Asymptotic Filter Behavior for High-Frequency Expert Opinions in a Market with Gaussian Drift
AAbdelali Gabih, Hakam Kondakji, and Ralf Wunderlich
Asymptotic Filter Behavior forHigh-Frequency Expert Opinions in aMarket with Gaussian Drift
Abstract:
This paper investigates a financial market where stock returns depend on a hidden Gaussianmean reverting drift process. Information on the drift is obtained from returns and expert opinions in theform of noisy signals about the current state of the drift arriving at the jump times of a homogeneous Poissonprocess. Drift estimates are based on Kalman filter techniques and described by the conditional mean andcovariance matrix of the drift given the observations. We study the filter asymptotics for increasing arrivalintensity of expert opinions and prove that the conditional mean is a consistent drift estimator, it convergesin the mean-square sense to the hidden drift. Thus, in the limit as the arrival intensity goes to infinityinvestors have full information about the drift.
Keywords:
Kalman filter, Ornstein-Uhlenbeck process, Partial information, Expert opinions, Black-Litterman model, Bayesian updating
MSC:
Primary 93E11; Secondary 60F17, 60G35
In this paper we investigate a hidden Gaussian model (HGM) for a financial market where asset pricesfollow a diffusion process with an unobservable Gaussian mean reverting drift modelled by an Ornstein-Uhlenbeck process. Such models are widely used in the study of portfolio optimization problems underpartial information on the drift. There are two popular model classes for the drift, the above-mentionedHGM and the hidden Markov model (HMM) in which the drift process is a continuous-time Markov chain.For utility maximization problems under HGM we refer to Lakner [15] and Brendle [4] while HMMs areused in Rieder and Bäuerle [18], Sass and Haussmann [20]. Both models are studied in Putschögl and Sass[17]. A generalization of these approaches and further references can be found in Björk et al. [1].For solving portfolio problems under partial information the drift has to be estimated from observablequantities such as stock returns. For the above two models, HGM and HMM, the conditional distributionof the drift process given the return observations can be described completely by finite-dimensional filterprocesses. This allows for efficient solutions to portfolio problems including the computation of an opti-
Abdelali Gabih:
Laboratoire Informatique et Mathematiques et leurs Applications (LIMA), Faculty of Science, ChouaibDoukkali University, El Jadida 24000, Morocco, [email protected]
Hakam Kondakji:
Institute of Computer Science, Brandenburg University of TechnologyCottbus - Senftenberg, Postbox 101344, 03013 Cottbus, Germany, [email protected]
Ralf Wunderlich:
Institute of Mathematics, Brandenburg University of TechnologyCottbus - Senftenberg, Postbox 101344, 03013 Cottbus, Germany, [email protected]
The authors thank Dorothee Westphal and Jörn Sass (TU Kaiserslautern) for valuable discussions that improved this pa-per. a r X i v : . [ q -f i n . M F ] M a r Gabih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions mal policy. For HGM and HMM finite-dimensional filters are known as the Kalman and Wonham filters,respectively, see e.g. Elliott, Aggoun and Moore [8], Liptser and Shiryaev [16].It is well known that the drift of a diffusion process is particularly hard to estimate. Even the estimationof a constant drift would require empirical data over an extremely large time horizon, see Rogers [19,Chapter 4.2]. Therefore, in practice filters computed from historical price observations lead to drift estimatesof quite poor precision since drifts tend to fluctuate randomly over time and drift effects are overshadowedby volatility. At the same time optimal investment strategies in dynamic portfolio optimization dependcrucially on the drift of the underlying asset price process. For these reasons, practitioners also utilizeexternal sources of information such as news, company reports, ratings or their own intuitive views onthe future asset performance for the construction of optimal portfolio strategies. These outside sources ofinformation are called expert opinions . The idea goes back to the celebrated Black-Litterman model whichis an extension of the classical one-period Markowitz model, see Black and Litterman [2]. It uses Bayesianupdating to improve drift estimates.Contrary to the classical static one-period model, we consider a continuous-time model for asset priceswhere additional information in the form of expert opinions arrives repeatedly over time. Davis and Lleo [7]termed that approach “Black-Litterman in Continuous-Time” (BLCT). The first papers addressing BLCTare Frey et al. (2012) [9] and their follow-up paper [10]. They consider an HMM for the drift and expertopinions arriving at the jump times of a Poisson process and study the maximization of expected powerutility of terminal wealth. An HGM and expert opinions arriving at fixed and known times have beeninvestigated in Gabih et al. [11] for a market with only one risky stock, and generalized in Sass et al. [21]for markets with multiple risky stocks. Here, the authors consider maximization of logarithmic utility. Davisand Lleo [6, 7] consider BLCT for power utility maximization under an HGM and expert opinions arrivingcontinuously over time. This allows for quite explicit solutions for the portfolio optimization problem. In[7], the authors also focus on the calibration of the model for expert opinions to real-world data.In a recent paper Sass et al. [22] consider an HGM with expert opinions both at fixed as well as randominformation dates and investigate the asymptotic behavior of the filter for increasing arrival frequency ofthe expert opinions. They assume that a higher frequency of expert opinions is only available at thecost of accuracy. In particular, the variance of expert opinions grows linearly with the arrival frequency.This assumption reflects that it is not possible for investors to gain unlimited information in a finitetime interval. Furthermore, it allows to find a certain asymptotic behavior that yields reasonable filterapproximations for investors observing discrete-time expert opinions arriving with a fixed and sufficientlylarge intensity. The authors derive limit theorems which state that the information obtained from observinghigh-frequency discrete-time expert opinions is asymptotically the same as that from observing a certaindiffusion process that has the same drift as the return process. The latter process can be interpreted asa continuous-time expert which permanently delivers noisy information about the drift. These so-calleddiffusion approximations show how the BLCT model of Davis and Lleo [6, 7] who work with continuous-timeexpert opinions can be obtained as a limit of BLCT models with discrete-time experts.The present paper can be considered as a companion paper to the above mentioned work of Sass etal. [22]. However, contrary to [22] we assume that the expert’s reliability expressed by its variance remainsbounded when the arrival intensity increases. For the sake of simplicity we restrict to the case of a constantvariance. This leads to a different asymptotic regime corresponding to the Law of Large Numbers whilethe results in [22] are in the sense of Functional Central Limit Theorems.When the arrival intensity increases, the investor receives more and more noisy signals about thecurrent state of the drift of the same precision. It is then expected that in the limit the drift estimate isperfectly accurate and equals the actual drift, i.e., the investor has full information about the drift. Whilethis statistical consistency of the estimator seems to be intuitively clear a rigorous proof is an open issueand will be addressed in this paper. Gabih et al. [11] and Sass et al. [21] provide such proof only for thecase of fixed and known information dates. However, their results and methods cannot be applied to thepresent model with random information dates. Note that also the methods for the proof of the diffusionlimits in [22] do not carry over to the present case of fixed expert’s reliability. To the best of our knowledgethe techniques for proving convergence constitute a new contribution to the literature. Compared to [11] abih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions and [21] we do not only give a rigorous convergence proof but we are also able to determine the rate ofconvergence and give explicit bounds for the estimation error.In this paper we concentrate on the asymptotic properties of drift estimates which are based onKalman filter techniques and described by the conditional mean and covariance matrix of the drift giventhe observations. We show that for increasing arrival intensity of expert opinions, the expectation of theconditional covariance goes to zero. This implies that the conditional mean is a consistent drift estimator,it converges to the hidden drift in the mean-square sense. We expect that these convergence results carryover to the value functions of portfolio optimization problems but do not include these studies in this paper.For the maximization of expected logarithmic utility, the convergence of value functions has already beenproven in Sass et al. [22]. The case of power utility will be addressed in our follow-up paper [12].The paper is organized as follows: In Section 2 we introduce the model for our financial market includingexpert opinions and define information regimes for investors with different sources of information. For eachof those information regimes, we state the dynamics of the corresponding conditional mean and conditionalcovariance process in Section 3. Section 4 contains our main contributions and studies the asymptoticfilter behavior for increasing arrival intensity of discrete-time expert opinions. First, Lemma 4.2 gives anestimate for the drift term in the semimartingale representation of the conditional covariance process.Based on this estimate, Theorem 4.3 shows that as the arrival intensity increases the expectation of theconditional covariance goes to zero. As a consequence, Theorem 4.6 states the mean-square convergenceof the conditional mean to the hidden drift. In Section 5, we study a related problem for continuous-timeexpert opinions that arises in the case of diffusion approximations of discrete-time expert opinions. Section 6illustrates the convergence results by some numerical experiments. In Appendix A we collect some auxiliaryresults and technical proofs needed for our main theorems. Notation.
