A Stochastic Model of Order Book Dynamics using Bouncing Geometric Brownian Motions
AA Stochastic Model of Order Book Dynamics usingBouncing Geometric Brownian Motions
Xin Liu ∗ , Vidyadhar G. Kulkarni † , and Qi Gong ‡ Department of Mathematical Sciences, Clemson University, Clemson, SC 29634. Department of Statistics and Operations Research, University of North Carolina, ChapelHill, NC 27599.March 28, 2016
Abstract
We consider a limit order book, where buyers and sellers register to trade a security atspecific prices. The largest price buyers on the book are willing to offer is called the market bidprice, and the smallest price sellers on the book are willing to accept is called the market askprice. Market ask price is always greater than market bid price, and these prices move upwardsand downwards due to new arrivals, market trades, and cancellations. We model these two priceprocesses as “bouncing geometric Brownian motions (GBMs)”, which are defined as exponentialsof two mutually reflected Brownian motions. We then modify these bouncing GBMs to constructa discrete time stochastic process of trading times and trading prices, which is parameterized bya positive parameter δ . Under this model, it is shown that the inter-trading times are inverseGaussian distributed, and the logarithmic returns between consecutive trading times follow anormal inverse Gaussian distribution. Our main results show that the logarithmic trading priceprocess is a renewal reward process, and under a suitable scaling, this process converges to astandard Brownian motion as δ →
0. We also prove that the modified ask and bid processesapproach the original bouncing GBMs as δ →
0. Finally, we derive a simple and effectiveprediction formula for trading prices, and illustrate the effectiveness of the prediction formulawith an example using real stock price data.
Keywords:
Order book dynamics; Geometric Brownian motions; Reflected Brownian mo-tions; Mutually reflected Brownian motions; Inverse Gaussian distributions; Normal inverseGaussian distributions; Renewal reward processes; Diffusion approximations; Scaling limits.
In a modern order-driven trading system, limit-sell and limit-buy orders arrive with specificprices and they are registered in a limit order book (LOB) . The price at which a buyer is willing tobuy is called the bid price and the price at which a seller is willing to sell is called the ask price. ∗ Email: [email protected]. † Email: [email protected]. ‡ Email: [email protected]. a r X i v : . [ q -f i n . T R ] M a r he order book organizes the orders by their prices and by their arrival times within each price.The highest bid price on the book is called the market bid price, and the lowest ask price on thebook is called the market ask price. In contrast to the limit orders, market orders have no prices:a market buy order is matched and settled against the sell order at the market ask price and amarket sell order is matched and settled against the buy order at the market bid price. (We areignoring the sizes of the orders in this simplified discussion.) When the market bid price equals themarket ask price, a trade occurs, and the two matched traders are removed from the LOB. Thusimmediately after the trade the market bid price decreases and the market ask price increases.Clearly the market ask price is always above the market bid price. Between two trading times, themarket ask and bid prices fluctuate due to new arrivals, cancellations, market trades, etc.There is an extensive literature on models of LOBs, including statistical analysis and stochasticmodeling. In particular, Markov models have been developed in [2, 3, 14, 12, 13, 16, 17, 22], toname a few. In such models, point processes are used to model arrival processes of limit andmarket orders, and the market bid and ask prices are formulated as complex jump processes. Tosimplify such complexity, one tries to develop suitable approximate models. Brownian motion typeapproximations are established, for example, in [2, 3, 12], and law of large numbers is recentlystudied in [16, 17].It is clear that the stochastic evolution of the market ask and bid prices is a result of a complexdynamics of the trader behavior and the market mechanism. It makes sense to ignore the detaileddynamics altogether and directly model the market ask and bid prices as stochastic processes. Welet A ( t ) and B ( t ) be the market ask and bid prices at time t , respectively, and model { A ( t ) , t ≥ } and { B ( t ) , t ≥ } as two stochastic processes with continuous sample paths that bounce off of eachother as follows. Initially A (0) > B (0). Intuitively, we assume that the market bid and ask pricesevolve according to two independent geometric Brownian motions (GBMs), and bounce off