 9. This multiplication makes the transaction costs for the three portfolios comparableto a model with a single risky asset (with three times the order flow and whence nine times thequadratic costs). In particular, our estimate Λ = diag { . , . , . } × − is of the sameorder of magnitude as the direct estimates obtained from a proprietary database of trades in [12]. To complete the model specification, it now remains to estimate the endowment volatilities ξ .This is difficult, since these are not observable. As a way out, we extend the approach developedin [20] for a single risky asset and calibrate these parameters to time series data for trading volume.To this end, recall from (5.4) that, at the leading order for small costs, the (signed) trading volume˙ ϕ approximately has the Ornstein-Uhlenbeck dynamics d ˙¯ ϕ t = − κ ˙¯ ϕ t dt + κ dW t , where κ = (cid:114) γ + γ − (cid:16) Λ αα (cid:62) (cid:17) , κ = − κ · (cid:16) αα (cid:62) (cid:17) − αξ. Since κ is positive definite, the stationary distribution of ˙ ϕ t has the density [37, Section 6.5] p ( x ) = (2 π ) − D/ (detΩ) − / exp (cid:18) − x (cid:62) Ω − x (cid:19) , where Ω satisfies the algebraic Riccati equation κ Ω + Ω κ (cid:62) = κ κ (cid:62) . By the ergodic theorem and the explicit formula for absolute moments of Gaussian distribution [34],it follows that the long-run averages averages of the second moments of the trading volumes havethe following closed-form expression:lim T →∞ T (cid:90) T | ( ˙ ϕ t ) i ( ˙ ϕ t ) j | dt = (cid:90) R | x i x j | p ij ( x i , x j ) dx i dx j = 2(Ω ii Ω jj ) / π Γ(1) H ( − / , − / , / , ρ ij ) for i (cid:54) = j, Ω ii for i = j. (6.1)(Here, ρ ij = Ω ij / (Ω ii Ω jj ) / and H is Gauss’ hypergeometric function.) We use this explicit formulato calibrate the (unobservable) volatility matrices ξ of the agents’ endowment streams as follows.We assume that this 3 × − I × ) to calculate κ , and in turn the left hand side of (6.1) for i, j = 1 , , , i ≤ j . Estimating the transaction costs using the Bachelier volatilities divided by volume as implied by Kyle’s model [27]gives comparable results: (0 . , . , . (cid:62) × − . Here, negative diagonal elements produce positive liquidity premia in line with the data.
12e then compare the result with the second moments of daily trading volume observed empirically.The parameters of the matrix ξ are in turn updated using the global optimizer GlobalSearch inMATLAB in order to find the parameter values that match the empirical data as well as possible.The result is ξ = − .
07 1 .
91 0 . . − . − . . − . − . × . This corresponds to the second moments of daily volumes(5 . , . , . , . , . , . (cid:62) × , which are very close to the second moments of the daily volumes observed in our dataset:(5 . , . , . , . , . , . (cid:62) × . With all parameters of the model specified, we can now calculate the leading-order adjust-ments of the equilibrium expected returns of the portfolios H, M and L due to transaction costs.To wit, equation (5.3) shows that the annualized (absolute) changes compared to the frictionlessversion of the model are ( − . , − . , . (cid:62) . After dividing by the corresponding aver-age prices, we obtain the following adjustments of the annualized relative (Black-Scholes) returns:( − . , − . , . (cid:62) . As a consequence, the expected return of the most liquid portfo-lio is indeed reduced, whereas the expected returns of the low liquidity portfolio is increased. Whenthe heterogeneity parameter is chosen (somewhat arbitrarily) as k = 2, the difference between thereturn adjustments is 0 .
74% annually, substantially smaller than the difference of 2 .
7% observedempirically. k A nnua li z ed li qu i d i t y p r e m i u m ( % ) Figure 1: Annualized liquidity premium (i.e., difference between the equilibrium relative returns ofthe L and H portfolios) plotted against the heterogeneity parameter k . The empirically observedliquidity premium is 2.69%.To study how this result depends on k , observe that (5.3) shows that the average return adjust-ments scale with k by a factor of ( k − k +1) − / k − / . To see this, note that γ = ¯ γ (1+ k ) /k, γ =13 γ (1 + k ), and thus κ scales with k by the factor (1 + k ) k − / . By calibrating ξ to match the samesecond moments of daily volumes, ξ has a factor of k / (1 + k ) − / , and (5.3) establishes the scalingfactor of the average return adjustments. Therefore, increasing the heterogeneity k increases theliquidity premium between the low and high liquidity portfolios. This is illustrated in Figure 1,which shows that to produce a realistic level of liquidity premia, our model requires a substantiallevel of heterogeneity in the agents’ preferences. This corroborates the partial equilibrium literatureon liquidity premia, which finds that additional features such as market closure [14], unobservableregime switches [10], or state-dependent transaction costs [1, 29] are needed to reproduce realis-tic levels of liquidity premia. Incorporating these effects into a general equilibrium analysis is animportant but challenging direction for future research. Proof of Proposition 3.3.
It is readily verified that the proposed price process solves the BSDEsystem (3.5). The corresponding covariance matrix αα (cid:62) is invertible by assumption. Whence, eachagent’s individually optimal trading strategy is given by (3.3). In view of (3.4), this simplifies to¯ ϕ nt = ¯ γγ n s − (cid:0) αα (cid:62) (cid:1) − αξ nt , t ∈ [0 , T ] . (7.1)In particular, these holdings are admissible because they are normally distributed. As the aggregateendowment is zero ( (cid:80) Nn =1 ξ n = 0), these strategies indeed sum to s as required for market clearing.For uniqueness, suppose there are two solutions with uniformly bounded volatilities. Then, bothof these solve the BSDE with truncated (and hence globally Lipschitz) generator, and thereforecoincide. The crucial tool for the proof of our main result on the existence of equilibria with transactioncosts is Lemma 4.4, which establishes wellposedness for the Riccati system (4.8) characterizingthis equilibrium. The proof of Lemma 4.4 is in turn based on a number of auxiliary estimates onmatrix-valued ODEs that we develop first.We start with the properties of the risk-aversion matrix Γ introduced in (4.7). Recall from (3.2)that, without loss of generality, agent N is supposed to be the most risk-averse one. Lemma 7.1.
The matrix Γ is positive definite and has only positive eigenvalues. Proof.
The second part of the assertion has been established in [21, Lemma A.5]. Therefore itremains to show that b (cid:62) Γ b > b ∈ R M \{ } . Observe that Γ is a “diagonal minus rank-1”matrix: Γ = diag { γ , · · · , γ N − } − N N − (cid:62) N − diag { γ − γ N , · · · , γ N − − γ N } . (4.7) In much of the literature, positive definite matrices are additionally required to be symmetric, because this isnecessary to derive many useful properties. However, the matrix Γ is generally not symmetric, and we in turn carryout the subsequent analysis without this convenient property. Notice that in the absence of symmetry, a squarematrix with positive eigenvalues can fail to be positive definite, and a positive definite matrix can fail to have realeigenvalues.
14o show that this matrix is positive definite, we define v := diag { γ , · · · , γ N − } N − − N − N − (cid:88) n =1 γ n N − , (7.2)and observe that v and N − are orthogonal: v (cid:62) N − = N − (cid:88) n =1 γ n − N − N − (cid:88) n =1 γ n (cid:62) N − N − = N − (cid:88) n =1 γ n − N − (cid:88) n =1 γ n = 0 . Whence, every vector b ∈ R N − has an orthogonal decomposition, in that there exist unique a , a v ∈ R and b ⊥ ∈ R N − , such that b = a N − + a v v + b ⊥ , where (cid:62) N − b ⊥ = 0 = v (cid:62) b ⊥ . With this notation, a direct calculation yields b (cid:62) diag { γ , · · · , γ N − } b = b (cid:62) (cid:0) a diag { γ , · · · , γ N − } N − + diag { γ , · · · , γ N − } ( a v v + b ⊥ ) (cid:1) = a b (cid:62) (cid:32) v + 1 N − N − (cid:88) n =1 γ n N − (cid:33) + b (cid:62) diag { γ , · · · , γ N − } ( a v v + b ⊥ )= a N − (cid:88) n =1 γ n + 2 a a v (cid:107) v (cid:107) + ( a v v + b ⊥ ) (cid:62) diag { γ , · · · , γ N − } ( a v v + b ⊥ ) ≥ a N − (cid:88) n =1 γ n + 2 a a v (cid:107) v (cid:107) . (Here, we have used (cid:62) N − diag { γ , · · · , γ N − } = ( v + N − (cid:80) N − n =1 γ n N − ) (cid:62) in the second to laststep.) Similarly, we can calculate1 N b (cid:62) N − (cid:62) N − diag { γ − γ N , · · · , γ N − − γ N } b = 1 N b (cid:62) N − (cid:62) N − diag { γ , · · · , γ N − } b − γ N N b (cid:62) N − (cid:62) N − b = 1 N a ( N − (cid:32) v + 1 N − N − (cid:88) n =1 γ n N − (cid:33) (cid:62) b − a ( N − N γ N = N − N a (cid:32) a v (cid:107) v (cid:107) + a N − (cid:88) n =1 γ n (cid:33) − a ( N − N γ N = N − N (cid:32) a a v (cid:107) v (cid:107) + a N − (cid:88) n =1 ( γ n − γ N ) (cid:33) .
15s diag { γ , · · · , γ N − } is positive definite and N ≥
2, these two identities lead to the estimate b (cid:62) Γ b = b (cid:62) diag { γ , · · · , γ N − } b − N b (cid:62) N − (cid:62) N − diag { γ − γ N , · · · , γ N − − γ N } b> N − N b (cid:62) diag { γ , · · · , γ N − } b − N b (cid:62) N − (cid:62) N − diag { γ − γ N , · · · , γ N − − γ N } b ≥ N − N (cid:32) a N − (cid:88) n =1 γ n + 2 a a v (cid:107) v (cid:107) (cid:33) − N − N (cid:32) a a v (cid:107) v (cid:107) + a N − (cid:88) n =1 ( γ n − γ N ) (cid:33) ≥ N − N a (cid:32) N − (cid:88) n =1 γ n + 2 N − (cid:88) n =1 ( γ N − γ n ) (cid:33) > , where we have taken into account (3.2) in the last step. Whence, Γ is indeed positive definite.For later use, we recall the definition of the operator norm, in which we will express our estimatesfor matrix ODEs below: Definition 7.2.
The operator norm of an M × M matrix A is defined by (cid:107) A (cid:107) op := sup {(cid:107) Ab (cid:107) : b ∈ R M , (cid:107) b (cid:107) = 1 } . Remark 7.3.