Throughout this paper, we use the notation 𝐼 𝑑 for the identity matrix in R 𝑑 × 𝑑 . For asymmetric and positive-semidefinite matrix 𝐴 ∈ R 𝑑 × 𝑑 we call a symmetric and positive-semidefinite matrix 𝐵 ∈ R 𝑑 × 𝑑 the square root of 𝐴 if 𝐵 = 𝐴 . The square root is unique and will be denoted by 𝐴 .For a vector 𝑋 we denote by ‖ 𝑋 ‖ the Euclidean norm. For a square matrix 𝐴 we denote by ‖ 𝐴 ‖ ageneric matrix norm, by ‖ 𝐴 ‖ 𝐹 = √︁∑︀ 𝑖,𝑗 ( 𝐴 𝑖𝑗 ) the Frobenius norm and by tr( 𝐴 ) = ∑︀ 𝑖 𝐴 𝑖𝑖 the trace of 𝐴 . The setting is based on Gabih et al. [11] and Sass et al. [21, 22]. For a fixed date
𝑇 > , 𝒢 , G , 𝑃 ), with filtration G = ( 𝒢 𝑡 ) 𝑡 ∈ [0 ,𝑇 ] satisfying the usual conditions. All processes are assumed to be G -adapted.We consider a market model for one risk-free bond with constant risk-free interest rate and 𝑑 riskysecurities whose return process 𝑅 = ( 𝑅 , . . . , 𝑅 𝑑 ) is defined by 𝑑𝑅 𝑡 = 𝜇 𝑡 𝑑𝑡 + 𝜎 𝑅 𝑑𝑊 𝑅𝑡 , (2.1)for a given 𝑑 -dimensional G -adapted Brownian motion 𝑊 𝑅 with 𝑑 ≥ 𝑑 . The constant volatility matrix 𝜎 𝑅 ∈ R 𝑑 × 𝑑 is assumed to be such that Σ 𝑅 := 𝜎 𝑅 𝜎 ⊤ 𝑅 is positive definite. In this setting the price process 𝑆 = ( 𝑆 , . . . , 𝑆 𝑑 ) of the risky securities reads as 𝑑𝑆 𝑡 = 𝑑𝑖𝑎𝑔 ( 𝑆 𝑡 ) 𝑑𝑅 𝑡 . (2.2) Gabih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions
Note that we can write log 𝑆 𝑖𝑡 = log 𝑠 𝑖 + 𝑡 ∫︁ 𝜇 𝑖𝑠 𝑑𝑠 + 𝑑 ∑︁ 𝑗 =1 (︁ 𝜎 𝑖𝑗𝑅 𝑊 𝑅,𝑗𝑡 −
12 ( 𝜎 𝑖𝑗𝑅 ) 𝑡 )︁ = log 𝑠 𝑖 + 𝑅 𝑖𝑡 − 𝑑 ∑︁ 𝑗 =1 ( 𝜎 𝑖𝑗𝑅 ) 𝑡, 𝑖 = 1 , . . . , 𝑑. (2.3)So we have the equality G 𝑅 = G log 𝑆 = G 𝑆 , where for a generic process 𝑋 we denote by G 𝑋 the filtrationgenerated by 𝑋 . This is useful since it allows to work with 𝑅 instead of 𝑆 in the filtering part.The dynamics of the drift process 𝜇 = ( 𝜇 𝑡 ) 𝑡 ∈ [0 ,𝑇 ] in (2.1) are given by the stochastic differentialequation (SDE) 𝑑𝜇 𝑡 = 𝜅 ( 𝜇 − 𝜇 𝑡 ) 𝑑𝑡 + 𝜎 𝜇 𝑑𝑊 𝜇𝑡 , (2.4)where 𝜅 ∈ R 𝑑 × 𝑑 , 𝜎 𝜇 ∈ R 𝑑 × 𝑑 and 𝜇 ∈ R 𝑑 are constants such that the matrices 𝜅 and Σ 𝜇 := 𝜎 𝜇 𝜎 ⊤ 𝜇 arepositive definite, and 𝑊 𝜇 is a 𝑑 -dimensional Brownian motion independent of 𝑊 𝑅 with 𝑑 ≥ 𝑑 . Here, 𝜇 is the mean-reversion level, 𝜅 the mean-reversion speed and 𝜎 𝜇 describes the volatility of 𝜇 . The initialvalue 𝜇 is assumed to be a normally distributed random variable independent of 𝑊 𝜇 and 𝑊 𝑅 with mean 𝑚 ∈ R 𝑑 and covariance matrix 𝑞 ∈ R 𝑑 × 𝑑 assumed to be symmetric and positive semidefinite. It is wellknown that SDE (2.4) has the closed-form solution 𝜇 𝑡 = 𝜇 + 𝑒 − 𝜅𝑡 [︁ ( 𝜇 − 𝜇 ) + 𝑡 ∫︁ 𝑒 𝜅𝑠 𝜎 𝜇 𝑑𝑊 𝜇𝑠 ]︁ , 𝑡 ≥ . (2.5)This is a Gaussian process and known as Ornstein-Uhlenbeck process. It has mean value and covariancefunction E [ 𝜇 𝑡 ] = 𝜇 + 𝑒 − 𝜅𝑡 ( 𝑚 − 𝜇 ) andCov( 𝜇 𝑠 , 𝜇 𝑡 ) = 𝑒 − 𝜅𝑠 (︃ 𝑞 + min { 𝑠,𝑡 } ∫︁ 𝑒 𝜅𝑢 Σ 𝜇 𝑒 𝜅 ⊤ 𝑢 𝑑𝑢 )︃ 𝑒 − 𝜅 ⊤ 𝑡 , 𝑠, 𝑡 ≥ . We assume that investors observe the return process 𝑅 but they neither observe the factor process 𝜇 northe Brownian motion 𝑊 𝑅 . They do however know the model parameters such as 𝜎 𝑅 , 𝜅, 𝜇, 𝜎 𝜇 and thedistribution 𝒩 ( 𝑚 , 𝑞 ) of the initial value 𝜇 . Information about the drift 𝜇 can be drawn from observingthe returns 𝑅 . A special feature of our model is that investors may also have access to additional informationabout the drift in form of expert opinions such as news, company reports, ratings or their own intuitiveviews on the future asset performance. The expert opinions provide noisy signals about the current state ofthe drift arriving at discrete points in time 𝑇 𝑘 . We model these expert opinions by a marked point process( 𝑇 𝑘 , 𝑍 𝑘 ) 𝑘 , so that at 𝑇 𝑘 the investor observes the realization of a random vector 𝑍 𝑘 whose distributiondepends on the current state 𝜇 𝑇 𝑘 of the drift process. The arrival dates 𝑇 𝑘 are modelled as jump timesof a standard Poisson process with intensity 𝜆 >
0, independent of both the Brownian motions 𝑊 𝑅 , 𝑊 𝐽 and the initial value of the drift 𝜇 , so that the timing of the information arrival does not carry any usefulinformation about the drift. For the sake of convenience we also write 𝑇 := 0 although no expert opinionarrives at time 𝑡 = 0.The signals or “the expert views” at time 𝑇 𝑘 are modelled by R 𝑑 -valued Gaussian random vectors 𝑍 𝑘 = ( 𝑍 𝑘 , · · · , 𝑍 𝑑𝑘 ) ⊤ with 𝑍 𝑘 = 𝜇 𝑇 𝑘 + Γ 𝜀 𝑘 , 𝑘 = 1 , , . . . , (2.6) abih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions where the matrix Γ ∈ R 𝑑 × 𝑑 is symmetric and positive definite. Further, ( 𝜀 𝑘 ) 𝑘 ≥ is a sequence of independentstandard normally distributed random vectors, i.e., 𝜀 𝑘 ∼ 𝒩 (0 , 𝐼 𝑑 ). It is also independent of the Brownianmotions 𝑊 𝑅 , 𝑊 𝜇 , the initial value 𝜇 and the arrival dates ( 𝑇 𝑘 ) 𝑘 ≥ . That means that, given 𝜇 𝑇 𝑘 , the expertopinion 𝑍 𝑘 is 𝒩 ( 𝜇 𝑇 𝑘 , Γ)-distributed. So, 𝑍 𝑘 can be considered as an unbiased estimate of the unknownstate of the drift at time 𝑇 𝑘 . The matrix Γ is a measure of the expert’s reliability. In a model with 𝑑 = 1risky asset Γ is just the variance of the expert’s estimate of the drift at time 𝑇 𝑘 : the larger Γ the lessreliable is the expert.Note that one may also allow for relative expert views where experts give an estimate for the differencein the drift of two stocks instead of absolute views. This extension is studied in Schöttle et al. [23] wherethe authors show how to switch between these two models for expert opinions by means of a pick matrix.Finally, we introduce expert opinions arriving continuously over time. This is motivated by the resultsof Sass et al. [22]. There the authors consider the information drawn from observing certain sequencesof expert opinions and show that for a large number of expert opinions it is essentially the same as theinformation resulting from observing another diffusion process. The interpretation of that diffusion processis an expert providing continuous-time estimates about the state of the drift. Let this estimate be given bythe diffusion process 𝑑𝐽 𝑡 = 𝜇 𝑡 𝑑𝑡 + 𝜎 𝐽 𝑑𝑊 𝐽𝑡 , (2.7)where 𝑊 𝐽 is an 𝑑 -dimensional Brownian motion with 𝑑 ≥ 𝑑 that is independent of all other Brownianmotions in the model and of the information dates 𝑇 𝑘 . The constant matrix 𝜎 𝐽 ∈ R 𝑑 × 𝑑 is assumed to besuch that Σ 𝐽 := 𝜎 𝐽 𝜎 ⊤ 𝐽 is positive definite. We consider various types of investors with different levels of information. The information available to aninvestor is described by the investor filtration F 𝐻 = ( ℱ 𝐻𝑡 ) 𝑡 ∈ [0 ,𝑇 ] . Here, 𝐻 denotes the information regimefor which we consider the cases 𝐻 = 𝑅, 𝑍, 𝐽, 𝐹 , where F 𝑅 = ( ℱ 𝑅𝑡 ) 𝑡 ∈ [0 ,𝑇 ] with ℱ 𝑅𝑡 = 𝜎 ( 𝑅 𝑠 , 𝑠 ≤ 𝑡 ) , F 𝑍 = ( ℱ 𝑍𝑡 ) 𝑡 ∈ [0 ,𝑇 ] with ℱ 𝑍𝑡 = 𝜎 ( 𝑅 𝑠 , 𝑠 ≤ 𝑡, ( 𝑇 𝑘 , 𝑍 𝑘 ) , 𝑇 𝑘 ≤ 𝑡 ) , F 𝐽 = ( ℱ 𝐽𝑡 ) 𝑡 ∈ [0 ,𝑇 ] with ℱ 𝐽𝑡 = 𝜎 ( 𝑅 𝑠 , 𝐽 𝑠 , 𝑠 ≤ 𝑡 ) , F 𝐹 = ( ℱ 𝐹𝑡 ) 𝑡 ∈ [0 ,𝑇 ] with ℱ 𝐹𝑡 = 𝜎 ( 𝑅 𝑠 , 𝜇 𝑠 , 𝑠 ≤ 𝑡 ) . We assume that the above 𝜎 -algebras ℱ 𝐻𝑡 are augmented by the null sets of 𝑃 . We call the investorwith filtration F 𝐻 = ( ℱ 𝐻𝑡 ) 𝑡 ∈ [0 ,𝑇 ] the 𝐻 -investor. The 𝑅 -investor observes only the return process 𝑅 , the 𝑍 -investor combines return observations with the discrete-time expert opinions 𝑍 𝑘 while the 𝐽 -investorobserves the return process together with the continuous-time expert 𝐽 . Finally, the 𝐹 -investor has fullinformation and can observe the drift process 𝜇 . For stochastic drift this case is not realistic, but we use itas a benchmark and in the next section it will serve as a limiting case for high-frequency expert opinions.We will denote an investor with investor filtration F 𝐻 as 𝐻 -investor.We assume that at 𝑡 = 0 the partially informed investors start with the same initial informationgiven by the 𝜎 -algebra ℱ 𝐼 , i.e., ℱ 𝐻 = ℱ 𝐼 ⊂ ℱ 𝐹 , 𝐻 = 𝑅, 𝑍, 𝐽 . This initial information ℱ 𝐼 models priorknowledge about the drift process at time 𝑡 = 0, e.g., from observing returns or expert opinions in thepast before the trading period [0 , 𝑇 ]. We assume that the conditional distribution of the initial drift value 𝜇 given ℱ 𝐻 is the normal distribution 𝒩 ( 𝑚 , 𝑞 ) with mean 𝑚 ∈ R 𝑑 and covariance matrix 𝑞 ∈ R 𝑑 × 𝑑 assumed to be symmetric and positive semidefinite. In this setting typical examples are:a) The investor has no information about the initial value of the drift 𝜇 . However, he knows the modelparameters, in particular the distribution 𝒩 ( 𝑚 , 𝑞 ) of 𝜇 with given parameters 𝑚 and 𝑞 . Thiscorresponds to ℱ 𝐼 = { ∅ , Ω } and 𝑚 = 𝑚 , 𝑞 = 𝑞 .b) The investor can fully observe the initial value of the drift 𝜇 , which corresponds to ℱ 𝐼 = ℱ 𝐹 and 𝑚 = 𝜇 ( 𝜔 ) and 𝑞 = 0. Gabih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions c) Between the above limiting cases we consider an investor who has some prior but no complete infor-mation about 𝜇 leading to { ∅ , Ω } ⊂ ℱ 𝐼 ⊂ ℱ 𝐹 . The trading decisions of investors are based on their knowledge about the drift process 𝜇 . While the 𝐹 -investor observes the drift directly, the 𝐻 -investor for 𝐻 = 𝑅, 𝑍, 𝐽 has to estimate it. This leads us to afiltering problem with hidden signal process 𝜇 and observations given by the returns 𝑅 and expert opinions( 𝑇 𝑘 , 𝑍 𝑘 ) or 𝐽 . The filter for the drift 𝜇 𝑡 is its projection on the ℱ 𝐻𝑡 -measurable random variables describedby the conditional distribution of the drift given ℱ 𝐻𝑡 . The mean-square optimal estimator for the drift attime 𝑡 , given the available information is the conditional mean 𝑀 𝐻𝑡 := E [ 𝜇 𝑡 |ℱ 𝐻𝑡 ] . The accuracy of that estimator can be described by the conditional covariance matrix 𝑄 𝐻𝑡 := E [( 𝜇 𝑡 − 𝑀 𝐻𝑡 )( 𝜇 𝑡 − 𝑀 𝐻𝑡 ) ⊤ |ℱ 𝐻𝑡 ] . (3.1)Since in our filtering problem the signal 𝜇 , the observations and the initial value of the filter are jointlyGaussian also the filter distribution is Gaussian and completely characterized by the conditional mean 𝑀 𝐻𝑡 and the conditional covariance 𝑄 𝐻𝑡 .In Section 4 we will study the asymptotic behavior of the filter for the 𝑍 -investor observing expertopinions arriving more and more frequently and derive limit theorems for the filter if the arrival intensity 𝜆 tends to infinity. Section 5 is devoted to a related problem and considers the asymptotics of the filterprocesses for the 𝐽 -investor with volatility 𝜎 𝐽 tending to zero. These results are based on the followingdynamics of the filters for 𝐻 = 𝑅, 𝑍, 𝐽 which already can be found in Sass et al. [21, 22]. 𝑅 - and 𝐽 -Investor The 𝑅 -investor only observes returns and has no access to additional expert opinions, the information isgiven by F 𝑅 . Then, we are in the classical case of the Kalman filter, see e.g. Liptser and Shiryaev [16],Theorem 10 .