awayfrom each other whenever they meet. Hence we call this the “bouncing GBMs” model of the LOB.To the best of our knowledge, this is the first time such a model is used to describe the dynamicsof the market ask and bid price processes in the LOB.Bouncing GBMs can be constructed from bouncing BMs (the detailed construction is givenSection 2). Bouncing BMs have been studied by Burdzy and Nualart in [10], and a related modelof bouncing Brownian balls has been studied by Saisho and Tanaka in [25]. Of these two papers,the one by Burdzy and Nuaalart is most relevant to our model. They study two Brownian motionsin which the lower one is reflected downward from the upper one. Thus the upper process isunperturbed by the lower process, while the lower process is pushed downward (by an appropriatereflection map) when it hits the upper process. We use a similar construction in our bouncingGBM model, except that in our case both processes reflect off of each other in opposite directionswhenever they meet. We assume that the reflection is symmetric, which will be made precise inSection 2.We would like to say that a transaction occurs when the market ask price process meets themarket bid price process, and the transaction price is the level where they meet. Unfortunately,the bouncing GBMs will meet at uncountably many times in any finite interval of time. This willcreate uncountably many transactions over a finite interval of time, which is not a good modelof the reality. In reality, transactions occur at discrete times. Denote by T n the n -th transactiontime, and P n the price at which the n -th transaction is settled. We are interested in studying thediscrete time process { ( P n , T n ) , n ≥ } . To define this correctly and conveniently, we assume a2rice separation parameter δ >
0, and construct two modified market ask and bid price processes A δ and B δ from the bouncing GBMs A and B . One can think of δ as representing the tick size ofthe LOB, typically one cent. The construction of A δ and B δ enables us to define a discrete timestochastic process { ( P δ,n , T δ,n ) , n ≥ } of transaction prices and times. The precise definitions of A δ , B δ and ( P δ,n , T δ,n ) are given in Section 3.We show that the inter-trading times T δ,n +1 − T δ,n follow an inverse Gaussian (IG) distribu-tion, and the logarithmic return between consecutive trading times ln( P δ,n +1 /P δ,n ) follow a normalinverse Gaussian (NIG) distribution. We then formulate the logarithmic trading price process asa renewal reward process in terms of inter-trading times and successive logarithmic returns. It isworth noting that δ is typically small, and in the numerical example in Section 6, δ = O (10 − ) . Finally, our main result shows that under a suitable scaling, the logarithmic trading price processconverges to a standard Brownian motion as δ →
0. We also study the limit of the modified marketask and bid price processes ( A δ , B δ ) as δ →
0, which is exactly the original bouncing GBMs (
A, B ).Using these asymptotics, we derive a simple and effective prediction formulas for trading prices.It is interesting to see that we get an asymptotic GBM model for the trading prices in the limit.The GBM model captures the intuition that the rates of returns over non-overlapping intervalsare independent of each other, and has been extensively used to model stock prices since thebreakthrough made by Black and Scholes [5] and Merton [23]. Another interesting observation isthe logarithmic returns between consecutive trading times are NIG distributed. In fact, empiricalstudies show that logarithmic returns of assets can be fitted very well by NIG distributions (see[6, 7, 24]) and Barndorff-Nielsen proposed NIG models in [8]. Thus our model of bouncing GBMsprovides another justification for the GBM model of trading prices.The rest of the paper is organized as follows. In Section 2, we introduce our model of bouncingGBMs in details. In Section 3 we construct the modified market ask and bid processes and the price-transaction process. All the main results about the distributions of transaction times and prices,and the limiting behaviors are summarized in Section 4. In Section 5, the estimators of the modelparameters are derived using the method of moments. In Section 6, we use asymptotic GBM modelobtained in Section 4 for trading prices, from which we derive a simple and effective forecastingformula. We also apply the formula to real data, and show that the estimated δ parameter is indeedvery small, and hence the asymptotic results are applicable, and work very well over short timehorizons. Finally, all proofs are given in Appendix. We consider a trading system, where buyers and sellers arrive with specific prices. Recall thatthe market bid price is the largest price at which buyers are willing to buy, and the market askprice is the smallest price at which sellers are willing to sell. The market ask price cannot be lessthan the market bid price, and a trade occurs when the market bid and ask prices are matched. Wewill model the market bid and ask prices as bouncing GBMs , which are defined as exponentials ofmutually reflected Brownian motions (BMs). More precisely, let A ( t ) and B ( t ) denote the marketask and bid prices at time t ≥