For the convenience of the reader, let us summarize the properties of the opera-tor norm and the Frobenius norm from [23, Chapter 5] and the properties of Kronecker productfrom [40, Chapter 2]. that we will use repeatedly and without further mention below:(i) (cid:107) A (cid:107) op = (cid:107) A (cid:62) (cid:107) op = (cid:107) A + A (cid:62) (cid:107) op .(ii) The operator norm is submultiplicative in that, for an M × M matrix A and an M × M matrix B , (cid:107) AB (cid:107) op ≤ (cid:107) A (cid:107) op (cid:107) B (cid:107) op . (iii) For an M × M matrix A , the corresponding operator norm and Frobenius norm are relatedby (cid:107) A (cid:107) op ≤ (cid:107) A (cid:107) ≤ (cid:112) M + M (cid:107) A (cid:107) op . (iv) For the Kronecker product of two matrices (of arbitrary dimension), we have (cid:107) A ⊗ B (cid:107) op = (cid:107) A (cid:107) op (cid:107) B (cid:107) op . (v) The transpose of the Kronecker product satisfies:( A ⊗ B ) (cid:62) = A (cid:62) ⊗ B (cid:62) . (vi) Bilinearity and associativity of Kronecker products: A ⊗ ( B + C ) = A ⊗ B + A ⊗ C, ( B + C ) ⊗ A = B ⊗ A + C ⊗ A,A ⊗ ⊗ A = 0 . (vii) The mixed-product property of Kronecker products: for matrices A , B , C and D of appro-priate dimensions, ( A ⊗ B )( C ⊗ D ) = AC ⊗ BD.
16e now verify that the Kronecker product preserves positive-semidefiniteness as long as itssecond argument is also symmetric:
Lemma 7.4.
If matrices A , B are positive semidefinite and B is symmetric, then the Kroneckerproduct A ⊗ B is also positive semidefinite.Proof. Notice that A + A (cid:62) is symmetric positive semidefinite. Thus, A + A (cid:62) and B are bothdiagonalizable, in that there exist orthogonal matrices P , Q and diagonal matrices D A , D B suchthat P D A P (cid:62) = A + A (cid:62) , QD B Q (cid:62) = B = B (cid:62) . Here, the diagonal elements of D A and D B are the eigenvalues of A + A (cid:62) and B , respectively.These are all nonnegative because these matrices are both positive semidefinite and symmetric.As a consequence, the Kronecker product D A ⊗ D B is also diagonal with nonnegative diagonalelements; in particular, it is also positive semidefinite. It follows that A ⊗ B + ( A ⊗ B ) (cid:62) is alsopositive semidefinite, because the symmetry of B and the properties of the Kronecker product allowus to rewrite this matrix as A ⊗ B + ( A ⊗ B ) (cid:62) = A ⊗ B + A (cid:62) ⊗ B (cid:62) = A ⊗ B + A (cid:62) ⊗ B = (cid:16) A + A (cid:62) (cid:17) ⊗ B = (cid:16) P D A P (cid:62) (cid:17) ⊗ (cid:16) QD B Q (cid:62) (cid:17) = ( P ⊗ Q ) ( D A ⊗ D B ) (cid:16) P (cid:62) ⊗ Q (cid:62) (cid:17) = ( P ⊗ Q ) ( D A ⊗ D B ) ( P ⊗ Q ) (cid:62) , and P , Q are orthogonal matrices. Whence, the matrix A ⊗ B is also positive semidefinite.With this toolbox, we now establish some properties of linear matrix ODEs that will be usedbelow to bound the Riccati system (4.8). Lemma 7.5.
Let A : R + → R M × M be a continuous function with A (0) = 0 , and let Y be theunique solution [11, Theorem 2.4, Definition 2.12] of the linear matrix ODE Y (cid:48) ( τ ) = A ( τ ) Y ( τ ) , Y (0) = I M . (7.3) Suppose that Y (cid:48)(cid:48) ( τ ) = B ( τ ) Y ( τ ) , where B ( τ ) is positive semidefinite for all τ ≥ . Then the matrix A ( τ ) is positive semidefinite for all τ ≥ as well.Proof. Differentiation and the ODE (7.3) give (cid:16) Y (cid:62) ( τ ) Y ( τ ) (cid:17) (cid:48) = (cid:0) Y (cid:48) ( τ ) (cid:1) (cid:62) Y ( τ ) + Y (cid:62) ( τ ) Y (cid:48) ( τ ) = Y (cid:62) ( τ ) (cid:16) A (cid:62) ( τ ) + A ( τ ) (cid:17) Y ( τ ) (7.4)and, in turn, (cid:16) Y (cid:62) ( τ ) Y ( τ ) (cid:17) (cid:48)(cid:48) = (cid:0) Y (cid:48)(cid:48) ( τ ) (cid:1) (cid:62) Y ( τ ) + 2 (cid:0) Y (cid:48) ( τ ) (cid:1) (cid:62) Y (cid:48) ( τ ) + Y (cid:62) ( τ ) Y (cid:48)(cid:48) ( τ )= Y (cid:62) ( τ ) (cid:16) B (cid:62) ( τ ) + B ( τ ) + 2 A (cid:62) ( τ ) A ( τ ) (cid:17) Y ( τ ) . b ∈ R M , we thus have (cid:16) b (cid:62) Y (cid:62) ( τ ) Y ( τ ) b (cid:17) (cid:48)(cid:48) = b (cid:62) Y (cid:62) ( τ ) (cid:16) B (cid:62) ( τ ) + B ( τ ) + 2 A (cid:62) ( τ ) A ( τ ) (cid:17) Y ( τ ) b = ( Y ( τ ) b ) (cid:62) (cid:16) B (cid:62) ( τ ) + B ( τ ) + 2 A (cid:62) ( τ ) A ( τ ) (cid:17) Y ( τ ) b ≥ , (7.5)because B ( τ ), B (cid:62) ( τ ) and A (cid:62) ( τ ) A ( τ ) are all positive semidefinite. Thus τ (cid:55)→ (cid:0) b (cid:62) Y (cid:62) ( τ ) Y ( τ ) b (cid:1) (cid:48) isincreasing on R + and (7.4) in turn yields2 b (cid:62) Y (cid:62) ( τ ) A ( τ ) Y ( τ ) b = b (cid:62) Y (cid:62) ( τ ) (cid:16) A (cid:62) ( τ ) + A ( τ ) (cid:17) Y ( τ ) b = (cid:16) b (cid:62) Y (cid:62) ( τ ) Y ( τ ) b (cid:17) (cid:48) ≥ (cid:16) b (cid:62) Y (cid:62) (0) Y (0) b (cid:17) (cid:48) = b (cid:62) Y (cid:62) (0) (cid:16) A (cid:62) (0) + A (0) (cid:17) Y (0) b = 0 . (7.6)By Liouville’s formula [11, Proposition 2.18], Y ( τ ) is invertible for every τ ≥
0. Hence, for every b ∈ R M , b (cid:62) A ( τ ) b = ( Y − ( τ ) b ) (cid:62) Y (cid:62) ( τ ) A ( τ ) Y ( τ ) Y − ( τ ) b ≥ .A ( τ ) therefore is indeed positive semidefinite for every τ ≥ Lemma 7.6.
Let Y be the unique solution of the linear matrix ODE (7.3) . If τ (cid:55)→ A ( τ ) iscontinuous and A ( τ ) is positive semidefinite for every τ ≥ , then τ (cid:55)→ (cid:107) Y ( τ ) (cid:107) op is increasing.Proof. For b ∈ R M with (cid:107) b (cid:107) = 1 and τ ≥ r ≥
0, (7.4) and (7.6) imply (cid:107) Y ( τ ) b (cid:107) = b (cid:62) Y (cid:62) ( τ ) Y ( τ ) b ≥ b (cid:62) Y (cid:62) ( r ) Y ( r ) b = (cid:107) Y ( r ) b (cid:107) ≥ . As a consequence, the operator norm of Y ( τ ) is indeed increasing in τ : (cid:107) Y ( τ ) (cid:107) op = sup {(cid:107) Y ( τ ) b (cid:107) : (cid:107) b (cid:107) = 1 } ≥ sup {(cid:107) Y ( r ) b (cid:107) : (cid:107) b (cid:107) = 1 } = (cid:107) Y ( r ) (cid:107) op . Corollary 7.7.
Let ( F, H ) be the unique local solution of the Riccati system (4.8) on its maximalinterval of existence [0 , T max ) . Then F ( τ ) is positive semidefinite for every τ ∈ [0 , T max ) .Proof. First, recall that Λ and Λ / are both symmetric and positive definite, and hence alsoinvertible. Let Φ F be the solution (on [0 , T max )) of the linear matrix ODEΦ (cid:48) F ( τ ) = (cid:16) I N − ⊗ Λ − / (cid:17) F (cid:62) ( τ ) (cid:16) I N − ⊗ Λ − / (cid:17) Φ F ( τ ) , Φ F (0) = I K ( N − . (7.7)Differentiation of this matrix function, the linear ODE (7.7) for Φ F , and the Riccati equation (4.8)for F implyΦ (cid:48)(cid:48) F = (cid:16) I N − ⊗ Λ − / (cid:17) (cid:0) F (cid:48) (cid:1) (cid:62) (cid:16) I N − ⊗ Λ − / (cid:17) Φ F + (cid:16) I N − ⊗ Λ − / (cid:17) F (cid:62) (cid:16) I N − ⊗ Λ − / (cid:17) Φ (cid:48) F = (cid:16) I N − ⊗ Λ − / (cid:17) (cid:0) F (cid:48) + F (cid:0) I N − ⊗ Λ − (cid:1) F (cid:1) (cid:62) (cid:16) I N − ⊗ Λ − / (cid:17) Φ F = (cid:16) I N − ⊗ Λ − / (cid:17) (cid:18) Γ ⊗ (cid:16) α + ( c ⊗ I K ) (cid:62) H (cid:17) (cid:16) α + ( c ⊗ I K ) (cid:62) H (cid:17) (cid:62) (cid:19) (cid:62) (cid:16) I N − ⊗ Λ − / (cid:17) Φ F . ⊗ [( α + ( c ⊗ I K ) (cid:62) H ( τ ))( α + ( c ⊗ I K ) (cid:62) H ( τ )) (cid:62) ] is positive semidefinite by Lemmas 7.4and 7.1. As Λ and in turn also I N − ⊗ Λ − / are symmetric and positive definite, it follows that (cid:0) I N − ⊗ Λ − / (cid:1) (cid:0) Γ ⊗ [( α + ( c ⊗ I K ) (cid:62) H ( τ ))( α + ( c ⊗ I K ) (cid:62) H ( τ )) (cid:62) ] (cid:1) (cid:0) I N − ⊗ Λ − / (cid:1) (cid:62) is also posi-tive semi-definite for every τ ∈ [0 , T max ). Together with Lemma 7.5, it follows that the matrix (cid:16) I N − ⊗ Λ − / (cid:17) F (cid:62) ( τ ) (cid:16) I N − ⊗ Λ − / (cid:17) is positive semidefinite (7.8)for every τ ∈ [0 , T max ) as well. The assertion now follows from (7.8) and the identity F (cid:62) ( τ ) = (cid:16) I N − ⊗ Λ / (cid:17) (cid:16) I N − ⊗ Λ − / (cid:17) F (cid:62) ( τ ) (cid:16) I N − ⊗ Λ − / (cid:17) (cid:16) I N − ⊗ Λ / (cid:17) = (cid:16) I N − ⊗ Λ / (cid:17) (cid:16) I N − ⊗ Λ − / (cid:17) F (cid:62) ( τ ) (cid:16) I N − ⊗ Λ − / (cid:17) (cid:16) I N − ⊗ Λ / (cid:17) (cid:62) . Corollary 7.8.