3, leading to the following dynamics of 𝑀 𝑅 and 𝑄 𝑅 . Lemma 3.1.
For the 𝑅 -investor the filter is Gaussian and the conditional distribution of the drift 𝜇 𝑡 given ℱ 𝑅𝑡 is the normal distribution 𝒩 (︀ 𝑀 𝑅𝑡 , 𝑄 𝑅𝑡 )︀ .The conditional mean 𝑀 𝑅 follows the dynamics 𝑑𝑀 𝑅𝑡 = 𝜅 ( 𝜇 − 𝑀 𝑅𝑡 ) 𝑑𝑡 + 𝑄 𝑅𝑡 Σ − 𝑅 (︀ 𝑑𝑅 𝑡 − 𝑀 𝑅𝑡 𝑑𝑡 )︀ . (3.2) The dynamics of the conditional covariance 𝑄 𝑅 is given by the Riccati differential equation 𝑑𝑄 𝑅𝑡 = (Σ 𝜇 − 𝜅𝑄 𝑅𝑡 − 𝑄 𝑅𝑡 𝜅 ⊤ − 𝑄 𝑅𝑡 Σ − 𝑅 𝑄 𝑅𝑡 ) 𝑑𝑡. (3.3) The initial values are 𝑀 𝑅 = 𝑚 and 𝑄 𝑅 = 𝑞 . Note that the conditional covariance matrix 𝑄 𝑅𝑡 satisfies an ordinary differential equation and is hencedeterministic, whereas the conditional mean 𝑀 𝑅𝑡 is a stochastic process defined by an SDE driven by thereturn process 𝑅 .Next, we consider the 𝐽 -investor who observes a 2 𝑑 -dimensional diffusion process with components 𝑅 and 𝐽 . That observation process is driven by a ( 𝑑 + 𝑑 )-dimensional Brownian motion with components 𝑊 𝑅 and 𝑊 𝐽 . Again, we can apply classical Kalman filter theory as in Liptser and Shiryaev [16] to deducethe dynamics of 𝑀 𝐽 and 𝑄 𝐽 . We also refer to Lemma 2.2 in the companion paper [22]. abih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions Lemma 3.2.
For the 𝐽 -investor the filter is Gaussian and the conditional distribution of the drift 𝜇 𝑡 given ℱ 𝐽𝑡 is the normal distribution 𝒩 (︀ 𝑀 𝐽𝑡 , 𝑄 𝐽𝑡 )︀ .The conditional mean 𝑀 𝐽 follows the dynamics 𝑑𝑀 𝐽𝑡 = 𝜅 ( 𝜇 − 𝑀 𝐽𝑡 ) 𝑑𝑡 + 𝑄 𝐽𝑡 (Σ − 𝑅 , Σ − 𝐽 ) (︂ 𝑑𝑅 𝑡 − 𝑀 𝐽𝑡 𝑑𝑡𝑑𝐽 𝑡 − 𝑀 𝐽𝑡 𝑑𝑡 )︂ . (3.4) The dynamics of the conditional covariance 𝑄 𝐽 is given by the Riccati differential equation 𝑑𝑄 𝐽𝑡 = (Σ 𝜇 − 𝜅𝑄 𝐽𝑡 − 𝑄 𝐽𝑡 𝜅 ⊤ − 𝑄 𝐽𝑡 (Σ − 𝑅 + Σ − 𝐽 ) 𝑄 𝐽𝑡 ) 𝑑𝑡. (3.5) The initial values are 𝑀 𝐽 = 𝑚 and 𝑄 𝐽 = 𝑞 . Note that, as in case of the 𝑅 -investor, the conditional covariance 𝑄 𝐽 is deterministic. 𝑍 -Investor Now we consider the filter for the 𝑍 -investor who combines continuous-time observations of stock returnsand expert opinions received at discrete points in time. Lemma 3.3.
For the 𝑍 -investor the filter is Gaussian and the conditional distribution of the drift 𝜇 𝑡 given ℱ 𝑍𝑡 is the normal distribution 𝒩 (︀ 𝑀 𝑍𝑡 , 𝑄 𝑍𝑡 )︀ . (i) Between two information dates 𝑇 𝑘 and 𝑇 𝑘 +1 , 𝑘 ∈ N , the conditional mean 𝑀 𝑍𝑡 satisfies SDE (3.2) ,i.e., 𝑑𝑀 𝑍𝑡 = 𝜅 ( 𝜇 − 𝑀 𝑍𝑡 ) 𝑑𝑡 + 𝑄 𝑍𝑡 Σ − 𝑅 (︀ 𝑑𝑅 𝑡 − 𝑀 𝑍𝑡 𝑑𝑡 )︀ for 𝑡 ∈ [ 𝑇 𝑘 , 𝑇 𝑘 +1 ) . The conditional covariance 𝑄 𝑍 satisfies the ordinary Riccati differential equation (3.3) , i.e., 𝑑𝑄 𝑍𝑡 = (Σ 𝜇 − 𝜅𝑄 𝑍𝑡 − 𝑄 𝑍𝑡 𝜅 ⊤ − 𝑄 𝑍𝑡 Σ − 𝑅 𝑄 𝑍𝑡 ) 𝑑𝑡. The initial values are 𝑀 𝑍𝑇 𝑘 and 𝑄 𝑍𝑇 𝑘 , respectively, with 𝑀 𝑍 = 𝑚 and 𝑄 𝑍 = 𝑞 . (ii) At the information dates 𝑇 𝑘 , 𝑘 ∈ N , the conditional mean and covariance 𝑀 𝑍𝑇 𝑘 and 𝑄 𝑍𝑇 𝑘 are obtainedfrom the corresponding values at time 𝑇 𝑘 − (before the arrival of the view) using the update formulas 𝑀 𝑍𝑇 𝑘 = 𝜌 𝑘 𝑀 𝑍𝑇 𝑘 − + ( 𝐼 𝑑 − 𝜌 𝑘 ) 𝑍 𝑘 ,𝑄 𝑍𝑇 𝑘 = 𝜌 𝑘 𝑄 𝑍𝑇 𝑘 − , with the update factor 𝜌 𝑘 = Γ( 𝑄 𝑍𝑇 𝑘 − + Γ) − .Proof. For a detailed proof we refer to Lemma 2.3 in [21] and Lemma 2.3 in [22]. (cid:3)
Note that the dynamics of 𝑀 𝑍 and 𝑄 𝑍 between information dates are the same as for the 𝑅 -investor, seeLemma 3.1. The values at an information date 𝑇 𝑘 are obtained from a Bayesian update.Recall that for the 𝑅 -investor the conditional mean 𝑀 𝑅 is a diffusion process and the conditionalcovariance 𝑄 𝑅 is deterministic. Contrary to that the conditional mean 𝑀 𝑍 of the 𝑍 -investor is a jump-diffusion process and the conditional covariance 𝑄 𝑍 is no longer deterministic since the updates lead tojumps at the random arrival dates 𝑇 𝑘 of the expert opinions. Hence, 𝑄 𝑍 is a piecewise deterministicstochastic process. The next lemma states in mathematical terms the intuitive property that additional information from theexpert opinions improves drift estimates. Since the accuracy of the filter is measured by the conditional
Gabih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions covariance it is expected that this quantity for the 𝑍 -investor who combines observations of returns andexpert opinions is “smaller” than for the 𝑅 -investor who observes returns only. Mathematically, this canbe expressed by the partial ordering of symmetric matrices. For symmetric matrices 𝐴, 𝐵 ∈ R 𝑑 × 𝑑 we write 𝐴 ⪯ 𝐵 if 𝐵 − 𝐴 is positive semidefinite. Note that 𝐴 ⪯ 𝐵 implies that ‖ 𝐴 ‖ ≤ ‖ 𝐵 ‖ . Proposition 3.4.
It holds 𝑄 𝑍𝑡 ⪯ 𝑄 𝑅𝑡 and 𝑄 𝐽𝑡 ⪯ 𝑄 𝑅𝑡 . In particular, there exists a constant 𝐶 𝑄 > suchthat ⃦⃦ 𝑄 𝑍𝑡 ⃦⃦ ≤ ⃦⃦ 𝑄 𝑅𝑡 ⃦⃦ ≤ 𝐶 𝑄 and ⃦⃦ 𝑄 𝐽𝑡 ⃦⃦ ≤ ⃦⃦ 𝑄 𝑅𝑡 ⃦⃦ ≤ 𝐶 𝑄 for all 𝑡 ∈ [0 , 𝑇 ] . For the proof we refer to [22], Lemma 2.4.
In the following we consider the 𝑍 -investor and its filter for increasing arrival intensity 𝜆 and study theasymptotic behavior of the conditional mean and conditional covariance for 𝜆 → ∞ . Then the averagenumber of expert opinions per unit of time goes to infinity, i.e., the 𝑍 -investor has more and more noisyestimates of the current state of the hidden drift at his disposal. This will lead to an increasing accuracy ofthe drift estimator. As a consequence of the Law of Large Numbers we expect that in the limit for 𝜆 → ∞ the drift estimator coincides with the drift. In fact we show in Theorem 4.6 that the drift estimator givenby the conditional mean 𝑀 𝑍 converges to the hidden drift 𝜇 in the mean-square sense with rate 1 / √ 𝜆 .Thus, 𝑀 𝑍 is a consistent estimator for 𝜇 and in the limit the 𝑍 -investor has full information about thedrift.Note that there is another asymptotic regime if additional expert opinions only come at the cost ofaccuracy described by the variance Γ. Assuming that this variance grows linearly in the arrival intensitySass et al. [22] show that the information the 𝑍 -investor obtains from observing discrete-time expertopinions is asymptotically the same as that from observing a certain diffusion process. The latter canbe interpreted as a continuous-time expert. The limit theorems obtained in [22] allow to derive so-calleddiffusion approximations of the filter for high-frequency discrete-time expert opinions. They constitute aFunctional Central Limit Theorem while the limit theorems obtained below for the case of fixed varianceΓ can be considered as a Functional Law of Large Numbers.In our notation we now want to emphasize the dependence of the filter processes and the investorfiltration on the intensity 𝜆 by adding the superscript 𝜆 . Thus, we write 𝑀 𝑍,𝜆𝑡 , 𝑄
𝑍,𝜆𝑡 and F 𝑍,𝜆 . We now show that the expectation of the conditional covariance process 𝑄 𝑍,𝜆𝑡 for 𝜆 → ∞ goes to zero.For this purpose, it will be useful to express the dynamics of 𝑄 𝑍,𝜆 given in Lemma 3.3 in a unified waythat comprises both the behavior between information dates and the jumps at times 𝑇 𝑘 . We thereforework with a Poisson random measure as in Cont and Tankov [5, Sec. 2.6]. Let 𝐸 = [0 , 𝑇 ] × R 𝑑 and let 𝑈 𝑘 , 𝑘 = 1 , , . . . , be a sequence of independent multivariate standard Gaussian random variables on R 𝑑 . Forany 𝐼 ∈ ℬ ([0 , 𝑇 ]) and 𝐵 ∈ ℬ ( R 𝑑 ) let 𝑁 ( 𝐼 × 𝐵 ) = ∑︁ 𝑘 : 𝑇 𝑘 ∈ 𝐼 { 𝑈 𝑘 ∈ 𝐵 } denote the number of jump times in 𝐼 where 𝑈 𝑘 takes a value in 𝐵 . Then 𝑁 defines a Poisson randommeasure with a corresponding compensated measure ̃︀ 𝑁 𝜆 ( 𝑑𝑠, 𝑑𝑢 ) = 𝑁 ( 𝑑𝑠, 𝑑𝑢 ) − 𝜆 𝑑𝑠 𝜙 ( 𝑢 ) 𝑑𝑢 , where 𝜙 is themultivariate standard normal density on R 𝑑 , see Cont and Tankov [5, Sec. 2.6.3].The next lemma rewrites the dynamics of 𝑄 𝑍,𝜆 given in Lemma 3.3 and provides a semimartingalerepresentation which is driven by the martingale ̃︀ 𝑁 𝜆 . For a detailed proof and further explanations werefer to Westphal [25, Prop. 8.14] and Kondakji [14, Sec. 3.1]. abih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions Lemma 4.1.