0, and assume that A (0) ≥ B (0). For t ≥
0, define X a ( t ) = ln A (0) + µ a t + σ a W a ( t ) , (2.1) X b ( t ) = ln B (0) + µ b t + σ b W b ( t ) , (2.2)3here W a , W b are independent standard BMs independent of A (0) and B (0), and µ a , µ b and σ a , σ b are the drift and variance parameters. We assume that µ a < µ b . We first define a pair of mutuallyreflected BMs ( Y a , Y b ) as follows. For t ≥
0, define Y a ( t ) = X a ( t ) + 12 L ( t ) , (2.3) Y b ( t ) = X b ( t ) − L ( t ) , (2.4)where { L ( t ) , t ≥ } is the unique continuous nondecreasing process such that(i) L (0) = 0;(ii) Y a ( t ) − Y b ( t ) ≥ t ≥ L ( t ) can increase only when Y a ( t ) − Y b ( t ) = 0, i.e., (cid:90) ∞ { Y a ( t ) − Y b ( t ) > } dL ( t ) = 0 . The existence and uniqueness of { L ( t ) , t ≥ } are from Skorohod lemma (see [20, Lemma 3.6.14]).In fact, L ( t ) has the following explicit formula L ( t ) = sup ≤ s ≤ t ( X a ( s ) − X b ( s )) − , (2.5)where for a ∈ R , a − = max {− a, } . Rougly speaking, the processes Y a ( t ) and Y b ( t ) behave like twoindependent BMs when Y a ( t ) > Y b ( t ), and whenever they meet, the process Y a ( t ) will be pushedup, while Y b ( t ) will be pushed down, to make Y a ( t ) ≥ Y b ( t ) for all t ≥
0. Here we assume thepushing effect for Y a ( t ) and Y b ( t ) are the same, and thus we have before the regulator process L ( t ) in both (2.3) and (2.4).Finally, A ( t ) and B ( t ) are defined as A ( t ) = e Y a ( t ) , (2.6) B ( t ) = e Y b ( t ) . (2.7)Thus A ( t ) and B ( t ) behave like two independent GBMs when A ( t ) > B ( t ), and whenever theybecome equal, they will be pushed away from each other such that A ( t ) ≥ B ( t ) for all t ≥ A ( t ) B ( t ) , which could reflect the ask-bid spread. We note that A ( t ) B ( t ) = e Y a ( t ) − Y b ( t ) = e X a ( t ) − X b ( t )+ L ( t ) , t ≥ . From (2.5), { Y a ( t ) − Y b ( t ) , t ≥ } is a reflected Brownian motion (RBM). It is well known that aRBM { R ( t ) , t ≥ } with mean µ and variance σ has the following transient cumulative distribution4unction (CDF) (see Section 1.8 in [15]). For x, y ∈ [0 , ∞ ) , Pr( R ( t ) ≤ y | R (0) = x ) = 1 − Φ (cid:18) − y + x + µtσ √ t (cid:19) − e µy/σ Φ (cid:18) − y − x − µtσ √ t (cid:19) , (2.8)where Φ( · ) is the CDF of the standard normal distribution. Thus for t ≥
0, the ratio A ( t ) B ( t ) has thefollowing CDF. Assuming A (0) and B (0) are deterministic constants, for y ≥ (cid:18) A ( t ) B ( t ) ≤ y (cid:19) = 1 − Φ − ln[ y ] + ln[ A (0) /B (0)] + ( µ a − µ b ) t (cid:113) ( σ a + σ b ) t − y − µb − µa ) σ a + σ b Φ − ln[ y ] − ln[ A (0) /B (0)] − ( µ a − µ b ) t (cid:113) ( σ a + σ b ) t . Consequently, under the condition that µ a < µ b , the stationary distribution of AB is power-lawdistributed with density function 2( µ b − µ a ) σ a + σ b y − − µb − µa ) σ a + σ b , y ≥ . (2.9)It is interesting to see that only stationary moments of order less than µ b − µ a ) σ a + σ b are finite. For finite t , a simple description of the k -th moment of A ( t ) B ( t ) with A (0) = B (0) is presented in the followinglemma, the proof of which is provided in Appendix. Lemma 2.1.
Assume A (0) = B (0) . Then for k ∈ N , E (cid:34)(cid:18) A ( t ) B ( t ) (cid:19) k (cid:35) = 1 + k ( σ a + σ b ) µ b − µ a (cid:90) ∞ exp (cid:26) k ( σ a + σ b ) xµ b − µ a − x (cid:27) F ( t ; x, dx, (2.10) where F ( t ; x,
0) = Φ (cid:18) t − x √ t (cid:19) + e x Φ (cid:18) − t − x √ t (cid:19) , x ≥ , is the CDF of the first-passage-time of { Y a ( t ) − Y b ( t ) , t ≥ } from x to . Note that lim t →∞ F ( t ; x,
0) = 1, and so lim t →∞ E [( A ( t ) B ( t ) ) k ] is finite only when k < µ b − µ a ) σ a + σ b .This result is consistent with the moments of the power law distribution in (2.9), and indeed onecan easily check that when k < µ b − µ a ) σ a + σ b ,lim t →∞ E (cid:34)(cid:18) A ( t ) B ( t ) (cid:19) k (cid:35) = E (cid:20) A ( ∞ ) B ( ∞ ) (cid:21) , (2.11)where A ( ∞ ) B ( ∞ ) is a random variable with density function (2.9). Other performance analysis canbe done by computing the joint distribution of ( A ( t ) , B ( t )). However, it is nontrivial to obtain asimple description of the transient behavior of ( A ( t ) , B ( t )) . Thus we would like to investigate such5roblems in a separate paper.