With the solution Φ F of the linear matrix ODE (7.7) , define Ψ F ( r ; τ ) = Φ F ( r )Φ − F ( τ ) , r, τ ∈ [0 , T max ) . (7.9) Then (cid:107) Ψ F ( r ; τ ) (cid:107) op ≤ for every ≤ r ≤ τ < T max .Proof. By the ODE for Φ F ( r ), we have ∂∂r Ψ F ( r ; τ ) = (cid:16) I N − ⊗ Λ − / (cid:17) F (cid:62) ( r ) (cid:16) I N − ⊗ Λ − / (cid:17) Ψ F ( r ; τ ) . In view of Lemma 7.4 and (7.8), (cid:0) I N − ⊗ Λ − / (cid:1) F (cid:62) ( r ) (cid:0) I N − ⊗ Λ − / (cid:1) is positive semidefinite forevery r ∈ [0 , T max ). Lemma 7.6 in turn yields (cid:107) Ψ F ( r ; τ ) (cid:107) op ≤ (cid:107) Ψ F ( τ ; τ ) (cid:107) op = (cid:107) Φ F ( τ )Φ − F ( τ ) (cid:107) op = 1 , for every 0 ≤ r ≤ τ ≤ T max , as asserted.After the above preparations, we now turn to the proof of Lemma 4.4. Proof of Lemma 4.4.
We show that the local solution (
F, H ) of the Riccati equation is in fact aglobal solution because it remains bounded on any finite time interval (so that T max = ∞ ).To this end, first observe that the ODEs (4.8) for F and (7.7) for Φ F give (cid:16) Φ (cid:62) F ( I N − ⊗ Λ − / ) F (cid:17) (cid:48) = (cid:0) Φ (cid:48) F (cid:1) (cid:62) (cid:16) I N − ⊗ Λ − / (cid:17) F + Φ (cid:62) F (cid:16) I N − ⊗ Λ − / (cid:17) F (cid:48) = Φ (cid:62) F (cid:16) I N − ⊗ Λ − / (cid:17) (cid:0) F (cid:0) I N − ⊗ Λ − (cid:1) F + F (cid:48) (cid:1) = Φ (cid:62) F (cid:16) I N − ⊗ Λ − / (cid:17) Γ ⊗ (cid:104) ( α + ( c ⊗ I K ) (cid:62) H )( α + ( c ⊗ I K ) (cid:62) H ) (cid:62) (cid:105) = Φ (cid:62) F (cid:18) Γ ⊗ Λ − / (cid:16) α + ( c ⊗ I K ) (cid:62) H (cid:17) (cid:16) α + ( c ⊗ I K ) (cid:62) H (cid:17) (cid:62) (cid:19) . Together with F (0) = 0, it follows thatΦ (cid:62) F ( τ ) (cid:16) I N − ⊗ Λ − / (cid:17) F ( τ ) (7.10)= (cid:90) τ Φ (cid:62) F ( r ) (cid:16) I N − ⊗ Λ − / (cid:17) (cid:18) Γ ⊗ (cid:16) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:17) (cid:16) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:17) (cid:62) (cid:19) dr.
19y Liouville’s Formula [11, Proposition 2.18], we havedet (Φ F ( τ )) = exp (cid:18)(cid:90) τ tr (cid:16)(cid:16) I N − ⊗ Λ − / (cid:17) F (cid:62) ( τ ) (cid:16) I N − ⊗ Λ − / (cid:17)(cid:17) dr (cid:19) det (Φ F (0)) > , so that Φ F ( τ ) is invertible for all τ < T max . We can in turn solve (7.10) for F by multiplying withthe inverse of Φ F and the inverse I ⊗ Λ / of I ⊗ Λ − / . With the notation from (7.9), this leads to F ( τ ) = (cid:90) τ (cid:16) I N − ⊗ Λ / (cid:17) Ψ (cid:62) F ( r ; τ ) (cid:18) Γ ⊗ Λ − / (cid:16) α + ( c ⊗ I K ) (cid:62) H (cid:17) (cid:16) α + ( c ⊗ I K ) (cid:62) H (cid:17) (cid:62) (cid:19) ( r ) dr. (7.11)Similarly, after multiplying Φ (cid:62) F ( τ ) (cid:0) I N − ⊗ Λ − / (cid:1) to the left of H , integrating, and then takinginto account the ODE (4.8) for H , we obtain H ( τ ) = (cid:90) τ (cid:16) I N − ⊗ Λ / (cid:17) Ψ (cid:62) F ( r ; τ ) (cid:16) Γ ⊗ Λ − / (cid:16) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:17)(cid:17) ξ dr. (7.12)As a consequence, α + ( c ⊗ I K ) (cid:62) H ( τ )= α + ( c ⊗ I K ) (cid:62) (cid:18)(cid:90) τ (cid:16) I N − ⊗ Λ / (cid:17) Ψ (cid:62) F ( r ; τ ) (cid:16) Γ ⊗ Λ − / (cid:16) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:17)(cid:17) ξ dr (cid:19) = α + (cid:90) τ (cid:16) c ⊗ Λ / (cid:17) Ψ (cid:62) F ( r ; τ ) (cid:16) Γ ⊗ Λ − / (cid:16) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:17)(cid:17) ξ dr. Recalling the definition of operator norm of a matrix from Definition 7.2 and the properties fromRemark 7.3, it follows that (cid:13)(cid:13)(cid:13) α + ( c ⊗ I K ) (cid:62) H ( τ ) (cid:13)(cid:13)(cid:13) op ≤ (cid:107) α (cid:107) op + (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) τ (cid:16) c ⊗ Λ / (cid:17) Ψ (cid:62) F ( r ; τ ) (cid:16) Γ ⊗ Λ − / (cid:16) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:17)(cid:17) ξ dr (cid:13)(cid:13)(cid:13)(cid:13) op ≤ (cid:107) α (cid:107) op + (cid:90) τ (cid:13)(cid:13)(cid:13)(cid:16) c ⊗ Λ / (cid:17) Ψ (cid:62) F ( r ; τ ) (cid:16) Γ ⊗ Λ − / (cid:16) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:17)(cid:17) ξ (cid:13)(cid:13)(cid:13) op dr ≤ (cid:107) α (cid:107) op + (cid:90) τ (cid:107) c (cid:107)(cid:107) Λ / (cid:107) op (cid:107) Ψ F ( r ; τ ) (cid:107) op (cid:107) Γ (cid:107) op (cid:107) Λ − / (cid:107) op (cid:13)(cid:13)(cid:13) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:13)(cid:13)(cid:13) op (cid:107) ξ (cid:107) op dr ≤ (cid:107) α (cid:107) + (cid:107) c (cid:107)(cid:107) Λ (cid:107) / (cid:107) Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107) (cid:90) τ (cid:13)(cid:13)(cid:13) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:13)(cid:13)(cid:13) op dr. (Here, the last step uses the estimate for (cid:107) Ψ F ( r ; τ ) (cid:107) op ≤ scalar function τ (cid:55)→ (cid:107) α + ( c ⊗ I K ) (cid:62) H ( τ ) (cid:107) op in turn yields (cid:13)(cid:13)(cid:13) α + ( c ⊗ I K ) (cid:62) H ( τ ) (cid:13)(cid:13)(cid:13) op ≤ (cid:107) α (cid:107) exp (cid:16) (cid:107) c (cid:107)(cid:107) Λ (cid:107) / (cid:107) Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107) τ (cid:17) , τ ∈ [0 , T max ) . (7.13)Together with (7.12), the fact that the Frobenius norm of a matrix is dominated by a constant20imes the operator norm, and D ≥ K , it now follows that (cid:107) H ( τ ) (cid:107) ≤ (cid:112) ( N − D + K ) (cid:107) H ( τ ) (cid:107) op ≤ (cid:112) N − D (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) τ (cid:16) I N − ⊗ Λ / (cid:17) Ψ (cid:62) F ( r ; τ ) (cid:16) Γ ⊗ Λ − / (cid:16) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:17)(cid:17) ξ dr (cid:13)(cid:13)(cid:13)(cid:13) op ≤ (cid:112) N − D (cid:90) τ (cid:107) Λ / (cid:107) op (cid:107) Ψ F ( r ; τ ) (cid:107) op (cid:107) Γ (cid:107) op (cid:107) Λ − / (cid:107) op (cid:13)(cid:13)(cid:13) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:13)(cid:13)(cid:13) op (cid:107) ξ (cid:107) op dr ≤ (cid:112) N − D (cid:90) τ (cid:107) Λ / (cid:107)(cid:107) Γ (cid:107)(cid:107) Λ − / (cid:107) (cid:13)(cid:13)(cid:13) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:13)(cid:13)(cid:13) op (cid:107) ξ (cid:107) dr ≤ (cid:112) N − D (cid:90) τ (cid:107) α (cid:107)(cid:107) Λ (cid:107) / (cid:107) Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107) exp (cid:16) (cid:107) c (cid:107)(cid:107) Λ (cid:107) / (cid:107) Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107) r (cid:17) dr. The integral in the upper bound is finite for any τ . All components of H therefore remain uniformlybounded on any finite time interval. Similarly, by (7.11), (cid:107) F ( τ ) (cid:107) ≤ (cid:112) N − K (cid:90) τ (cid:107) Λ (cid:107) / (cid:107) Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) α (cid:107) exp (cid:16) (cid:107) c (cid:107)(cid:107) Λ (cid:107) / (cid:107) Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107) r (cid:17) dr. Whence, all elements of F also remain uniformly bounded on any finite time interval. The localsolution of the Riccati system (4.8) therefore also is a global solution on any finite time interval. Having established wellposedness for the Riccati system (4.8), we now turn to the proof of our mainresult on the global existence of equilibria with transaction costs.
Proof of Theorem 4.5.