The dynamics of the conditional covariance matrix 𝑄 𝑍,𝜆 are given by 𝑑𝑄 𝑍,𝜆𝑡 = 𝛼 𝑍,𝜆 ( 𝑄 𝑍,𝜆𝑡 ) 𝑑𝑡 + ∫︁ R 𝑑 𝛾 ( 𝑄 𝑍,𝜆𝑠 − ) ̃︀ 𝑁 𝜆 ( 𝑑𝑠, 𝑑𝑢 ) , 𝑄 𝑍,𝜆 = 𝑞 , (4.1) where 𝛼 𝑍,𝜆 ( 𝑞 ) = Σ 𝜇 − 𝜅𝑞 − 𝑞𝜅 ⊤ − 𝑞 Σ − 𝑅 𝑞 − 𝜆𝑞 ( 𝑞 + Γ) − 𝑞, (4.2) 𝛾 ( 𝑞 ) = − 𝑞 ( 𝑞 + Γ) − 𝑞. (4.3)We rewrite the above F 𝑍,𝜆 -semimartingale decomposition of 𝑄 𝑍,𝜆 in integral form and obtain 𝑄 𝑍,𝜆𝑡 = 𝒜 𝜆𝑡 + 𝒦 𝜆𝑡 for 𝑡 ∈ [0 , 𝑇 ] , (4.4)with 𝒜 𝜆𝑡 := 𝑞 + 𝑡 ∫︁ 𝛼 𝑍,𝜆 ( 𝑄 𝑍,𝜆𝑠 ) 𝑑𝑠 and 𝒦 𝜆𝑡 := 𝑡 ∫︁ ∫︁ R 𝑑 𝛾 ( 𝑄 𝑍,𝜆𝑠 − ) ̃︀ 𝑁 𝜆 ( 𝑑𝑠, 𝑑𝑢 ) . Since by Proposition 3.4 the conditional covariance 𝑄 𝑍,𝜆𝑠 is bounded on [0 , 𝑡 ] also 𝛾 is bounded and thejump process 𝒦 𝜆 is an F 𝑍 -martingale and hence E [ 𝒦 𝜆𝑡 ] = 0 and E [ 𝑄 𝑍,𝜆𝑡 ] = E [ 𝒜 𝜆𝑡 ] . (4.5)For the study of the asymptotic behavior of the conditional covariance 𝑄 𝑍,𝜆 we investigate the drift ofthe process 𝒜 𝜆 which is given by the non-linear matrix-valued function 𝛼 𝑍,𝜆 . The following lemma givesan estimate of the trace of 𝛼 𝑍,𝜆 ( 𝑞 ) in terms of a linear function of the trace of 𝑞 . That estimate will playa crucial role for deriving the convergence result in Theorem 4.3. Lemma 4.2. ( Properties of 𝛼 𝑍,𝜆 )For the function 𝛼 𝑍,𝜆 given in (4.2) there exist constants 𝑎 𝛼 , 𝑏 𝛼 > independent of 𝜆 and there exists 𝜆 > such that for all symmetric and positive semidefinite 𝑞 ∈ R 𝑑 × 𝑑 tr (︀ 𝛼 𝑍,𝜆 ( 𝑞 ) )︀ ≤ 𝑎 𝛼 − √ 𝜆 𝑏 𝛼 tr( 𝑞 ) , for 𝜆 ≥ 𝜆 . (4.6) The above estimate holds for every 𝑎 𝛼 > tr(Σ 𝜇 ) ,𝑏 𝛼 < 𝑏 𝛼 = 𝑏 𝛼 ( 𝑎 𝛼 ) := 2 √︃ 𝑎 𝛼 − tr(Σ 𝜇 )tr(Γ) , (4.7) 𝜆 = 𝜆 ( 𝑎 𝛼 , 𝛽 𝛼 ) := (︂ 𝑑 ( 𝑎 𝛼 − tr(Σ 𝜇 ))2 √︀ tr(Γ)( 𝑎 𝛼 − tr(Σ 𝜇 )) − 𝑏 𝛼 tr(Γ) )︂ . The quite technical proof is given in Appendix A.2. The following main theorem gives an upper bound forthe expectation of the trace of 𝑄 𝑍,𝜆 from which the convergence to zero can be deduced.
Theorem 4.3.
For every 𝛿 ∈ (0 , 𝑇 ] there exists 𝜆 𝑄 > such that E [︀ tr (︀ 𝑄 𝑍,𝜆𝑡 )︀]︀ ≤ 𝐾 𝑍 √ 𝜆 for 𝜆 ≥ 𝜆 𝑄 , 𝑡 ∈ [ 𝛿, 𝑇 ] and (4.8) 𝐾 𝑍 = 𝐾 𝑍 ( 𝛿 ) = (︀ tr(Γ)[tr(Σ 𝜇 ) + tr( 𝑞 )( 𝑒 𝛿 ) − ] )︀ / , (4.9) where 𝑒 = exp(1) denotes Euler’s number.In particular, it holds E [︀ tr (︀ 𝑄 𝑍,𝜆𝑡 )︀]︀ → as 𝜆 → ∞ for all 𝑡 ∈ (0 , 𝑇 ] . Gabih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions
Proof.
Let us define the function 𝑔 ( 𝑡 ) := E [︀ tr( 𝑄 𝑍,𝜆𝑡 ) ]︀ for 𝑡 ∈ [0 , 𝑇 ]. Then using (4.4), (4.5) and the linearityof the expectation and the trace operator yields 𝑔 ( 𝑡 ) = tr( E [ 𝑄 𝑍,𝜆𝑡 ]) = tr( 𝑞 ) + 𝑡 ∫︁ E [︀ tr( 𝛼 𝑍,𝜆 ( 𝑄 𝑍,𝜆𝑠 )) ]︀ 𝑑𝑠. Since according to Proposition 3.4 the conditional covariance 𝑄 𝑍,𝜆 is bounded and piecewise continuousthe function 𝑔 is piecewise differentiable and for its derivative it holds 𝑔 ′ ( 𝑡 ) = E [︀ tr( 𝛼 𝑍,𝜆 ( 𝑄 𝑍,𝜆𝑡 )) ]︀ . Furtherwe have 𝑔 (0) = tr( 𝑞 ). Lemma 4.2 implies that there are constants 𝑎 𝛼 , 𝑏 𝛼 , 𝜆 > 𝑔 ′ ( 𝑡 ) ≤ E [︀ 𝑎 𝛼 − √ 𝜆 𝑏 𝛼 tr( 𝑄 𝑍,𝜆𝑡 ) ]︀ = 𝑎 𝛼 − √ 𝜆 𝑏 𝛼 E [︀ tr( 𝑄 𝑍,𝜆𝑡 ) ]︀ = 𝑎 𝛼 − √ 𝜆 𝑏 𝛼 𝑔 ( 𝑡 ) for 𝜆 ≥ 𝜆 . (4.10)We now apply Gronwall’s Lemma in differential form to obtain for 𝑡 ∈ [ 𝛿, 𝑇 ] and 𝜆 ≥ 𝜆 𝑔 ( 𝑡 ) ≤ 𝑔 (0) 𝑒 −√ 𝜆 𝑏 𝛼 𝑡 + 𝑎 𝛼 √ 𝜆 𝑏 𝛼 (1 − 𝑒 −√ 𝜆 𝑏 𝛼 𝑡 ) ≤ √ 𝜆 (︀ ℎ ( 𝛿, 𝜆, 𝑏 𝛼 ) + 𝑎 𝛼 𝑏 𝛼 )︀ , (4.11)where ℎ ( 𝛿, 𝜆, 𝑏 𝛼 ) := tr( 𝑞 ) √ 𝜆 𝑒 −√ 𝜆 𝑏 𝛼 𝛿 . Next we show how for given 𝛿 ∈ (0 , 𝑇 ] we can choose the constants 𝑎 𝛼 , 𝑏 𝛼 , 𝜆 𝑄 > ℎ ( 𝛿, 𝜆, 𝑏 𝛼 ) + 𝑎 𝛼 /𝑏 𝛼 ≤ 𝐾 𝑍 ( 𝛿 ) for 𝜆 ≥ 𝜆 𝑄 with the constant 𝐾 𝑍 ( 𝛿 ) given in (4.9).Consider for 𝜆 ≥ 𝜆 ↦→ 𝑓 ( 𝜆 ) = ℎ ( 𝛿, 𝜆, 𝑏 𝛼 ) for fixed 𝛿 ∈ (0 , 𝑇 ] and 𝑏 𝛼 ∈ (0 , 𝑏 𝛼 ), where 𝑏 𝛼 isgiven in (4.7). The function 𝑓 is non-negative, it holds 𝑓 (0) = 0 and 𝑓 ( 𝜆 ) → 𝜆 → ∞ . There is aunique maximum at 𝜆 * = ( 𝑏 𝛼 𝛿 ) − with 𝑓 ( 𝜆 * ) = ( 𝑒 𝑏 𝛼 𝛿 ) − tr( 𝑞 ). Hence for the last term on the r.h.s. of(4.11) we obtain ℎ ( 𝛿, 𝜆, 𝑏 𝛼 ) + 𝑎 𝛼 𝑏 𝛼 ≤ 𝑏 𝛼 (tr( 𝑞 )( 𝑒 𝛿 ) − + 𝑎 𝛼 ) for 𝜆 ≥ 𝜆 . (4.12)The latter expression is decreasing in 𝑏 𝛼 and the minimum on (0 , 𝑏 𝛼 ] is attained for 𝑏 𝛼 = 𝑏 𝛼 . Accordingto (4.7) this selection leads to 𝜆 = ∞ which is not feasible and we have to restrict to values 𝑏 𝛼 < 𝑏 𝛼 .However, we can achieve the above mentioned minimal value by choosing 𝑏 𝛼 = 𝑏 𝛼 − 𝜂 with a sufficientlysmall 𝜂 > 𝜆 𝑄 ≥ min( 𝜆 , 𝜆 * ) such that ℎ ( 𝛿, 𝜆, 𝑏 𝛼 ) + 𝑎 𝛼 𝑏 𝛼 = ℎ ( 𝛿, 𝜆, 𝑏 𝛼 − 𝜂 ) + 𝑎 𝛼 𝑏 𝛼 − 𝜂 ≤ 𝑏 𝛼 (tr( 𝑞 )( 𝑒 𝛿 ) − + 𝑎 𝛼 ) for 𝜆 ≥ 𝜆 𝑄 . To see this estimate we note that 𝑓 ( 𝜆 ) is decreasing on ( 𝜆 * , ∞ ) and tends to zero for 𝜆 → ∞ . Hence ℎ ( 𝛿, 𝜆, 𝑏 𝛼 − 𝜂 ) can be made arbitrarily small by selecting 𝜆 large enough.Finally, we study the dependence of the above estimate on 𝑎 𝛼 and take into account the definition of 𝑏 𝛼 given in (4.7), i.e. we consider the function 𝑎 𝛼 ↦→ 𝑏 𝛼 (tr( 𝑞 )( 𝑒 𝛿 ) − + 𝑎 𝛼 ) = √︀ tr(Γ)2 √︀ 𝑎 𝛼 − tr(Σ 𝜇 ) (tr( 𝑞 )( 𝑒 𝛿 ) − + 𝑎 𝛼 )for 𝑎 𝛼 > tr(Σ 𝜇 ). There is a unique minimizer at 𝑎 * 𝛼 = 2 tr(Σ 𝜇 ) + tr( 𝑞 )( 𝑒 𝛿 ) − and the minimal value isgiven by 𝐾 𝑍 defined in (4.9). This proves the first claim.Since that inequality holds for all 𝛿 ∈ (0 , 𝑇 ], the convergence E [︀ tr (︀ 𝑄 𝑍,𝜆𝑡 )︀]︀ → 𝜆 → ∞ holds forall 𝑡 ∈ (0 , 𝑇 ]. (cid:3) From the above asymptotic properties for the expectation of the trace of 𝑄 𝑍,𝜆 we can easily deduceanalogous results for the expectation of the norm ‖ 𝑄 𝑍,𝜆 ‖ of the conditional covariance. Corollary 4.4.