Assuming that A (0) > B (0), the first trading time is defined to be the first time that the marketask and bid prices become equal, and we would like to define the n th trading time to be the n thtime the two prices become equal. However, the zero set { t ≥ A ( t ) − B ( t ) = 0 } is uncountablyinfinite, and we cannot define the n th trading time as conveniently as the first one. Also notethat in practice every time the market ask and bid prices become equal, they will separate fromeach other by at least one cent. Thus we consider the following modified market ask and bid priceprocesses A δ and B δ , where the positive constant δ represents the tick size. We then use A δ and B δ to define the trading times and trading prices. More precisely, let δ be a strictly positive constant,and recall that A (0) and B (0) are the initial values of the market ask and bid price processes A and B , and X a and X b are two independent BMs defined in (2.1) and (2.2). For n ≥
1, define thefollowing stopping times: T δ, = 0, and T δ,n = inf { t ≥ X a ( t ) − X b ( t ) = − n − δ } . (3.1)Then T δ,n ≥ T δ,n − and T δ,n → ∞ almost surely as n → ∞ . We next define the modified marketask and bid price processes. For t ≥ ,A δ ( t ) = exp (cid:40) X a ( t ) + ∞ (cid:88) n =1 ( n − δ { t ∈ [ T δ,n − ,T δ,n ) } (cid:41) , (3.2) B δ ( t ) = exp (cid:40) X b ( t ) − ∞ (cid:88) n =1 ( n − δ { t ∈ [ T δ,n − ,T δ,n ) } (cid:41) . (3.3)Thus the first trade occurs at T δ, , which is the first time the modified market ask and bid pricesbecome equal, and the first trading price is defined as P δ, = A δ ( T δ, − ) = B δ ( T δ, − ) = e X a ( T δ, ) = e X b ( T δ, ) . (Note that T δ, and P δ, don’t depend on δ if the initials A (0) and B (0) are independent of δ .)Right after the first trade occurs, the market ask and bid prices will separate in the following way. A δ ( T δ, ) = P δ, e δ > P δ, , B δ ( T δ, ) = P δ, e − δ < P δ, . Starting from T δ, , the processes A δ and B δ evolves as two independent GMB’s with initial values P δ, e δ and P δ, e − δ until they meet again at T δ, . Recursively, for n ≥
1, the stopping time T δ,n willbe the n th meeting time of A δ and B δ , and the n th trading price is defined as P δ,n = A δ ( T δ,n − ) = B δ ( T δ,n − ) , (3.4)and the modified market ask and bid prices at T δ,n move to A δ ( T δ,n ) = P δ,n e δ > P δ,n , B δ ( T δ,n ) = P δ,n e − δ < P δ,n . (3.5)6 TimePrice T δ, T δ, P δ, P δ, A δ B δ Figure 1: Dynamics of the modified market ask and bid prices ( A δ , B δ ).Right after T δ,n , the processes A δ and B δ evolve as two independent GBMs with initials P δ,n e δ and P δ,n e − δ until they meet again at T δ,n +1 . The dynamics of the market ask and bid prices is shownin Figure 1.The relationship between the modified market ask and bid prices ( A δ , B δ ) and the originalmarket ask and bid prices ( A, B ) is summarized in the following proposition. Its proof can be foundin Appendix. In particular, it shows that { T δ,n } n ∈ N are also the meeting times of the original priceprocesses A and B , and that ( A δ ( t ) , B δ ( t )) converges to ( A ( t ) , B ( t )) almost surely and uniformlyon compact sets of [0 , ∞ ) as δ → Proposition 3.1. (i)
For δ > and n ∈ N , A ( T δ,n ) = B ( T δ,n ) . (ii) For δ > and t ≥ , A δ ( t ) ≥ A ( t ) , and B δ ( t ) ≤ B ( t ) . (iii) For t ≥ , sup ≤ s ≤ t A δ ( t ) A ( t ) → , and sup ≤ s ≤ t B ( t ) B δ ( t ) → , almost surely as δ → . For convenience, we denote U δ,n +1 = ln( P δ,n +1 /P δ,n ) , V δ,n +1 = T δ,n +1 − T δ,n , n ≥ ,U δ, = ln P δ, , V δ, = T δ, .
7e are interested in the evolution of the trading prices. Define for t ≥ N δ ( t ) = max { n ≥ T δ,n ≤ t } , (3.6)which gives the number of trades up to time t . Now the latest trading price can be formulated as P δ ( t ) = P δ,N δ ( t ) , for t ≥ T δ, . (3.7)For t ≥ T δ, , let Z δ ( t ) = ln( P δ ( t )), and so Z δ ( t ) = (cid:80) N δ ( t ) n =1 U δ,n . When 0 ≤ t < T δ, , we simply let Z δ ( t ) = 0. Thus we have Z δ ( t ) = N δ ( t ) (cid:88) n =1 U δ,n , (3.8)with the convention that (cid:80) n =1 U δ,n = 0 . We will see in Lemma 4.5 that { Z δ ( t ) , t ≥ } is a renewalreward process. Our goal is to establish a scaling limit theorem for Z as δ →
0, and develop anasymptotic model for real financial data.
We present our main results in this section. In particular, it is shown that ( U δ,n , V δ,n ) , n ≥ , are i.i.d. random variables (see Lemma 4.1), and U δ,n follows a NIG distribution and V δ,n is IGdistributed (see Corollary 4.3). Using these results, it is clear that { Z δ ( t ) , t ≥ } is a renewal rewardprocess, and the scaling limit theorem is established in Theorem 4.7. All the proofs are providedin Appendix. ( U δ,n , V δ,n ) We derive the joint distribution of ( U δ,n , V δ,n ) for each n ≥ U δ, , V δ, ) doesn’t depend on δ if the intial values A (0) and B (0) are independent of δ . Lemma 4.1. (i)
Assume A (0) = e α , B (0) = e β , and α > β . For t ≥ and x ∈ R , we have Pr( U δ, ∈ dx, V δ, ∈ dt ) = α − β πt σ a σ b exp − (cid:104) σ b σ a ( x − α − µ a t ) + σ a σ b ( x − β − µ b t ) (cid:105) + [ α − β − ( µ b − µ a ) t ] σ a + σ b ) t dxdt. (4.1) In particular, V δ, follows IG distribution with the following density function Pr( V δ, ∈ dt ) = α − β (cid:113) π ( σ a + σ b ) t exp (cid:26) − [ α − β − ( µ b − µ a ) t ] σ a + σ b ) t (cid:27) dt, (4.2) and given V δ, = t , U δ, follows normal distribution with mean σ b ( α + µ a t )+ σ a ( β + µ b t ) σ a + σ b and vari- nce σ a σ b tσ a + σ b . (ii) The sequence ( U δ,n , V δ,n ) n ≥ is an i.i.d. sequence, which is independent of ( U δ, , V δ, ) and hasthe same distribution as in (i) with α = δ and β = − δ. To derive the marginal distributions of U n , n ≥
1, we introduce the following definitions of IGand NIG distributions (see [26]).