In view of Lemma 4.2, we have to verify that the candidate processes fromTheorem 4.5 indeed solve the FBSDE system from Lemma 4.2.First, recall that F is positive definite by Corollary 7.7. Lemma 7.6 applied to the linearmatrix Riccati equation (4.9) in turn shows that Φ( t ) is well-defined on [0 , T ] and has increasingoperator norm. Together with the proof of Lemma 4.4, it follows that the functions F , H , Φ are alluniformly bounded on [0 , T ], and it is in turn straightforward to verify using the Gaussian law ofthe driving Brownian motion that the candidate solution ( ϕ, ˙ ϕ, ˙ Z, ¯ S + Y − ( c ⊗ Λ) (cid:62) ˙ ϕ, ¯ σ − ( c ⊗ Λ) (cid:62) ˙ Z )indeed belongs to H ( R K ( N − ) × H ( R K ( N − ) × H ( R K ( N − × D ) × S ( R K ) × H ( R K × D ). Hence,it remains to verify that these processes also satisfy the dynamics and initial/terminal conditionsfrom Lemma 4.2.To this end, recall that by Liouville’s formula [11, Proposition 2.18], the matrix Φ( t ) is invertiblefor each t ∈ [0 , T ]. Differentiation of this matrix function and the ODE (4.9) give (cid:0) Φ − ( t ) (cid:1) (cid:48) = − Φ − ( t )Φ (cid:48) ( τ )Φ − ( τ ) = − Φ − ( t ) (cid:16) I N − ⊗ Λ − / (cid:17) F (cid:62) ( T − t ) (cid:16) I N − ⊗ Λ − / (cid:17) . (7.14)Moreover, by definition of the function Ψ in (4.10),Φ (cid:62) ( t ) (cid:16) I N − ⊗ Λ / (cid:17) Ψ (cid:62) ( r ; t ) = (Ψ( r ; t ) (cid:16) I N − ⊗ Λ / (cid:17) Φ( t )) (cid:62) = Φ (cid:62) ( r ) (cid:16) I N − ⊗ Λ / (cid:17) . With these observations, we can rewrite (4.11) asΦ (cid:62) ( t ) (cid:16) I N − ⊗ Λ / (cid:17) ( ϕ t − ¯ ϕ )= (cid:16) I N − ⊗ Λ / (cid:17) ( ϕ − − ¯ ϕ ) − (cid:90) t Φ (cid:62) ( r ) (cid:16) I N − ⊗ Λ − / (cid:17) H ( T − r ) W r dr,
21o that d Φ (cid:62) ( t ) (cid:16) I N − ⊗ Λ / (cid:17) ( ϕ t − ¯ ϕ ) = − Φ (cid:62) ( t ) (cid:16) I N − ⊗ Λ − / (cid:17) H ( T − t ) W t dt. (7.15)Integration by parts and the dynamics (7.14)-(7.15) in turn give dϕ t = d (cid:20)(cid:16) I N − ⊗ Λ − / (cid:17) (cid:16) Φ (cid:62) ( t ) (cid:17) − Φ (cid:62) ( t ) (cid:16) I N − ⊗ Λ / (cid:17) ( ϕ t − ¯ ϕ ) (cid:21) = − (cid:0) I N − ⊗ Λ − (cid:1) F ( T − t ) (cid:16) I N − ⊗ Λ − / (cid:17) (cid:16) Φ (cid:62) ( t ) (cid:17) − Φ (cid:62) ( t ) (cid:16) I N − ⊗ Λ / (cid:17) ( ϕ t − ¯ ϕ ) dt − (cid:16) I N − ⊗ Λ − / (cid:17) (cid:16) Φ (cid:62) ( t ) (cid:17) − Φ (cid:62) ( t ) (cid:16) I N − ⊗ Λ − / (cid:17) H ( T − t ) W t dt = − (cid:0) I N − ⊗ Λ − (cid:1) [ F ( T − t ) ( ϕ t − ¯ ϕ ) + H ( T − t ) W t ] dt (7.16)= ˙ ϕ t dt. Moreover, ϕ = ¯ ϕ + Ψ (cid:62) (0; 0) ( ϕ − − ¯ ϕ ) − ϕ − , so that the first equation of the FBSDE system in Lemma 4.2 is indeed satisfied.To verify that the other two equations from Lemma 4.2 are satisfied as well, we first observethe following identities for the matrix Γ from (4.7) and the vector c defined in Lemma 4.4:Γ (cid:18) c + ¯ γγ N N − (cid:19) = γ N N N − , (7.17)Γ (cid:62) c = 1 N (cid:2) γ N − γ n · · · γ N − γ N − (cid:3) (cid:62) , (7.18) (cid:62) N − c = 1 − ¯ γ Nγ N . (7.19)With the (constant) frictionless equilibrium volatility ¯ σ = α from Lemma 3.3 and the process˙ Z from (4.14), the candidate for the frictional equilibrium volatility is σ t = ¯ σ t − ( c ⊗ Λ) (cid:62) ˙ Z t = α + ( c ⊗ Λ) (cid:62) (cid:0) I N − ⊗ Λ − (cid:1) H ( T − t ) = α + ( c ⊗ I K ) (cid:62) H ( T − t ) . (7.20)The definition of ˙ ϕ in (4.12), integration by parts, the Riccati equations (4.8) for F , H , and (7.20)in turn lead to d ˙ ϕ t = − (cid:0) I N − ⊗ Λ − (cid:1) d [ F ( T − t ) ( ϕ t − ¯ ϕ ) + H ( T − t ) W t ]= (cid:0) I N − ⊗ Λ − (cid:1) (cid:2)(cid:0) F (cid:48) ( T − t ) ( ϕ t − ¯ ϕ ) + H (cid:48) ( T − t ) W t (cid:1) dt − F ( T − t ) d ( ϕ t − ¯ ϕ ) − H ( T − t ) dW t (cid:3) = (cid:0) I N − ⊗ Λ − (cid:1) (cid:0) F (cid:48) ( T − t ) + F ( T − t ) (cid:0) I N − ⊗ Λ − F ( t − t ) (cid:1)(cid:1) ( ϕ t − ¯ ϕ ) dt + (cid:0) I N − ⊗ Λ − (cid:1) (cid:0) H (cid:48) ( T − t ) + F ( T − t ) (cid:0) I N − ⊗ Λ − (cid:1) H ( T − t ) (cid:1) W t dt − (cid:0) I N − ⊗ Λ − (cid:1) H ( T − t ) dW t = (cid:0) I N − ⊗ Λ − (cid:1) (cid:16)(cid:16) Γ ⊗ σ t σ (cid:62) t (cid:17) ( ϕ t − ¯ ϕ ) + (Γ ⊗ σ t ) ξW t (cid:17) dt + ˙ Z t dW t = (cid:16)(cid:16) Γ ⊗ Λ − σ t σ (cid:62) t (cid:17) ϕ t + (cid:0) Γ ⊗ Λ − σ t (cid:1) ξ t − (cid:16) Γ ⊗ Λ − σ t σ (cid:62) t (cid:17) ¯ ϕ (cid:17) dt + ˙ Z t dW t . (7.21)With the explicit form of ¯ ϕ n from (7.1), we can write ¯ ϕ = ( c + ¯ γγ N N − ) ⊗ s . Together with (7.30),it follows that (cid:16) Γ ⊗ Λ − σ t σ (cid:62) t (cid:17) ¯ ϕ = Γ (cid:18) c + ¯ γγ N N − (cid:19) ⊗ Λ − σ t σ (cid:62) t s = γ N N N − ⊗ Λ − σ t σ (cid:62) t s. ϕ T = − (cid:0) I N − ⊗ Λ − (cid:1) [ F (0) ( ϕ T − ¯ ϕ ) + H (0) W T ] = 0 . Finally, for the frictionless equilibrium price ¯ S from Lemma 3.3 and Y defined in (4.13), d (cid:0) ¯ S t + Y t (cid:1) = ¯ γ (cid:16) αα (cid:62) + ( c ⊗ I K ) (cid:62) Hα (cid:62) + αH (cid:62) ( c ⊗ I K ) + ( c ⊗ I K ) (cid:62) HH (cid:62) ( c ⊗ I K ) (cid:17) ( T − t ) sdt + αdW t = ¯ γ (cid:16) α + ( c ⊗ I K ) (cid:62) H ( T − t ) (cid:17) (cid:16) α + ( c ⊗ I K ) (cid:62) H ( T − t ) (cid:17) (cid:62) sdt + αdW t = ¯ γσ t σ (cid:62) t sdt + αdW t . (Here, we have used (7.20) in the last step.) Next, observe that the dynamics of ˙ ϕ computed in(7.21) and the identities (7.18), (7.19) give( c ⊗ Λ) (cid:62) d ˙ ϕ t = (cid:18)(cid:16) c (cid:62) Γ ⊗ σ t σ (cid:62) t (cid:17) ϕ t + (cid:16) c (cid:62) Γ ⊗ σ t (cid:17) ξ t − γ N N c (cid:62) N − ⊗ σ t σ (cid:62) t s (cid:19) dt + ( c ⊗ Λ) (cid:62) ˙ Z t dW t = (cid:32) N σ t N − (cid:88) n =1 ( γ N − γ n ) (cid:16) σ (cid:62) t ϕ nt + ξ nt (cid:17) + (cid:18) ¯ γ − γ N N (cid:19) σ t σ (cid:62) t s (cid:33) dt + ( c ⊗ Λ) (cid:62) ˙ Z t dW t . For S t = ¯ S t + Y t − ( c ⊗ Λ) (cid:62) ˙ ϕ t from Theorem 4.5, the dynamics we have just computed as wellas (7.20) show dS t = ¯ γσ t σ (cid:62) t sdt − (cid:32) N σ t N − (cid:88) n =1 ( γ N − γ n ) (cid:16) σ (cid:62) t ϕ nt + ξ nt (cid:17) + (cid:18) ¯ γ − γ N N (cid:19) σ t σ (cid:62) t s (cid:33) dt + σ t dW t = (cid:32) γ N N σ t σ (cid:62) t s + 1 N σ t N − (cid:88) n =1 ( γ n − γ N ) (cid:16) σ (cid:62) t ϕ nt + ξ nt (cid:17)(cid:33) dt + σ t dW t . The third equation in Lemma 4.2 is therefore also satisfied, because the corresponding terminalcondition is matched as well: S T = ¯ S T + Y T − ( c ⊗ Λ) (cid:62) ˙ ϕ T = S − − S , This completes the proof.
The rigorous convergence proof for the asymptotics approximations is based on estimates for thelargest and smallest singular values of the involved matrices. We first recall the definition and theproperties of singular values of matrices. Then, we establish bounds on the singular values of thesolutions of linear matrix ODEs in Lemma (7.11). Using this tool and a matrix version of thevariation on of constants formula, we then derive estimates for the solution F λ , H λ of the RiccatiODEs (4.8) as a function of the asymptotic parameter λ . These in turn allow us to show that thefunctions can be approximated by constant matrices that solve some algebraic Riccati equations.With these approximations at hand, we then proof the asymptotic expansions of the equilibriumprice and trading volume from Theorem 5.1. 23 efinition 7.9.
The singular values of a real-valued M × M matrix A are the square roots of thenon-negative eigenvalues of AA (cid:62) . For the convenience of the reader, we summarize the properties of singular values from [23,Chapter 2] that we henceforth use without further mention.
Remark 7.10.
Let A be a real-valued M × M matrix.(i) A and A (cid:62) have the same non-zero singular values, but not the same as ( A + A (cid:62) ).(ii) If A is symmetric and M = M , then the absolute value of the eigenvalues of A coincide withthe singular values.(iii) Minimax representation for singular values: Let σ max ( A ) and σ min ( A ) denote the largest andsmallest singular value of A , respectively. Then, σ max ( A ) = sup {(cid:107) A (cid:62) b (cid:107) : b ∈ R M , (cid:107) b (cid:107) = 1 } , σ min ( A ) = inf {(cid:107) A (cid:62) b (cid:107) : b ∈ R M , (cid:107) b (cid:107) = 1 } . In particular, (cid:107) A (cid:107) op = σ max ( A ).We now consider the linear matrix ODE (7.3), which is the key ingredient for the matrix versionof the variation of constants formula that we use to prove our asymptotic expansions below. Thefollowing lemma shows that bounds on the singular values of the matrix function on the right-handside of (7.3) are inherited by the largest and smallest singular values of the solution: Lemma 7.11.