For every 𝛿 ∈ (0 , 𝑇 ] and any matrix norm ‖ · ‖ there exist constants 𝐶, 𝜆 𝑄 > such that E [︀ ⃦⃦ 𝑄 𝑍,𝜆𝑡 ⃦⃦ 𝑝 ]︀ ≤ 𝐶 √ 𝜆 for 𝜆 ≥ 𝜆 𝑄 , 𝑡 ∈ [ 𝛿, 𝑇 ] and 𝑝 ≥ . (4.13) abih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions For the Frobenius norm ‖ · ‖ 𝐹 the constant 𝐶 can be chosen as 𝐶 = 𝐾 𝑍 𝐶 𝑝 − 𝐹 where 𝐾 𝑍 is given in (4.9) and 𝐶 𝐹 denotes the upper bound from Proposition 3.4 for the Frobenius norm ‖ 𝑄 𝑍,𝜆 ‖ 𝐹 .In particular, it holds E [︀ ⃦⃦ 𝑄 𝑍,𝜆𝑡 ⃦⃦ 𝑝 ]︀ → as 𝜆 → ∞ for all 𝑡 ∈ (0 , 𝑇 ] . Proof.
For the Frobenius norm of a symmetric and positive semidefinite matrix 𝐴 it holds ‖ 𝐴 ‖ 𝐹 ≤ tr( 𝐴 )(see Lemma A.1, Inequality (A.5)). Further, Proposition 3.4 implies ‖ 𝐴 ‖ 𝐹 ≤ 𝐶 𝐹 . Hence ‖ 𝑄 𝑍,𝜆 ‖ 𝑝𝐹 ≤ 𝐶 𝑝 − 𝐹 ‖ 𝑄 𝑍,𝜆 ‖ 𝐹 ≤ 𝐶 𝑝 − 𝐹 tr( 𝑄 𝑍,𝜆 )and Theorem 4.3 with inequality (4.8) proves the claim. The equivalence of matrix norms implies theassertion for other norms. (cid:3)
We are now in a position to state and prove a similar convergence result for the asymptotic behavior ofthe filter 𝑀 . The proof is based on the following identity which relates the mean-square error of the filterestimate to the conditional covariance. Lemma 4.5.
It holds E [︀ ⃦⃦ 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 ⃦⃦ ]︀ = tr (︀ E [︀ 𝑄 𝑍,𝜆𝑡 ]︀)︀ . (4.14) Proof.
For the mean-square criterion from (4.14) it holds E [︀⃦⃦ 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 ⃦⃦ ]︀ = E [︀ ( 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 ) ⊤ ( 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 ) ]︀ = tr (︀ E [︀ ( 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 )( 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 ) ⊤ ]︀)︀ . (4.15)For the expectation in the last term the tower law of conditional expectation and the definition of theconditional covariance in (3.1) yields E [︀ ( 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 )( 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 ) ⊤ ]︀ = E [︁ E [︀ ( 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 )( 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 ) ⊤ ⃒⃒ ℱ 𝑍,𝜆𝑡 ]︀]︁ = E [︀ 𝑄 𝑍,𝜆𝑡 ]︀ . Substituting into (4.15) yields the assertion (4.14). (cid:3)
Theorem 4.6.
Let 𝐾 𝑍 , 𝜆 𝑄 be the constants given in Theorem 4.3. Then for every 𝛿 ∈ (0 , 𝑇 ] E [︀ ⃦⃦ 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 ⃦⃦ ]︀ ≤ 𝐾 𝑍 √ 𝜆 for 𝜆 ≥ 𝜆 𝑄 , 𝑡 ∈ [ 𝛿, 𝑇 ] . (4.16) In particular, it holds E [︀ ⃦⃦ 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 ⃦⃦ ]︀ → as 𝜆 → ∞ for all 𝑡 ∈ (0 , 𝑇 ] . Proof.
Using identity (4.14) from Lemma 4.5 and applying inequality (4.8) of Theorem 4.3 we obtain E [︀⃦⃦ 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 ⃦⃦ ]︀ = tr (︀ E [︀ 𝑄 𝑍,𝜆𝑡 ]︀)︀ ≤ 𝐾 𝑍 √ 𝜆 for 𝜆 ≥ 𝜆 , 𝑡 ∈ [ 𝛿, 𝑇 ] . Since the above inequality holds for all 𝛿 ∈ (0 , 𝑇 ] we finally obtain the desired convergence of the filter 𝑀 𝑍,𝜆𝑡 for 𝑡 ∈ (0 , 𝑇 ]) as 𝜆 → ∞ , i.e. E [︀⃦⃦ 𝑀 𝑍,𝜆𝑡 − 𝜇 𝑡 ⃦⃦ ]︀ → 𝜆 → ∞ . (cid:3) Gabih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions
In the preceding section we already mentioned that there is another asymptotic regime if the variance ofthe expert opinions Γ is not independent of the arrival intensity 𝜆 but grows linearly in 𝜆 . We now wantto establish some relations to the case of constant Γ studied above.Suppose that Γ = Γ 𝜆 = 𝜆𝜎 𝐽 𝜎 ⊤ 𝐽 where 𝜎 𝐽 is the volatility matrix of the continuous-time expertopinion process 𝑑𝐽 𝑡 = 𝜇 𝑡 𝑑𝑡 + 𝜎 𝐽 𝑑𝑊 𝐽𝑡 defined in (2.7). There we introduced the 𝐽 -investor who combinesobservations of stock returns with those of 𝐽 (instead of discrete-time expert opinions). For that setting Sasset al. [22] show that the information the 𝑍 -investor obtains from observing discrete-time expert opinionsis asymptotically the same as the information of the 𝐽 -investor extracting from observing the diffusionprocess 𝐽 if the model of the expert’s views 𝑍 𝑘 given in (2.6) uses standard normally distributed randomvariables 𝜀 𝑘 defined by the increments of 𝑊 𝐽 in the form 𝜀 𝑘 = √ 𝜆 ( 𝑊 𝐽𝑘/𝜆 − 𝑊 𝐽 ( 𝑘 − /𝜆 ) , 𝑘 ∈ N . In particularthey prove the mean-square convergence of filter processes 𝑀 𝑍 , 𝑄 𝑍 to the corresponding filter processes 𝑀 𝐽 , 𝑄 𝐽 of the 𝐽 -investor and also provide the corresponding error estimates. These limit theorems justifyso-called diffusion approximations of the filter for high-frequency discrete-time expert opinions to fixed andsufficiently large variance Γ of the expert stating that the filter for the 𝑍 -investor can be approximated bythe filter of a 𝐽 -investor with volatility matrix 𝜎 𝐽 = 𝜎 𝜆𝐽 = √ 𝜆 Γ / .Motivated by the results of the preceding section where we studied the filter asymptotics of the 𝑍 -investor with fixed expert’s variance Γ = Γ for 𝜆 → ∞ we now want to study the asymptotics of theassociated diffusion approximations. We therefore introduce a family of diffusion processes ( 𝐽 𝜆 ) 𝜆> definedby 𝑑𝐽 𝜆𝑡 = 𝜇 𝑡 𝑑𝑡 + 1 √ 𝜆 𝜎 𝐽 𝑑𝑊 𝐽𝑡 (5.1)with a constant matrix 𝜎 𝐽 ∈ R 𝑑 × 𝑑 chosen such that Σ 𝐽 := 𝜎 𝐽 𝜎 ⊤ 𝐽 is positive definite. Then it holdsΣ 𝐽 = Σ 𝜆𝐽 = 𝜆 Σ 𝐽 . Since = √ 𝜆 𝜎 𝐽 → 𝜆 → ∞ the limit case is not covered by the limit theorems in [22]and the diffusion approximation degenerates. Nevertheless, from a statistical point of view there is a clearinterpretation. In the limit the 𝐽 -investor can perfectly reconstruct the hidden drift 𝜇 from observing thelimiting process 𝐽 ∞ defined by the (deterministic) ODE 𝑑𝐽 ∞ 𝑡 = 𝜇 𝑡 𝑑𝑡 and has thus full information on thedrift.Below we provide a precise mathematical meaning to that convergence to full information and provethe corresponding limit theorems for the filter processes which are analogues to their counterparts forhigh-frequency discrete-time experts in Section 4. We also provide the associated error bounds for the driftestimates of the 𝐽 -investor. It turns out that we can benefit a lot from the techniques developed in theproofs of Section 4.Starting point is the conditional covariance 𝑄 𝐽 = 𝑄 𝐽,𝜆 . Note that, contrary to the stochastic conditionalcovariance 𝑄 𝑍,𝜆 of the 𝑍 -investor, 𝑄 𝐽,𝜆 is deterministic. According to Lemma 3.5 it satisfies the Riccatidifferential equation (3.5) which we rewrite as 𝑑𝑄 𝐽,𝜆𝑡 = 𝛼 𝐽,𝜆 ( 𝑄 𝐽,𝜆𝑡 ) 𝑑𝑡, 𝑄 𝐽,𝜆 = 𝑞 , where (5.2) 𝛼 𝐽,𝜆 ( 𝑞 ) = Σ 𝜇 − 𝜅𝑞 − 𝑞𝜅 ⊤ − 𝑞 (Σ − 𝑅 + 𝜆 Σ − 𝐽 ) 𝑞. (5.3) Lemma 5.1. ( Properties of 𝛼 𝐽,𝜆 )For the function 𝛼 𝐽,𝜆 given in (5.3) there exist constants 𝑎 𝛼 , 𝑏 𝛼 > independent of 𝜆 such that for allsymmetric and positive semidefinite 𝑞 ∈ R 𝑑 × 𝑑 tr (︀ 𝛼 𝐽,𝜆 ( 𝑞 ) )︀ ≤ 𝑎 𝛼 − √ 𝜆 𝑏 𝛼 tr( 𝑞 ) , for 𝜆 > . (5.4) The above estimate holds for 𝑎 𝛼 = tr(Σ 𝜇 ) + ( 𝑑 tr(Σ 𝐽 ) 𝑟 ) − and 𝑏 𝛼 = 2( 𝑑 tr(Σ 𝐽 ) √ 𝑟 ) − (5.5) and every 𝑟 > . abih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions The proof is given in Appendix A.3. Note that contrary to the corresponding estimate for 𝛼 𝑍,𝜆 given inLemma 4.2 the above estimate for 𝛼 𝐽,𝜆 is valid not only for sufficiently large 𝜆 ≥ 𝜆 > 𝜆 > 𝑄 𝐽,𝜆 from which the convergence to zero can be deduced.