Definition 4.2. (i)
An inverse Gaussian (IG) distribution with parameters a and a has densityfunction f ( x ; a , a ) = a √ πx exp (cid:26) − ( a − a x ) x (cid:27) , x > , which is usually denoted by IG( a , a ) . (ii) A random variable X follows a normal inverse Gaussian (NIG) distribution with parameters ¯ α, ¯ β, ¯ µ, ¯ δ with notation NIG( ¯ α, ¯ β, ¯ µ, ¯ δ ) if Y | X = x ∼ N (¯ µ + ¯ βx, x ) , and X ∼ IG(¯ δ, (cid:113) ¯ α − ¯ β ) . The density function of Y is given as f ( y ; ¯ α, ¯ β, ¯ µ, ¯ δ ) = ¯ απ ¯ δ exp (cid:26)(cid:113) ¯ α − ¯ β + ¯ β ¯ δ ( y − ¯ µ ) (cid:27) K (cid:16) ¯ α (cid:113) y − ¯ µ ¯ δ ) (cid:17)(cid:113) y − ¯ µ ¯ δ ) , where K ( z ) = (cid:82) ∞ e − z ( t + t − ) / dt is the modified Bessel function of the third kind with index . Using the above definitions, we have the following conclusion on the marginal distributions of( U n , V n ) , n ≥ Corollary 4.3. (i)
Assume A (0) = e α , B (0) = e β , and α > β . Then V δ, ∼ IG α − β (cid:113) σ a + σ b , µ b − µ a (cid:113) σ a + σ b , and U δ, ∼ NIG (cid:113) ( σ a + σ b )( µ a σ b + µ b σ a ) σ a σ b , µ a σ b + µ b σ a σ a σ b , ασ b + βσ a σ a + σ b , ( α − β ) σ a σ b σ a + σ b . (ii) For n ≥ , V δ,n and U δ,n follow the same IG and NIG distributions as in (i) with α = δ and β = − δ. Let ( U δ , V δ ) be a generic random variable with the same joint distribution as ( U δ,n , V δ,n ) , n ≥ U δ , V δ ), which will be used in the proof of Theorem4.7 and Section 5. 9 emma 4.4. There exists h > such that the moment generating function of ( U δ , V δ ) exists for | ( s, t ) | ≤ h , and is given by φ δ ( s, t ) = E [exp { sU δ + tV δ } ] = exp { [2 θ ( s, t ) − s ] δ } , (4.3) where θ ( s, t ) is defined as follows. θ ( s, t ) = ( µ b − µ a + sσ b ) − (cid:113) ( µ b − µ a + sσ b ) − ( σ a + σ b )( s σ b + 2 t + 2 sµ b ) σ a + σ b . (4.4) In particular, the first two moments of ( U δ , V δ ) are as given below: E ( V δ ) = 2 δµ b − µ a , E ( U δ ) = δ ( µ b + µ a ) µ b − µ a , Var ( V δ ) = 2( σ a + σ b ) δ ( µ b − µ a ) , Var ( U δ ) = 2( µ b σ a + µ a σ b ) δ ( µ b − µ a ) , Cov ( U δ , V δ ) = 2( µ b σ a + µ a σ b ) δ ( µ b − µ a ) . Furthermore, for k, l ∈ N ∪ { } and k + l ≥ , there exists some constant c such that E ( U kδ V lδ ) δ → c , as δ → . (4.5) { Z δ ( t ) , t ≥ } In this section we study the behaviors of the { Z δ ( t ) , t ≥ } process as either t → ∞ or δ → δ , { Z δ ( t ) , t ≥ } is a renewal reward process, andwe summarize it in the following lemma. Lemma 4.5.
For δ > , { Z δ ( t ) , t ≥ } is a renewal reward process and { P δ ( t ) , t ≥ } is a semi-Markov process. The next result from Brown and Solomon [9] characterizes the asymptotic first and secondmoments of Z δ ( t ) as t → ∞ , and is also helpful to identify the proper scaling in Theorem 4.7. Theorem 4.6 (Brown and Solomon [9]) . We have E ( Z δ ( t )) = mt + O (1) , (4.6) where m = 12 ( µ a + µ b ) , (4.7) and Var ( Z δ ( t )) = st + O (1) , (4.8) where s = 14 ( σ a + σ b ) . (4.9)10 ere O (1) is a function that converges to a finite constant as t → ∞ . The main result is given in the following theorem. For t ≥
0, defineˆ Z δ ( t ) = δZ δ ( t/δ ) − mt √ sδ , where m and s are as given in (4.7) and (4.9). Theorem 4.7.