Let Y be the unique solution of the linear matrix ODE (7.3) . Suppose τ (cid:55)→ A ( τ ) iscontinuous and A ( τ ) is positive semidefinite for every τ ≥ , with a max > a min > such that forevery τ ∈ [0 , T ] : a max ≥ σ max ( A ( τ ) + A (cid:62) ( τ )) ≥ σ min ( A ( τ ) + A (cid:62) ( τ )) ≥ a min > . Then for every ≤ r ≤ τ ≤ T , e − a max ( τ − r ) ≤ σ min (cid:0) Y ( r ) Y − ( τ ) (cid:1) ≤ σ max (cid:0) Y ( r ) Y − ( τ ) (cid:1) ≤ e − a min ( τ − r ) . (7.22) Proof.
By Liouville’s formula [11, Proposition 2.18], both Y ( r ) and Y ( τ ) are invertible, hence forevery b ∈ R M \ { } , we have (cid:13)(cid:13) Y ( r ) Y − ( τ ) b (cid:13)(cid:13) >
0. By (7.4), we then have ∂∂ ˜ r (cid:13)(cid:13) Y (˜ r ) Y − ( τ ) b (cid:13)(cid:13) = ∂∂ ˜ r (cid:16) b (cid:62) ( Y − ( τ )) (cid:62) Y (cid:62) (˜ r ) Y (˜ r ) Y − ( τ ) b (cid:17) = (cid:0) Y − ( τ ) b (cid:1) (cid:62) ( Y (cid:62) (˜ r ) Y (˜ r )) (cid:48) (cid:0) Y − ( τ ) b (cid:1) = (cid:0) Y (˜ r ) Y − ( τ ) b (cid:1) (cid:62) (cid:16) A (˜ r ) + A (cid:62) (˜ r ) (cid:17) (cid:0) Y (˜ r ) Y − ( τ ) b (cid:1) ≥ a min (cid:13)(cid:13) Y (˜ r ) Y − ( τ ) b (cid:13)(cid:13) . Now divide by (cid:13)(cid:13) Y (˜ r ) Y − ( τ ) b (cid:13)(cid:13) and integrate from r to τ , obtaining e a min ( τ − r ) (cid:13)(cid:13) Y ( r ) Y − ( τ ) b (cid:13)(cid:13) ≤ (cid:13)(cid:13) Y ( τ ) Y − ( τ ) b (cid:13)(cid:13) = (cid:107) b (cid:107) . Hence, (cid:13)(cid:13) Y ( r ) Y − ( τ ) b (cid:13)(cid:13) / (cid:107) b (cid:107) ≤ e − a min ( τ − r ) , and in turn σ max ( Y ( r ) Y − ( τ )) ≤ e − a min ( τ − r ) .24imilarly, for every b ∈ R M \ { } , ∂∂ ˜ r (cid:13)(cid:13) Y (˜ r ) Y − ( τ ) b (cid:13)(cid:13) ≤ a max (cid:13)(cid:13) Y (˜ r ) Y − ( τ ) b (cid:13)(cid:13) . Dividing by (cid:13)(cid:13) Y (˜ r ) Y − ( τ ) b (cid:13)(cid:13) and integrating from r to τ in turn yields (cid:107) b (cid:107) = (cid:13)(cid:13) Y ( τ ) Y − ( τ ) b (cid:13)(cid:13) ≤ e a max ( τ − r ) (cid:13)(cid:13) Y ( r ) Y − ( τ ) b (cid:13)(cid:13) . Hence, (cid:13)(cid:13) Y ( r ) Y − ( τ ) b (cid:13)(cid:13) / (cid:107) b (cid:107) ≥ e − a max ( τ − r ) , and thus σ min ( Y ( r ) Y − ( τ )) ≥ e − a max ( τ − r ) as asserted.Using this lemma, we now approximate the solution to the Riccati system (4.8). Recall thatthe (normalized) transaction cost matrix ¯Λ is symmetric and positive definite and the risk aversionmatrix Γ only has positive eigenvalues, so their square roots ¯Λ / and Γ / are well defined. Alsonote that ¯Λ, ¯Λ − , ¯Λ / and ¯Λ − / commute. Lemma 7.12.
Let ( F λ , H λ ) be the solution of the Riccati system (4.8) for small transaction costs Λ λ = λ ¯Λ . Define the constant matrix: (cid:98) F := Γ / ⊗ (cid:16) ¯Λ αα (cid:62) (cid:17) , (7.23) and recall the definition of M from Theorem 5.1, M := (cid:18) c (cid:62) Γ / ⊗ ¯Λ (cid:16) ¯Λ αα (cid:62) (cid:17) − α (cid:19) . (7.24) Then, as λ ↓ , the following estimates hold: (cid:107) F λ ( τ ) (cid:107) op = O ( λ / ) , τ ∈ [0 , T ] , (cid:90) T (cid:107) F λ ( τ ) − λ / (cid:98) F (cid:107) op dτ = O ( λ ) , (7.25) (cid:107) H λ ( τ ) (cid:107) op = O ( λ / ) , τ ∈ [0 , T ] , (cid:90) T (cid:107) ( c ⊗ I K ) (cid:62) H λ ( τ ) − λ / M ξ (cid:107) op dτ = O ( λ ) . (7.26) Proof.
The asserted bounds will be derived from a matrix version of the variation of constantformula below. Compared to the one-dimensional case treated in [39, Chapter 4], this is complicatedby the fact that the involved matrices generally do not commute. To overcome this difficulty, weintroduce the unique solutions Φ F λ and Φ (cid:98) F on [0 , T ] of the following linear matrix ODEs:Φ (cid:48) F λ ( τ ) = 1 λ (cid:16) I N − ⊗ ¯Λ − / (cid:17) F λ (cid:62) ( τ ) (cid:16) I N − ⊗ ¯Λ − / (cid:17) Φ F λ ( τ ) , Φ F λ (0) = I K ( N − , (7.27)Φ (cid:48) (cid:98) F ( τ ) = 1 λ / (cid:16) I N − ⊗ ¯Λ − / (cid:17) (cid:98) F (cid:16) I N − ⊗ ¯Λ − / (cid:17) Φ (cid:98) F ( τ ) , Φ (cid:98) F (0) = I K ( N − . (7.28)Moreover, for 0 ≤ r ≤ τ ≤ T , defineΨ F λ ( r ; τ ) := Φ F λ ( r )Φ − F λ ( τ ) , Ψ (cid:98) F ( r ; τ ) := Φ (cid:98) F ( r )Φ − (cid:98) F ( τ ) , (7.29)The proof of the asymptotic expansions then proceeds along the following steps: Recall that the square-root of Γ is well defined by [22, Theorem 1.29] even though this matrix has positiveeigenvalues that is generally only positive semidefinite but not symmetric, compare [8, Lemma A.5]. tep 1: Show that for every τ ∈ [0 , T ], (cid:107) F λ ( τ ) (cid:107) op = O ( λ / ). Step 2:
Show that for every τ ∈ [0 , T ], (cid:107) ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:107) ≤ σ min ( α ) (cid:16) − e −(cid:107) c (cid:107)(cid:107) ¯Λ (cid:107) / (cid:107) ¯Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107) T (cid:17) . Step 3:
Show that for every 0 ≤ r ≤ τ ≤ T , (cid:82) τ Ψ F λ ( r ; τ ) dr = O ( λ / ). Step 4:
Show that for every τ ∈ [0 , T ], (cid:107) H λ ( τ ) (cid:107) op = O ( λ / ). Step 5:
Show that the approximations of F λ and H λ in (7.25) and (7.26) are valid at the assertedorders. Step 1:
Notice that (cid:98) F is the solution of the algebraic Riccati equation (cid:98) F (cid:0) I N − ⊗ ¯Λ − (cid:1) (cid:98) F = Γ ⊗ αα (cid:62) . To simplify notation, set G λ = α ( H λ ) (cid:62) ( c ⊗ I K ) + ( c ⊗ I K ) (cid:62) H λ ( α + ( c ⊗ I K ) (cid:62) H λ ) (cid:62) . The differencebetween the function F λ and the constant λ / (cid:98) F satisfies (cid:16) F λ − λ / (cid:98) F (cid:17) (cid:48) = (cid:16) F λ (cid:17) (cid:48) = Γ ⊗ (cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17) (cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17) (cid:62) − F λ (cid:0) I N − ⊗ ¯Λ − (cid:1) F λ λ = 1 λ (cid:16) λ (cid:98) F (cid:0) I N − ⊗ ¯Λ − (cid:1) (cid:98) F − F λ (cid:0) I N − ⊗ ¯Λ − (cid:1) F λ (cid:17) + Γ ⊗ G λ = 1 λ (cid:16) λ / (cid:98) F (cid:0) I N − ⊗ ¯Λ − (cid:1) ( λ / (cid:98) F − F λ ) + ( λ / (cid:98) F − F λ ) (cid:0) I N − ⊗ ¯Λ − (cid:1) F λ (cid:17) + Γ ⊗ G λ . We now want to apply a version of the variation of constant formula to obtain explicit estimateseven though the matrices involved generally do not commute. To this end, multiply Φ (cid:62) F λ and Φ (cid:98) F on the left and right of F λ − (cid:98) F , respectively. Then, taking derivatives and plugging in (cid:0) F λ (cid:1) (cid:48) yields (cid:16) Φ (cid:62) F λ (cid:16) I N − ⊗ ¯Λ − / (cid:17) (cid:16) F λ − λ / (cid:98) F (cid:17) (cid:16) I N − ⊗ ¯Λ − / (cid:17) Φ (cid:98) F (cid:17) (cid:48) = Φ (cid:62) F λ (cid:16) Γ ⊗ ¯Λ − / G λ ¯Λ − / (cid:17) Φ (cid:98) F . Now recall the initial condition F λ (0) = 0 and integrate both sides, obtainingΦ (cid:62) F λ ( τ ) (cid:16) I N − ⊗ ¯Λ − / (cid:17) (cid:16) F λ ( τ ) − λ / (cid:98) F (cid:17) (cid:16) I N − ⊗ ¯Λ − / (cid:17) Φ (cid:98) F ( τ )= (cid:90) τ Φ (cid:62) F λ ( r ) (cid:16) Γ ⊗ ¯Λ − / G λ ( r ) ¯Λ − / (cid:17) Φ (cid:98) F ( r ) dr − λ / (cid:16) I N − ⊗ ¯Λ − / (cid:17) (cid:98) F (cid:16) I N − ⊗ ¯Λ − / (cid:17) . (Here, the arguments are dropped to ease notation.) By definition of Ψ F λ ( r ; τ ) and Ψ (cid:98) F ( r ; τ )in (7.29), we have F λ ( τ ) − λ / (cid:98) F = − λ / (cid:16) I N − ⊗ ¯Λ / (cid:17) Ψ (cid:62) F λ (0; τ ) (cid:16) I N − ⊗ ¯Λ − / (cid:17) (cid:98) F (cid:16) I N − ⊗ ¯Λ − / (cid:17) Ψ (cid:98) F (0; τ ) (cid:16) I N − ⊗ ¯Λ / (cid:17)(cid:90) τ (cid:16) I N − ⊗ ¯Λ / (cid:17) Ψ (cid:62) F λ ( r ; τ ) (cid:16) Γ ⊗ ¯Λ − / G λ ( r ) ¯Λ − / (cid:17) Ψ (cid:98) F ( r ; τ ) (cid:16) I N − ⊗ ¯Λ / (cid:17) dr. (7.30)With C := (cid:107) c (cid:107)(cid:107) ¯Λ (cid:107) / (cid:107) ¯Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107) = O (1), the representation (7.13) for α + ( c ⊗ I K ) (cid:62) H