Theorem 5.2.
For every 𝛿 ∈ (0 , 𝑇 ] and 𝜆 > it holds tr (︀ 𝑄 𝐽,𝜆𝑡 )︀ ≤ 𝐾 𝐽 √ 𝜆 for 𝑡 ∈ [ 𝛿, 𝑇 ] where (5.6) 𝐾 𝐽 = 𝐾 𝐽 ( 𝛿 ) = (︀ 𝑑 tr(Σ 𝐽 )[tr(Σ 𝜇 ) + tr( 𝑞 )( 𝑒 𝛿 ) − ] )︀ / , (5.7) where 𝑒 = exp(1) denotes Euler’s number.In particular, it holds tr (︀ 𝑄 𝐽,𝜆𝑡 )︀ → as 𝜆 → ∞ for all 𝑡 ∈ (0 , 𝑇 ] . Proof.
Let us define the function 𝑔 ( 𝑡 ) := tr( 𝑄 𝐽,𝜆𝑡 ) for 𝑡 ∈ [0 , 𝑇 ]. Then using (5.2) and the linearity of thetrace operator yields 𝑔 ( 𝑡 ) = tr( 𝑞 ) + ∫︀ 𝑡 tr( 𝛼 𝐽,𝜆 ( 𝑄 𝐽,𝜆𝑠 )) 𝑑𝑠 = tr( 𝑞 ) + ∫︀ 𝑡 𝑔 ( 𝑠 ) 𝑑𝑠. Analogous to the proof of(4.12) in Theorem 4.3, i.e., applying Lemma 5.1 and Gronwall’s Lemma we obtain for 𝑡 ∈ [ 𝛿, 𝑇 ] and 𝜆 > 𝑔 ( 𝑡 ) ≤ √ 𝜆 𝑏 𝛼 (tr( 𝑞 )( 𝑒 𝛿 ) − + 𝑎 𝛼 ) . (5.8)Recall (5.5) stating that the constants 𝑎 𝛼 , 𝑏 𝛼 can be chosen as 𝑎 𝛼 = 𝑎 𝛼 ( 𝑟 ) = tr(Σ 𝜇 ) + ( 𝑑 tr(Σ 𝐽 ) 𝑟 ) − and 𝑏 𝛼 = 𝑏 𝛼 ( 𝑟 ) = 2( 𝑑 tr(Σ 𝐽 ) √ 𝑟 ) − (5.9)for any 𝑟 >
0. We now choose 𝑟 such that the r.h.s. of (5.8) attains its minimum. The unique minimizeris found as 𝑟 * = ( 𝑑 tr(Σ 𝐽 )[tr( 𝑞 )( 𝑒 𝛿 ) − + tr(Σ 𝜇 )]) − and the minimal value is given by 𝐾 𝐽 / √ 𝜆 with 𝐾 𝐽 defined in (5.7). This proves the first claim. Since that inequality holds for all 𝛿 ∈ (0 , 𝑇 ] the convergencetr( 𝑄 𝑍,𝜆𝑡 ) → 𝜆 → ∞ holds for all 𝑡 ∈ (0 , 𝑇 ]. (cid:3) As in Section 4.1 the above asymptotic properties for the trace of 𝑄 𝐽,𝜆 imply analogous results for itsnorm. The proof is analogous to the proof of Corollary 4.4.
Corollary 5.3.
For every 𝛿 ∈ (0 , 𝑇 ] and any matrix norm ‖ · ‖ there exists a constant 𝐶 > such that ⃦⃦ 𝑄 𝐽,𝜆𝑡 ⃦⃦ 𝑝 ≤ 𝐶 √ 𝜆 for 𝜆 > , 𝑡 ∈ [ 𝛿, 𝑇 ] and 𝑝 ≥ . (5.10) For the Frobenius norm ‖ · ‖ 𝐹 the constant 𝐶 can be chosen as 𝐶 = 𝐾 𝐽 𝐶 𝑝 − 𝐹 where 𝐾 𝐽 is given in (5.7) and 𝐶 𝐹 denotes the upper bound from Proposition 3.4 for the Frobenius norm ‖ 𝑄 𝐽,𝜆 ‖ 𝐹 .In particular, it holds ⃦⃦ 𝑄 𝐽,𝜆𝑡 ⃦⃦ 𝑝 → as 𝜆 → ∞ for all 𝑡 ∈ (0 , 𝑇 ] . Based on the limit theorem for the conditional variance we now can state the corresponding result for theconvergence of the conditional mean 𝑀 𝐽 = 𝑀 𝐽,𝜆 . The proof is analogous to the proof of Theorem 4.6.
Theorem 5.4.
Let 𝐾 𝐽 be the constant given in Theorem 5.2. Then for every 𝛿 ∈ (0 , 𝑇 ] E [︀ ⃦⃦ 𝑀 𝐽,𝜆𝑡 − 𝜇 𝑡 ⃦⃦ ]︀ ≤ 𝐾 𝐽 √ 𝜆 for 𝜆 > , 𝑡 ∈ [ 𝛿, 𝑇 ] . (5.11) In particular, it holds E [︀ ⃦⃦ 𝑀 𝐽,𝜆𝑡 − 𝜇 𝑡 ⃦⃦ ]︀ → as 𝜆 → ∞ for all 𝑡 ∈ (0 , 𝑇 ] . Note that contrary to the corresponding estimate for 𝑀 𝑍,𝜆 given in Theorem 4.6 the above estimate for 𝑀 𝐽,𝜆 is valid not only for sufficiently large 𝜆 ≥ 𝜆 𝑄 > 𝜆 > Gabih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions
In this section we illustrate the theoretical findings of the previous sections by results of some numericalexperiments. These experiments are based on a stock market model where the unobservable drift 𝜇 followsan Ornstein-Uhlenbeck process as given in (2.4) and (2.5) whereas the volatility is known and constant.For simplicity, we assume that there is only one risky asset in the market, i.e. 𝑑 = 1. For our numericalexperiments we use the model parameters given in Table 1.The distribution of the initial value 𝜇 of the drift process is assumed to be the stationary distributionof the Ornstein-Uhlenbeck process, i.e., the limit of the marginal distribution of 𝜇 𝑡 for 𝑡 → ∞ which isknown to be Gaussian with mean 𝑚 = 𝜇 and variance 𝑞 = 𝜎 𝜇 𝜅 . Mean reversion level 𝜇 . Time horizon 𝑇 yearMean reversion speed 𝜅 Stock volatility 𝜎 𝑅 . Volatility 𝜎 𝜇 Expert’s variance
Γ = Σ 𝐽 . Initial value 𝜇 : mean 𝑚 = 𝜇 . Filter: initial values 𝑚 = 𝑚 . variance 𝑞 = 𝜎 𝜇 𝜅 . 𝑞 = 𝑞 . Table 1.
Model parameters for numerical experiments
The arrival dates of the expert opinions are modelled as jump times of a Poisson process with intensity 𝜆 . Then the waiting times between two information dates are exponentially distributed with parameter 𝜆 and the investor receives until time 𝑇 on average 𝜆𝑇 expert opinions. Recall that the expert’s views aremodelled by 𝑍 𝑘 = 𝜇 𝑇 𝑘 + Γ 𝜀 𝑘 , 𝑘 ∈ N , where ( 𝜀 𝑘 ) 𝑘 ≥ is a sequence of independent standard normally distributed random variables.At initial time 𝑡 = 0 all partially informed investors have the same information about the hidden drift.For the experiment we assume that they only know the model parameters described by ℱ 𝐻 = { ∅ , Ω } .Then the initial values for the filter processes 𝑀 𝐻 and 𝑄 𝐻 are the parameters of the Gaussian distributionof 𝜇 , i.e. 𝑚 = 𝑚 = 𝜇 and 𝑞 = 𝑞 = 𝜎 𝜇 𝜅 , respectively.In Figure 1 we plot the filters given by conditional mean 𝑀 𝐻 and conditional variance 𝑄 𝐻 of the 𝑅 -investor (blue), the 𝑍 -investor together with the associated 𝐽 -investor against time. For the 𝑍 -investorwe consider the arrival intensities 𝜆 = 5 , , 𝜎 𝜆𝐽 = √︀ Γ /𝜆 . In the upper plot one can see the conditionalvariances 𝑄 𝑅 , 𝑄 𝑍,𝜆 , 𝑄 𝐽,𝜆 and we also highlight (in green) the zero level corresponding to the limit processfor 𝜆 → ∞ . The lower plot shows a realization of the unobservable drift process 𝜇 (in green) together withits estimates given by the conditional means 𝑀 𝑅 (blue) and 𝑀 𝑍,𝜆 (yellow, orange, red). We omit plottingthe paths of 𝑀 𝐽,𝜆 .Since the filter processes for the 𝑅 - and 𝑍 -investor start with the same initial value their paths areidentical until the arrival of the first expert opinion leading to a filter update. This can be nicely seenfor 𝜆 = 5 and also for 𝜆 = 20 while for 𝜆 = 2000 the first update is almost immediately after the initialtime 𝑡 = 0. At the information dates the updates decrease the conditional variance and lead to a jump ofthe conditional mean. The updates of the conditional mean typically decrease the distance of 𝑀 𝑍,𝜆 to thehidden drift 𝜇 , of course this depends on the actual value of the expert’s view. Note that the drift estimate 𝑀 𝑅 of the 𝑅 -investor is quite poor and fluctuates just around the mean-reversion level 𝜇 . However, theexpert opinions visibly improve the drift estimate. abih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions Fig. 1.