Assume that E (ln [ A (0) /B (0)]) < ∞ . Then the process ˆ Z δ converges weakly to astandard Brownian Motion as δ → . Remark 4.8.
We note that Z δ ( t ) = (cid:114) sδ ˆ Z δ ( δt ) + mt, t ≥ . From Theorem 4.7, for small δ , we will use the following asymptotic model for logarithmic tradingprices Z ( t ) in Section 6: (cid:114) sδ B ( δt ) + mt, (4.10) where { B ( t ) , t ≥ } is a standard Brownian motion. We note that (4.10) is normal distributed withmean mt and variance st . The process { P ( t ) , t ≥ } is observable, while { ( A ( t ) , B ( t )) , t ≥ } may not be publicly ob-servable. The market ask and bid processes may be accessible to the brokers and dealers, but notto common traders. The question becomes how to find the parameters of { ( A ( t ) , B ( t )) , t ≥ } byobserving { P ( t ) , t ≥ } . In this section we will estimate the parameters µ a , µ b , σ a , σ b and δ usingthe method of moments.Suppose that the sample data for the i th trading time t i and the i th trading price p i are givenfor i = 1 , , · · · , n . Let u = ln P , v = t , and u i +1 = ln( p i +1 /p i ) , v i +1 = t i +1 − t i , i ≥ . Then the sample data is given by { ( u i , v i ) } ni =1 . Let x = n (cid:88) i =1 v i n , x = n (cid:88) i =1 u i n , x = n (cid:88) i =1 v i n , x = n (cid:88) i =1 u i n , x = n (cid:88) i =1 v i u i n . We aim to derive explicit estimators of the five parameters µ a , µ b , σ a , σ b , δ using moment estima-11ions. Define the estimators of µ a , µ b , σ a , σ b , δ as follows.ˆ µ na = y − (cid:113) y − y y − y ) y , ˆ µ nb = y + (cid:113) y − y y − y ) y , ˆ σ na = (cid:113) ( y − ˆ µ na y )(ˆ µ nb − ˆ µ na ) , ˆ σ nb = (cid:113) (ˆ µ nb y − y )(ˆ µ nb − ˆ µ na ) , ˆ δ n = (ˆ µ nb − ˆ µ na ) x , (5.1)where y = 2 x x , y = x − x x , y = x − x x , y = x − x x x . For convinence, denote Θ = ( µ a , µ b , σ a , σ b , δ ) and ˆΘ n = (ˆ µ na , ˆ µ nb , ˆ σ na , ˆ σ nb , ˆ δ n ) . Let g : R → R be thedifferentiable function such that ˆΘ n = g ( x , x , x , x , x ) . Note that g can be uniquely determined by (5.1) and has an explicit expression. Lemma 5.1.
The estimators ˆΘ n is well defined, i.e., y − y y − y ) y ≥ , ( y − ˆ µ na y )(ˆ µ nb − ˆ µ na ) ≥ , (ˆ µ nb y − y )(ˆ µ nb − ˆ µ na ) ≥ , (5.2) and as n → ∞ , ˆΘ n → Θ , almost surely. (5.3) Furthermore, √ n ( ˆΘ n − Θ) converges weakly to a five dimensional normal distribution with zeromean and covariance matrix ∇ g (Θ)Σ , where Σ is the covariance matrix of ( V δ , U δ , V δ , U δ , U δ V δ ) ,and ∇ g is the gradient of g . In this section we apply our model to the real data, with an aim to forecast the trading pricemovement over a short period. We develop an asymptotic GBM model for trading prices as follows.Given the sample data { ( u i , v i ) } ni =1 , we first estimate the parameters µ a , µ b , σ a , σ b and δ as in (5.1),and use the estimators ˆ µ na , ˆ µ nb , ˆ σ na and ˆ σ nb to compute m and s by substituting µ a , µ b , σ a , σ b withˆ µ na , ˆ µ nb , ˆ σ na , ˆ σ nb , respectively, in (4.7) and (4.9). Typically, the estimator ˆ δ n is small (see Figures 6- 9) and so from Theorem 4.7, we approximate Z ( t ) by a N ( st, mt ) random variable. Hence theprediction formula for ln P ( t ) − ln P (0) is (ˆ µ na + ˆ µ nb ) t , µ na + ˆ µ nb ) t (cid:112) [(ˆ σ na ) + (ˆ σ nb ) ] t , (ˆ µ na + ˆ µ nb ) t − (cid:112) [(ˆ σ na ) + (ˆ σ nb ) ] t . We next apply the above formulas to real data. Here we select the stock SUSQ (SusquehannaBancshares Inc). The data is chosen from 01/04/2010 9:30AM to 01/04/2010 4:00PM, including thetrading prices and trading times. The unit of trading prices is dollars and the unit of the differenceof consecutive trading times is seconds. We perform the back test to evaluate the performance ofthe prediction. To be precise, we predict the logarithmic trading price at each trading time usingthe 10-minute data 1-minute before the trading time. For example, observing that there is a tradeat 10:34:56, we then use the data from 10:23:56 to 10:33:56 to estimate the parameters and predictthe logarithmic trading price at 10:34:56, and the last trading price during the time interval from10:23:56 to 10:33:56 is regarded as P (0). At the same time we calculate the upper and lower boundsof the prediction at that trading time. We note that even though the drift and volatility parametersin the asymptotic model (4.10) is constant, the estimated parameters for predictions are actuallytime-varying. We compare this predicted logarithmic trading prices with the real trading prices inFigure 2. We do the similar prediction for each trading time but using the 10-minute data 2-, 5-,10-minute before the trading time respectively. The comparisons are shown in Figures 3-5. DefineRelative error (RE) = Real price - Predicted priceReal price . For the predictions 1-, 2-, 5-, 10-minute into the future, the maximum absolute REs are 0 . . . . δ is small. We present the values of ˆ δ n for all four predictions in Figures 6 - 9, and observe that allvalues are O (10 − ). Thus it is reasonable to use the asymptotic results in the regime δ → . Figure 2: Predictions of trading prices using 10-minute data 1-minute before each trading time.13igure 3: Predictions of trading prices using 10-minute data 2-minute before each trading time.Figure 4: Predictions of trading prices using 10-minute data 5-minute before each trading time.14igure 5: Predictions of trading prices using 10-minute data 10-minute before each trading time.Figure 6: Values of ˆ δ n when using 10-minute data 1-minute before each trading time.Figure 7: Values of ˆ δ n when using 10-minute data 2-minute before each trading time.15igure 8: Values of ˆ δ n when using 10-minute data 5-minute before each trading time.Figure 9: Values of ˆ δ n when using 10-minute data 10-minute before each trading time. Appendix
Proof of Lemma 2.1.
We note that A ( t ) B ( t ) = e Y a ( t ) − Y b ( t ) , and Y a ( t ) − Y b ( t ) is a RBM with mean µ a − µ b < σ a + σ b . Thus using Taylor series expansion and Fubini’s theorem, wehave E (cid:34)(cid:18) A ( t ) B ( t ) (cid:19) k (cid:35) = E (cid:104) e k ( Y a ( t ) − Y b ( t )) (cid:105) = 1 + E ∞ (cid:88) j =1 k j ( Y a ( t ) − Y b ( t )) j j ! = 1 + ∞ (cid:88) j =1 k j E [( Y a ( t ) − Y b ( t )) j ] j ! , (6.1)where from Theorem 1.3 in [1], E [( Y a ( t ) − Y b ( t )) j ] = E [( Y a ( ∞ ) − Y b ( ∞ )) j ] (cid:90) ∞ g j ( x ) F ( t ; x, dx. (6.2)Here Y a ( ∞ ) − Y b ( ∞ ) is the weak limit of Y a ( t ) − Y b ( t ) as t → ∞ , and g k ( x ) is a gamma densitywith mean k/ k/
4. We note that from (2.8), Y a ( ∞ ) − Y b ( ∞ ) follows an exponentialdistribution with mean µ ≡ ( σ a + σ b ) / [2( µ b − µ a )], and thus E [( Y a ( ∞ ) − Y b ( ∞ )) j ] = j ! µ j . (6.3)16urthermore, the gamma density function is given by g j ( x ) = 2 j x j − ( j − e − x , x ≥ . (6.4)Putting (6.3),(6.4) and (6.2) into (6.1), using Fubini’s theorem again, we have E (cid:34)(cid:18) A ( t ) B ( t ) (cid:19) k (cid:35) = 1 + ∞ (cid:88) j =1 (cid:90) ∞ k j µ j j x j − ( j − e − x F ( t ; x, dx = 1 + (cid:90) ∞ ∞ (cid:88) j =1 k j µ j j x j − ( j − e − x F ( t ; x, dx = 1 + 2 kµ (cid:90) ∞ e kµx − x F ( t ; x, dx. Proof of Proposition 3.1.