λ implies that, for every τ ∈ [0 , T ], (cid:13)(cid:13)(cid:13) α + ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:13)(cid:13)(cid:13) op ≤ (cid:107) α (cid:107) exp (cid:16) (cid:107) c (cid:107)(cid:107) λ ¯Λ (cid:107) / (cid:107) λ ¯Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107) τ (cid:17) = (cid:107) α (cid:107) e C τ ≤ (cid:107) α (cid:107) e C T . (cid:107) Ψ F λ ( r ; τ ) (cid:107) op ≤ . To derive a similar bound for (cid:107) Ψ (cid:98) F ( r ; τ ) (cid:107) op , notice that (cid:16) I N − ⊗ ¯Λ − / (cid:17) (cid:98) F (cid:16) I N − ⊗ ¯Λ − / (cid:17) = Γ / ⊗ (cid:16) ¯Λ − / αα (cid:62) ¯Λ − / (cid:17) / , where (cid:0) ¯Λ − / αα (cid:62) ¯Λ − / (cid:1) / is a symmetric positive definite matrix. By Lemma 7.1, the smallesteigenvalue of Γ + Γ (cid:62) is strictly positive, so (cid:98) F min := 12 σ min (Γ + Γ (cid:62) ) σ min (cid:16) ¯Λ − / αα (cid:62) ¯Λ − / (cid:17) / > . Lemma 7.11 therefore yields the following upper bound for Ψ (cid:98) F ( r ; τ ), valid for every 0 ≤ r ≤ τ ≤ T : (cid:107) Ψ (cid:98) F ( r ; τ ) (cid:107) op ≤ e − (cid:98) F min λ / ( τ − r ) . Moreover, with the help of (7.13), direct calculation yields (cid:13)(cid:13)(cid:13) Γ ⊗ ¯Λ − / G λ ( r ) ¯Λ − / (cid:13)(cid:13)(cid:13) op ≤ (cid:107) Γ (cid:107)(cid:107) ¯Λ − (cid:107) (cid:18) (cid:107) α (cid:107) + (cid:13)(cid:13)(cid:13) α + ( c ⊗ I K ) (cid:62) H ( r ) (cid:13)(cid:13)(cid:13) op (cid:19) (cid:107) ( c ⊗ I K ) (cid:62) H λ ( r ) (cid:107) op ≤ e C T (cid:107) Γ (cid:107)(cid:107) ¯Λ − (cid:107) α (cid:107)(cid:107) ( c ⊗ I K ) (cid:62) H λ ( r ) (cid:107) op . After taking into account the above estimates, (7.30) leads to the following bound for the differencebetween the solution of the Riccati system and its constant approximation: (cid:107) F λ ( τ ) − λ / (cid:98) F (cid:107) op ≤ e C T (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107) Γ (cid:107)(cid:107) α (cid:107) (cid:90) τ (cid:107) Ψ F λ ( r ; τ ) (cid:107) op (cid:107) Ψ (cid:98) F ( r ; τ ) (cid:107) op (cid:107) ( c ⊗ I K ) (cid:62) H λ ( r ) (cid:107) op dr + λ / (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) (cid:98) F (cid:107)(cid:107) Ψ (cid:98) F (0; τ ) (cid:107) op (cid:107) Ψ F λ (0; τ ) (cid:107) op ≤ e C T (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) Γ (cid:107)(cid:107) α (cid:107) (cid:90) τ e − (cid:98) F min λ / ( τ − r ) (cid:107) ( c ⊗ I K ) (cid:62) H λ ( r ) (cid:107) op dr + λ / (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) (cid:98) F (cid:107)(cid:107) Ψ (cid:98) F (0; τ ) (cid:107) op (cid:107) Ψ F λ (0; τ ) (cid:107) op . (7.31)Recalling (7.12) and (7.13), and taking into account that (cid:107) Ψ F λ ( r ; τ ) (cid:107) op ≤ ≤ r ≤ τ ≤ T ,we have the following bound for ( c ⊗ I K ) (cid:62) H λ ( τ ) for small costs ¯Λ λ = λ ¯Λ: (cid:107) ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:107) op ≤ (cid:90) τ (cid:107) c (cid:107)(cid:107) λ ¯Λ (cid:107) / (cid:107) λ ¯Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107)(cid:107) α (cid:107) e C r dr ≤ (cid:90) τ (cid:107) c (cid:107)(cid:107) ¯Λ (cid:107) / (cid:107) ¯Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107)(cid:107) α (cid:107) e C r dr ≤ (cid:107) α (cid:107) e C τ ≤ (cid:107) α (cid:107) e C T . The triangle inequality and (7.31) in turn give (cid:107) F λ ( τ ) (cid:107) op ≤ λ / (cid:107) (cid:98) F (cid:107) op + (cid:107) F λ ( τ ) − λ / (cid:98) F (cid:107) op ≤ λ / (cid:107) (cid:98) F (cid:107) + λ / (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) (cid:98) F (cid:107)(cid:107) Ψ (cid:98) F (0; τ ) (cid:107) op + 2 e C T (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) Γ (cid:107)(cid:107) α (cid:107) (cid:90) τ e − (cid:98) F min λ / ( τ − r ) dr ≤ λ / (cid:18) (cid:107) (cid:98) F (cid:107) + (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) (cid:98) F (cid:107) + 2 e (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) Γ (cid:107)(cid:107) α (cid:107) (cid:98) F min (cid:19) =: λ / F max = O ( λ / ) . σ max (cid:16)(cid:16) I N − ⊗ ¯Λ − / (cid:17) (cid:16) F λ ( τ ) + F λ (cid:62) ( τ ) (cid:17) (cid:16) I N − ⊗ ¯Λ − / (cid:17)(cid:17) ≤ λ / (cid:107) ¯Λ − (cid:107) op F max . By Lemma 7.11, it follows that the smallest singular value of Ψ F λ ( r ; τ ) for τ ≤ r ≤ τ ≤ τ satisfies σ min (Ψ F λ ( r ; τ )) ≥ e − F max (cid:107) ¯Λ − (cid:107) op λ / ( τ − r ) . (7.32) Step 2:
Recall C = (cid:107) c (cid:107)(cid:107) ¯Λ (cid:107) / (cid:107) ¯Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107) = O (1). We prove the claim by contradiction.To this end, suppose there exists a time τ ∈ [0 , T ] such that (cid:107) ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:107) > σ min ( α (cid:62) )(1 − e − C T ) . Notice that the Frobenius norm τ (cid:55)→ (cid:107) ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:107) is continuous, and (cid:107) ( c ⊗ I K ) (cid:62) H λ (0) (cid:107) = 0.By the intermediate value theorem, there exists τ ∈ [0 , τ ] (depending on λ ) such that (cid:107) ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:107) = σ min ( α (cid:62) )(1 − e − C T ) (7.33)and (cid:107) ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:107) < σ min ( α (cid:62) )(1 − e − C T ) , for τ ∈ [0 , τ ).On [0 , τ ], we then have σ min (cid:16) α + ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:17) ≥ σ min ( α ) − (cid:107) ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:107) op ≥ σ min ( α ) (cid:0) − (1 − e − C T ) (cid:1) = σ min ( α ) e − C T . (7.34)Define τ = λ / /C . Then for τ ∈ [0 , τ ], the representation (7.12) for H λ yields (cid:107) ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:107) ≤ √ K + D (cid:107) ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:107) op ≤ √ K + D (cid:107) α (cid:107) (cid:90) τ (cid:107) c (cid:107)(cid:107) ¯Λ (cid:107) / (cid:107) ¯Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107) e (cid:107) c (cid:107)(cid:107) ¯Λ (cid:107) / (cid:107) ¯Λ − (cid:107) / (cid:107) Γ (cid:107)(cid:107) ξ (cid:107) r dr ≤ √ K + D (cid:107) α (cid:107) (cid:0) e C τ − (cid:1) ≤ √ K + D (cid:107) α (cid:107) (cid:0) e C τ − (cid:1) = √ K + D (cid:107) α (cid:107) (cid:16) e λ / − (cid:17) = O ( λ / ) . For sufficiently small λ , we thus have τ < τ .We now derive an upper bound of (cid:107) ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:107) on [ τ , τ ] that will lead to the desiredcontradiction to (7.33). To this end, we first develop some upper and lower bounds for Ψ F λ ( r ; τ )and F λ . By the identity (7.5) and the initial condition F λ (0) = 0, (cid:16) I N − ⊗ ¯Λ − / (cid:17) (cid:16) F λ ( τ ) + F λ (cid:62) ( τ ) (cid:17) (cid:16) I N − ⊗ ¯Λ − / (cid:17) = (cid:90) τ Ψ (cid:62) F λ ( r ; τ ) (cid:16) (Γ + Γ (cid:62) ) ⊗ ( α + ( c ⊗ I K ) (cid:62) H λ ( r ))( α + ( c ⊗ I K ) (cid:62) H λ ( r )) (cid:62) (cid:17) Ψ F λ ( r ; τ ) dr + 2 (cid:90) τ Ψ (cid:62) F λ ( r ; τ ) (cid:16) I N − ⊗ ¯Λ − / (cid:17) F λ ( r ) (cid:0) I N − ⊗ ¯Λ − (cid:1) F λ (cid:62) ( r ) (cid:16) I N − ⊗ ¯Λ − / (cid:17) Ψ F λ ( r ; τ ) dr. σ min (cid:16)(cid:16) I N − ⊗ ¯Λ − / (cid:17) (cid:16) F λ ( τ ) + F λ (cid:62) ( τ ) (cid:17) (cid:16) I N − ⊗ ¯Λ − / (cid:17)(cid:17) ≥ (cid:90) τ σ min (Γ + Γ (cid:62) ) σ min ( ¯Λ − ) σ ( α + ( c ⊗ I K ) (cid:62) H λ ( r )) σ (Ψ F λ ( r ; τ )) dr ≥ (cid:90) τ σ min (Γ + Γ (cid:62) ) σ min ( ¯Λ − ) σ ( α ) e − C T e − F max (cid:107) ¯Λ − (cid:107) op λ / ( τ − r ) dr = λ / σ min (Γ + Γ (cid:62) ) σ min ( ¯Λ − ) σ ( α ) e C T F max (cid:107) ¯Λ − (cid:107) op (cid:32) − e − F max (cid:107) ¯Λ − (cid:107) op λ / τ (cid:33) ≥ λ / σ min (Γ + Γ (cid:62) ) σ min ( ¯Λ − ) σ ( α ) e C T F max (cid:107) ¯Λ − (cid:107) op (cid:32) − e − F max (cid:107) ¯Λ − (cid:107) op λ / τ (cid:33) = λ / σ min (Γ + Γ (cid:62) ) σ min ( ¯Λ − ) σ ( α ) e C T F max (cid:107) ¯Λ − (cid:107) op (cid:18) − e − F max (cid:107) ¯Λ − (cid:107) op C (cid:19) := 2 λ / F min = O ( λ / ) . Again by Lemma 7.11, we can estimate the largest singular value of Ψ F λ ( r ; τ ) for every τ ≤ r ≤ τ ≤ τ as follows: (cid:107) Ψ F λ ( r ; τ ) (cid:107) op = σ max (Ψ F λ ( r ; τ )) ≤ e − F min λ / ( τ − r ) . (7.35)Therefore, after plugging in (7.12) and (7.13), we can estimate the Frobenius norm of ( c ⊗ I K ) (cid:62) H λ ( τ ) as (cid:107) ( c ⊗ I K ) (cid:62) H λ ( τ ) (cid:107) ≤ √ K + D (cid:90) τ (cid:107) c (cid:107)(cid:107) Γ (cid:107)(cid:107) ¯Λ / (cid:107)(cid:107) ¯Λ − / (cid:107)(cid:107) ξ (cid:107)(cid:107) Ψ F λ ( r ; τ ) (cid:107) op (cid:107) α + c ⊗ I K ) (cid:62) H λ ( r ) (cid:107) op dr ≤ √ K + D (cid:107) c (cid:107)(cid:107) α (cid:107)(cid:107) Γ (cid:107)(cid:107) ¯Λ / (cid:107)(cid:107) ¯Λ − / (cid:107)(cid:107) ξ (cid:107) e C T (cid:90) τ (cid:107) Ψ F λ ( r ; τ ) (cid:107) op dr ≤ √ K + D (cid:107) c (cid:107)(cid:107) α (cid:107)(cid:107) Γ (cid:107)(cid:107) ¯Λ / (cid:107)(cid:107) ¯Λ − / (cid:107)(cid:107) ξ (cid:107) e C T (cid:18)(cid:90) τ dr + (cid:90) τ τ e − F min λ / ( τ − r ) dr (cid:19) ≤ √ K + D (cid:107) c (cid:107)(cid:107) α (cid:107)(cid:107) Γ (cid:107)(cid:107) ¯Λ / (cid:107)(cid:107) ¯Λ − / (cid:107)(cid:107) ξ (cid:107) e C T (cid:32) λ / C + λ / F min e − F min λ / ( τ − r ) (cid:33) = O ( λ / ) . For sufficiently small λ , this contradicts (7.33) and therefore completes the proof of Step 2. Step 3:
From Step 2, we know that the estimate (7.35) holds for λ / /C = τ ≤ r ≤ τ ≤ T .This upper bound in turn implies (cid:90) ττ (cid:107) Ψ F λ ( r ; τ ) (cid:107) op dr ≤ (cid:90) ττ e − F min λ / ( τ − r ) dr = λ / F min (cid:18) − e − F min λ / ( τ − τ ) (cid:19) ≤ λ / F min . Together with the coarser upper bound (cid:107) Ψ F λ ( r ; τ ) (cid:107) op ≤ ≤ r ≤ τ ≤ T ), the desired estimatenow follows: (cid:90) τ (cid:107) Ψ F λ ( r ; τ ) (cid:107) op dr = (cid:90) τ (cid:107) Ψ F λ ( r ; τ ) (cid:107) op dr + (cid:90) ττ (cid:107) Ψ F λ ( r ; τ ) (cid:107) op dr ≤ τ + λ / F min = λ / (cid:18) C + 1 F min (cid:19) . tep 4. The representation (7.12) for H λ , the estimate (7.13) and the bound for the integralof Ψ F λ ( r ; τ ) from Step 3, lead to the following upper bound for the operator norm of H λ : (cid:107) H λ ( τ ) (cid:107) op ≤ (cid:90) τ (cid:107) Γ (cid:107)(cid:107) ¯Λ / (cid:107)(cid:107) ¯Λ − / (cid:107)(cid:107) ξ (cid:107)(cid:107) Ψ F λ ( r ; τ ) (cid:107) op (cid:107) α + c ⊗ I K ) (cid:62) H λ ( r ) (cid:107) op dr ≤ (cid:107) α (cid:107)(cid:107) Γ (cid:107)(cid:107) ¯Λ / (cid:107)(cid:107) ¯Λ − / (cid:107)(cid:107) ξ (cid:107) e C T (cid:18)(cid:90) τ (cid:107) Ψ F λ ( r ; τ ) (cid:107) op dr (cid:19) = O ( λ / ) . Step 5:
With the estimate from Steps 1-4, we can now complete the proof of Lemma 7.12. Forthe approximation of F λ , insert the bounds for H λ from Step 4 into (7.31), obtaining (cid:107) F λ ( τ ) − λ / (cid:98) F (cid:107) op ≤ λ / (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) (cid:98) F (cid:107)(cid:107) Ψ (cid:98) F (0; τ ) (cid:107) op (cid:107) Ψ F λ (0; τ ) (cid:107) op + 2 e C T (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) Γ (cid:107)(cid:107) α (cid:107) (cid:90) τ e − (cid:98) F min λ / ( τ − r ) (cid:107) c (cid:107)(cid:107) H λ ( r ) (cid:107) op dr = λ / (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) (cid:98) F (cid:107)(cid:107) Ψ (cid:98) F (0; τ ) (cid:107) op (cid:107) Ψ F λ (0; τ ) (cid:107) op + O ( λ ) . Now, recall that (cid:107) Ψ F λ (0; τ ) (cid:107) op ≤
1; integrating (7.31) in turn yields the desired approximation of F λ : (cid:90) T (cid:107) F λ ( τ ) − λ / (cid:98) F (cid:107) op dτ = λ / (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) (cid:98) F (cid:107) (cid:90) T (cid:107) Ψ (cid:98) F (0; τ ) (cid:107) op dτ + O ( λ ) ≤ λ / (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) (cid:98) F (cid:107) λ / (cid:98) F min + O ( λ ) = O ( λ ) . To derive an analogous result for H λ , define (cid:98) H := (cid:18) Γ / ⊗ ¯Λ (cid:16) ¯Λ αα (cid:62) (cid:17) − α (cid:19) ξ. Observe that (cid:98) H is the solution of the linear algebraic equation (cid:98) F ( I N − ⊗ ¯Λ − ) (cid:98) H = Γ ⊗ α . Whence wecan express the difference between λ / (cid:98) H and the solution H λ ( τ ) of the linear Riccati equation (4.8)as (cid:16) H λ − λ / (cid:98) H (cid:17) (cid:48) = (cid:16) H λ (cid:17) (cid:48) = (cid:16) Γ ⊗ (cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17)(cid:17) ξ − λ F λ (cid:0) I N − ⊗ ¯Λ − (cid:1) H λ = (cid:16) Γ ⊗ ( c ⊗ I K ) (cid:62) H λ (cid:17) ξ + 1 λ (cid:16) λ (cid:98) F (cid:0) I N − ⊗ ¯Λ − (cid:1) (cid:98) H − F λ (cid:0) I N − ⊗ ¯Λ − (cid:1) H λ (cid:17) = (cid:16) Γ ⊗ ( c ⊗ I K ) (cid:62) H λ (cid:17) ξ + 1 λ F λ (cid:0) I N − ⊗ ¯Λ − (cid:1) (cid:16) λ / (cid:98) H − H λ (cid:17) + 1 λ / (cid:16) λ / (cid:98) F − F λ (cid:17) (cid:0) I N − ⊗ ¯Λ − (cid:1) (cid:98) H. Similarly as above, a matrix version of variation of constants now yields H λ ( τ ) − λ / (cid:98) H = − λ / Ψ (cid:62) F λ (0; τ ) (cid:98) H + (cid:90) τ Ψ (cid:62) F λ ( r ; τ ) (cid:16) Γ ⊗ ( c ⊗ I K ) (cid:62) H λ ( r ) (cid:17) ξdr + 1 λ / (cid:90) τ Ψ (cid:62) F λ ( r ; τ ) (cid:16) λ / (cid:98) F − F λ ( r ) (cid:17) (cid:0) I N − ⊗ ¯Λ − (cid:1) (cid:98) Hdr.
The first term is of order O ( λ / ) (cid:107) Ψ F λ (0; τ ) (cid:107) op = O ( λ ). The estimates from Step 3 and 4, and adirect calculation in turn show that the second term is of order O ( λ ) as well: (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) τ Ψ (cid:62) F λ ( r ; τ ) (cid:16) Γ ⊗ ( c ⊗ I K ) (cid:62) H λ ( r ) (cid:17) ξdr (cid:13)(cid:13)(cid:13)(cid:13) op ≤ (cid:107) c (cid:107)(cid:107) Γ (cid:107)(cid:107) ξ (cid:107) (cid:90) τ (cid:107) H λ ( r ) (cid:107) op (cid:107) Ψ F λ ( r ; τ ) (cid:107) op dr = O ( λ ) . H λ − λ / (cid:98) H , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) τ Ψ (cid:62) F λ ( r ; τ ) (cid:16) λ / (cid:98) F − F λ ( r ) (cid:17) (cid:0) I N − ⊗ ¯Λ − (cid:1) (cid:98) Hdr (cid:13)(cid:13)(cid:13)(cid:13) op ≤ (cid:90) τ (cid:107) Ψ F λ ( r ; τ ) (cid:107) op (cid:107) ¯Λ − (cid:107)(cid:107) (cid:98) H (cid:107)(cid:107) λ / (cid:98) F − F λ ( r ) (cid:107) op dr ≤ (cid:90) τ (cid:107) Ψ F λ ( r ; τ ) (cid:107) op (cid:107) ¯Λ − (cid:107)(cid:107) (cid:98) H (cid:107) (cid:16) λ / (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) (cid:98) F (cid:107)(cid:107) Ψ (cid:98) F (0; r ) (cid:107) op (cid:107) Ψ F λ (0; r ) (cid:107) op + O ( λ ) (cid:17) dr = O ( λ / ) (cid:90) τ (cid:107) Ψ F λ ( r ; τ ) (cid:107) op (cid:107) Ψ (cid:98) F (0; r ) (cid:107) op (cid:107) Ψ F λ (0; r ) (cid:107) op dr + O ( λ ) (cid:90) τ (cid:107) Ψ F λ ( r ; τ ) (cid:107) op dr = O ( λ / ) (cid:107) Ψ F λ (0; τ ) (cid:107) op (cid:90) τ (cid:107) Ψ (cid:98) F (0; r ) (cid:107) op dr + O ( λ / ) . = O ( λ ) (cid:107) Ψ F λ (0; τ ) (cid:107) op + O ( λ / ) . Together with the estimate from Step 3, it follows that (cid:90) T (cid:107) H λ ( τ ) − λ / (cid:98) H (cid:107) op dτ ≤ O ( λ ) + O ( λ / ) (cid:90) T (cid:107) Ψ F λ (0; τ ) (cid:107) op dτ = O ( λ ) . Therefore, the assertion follows after recalling that
M ξ = ( c ⊗ I K ) (cid:62) (cid:98) H by definition.With the above approximations of the Riccati system (4.8) at hand, we can now carry out therigorous convergence proof for the asymptotic expansions from Theorem 5.1. Proof of Theorem 5.1.