Simulation of the filter processes 𝑄 𝐻 and 𝑀 𝐻 . The upper subplot shows realizations of the conditional variances 𝑄 𝑅 , 𝑄 𝑍,𝜆 (solid) and 𝑄 𝐽,𝜆 (dotted) for various intensities 𝜆 . The volatility of the continuous-time expert opinions is cho-sen as 𝜎 𝜆𝐽 = √︀ Γ /𝜆 . The lower subplot shows realizations of the corresponding conditional means 𝑀 𝑅 and 𝑀 𝑍,𝜆 togetherwith the path of the drift process 𝜇 (green). After an update the conditional variance 𝑄 𝑍,𝜆 increases and if the waiting time to the next informationdate is sufficiently large then it almost approaches the level of 𝑄 𝑅 . Again, this can nicely be observed for 𝜆 = 5. During such long periods without new expert opinions the conditional mean of the 𝑍 -investor 𝑀 𝑍,𝜆 tends to move towards the path of 𝑀 𝑅 .Looking at the paths of the conditional variance it can be seen that 𝑄 𝑅𝑡 dominates 𝑄 𝑍,𝜆𝑡 and 𝑄 𝐽,𝜆𝑡 forall 𝑡 ∈ (0 , 𝑇 ] which confirms the corresponding property stated in Proposition 3.4 and illustrates the factthat additional information by expert opinions leads to improved drift estimates. Note that for increasing 𝑡 the conditional variances 𝑄 𝑅𝑡 and 𝑄 𝐽,𝜆𝑡 quickly approach a constant which is the limit for 𝑡 → ∞ . Thatconvergence 𝑄 𝑅 has been proven in Proposition 4.6 of Gabih et al. [11] for markets with a single stockand generalized in Theorem 4.1 of Sass et al. [21] for markets with multiple stocks. The proof for 𝑄 𝐽,𝜆 isanalogous.Comparing the paths of the filter processes of the 𝑍 - and 𝐽 -investor for increasing arrival intensity 𝜆 it can be observed that the conditional variances 𝑄 𝑍,𝜆 and 𝑄 𝐽,𝜆 approach zero for any 𝑡 ∈ (0 , 𝑇 ]. Thisfact illustrates our findings in Theorems 4.3 and 5.2. Further, with increasing 𝜆 the path of the conditionalmean 𝑀 𝑍,𝜆 approaches the path of the hidden drift 𝜇 which confirms the mean-square convergence statedin Theorem 4.6.Finally, we want to examine the goodness of the upper bounds 𝐾 𝑍 / √ 𝜆 and 𝐾 𝐽 / √ 𝜆 for the conditionalvariances of the 𝑍 - and 𝐽 -investor given in Theorems 4.3 and 5.2, respectively. Note that in the presentexample with 𝑑 = 1 stock the two constants 𝐾 𝑍 , 𝐾 𝐽 coincide, it holds 𝐾 𝑍 = 𝐾 𝐽 = (︀ Γ[ 𝜎 𝜇 + 𝑞 ( 𝑒 𝛿 ) − ] )︀ / . In order to facilitate the visual comparison of the conditional variances and their upper bounds we focuson the information regime 𝐻 = 𝐽 and rewrite the estimate (5.6) as √ 𝜆𝑄 𝐽,𝜆𝑡 ≤ 𝐾 𝐽 = 𝐾 𝐽 ( 𝛿 ) for 𝑡 ∈ [ 𝛿, 𝑇 ].Fig. 2 shows for 𝜆 = 5 , , 𝑄 𝐽,𝜆𝑡 scaled by √ 𝜆 together with the upper bounds 𝐾 𝐽 ( 𝛿 ) (green) for two values of 𝛿 . It can be seen that the upper boundsare quite close to the actual values on [ 𝛿, 𝑇 ], in particular for larger 𝛿 .We also plot realizations of √ 𝜆𝑄 𝑍,𝜆𝑡 for the associated 𝑍 -investor (dashed lines). Note that estimate(4.8) does hold for the expected variance E [ 𝑄 𝑍,𝜆𝑡 ] but not for the realizations. Gabih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions
Fig. 2.
Conditional variances scaled by √ 𝜆 of the 𝐽 -investor, i.e. √ 𝜆 𝑄 𝐽,𝜆 for 𝜆 = 5 , , (solid), and of the 𝑍 -investor √ 𝜆 𝑄 𝑍,𝜆 (dotted) for 𝜆 = 5 , . The volatility of the continuous-time expert opinions is chosen as 𝜎 𝜆𝐽 = √︀ Γ /𝜆 .The green lines represent the upper bounds 𝐾 𝐽 = 𝐾 𝐽 ( 𝛿 ) given in Theorem 5.2 for 𝛿 = 𝛿 = 0 . and 𝛿 = 𝛿 = 0 . . A Proofs
A.1 Auxiliary Results
The proof of Lemma 4.2 which is given in Appendix A.2 is based on various properties of symmetric andpositive semidefinite matrices which we collect in the next lemma.
Lemma A.1. ( Properties of symmetric and positive semidefinite matrices )Let
𝐴, 𝐵 ∈ R 𝑑 × 𝑑 , 𝑑 ∈ N , symmetric and positive semidefinite matrices. Then it holds 𝐴 + 𝐵 is symmetric positive semidefinite. The eigenvalues 𝜚 𝑖 = 𝜚 𝑖 ( 𝐴 ) of 𝐴 are nonnegative, and there exists an orthogonal matrix 𝑉 such that 𝐴 = 𝑉 𝐷𝑉 ⊤ with 𝐷 = diag( 𝜚 , · · · , 𝜚 𝑑 ) , (A.1) i.e., 𝐴 is diagonalizable. If 𝐴 is positive definite then it is nonsingular and the inverse 𝐴 − is symmetric and positive definite. 𝜚 min ( 𝐴 ) tr (︀ 𝐵 )︀ ≤ tr( 𝐴𝐵 ) ≤ 𝜚 max ( 𝐴 ) tr( 𝐵 ) (A.2) where 𝜚 min ( 𝐴 ) and 𝜚 max ( 𝐴 ) denote the smallest and largest eigenvalue of 𝐴 , respectively.
5. tr( 𝐵 )tr (︀ 𝐴 − )︀ tr( 𝐴𝐵 ) ≤ tr( 𝐴 ) tr( 𝐵 ) (A.3) where for the first inequality 𝐴 is assumed to be positive definite.
6. tr ( 𝐴 ) ≥ tr (︀ 𝐴 )︀ ≥ 𝑑 tr ( 𝐴 ) (A.4)7. ‖ 𝐴 ‖ 𝐹 = √︁ tr (︀ 𝐴 ) ≤ tr( 𝐴 ) (A.5) where ‖ 𝐴 ‖ 𝐹 denotes the Frobenius norm of 𝐴 .Proof. The first three properties are standard and we refer to Horn and Johnson [13, Chapter 7]. The proofof (A.2) is given in Wang et al. [24, Lemma 1]. abih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions
5. From (A.1) we have 𝐴 = 𝑉 𝐷𝑉 ⊤ with an orthogonal matrix 𝑉 and 𝐷 = diag( 𝜚 , · · · , 𝜚 𝑑 ). If 𝐴 is positivedefinite then 𝜚 min ( 𝐴 ) > 𝐴 − exists, see property 3. It holds tr( 𝐴 ) = ∑︀ 𝑑𝑖 =1 𝜚 𝑖 ( 𝐴 ) ≥ 𝜚 max ( 𝐴 ) and tr (︀ 𝐴 − )︀ = tr( 𝑉 𝐷 − 𝑉 ⊤ ) = tr( 𝐷 − )︀ = 𝑑 ∑︁ 𝑖 =1 𝜚 𝑖 ( 𝐴 ) ≥ 𝜚 min ( 𝐴 ) . The above inequalities together with (A.2) imply (A.3).6. As above we use 𝐴 = 𝑉 𝐷𝑉 ⊤ with an orthogonal matrix 𝑉 and deduce 𝐴 = 𝑉 𝐷 𝑉 ⊤ andtr (︀ 𝐴 )︀ = tr (︀ 𝑉 ⊤ 𝑉 𝐷 )︀ = tr (︀ 𝐷 )︀ = 𝑑 ∑︁ 𝑖 =1 𝜚 𝑖 ≥ 𝑑 (︁ 𝑑 ∑︁ 𝑖 =1 𝜚 𝑖 )︁ = 1 𝑑 tr (︀ 𝐴 )︀ , where we have applied the Cauchy-Schwarz inequality. The first inequality in (A.4) follows from (A.3)with 𝐴 = 𝐵 .7. Let 𝐶 = 𝐴 , then 𝐶 𝑖𝑖 = ∑︀ 𝑑𝑘 =1 ( 𝐴 𝑖𝑘 ) andtr( 𝐴 ) = tr( 𝐶 ) = 𝑑 ∑︁ 𝑖 =1 𝐶 𝑖𝑖 = 𝑑 ∑︁ 𝑖,𝑘 =1 ( 𝐴 𝑖𝑘 ) = ‖ 𝐴 ‖ 𝐹 yielding the first equality. The inequality follows from (A.4). (cid:3) A.2 Proof of Lemma 4.2
For the convenience of the reader we recall the statement of Lemma 4.2:
For the function 𝛼 𝑍,𝜆 given in (4.2) there exist constants 𝑎 𝛼 , 𝑏 𝛼 > independent of 𝜆 and there exists 𝜆 > such that for all symmetric and positive semidefinite 𝑞 ∈ R 𝑑 × 𝑑 tr (︀ 𝛼 𝑍,𝜆 ( 𝑞 ) )︀ ≤ 𝑎 𝛼 − √ 𝜆 𝑏 𝛼 tr( 𝑞 ) , for 𝜆 ≥ 𝜆 . The above estimate holds for every 𝑎 𝛼 > tr(Σ 𝜇 ) , 𝑎 𝛼 > tr(Σ 𝜇 ) ,𝑏 𝛼 < 𝑏 𝛼 = 𝑏 𝛼 ( 𝑎 𝛼 ) := 2 √︃ 𝑎 𝛼 − tr(Σ 𝜇 )tr(Γ) ,𝜆 = 𝜆 ( 𝑎 𝛼 , 𝛽 𝛼 ) := (︂ 𝑑 ( 𝑎 𝛼 − tr(Σ 𝜇 ))2 √︀ tr(Γ)( 𝑎 𝛼 − tr(Σ 𝜇 )) − 𝑏 𝛼 tr(Γ) )︂ . Proof.
Using the definition of 𝛼 𝑍,𝜆 in (4.2), the linearity of tr( · ) and that 𝑞 and Σ 𝑅 and therefore Σ − 𝑅 aresymmetric positive definite, and that 𝜅 is positive definite we findtr (︀ 𝛼 𝑍,𝜆 ( 𝑞 ) )︀ = tr (︁ Σ 𝜇 − 𝜅𝑞 − 𝑞𝜅 ⊤ − 𝑞 Σ − 𝑅 𝑞 − 𝜆𝑞 (Γ + 𝑞 ) − 𝑞 )︁ ≤ tr (︀ 𝛼 𝜆 ( 𝑞 ) )︀ , (A.6)where 𝛼 𝜆 ( 𝑞 ) := Σ 𝜇 − 𝜆𝑞 (Γ + 𝑞 ) − 𝑞. The inequality follows from properties of symmetric positive definite matrices, see (A.2), (A.3) and (A.4)from which we deduce tr( 𝜅𝑞 + 𝑞𝜅 ⊤ ) = tr(( 𝜅 + 𝜅 ⊤ ) 𝑞 ) ≥ 𝜚 min ( 𝜅 + 𝜅 ⊤ ) tr( 𝑞 ) ≥ , tr( 𝑞 Σ − 𝑅 𝑞 ) = tr( 𝑞 Σ − 𝑅 ) ≥ tr( 𝑞 )tr(Σ 𝑅 ) ≥ 𝑑 tr ( 𝑞 )tr(Σ 𝑅 ) ≥ . Gabih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions
Here, 𝜚 min ( · ) denotes the the smallest eigenvalue of a positive definite symmetric matrix, which are allpositive. Note that since 𝜅 is positive definite 𝜅 + 𝜅 ⊤ is symmetric and positive definite. Further, 𝑞 issymmetric and positive semidefinite and according to property 3 of Lemma A.1 Σ − 𝑅 is symmetric andpositive definite.Inequality (A.6) implies that it suffices to prove the claim for 𝛼 𝜆 , i.e.,tr( 𝛼 𝜆 ( 𝑞 )) ≤ 𝑎 𝛼 − √ 𝜆𝑏 𝛼 tr( 𝑞 ) for 𝜆 ≥ 𝜆 . (A.7)For the proof of (A.7) we set 𝜀 = √ 𝜆 , 𝑞 = 𝜀𝑧 , 𝑎 𝜇 = tr (︀ Σ 𝜇 )︀ and consider the function 𝐻 𝜀 : R 𝑑 × 𝑑 → R with 𝐻 𝜀 ( 𝑧 ) := − tr( 𝛼 /𝜀 ( 𝜀𝑧 )) + 𝑎 𝛼 − 𝜀 𝑏 𝛼 tr( 𝜀𝑧 )= tr( 𝑧 (Γ + 𝜀𝑧 ) − 𝑧 ) − 𝑏 𝛼 tr( 𝑧 ) + 𝑎 𝛼 − 𝑎 𝜇 (A.8)for 𝑎 𝛼 , 𝑏 𝛼 , 𝜀 > 𝑧 . Below we show that there existpositive constants 𝑎 𝛼 , 𝑏 𝛼 , 𝜀 such that for all 𝑧 it holds 𝐻 𝜀 ( 𝑧 ) ≥ 𝜀 ≤ 𝜀 . (A.9)That inequality implies for 𝑧 = 𝜀 𝑞 = √ 𝜆𝑞 ≤ 𝐻 𝜀 ( 𝑧 ) = 𝐻 𝜀 ( √ 𝜆𝑞 ) = − tr( 𝛼 𝜆 ( 𝑞 )) − √ 𝜆𝑏 𝛼 tr( 𝑞 ) + 𝑎 𝛼 , and (A.6) yields for 𝜆 ≥ 𝜆 = ( 𝜀 ) tr( 𝛼 𝑍,𝜆 ( 𝑞 )) ≤ tr( 𝛼 𝜆 ( 𝑞 )) ≤ 𝑎 𝛼 − √ 𝜆𝑏 𝛼 tr( 𝑞 ) , which proves the assertion.In the remainder of the proof we show inequality (A.9). The matrices 𝑧 and Γ are symmetric, 𝑧 is positivesemidefinite and Γ is strictly positive definite. Then Γ + 𝜀𝑧 is strictly positive definite and according toproperties 1. and 3. of Lemma A.1, the matrix (Γ + 𝜀𝑧 ) − is symmetric and strictly positive definite.Further, 𝑧 is symmetric and positive semidefinite. Inequality (A.3) implies tr( 𝐴𝐵 ) ≥ tr( 𝐵 ) / tr (︀ 𝐴 − )︀ andwith 𝐴 = (Γ + 𝜀𝑧 ) − and 𝐵 = 𝑧 we findtr( 𝑧 (Γ + 𝜀𝑧 ) − 𝑧 ) = tr( 𝑧 (Γ + 𝜀𝑧 ) − ) ≥ tr( 𝑧 )tr(Γ + 𝜀𝑧 ) = tr( 𝑧 )tr(Γ) + 𝜀 tr( 𝑧 ) . Inequality (A.4) yields tr( 𝑧 ) ≥ 𝑑 tr ( 𝑧 ), and hence we obtaintr( 𝑧 (Γ + 𝜀𝑧 ) − 𝑧 ) ≥ 𝑑 tr ( 𝑧 )tr(Γ) + 𝜀 tr( 𝑧 ) = tr ( 𝑧 ) 𝜓 + 𝜀𝑑 tr( 𝑧 ) , (A.10)where 𝜓 = 𝑑 tr(Γ). Set 𝑥 = tr( 𝑧 ) ≥ 𝑔 ( 𝑥 ) = 𝑥 𝜓 + 𝜀𝑑𝑥 − 𝑏 𝛼 𝑥 + 𝑎 𝛼 − 𝑎 𝜇 then (A.10) implies 𝐻 𝜀 ( 𝑧 ) ≥ 𝑔 ( 𝑥 ) . Now it remains to choose constants 𝑎 𝛼 , 𝑏 𝛼 , 𝜀 > 𝑔 ( 𝑥 ) ≥ 𝑥 ≥ 𝜀 ≤ 𝜀 . (A.11)Let 𝑎 𝛼 > 𝑎 𝜇 = tr(Σ 𝜇 ). Then 𝑎 := 𝑔 (0) = 𝑎 𝛼 − 𝑎 𝜇 >
0. Since 𝜓 + 𝜀𝑑𝑥 > 𝑔 ( 𝑥 ) ≥ ≤ 𝑓 ( 𝑥 ) := ( 𝜓 + 𝜀𝑑𝑥 ) 𝑔 ( 𝑥 ) = 𝐴 𝜀 𝑥 + 𝐵 𝜀 𝑥 + 𝐶, abih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions where 𝐴 𝜀 = 1 − 𝜀𝑑𝑏 𝛼 , 𝐵 𝜀 = 𝜀𝑑𝑎 − 𝑏 𝛼 𝜓 , 𝐶 = 𝜓𝑎 . Let 𝜀 > 𝐴 𝜀 >
0, then 𝑓 ( 𝑥 ) = 𝐴 𝜀 (︁ 𝑥 + 𝐵 𝜀 𝐴 𝜀 𝑥 + 𝐶𝐴 𝜀 )︁ = 𝐴 𝜀 (︁ 𝑥 − 𝐾 𝜀 )︁ + 𝐷 𝜀 with 𝐾 𝜀 = 𝐵 𝜀 𝐴 𝜀 and 𝐷 𝜀 := 𝐶 −
14 ( 𝐵 𝜀 ) 𝐴 𝜀 = 4 𝐶𝐴 𝜀 − ( 𝐵 𝜀 ) 𝐴 𝜀 . We choose 𝑎 𝛼 > 𝑎 𝜇 , i.e., 𝑎 = 𝑎 𝛼 − 𝑎 𝜇 >
0, then we have 𝑓 ( 𝑥 ) ≥ 𝐷 𝜀 ≥ 𝑃 ( 𝜀 ) := 4 𝐶𝐴 𝜀 − ( 𝐵 𝜀 ) ≥ .𝑃 ( 𝜀 ) is a quadratic function and it holds 𝑃 ( 𝜀 ) = 4 𝜓𝑎 − ( 𝜀𝑑𝑎 + 𝑏 𝛼 𝜓 ) , hence 𝑃 (0) = 4 𝜓𝑎 − 𝑏 𝛼 𝜓 and 𝑃 isdecreasing for 𝜀 >
0. Thus we have to require 𝑃 (0) > < 𝑏 𝛼 ≤ 𝑏 𝛼 = 𝑏 𝛼 ( 𝑎 𝛼 ) = 2 √︂ 𝑎 𝛼 − 𝑎 𝜇 𝜓 . Then 𝑃 ( 𝜀 ) ≥ 𝜀 ∈ (0 , 𝜀 ] where 𝜀 is the positive zero of 𝑃 given by 𝜀 = 𝜀 ( 𝑎 𝛼 , 𝛽 𝛼 ) = 1 𝑑 ( 𝑎 𝛼 − 𝑎 𝜇 ) (︁ √︀ 𝜓 ( 𝑎 𝛼 − 𝑎 𝜇 ) − 𝑏 𝛼 𝜓 )︁ . It is not difficult to check that for 𝜀 < 𝜀 it holds 𝐴 𝜀 = 1 − 𝜀𝑑𝑏 𝛼 > (︁ − 𝑏 𝛼 √︃ 𝜓𝑎 𝛼 − 𝑎 𝜇 )︁ ≥ . Note that for 𝑏 𝛼 = 𝑏 𝛼 it holds 𝜀 = 0 which is not feasible. Hence for 𝑎 𝛼 > 𝑎 𝜇 , 𝑏 𝛼 ∈ (0 , 𝑏 𝛼 ( 𝑎 𝛼 )) and for 𝜀 ≤ 𝜀 = 𝜀 ( 𝑎 𝛼 , 𝑏 𝛼 ) or equivalently 𝜆 ≥ 𝜆 = 1 /𝜀 it holds (A.11) and therefore 𝐻 𝜀 ( 𝑧 ) ≥ (cid:3) A.3 Proof of Lemma 5.1
For the convenience of the reader we recall the statement of Lemma 5.1:
For the function 𝛼 𝐽,𝜆 given in (5.3) there exist constants 𝑎 𝛼 , 𝑏 𝛼 > independent of 𝜆 such that for allsymmetric and positive semidefinite 𝑞 ∈ R 𝑑 × 𝑑 tr (︀ 𝛼 𝐽,𝜆 ( 𝑞 ) )︀ ≤ 𝑎 𝛼 − √ 𝜆 𝑏 𝛼 tr( 𝑞 ) , for 𝜆 > . The above estimate holds for 𝑎 𝛼 = tr(Σ 𝜇 ) + ( 𝑑 tr(Σ 𝐽 ) 𝑟 ) − and 𝑏 𝛼 = 2( 𝑑 tr(Σ 𝐽 ) √ 𝑟 ) − and every 𝑟 > .Proof. Using the definition of 𝛼 𝐽,𝜆 in (5.3) and the linearity of tr( · ) we findtr (︀ 𝛼 𝐽,𝜆 ( 𝑞 ) )︀ = tr (︀ Σ 𝜇 )︀ − tr (︀ 𝜅𝑞 + 𝑞𝜅 ⊤ )︀ − tr (︀ 𝑞 (Σ − 𝑅 + 𝜆 Σ − 𝐽 ) 𝑞 )︀ . (A.12)For the second term on the r.h.s. (A.2) implies tr( 𝜅𝑞 + 𝑞𝜅 ⊤ ) = tr(( 𝜅 + 𝜅 ⊤ ) 𝑞 ) ≥ 𝛽 tr( 𝑞 ) where 𝛽 := 𝜚 min ( 𝜅 + 𝜅 ⊤ ) > 𝜅 + 𝜅 ⊤ . That matrix is symmetric and positive definite since 𝜅 is positive definite. Using (A.3) and (A.4) we deducetr( 𝑞 (Σ − 𝑅 + 𝜆 Σ − 𝐽 ) 𝑞 ) ≥ 𝜆 tr( 𝑞 Σ − 𝐽 𝑞 ) = 𝜆 tr( 𝑞 Σ − 𝐽 ) ≥ 𝜆 tr( 𝑞 )tr(Σ 𝐽 )) ≥ 𝜆 𝑑 tr ( 𝑞 )tr(Σ 𝐽 )) = 𝜆𝜓 tr ( 𝑞 )where 𝜓 := ( 𝑑 tr(Σ 𝐽 )) − >
0. Substituting the above estimates into (A.12) we obtaintr (︀ 𝛼 𝐽,𝜆 ( 𝑞 ) )︀ ≤ 𝑓 (tr( 𝑞 )) with 𝑓 ( 𝑥 ) := 𝑎 𝜇 − 𝛽𝑥 − 𝜆𝜓𝑥 , 𝑥 ≥ , (A.13)where we set 𝑎 𝜇 = tr(Σ 𝜇 ). The quadratic function 𝑓 is strictly concave, thus for any 𝑥 ≥ 𝑓 ( 𝑥 ) ≤ 𝑓 ( 𝑥 ) + 𝑓 ′ ( 𝑥 )( 𝑥 − 𝑥 ). Choosing 𝑥 = 1 / √ 𝜆𝑟 for some 𝑟 > 𝑓 ( 𝑥 ) ≤ 𝑎 𝜇 + 𝜓𝑟 − √ 𝜆 𝜓 √ 𝑟 𝑥 = 𝑎 𝛼 − √ 𝜆𝑏 𝛼 𝑥 where we used the definition of 𝑎 𝛼 , 𝑏 𝛼 in (5.5). Substituting this estimate into (A.13) proves the claim. (cid:3) Gabih, Kondakji and Wunderlich, Asymptotic Filter Behavior for High-Frequency Expert Opinions
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