From (4.14) in Theorem 5.1, we have σ t − ¯ σ t = ( c ⊗ I K ) H λ ( T − t ). Hencethe approximation (5.1) of the volatility correction due to small transaction costs follows directlyfrom (7.26).Next, we turn to the trading rate ˙ ϕ . To this end, we first need a further estimation. Noticethat (the arguments are dropped here to ease notation) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17) − (cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17) (cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17) (cid:62) (cid:16) αα (cid:62) (cid:17) − α (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17) I D − (cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17) (cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17) (cid:62) (cid:16) αα (cid:62) (cid:17) − α (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17) (cid:18) I D − (cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17) (cid:62) (cid:16) αα (cid:62) (cid:17) − α (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) ( c ⊗ I K ) (cid:62) H λ (cid:18) I D − (cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17) (cid:62) (cid:16) αα (cid:62) (cid:17) − α (cid:19) − α (cid:16) H λ (cid:17) (cid:62) ( c ⊗ I K ) (cid:16) αα (cid:62) (cid:17) − α (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:107) c (cid:107)(cid:107) H λ (cid:107) (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) I D − (cid:16) α + ( c ⊗ I K ) (cid:62) H λ (cid:17) (cid:62) (cid:16) αα (cid:62) (cid:17) − α (cid:13)(cid:13)(cid:13)(cid:13) + (cid:107) α (cid:107) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) αα (cid:62) (cid:17) − α (cid:13)(cid:13)(cid:13)(cid:13)(cid:19) = O ( λ / ) . Define E λ ( τ ) = H λ ( τ ) − F λ ( τ ) (cid:18) I N − ⊗ (cid:16) αα (cid:62) (cid:17) − α (cid:19) ξ. By the representations (7.11) and (7.12) for F λ and H λ , we have (cid:13)(cid:13)(cid:13) E λ ( τ ) (cid:13)(cid:13)(cid:13) op ≤ O ( λ / ) (cid:90) τ (cid:107) ¯Λ (cid:107)(cid:107) ¯Λ − (cid:107)(cid:107) Φ F λ ( r ; τ ) (cid:107) dr = O ( λ ) . d (cid:104) Φ (cid:62) ( t ) (cid:16) I N − ⊗ ¯Λ / (cid:17) ( ϕ t − ¯ ϕ t ) (cid:105) = Φ (cid:62) ( t ) (cid:16) I N − ⊗ ¯Λ / (cid:17) (cid:0) λ − (cid:0) I N − ⊗ ¯Λ − (cid:1) F ( T − t ) ( ϕ t − ¯ ϕ t ) dt + d ( ϕ t − ¯ ϕ t ) (cid:1) = − λ − Φ (cid:62) ( t ) (cid:16) I N − ⊗ ¯Λ − / (cid:17) E λ ( T − t ) W t dt + Φ (cid:62) ( t ) (cid:18) I N − ⊗ ¯Λ / (cid:16) αα (cid:62) (cid:17) − α (cid:19) ξdW t . Now, we show that the deviation ϕ − ¯ ϕ is approximated by the following ( K ( N − d ∆ t := − λ − / (cid:16) Γ / ⊗ ¯Λ − (cid:16) ¯Λ αα (cid:62) (cid:17)(cid:17) ∆ t dt + (cid:18) I N − ⊗ ¯Λ / (cid:16) αα (cid:62) (cid:17) − α (cid:19) ξdW t = − λ − / (cid:0) I N − ⊗ ¯Λ − (cid:1) (cid:98) F ∆ t dt + (cid:18) I N − ⊗ ¯Λ / (cid:16) αα (cid:62) (cid:17) − α (cid:19) ξdW t , ∆ = 0 . (7.36)First, again by our matrix-version of variation of constants, we can have the explicit solution of theSDE (7.36) can be written as∆ t = (cid:90) t (cid:16) I N − ⊗ ¯Λ − / (cid:17) Ψ (cid:62) (cid:98) F (cid:62) ( r ; t ) (cid:18) I N − ⊗ ¯Λ / (cid:16) αα (cid:62) (cid:17) − α (cid:19) ξdW r , where Ψ (cid:98) F (cid:62) ( r ; t ) = Φ (cid:98) F (cid:62) ( r )Φ − (cid:98) F (cid:62) ( t ), and Φ (cid:98) F (cid:62) is the solution to the following matrix linear ODE:Φ (cid:48) (cid:98) F (cid:62) ( τ ) = 1 λ / (cid:16) I N − ⊗ ¯Λ − / (cid:17) (cid:98) F (cid:62) (cid:16) I N − ⊗ ¯Λ − / (cid:17) Φ (cid:98) F ( τ ) , Φ (cid:98) F (0) = I K ( N − . The process ∆ is a Gaussian with mean 0; moreover, all eigenvalues of its covariance matrix are oforder O ( λ / ). As a consequence, E [ (cid:107) ∆ t (cid:107) ] = O ( λ / ).To assess the accuracy of the asserted asymptotic approximation, consider the (rescaled) differ-ence between ϕ − ¯ ϕ and ∆: d (cid:104) Φ (cid:62) ( t ) (cid:16) I N − ⊗ ¯Λ / (cid:17) ( ϕ t − ¯ ϕ t − ∆ t ) (cid:105) = λ − Φ (cid:62) ( t ) (cid:16) I N − ⊗ ¯Λ − / (cid:17) (cid:16)(cid:16) λ / (cid:98) F − F λ ( T − t ) (cid:17) ∆ t − E λ ( T − r ) W t (cid:17) dt. As the initial value of the difference vanishes by assumption, it follows that ϕ t − ¯ ϕ t − ∆ t = λ − (cid:90) t Ψ (cid:62) ( r ; t ) (cid:0) I N − ⊗ ¯Λ − (cid:1) (cid:16)(cid:16) λ / (cid:98) F − F ( T − r ) (cid:17) ∆ r − E λ ( T − r ) W r (cid:17) dr. With similar argument on (cid:107) Ψ (cid:107) op and || λ / (cid:98) F − F λ ( T − r ) || op as in the approximation of H λ , weobtain (cid:107) ϕ − ¯ ϕ − ∆ (cid:107) H p ≤ (cid:18)(cid:90) T E (cid:2) (cid:107) ϕ t − ¯ ϕ t − ∆ t (cid:107) p op (cid:3) dt (cid:19) / p ≤ λ − (cid:90) T (cid:18)(cid:90) t (cid:107) Ψ( r ; t ) (cid:107) op (cid:13)(cid:13)(cid:13) λ / (cid:98) F − F λ ( T − r ) (cid:13)(cid:13)(cid:13) op O ( λ / ) dr + O ( λ / ) (cid:19) dt = O ( λ / ) (cid:90) T (cid:107) Ψ(0; t ) (cid:107) op dt + O ( λ / ) = O ( λ / ) . ϕ from (4.12), which we can rewrite as˙ ϕ t = − λ − (cid:0) I N − ⊗ ¯Λ − (cid:1) (cid:104) F λ ( T − t ) ( ϕ t − ¯ ϕ t ) + E λ ( T − t ) W t (cid:105) = − λ − / (cid:0) I N − ⊗ ¯Λ − (cid:1) (cid:98) F ∆ t + O H p (1)= − λ − / (cid:16) Γ / ⊗ ¯Λ − (cid:16) ¯Λ αα (cid:62) (cid:17)(cid:17) ∆ t + O H p (1) . Setting ˙¯ ϕ := − λ − / (cid:0) Γ / ⊗ ¯Λ − (cid:0) ¯Λ αα (cid:62) (cid:1)(cid:1) ∆, we then have d ˙¯ ϕ t = − λ − / (cid:16) Γ / ⊗ ¯Λ − (cid:16) ¯Λ αα (cid:62) (cid:17)(cid:17) d ∆ t = − λ − / (cid:16) Γ / ⊗ ¯Λ − (cid:16) ¯Λ αα (cid:62) (cid:17)(cid:17) (cid:18) ˙¯ ϕ t dt + (cid:18) I N − ⊗ ¯Λ / (cid:16) αα (cid:62) (cid:17) − α (cid:19) ξdW t (cid:19) , (7.37)which established the desired approximation from Theorem 5.1.To derive the corresponding result for the equilibrium prices, recall from (4.11)-(4.13) in Theo-rem 4.5 that the difference of frictional and frictionless price level is S t − ¯ S t = Y t − λ (cid:0) c ⊗ ¯Λ (cid:1) (cid:62) ˙ ϕ t = Y t + λ / (cid:16) c (cid:62) Γ / ⊗ (cid:16) ¯Λ αα (cid:62) (cid:17)(cid:17) ∆ t + O H p ( λ ) . At the initial time t = 0, ∆ = 0, the definition of Y in (4.13) and the estimates (7.26) fromLemma 7.12 give S − ¯ S = − ¯ γ (cid:18)(cid:90) T ( c ⊗ I K ) (cid:62) H λ ( r ) α (cid:62) + α (cid:16) H λ ( r ) (cid:17) (cid:62) ( c ⊗ I K ) + ( c ⊗ I K ) (cid:62) H λ ( r ) (cid:16) H λ ( r ) (cid:17) (cid:62) ( c ⊗ I K ) dr (cid:19) s = − ¯ γ (cid:18)(cid:90) T ( c ⊗ I K ) (cid:62) H λ ( r ) α (cid:62) + α (cid:16) H λ ( r ) (cid:17) (cid:62) ( c ⊗ I K ) dr (cid:19) s + O H p ( λ )= − ¯ γ (cid:18)(cid:90) T (cid:16) ( c ⊗ I K ) (cid:62) H λ ( r ) − λ / M ξ (cid:17) α (cid:62) + α (cid:16) H λ ( r ) − λ / M ξ (cid:17) (cid:62) ( c ⊗ I K ) dr (cid:19) s − λ / ¯ γ (cid:16) M ξα (cid:62) + αξ (cid:62) M (cid:62) (cid:17) sT + O H p ( λ )= − λ / ¯ γ (cid:16) M ξα (cid:62) + αξ (cid:62) M (cid:62) (cid:17) sT + O H p ( λ ) . A straightforward but tedious computation shows that the drift term of Y from (4.13) can bewritten as d Y t dt = ¯ γ (cid:18) ( c ⊗ I K ) (cid:62) H λ α (cid:62) + α (cid:16) H λ (cid:17) (cid:62) ( c ⊗ I K ) + ( c ⊗ I K ) (cid:62) H λ (cid:16) H λ (cid:17) (cid:62) ( c ⊗ I K ) ( T − t ) (cid:19) s = λ / ¯ γ (cid:16) M ξα (cid:62) + αξ (cid:62) M (cid:62) (cid:17) s + O H p ( λ ) . For the drift term of ˙ ϕ from (7.37) we have − λ − / (cid:16) Γ / ⊗ ¯Λ − (cid:16) ¯Λ αα (cid:62) (cid:17)(cid:17) ˙¯ ϕ + O H p (1) . In summary, we therefore obtain the following approximation for the frictional equilibrium expectedreturns: µ t = ¯ µ t + λ / (cid:16) c (cid:62) Γ / ⊗ (cid:16) ¯Λ αα (cid:62) (cid:17)(cid:17) ˙ ϕ t + λ / ¯ γ (cid:16) M ξα (cid:62) + αξ (cid:62) M (cid:62) (cid:17) s + O H p ( λ )= ¯ γαα (cid:62) s + λ / (cid:16) c (cid:62) Γ / ⊗ (cid:16) ¯Λ αα (cid:62) (cid:17)(cid:17) ˙ ϕ t + λ / ¯ γ (cid:16) M ξα (cid:62) + αξ (cid:62) M (cid:62) (cid:17) s + O H p ( λ ) . 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