For (i), recall that for t ≥ A ( t ) = exp (cid:26) X a ( t ) + 12 L ( t ) (cid:27) ,B ( t ) = exp (cid:26) X b ( t ) − L ( t ) (cid:27) , where L ( t ) = sup ≤ s ≤ t ( X a ( s ) − X b ( s )) − . So it suffices to show that X a ( T δ,n ) − X b ( T δ,n ) = L ( T δ,n ) . Now recall that T δ, = 0 and T δ,n = inf { t ≥ X a ( t ) − X b ( t ) = − n − δ } , and so X a ( T δ,n ) − X b ( T δ,n ) = − n − δ , and X a ( t ) − X b ( t ) ≥ − n −
1) for t ∈ [0 , T δ,n ]. Thus L ( T n,δ ) = 2( n − δ, and so X a ( T δ,n ) − X b ( T δ,n ) = L ( T δ,n ) = 2( n − δ. To show (ii) and (iii), we first recall that A δ ( t ) = exp (cid:40) X a ( t ) + ∞ (cid:88) n =1 ( n − δ { t ∈ [ T δ, ( n − ,T δ,n ) } (cid:41) B δ ( t ) = exp (cid:40) X b ( t ) − ∞ (cid:88) n =1 ( n − δ { t ∈ [ T δ, ( n − ,T δ,n ) } (cid:41) . Now for t ∈ [ T δ,n − , T δ,n ) , n = 1 , , . . . , noting that X a ( T δ,n − ) − X b ( T δ,n − ) = − n − δ and that17 a ( t ) − X b ( t ) must be great than − n − δ , we have that L ( t ) = sup ≤ s ≤ t ( X a ( s ) − X b ( s )) − ∈ [2( n − δ, n − δ ) . (6.5)Thus for t ≥ A δ ( t ) ≥ A ( t ), andsup ≤ s ≤ t A δ ( s ) A ( s ) = sup ≤ s ≤ t ∞ (cid:88) n =1 s ∈ [ T δ,n − ,T δ,n ) exp (cid:26) ( n − δ − L ( s ) (cid:27) ∈ [ e δ , , (6.6)which yields sup ≤ s ≤ t A δ ( s ) A ( s ) → , as δ → . (6.7)Similarly, it can be shown that for t ≥ , B ( t ) ≥ B δ ( t ), andsup ≤ s ≤ t B ( s ) B δ ( s ) → , as δ → . (6.8) Proof of Lemma 4.1.
Let for t ≥ X ( t ) = (cid:18) X a ( t ) X b ( t ) (cid:19) = (cid:18) αβ (cid:19) + (cid:18) µ a tµ b t (cid:19) + (cid:18) σ a W a ( t ) σ b W b ( t ) (cid:19) . Then V δ, and U δ, are the first meeting time and point of X a and X b . Let θ = arctan( σ a σ − b ), anddefine M = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) σ − a σ − b (cid:19) , and for t ≥ , ˇ X ( t ) = M X ( t )= (cid:18) ασ − a cos θ + βσ − b sin θ − ασ − a sin θ + βσ − b cos θ (cid:19) + (cid:18) µ a σ − a cos θ + µ b σ − b sin θ − µ a σ − a sin θ + µ b σ − b cos θ (cid:19) t + (cid:18) W a ( t ) cos θ + W b ( t ) sin θ − W a ( t ) sin θ + W b ( t ) cos θ (cid:19) . Let ˇ a = ασ − a cos θ + βσ − b sin θ, ˇ b = − ασ − a sin θ + βσ − b cos θ, ˇ µ a = µ a σ − a cos θ + µ b σ − b sin θ, ˇ µ b = − µ a σ − a sin θ + µ b σ − b cos θ, ˇ B a ( t ) = W a ( t ) cos θ + W b ( t ) sin θ, ˇ B b ( t ) = − W a ( t ) sin θ + W b ( t ) cos θ.
18e note that V δ, = inf { t ≥ X a ( t ) = X b ( t ) } = inf (cid:8) t ≥ X ( t ) ∈ { ( x, y ) : y = 0 } (cid:9) = inf { t ≥ B b ( t ) + ˇ µ b t = − ˇ b } . Noting that ˇ B b is a standard Brownian motion, using Girsonov theorem, and from (5.12) in [20,Chapter 3.5.C], we have for t ≥ , Pr( V δ, ∈ dt ) = | ˇ b |√ πt exp (cid:26) − ( − ˇ b − ˇ µ b t ) t (cid:27) dt = α − β (cid:113) π ( σ a + σ b ) t exp (cid:26) − [ α − β − ( µ b − µ a ) t ] σ a + σ b ) t (cid:27) dt. Next noting that ˇ B a and V δ, are independent, we have for t ≥ x ∈ R ,Pr( V δ, ∈ dt, U δ, ∈ dx )= Pr( V δ, ∈ dt, X a ( V δ, ) = X b ( V δ, ) ∈ dx )= Pr (cid:32) V δ, ∈ dt, ˇ a + ˇ µ a V δ, + ˇ B a ( V δ, ) σ − a cos θ + σ − b sin θ ∈ dx (cid:33) = Pr (cid:32) ˇ a + ˇ µ a t + ˇ B a ( t ) σ − a cos θ + σ − b sin θ ∈ dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V δ, ∈ dt (cid:33) Pr( V δ, ∈ dt )= σ − a cos θ + σ − b sin θ √ πt exp (cid:40) − (( σ − a cos θ + σ − b sin θ ) x − ˇ a − ˇ µ a t ) t (cid:41) dx Pr( V δ, ∈ dt )= α − β πt σ a σ b exp − (cid:104) σ b σ a ( x − α − µ a t ) + σ a σ b ( x − β − µ b t ) (cid:105) + [ α − β − ( µ b − µ a ) t ] σ a + σ b ) t dxdt, where the last equality follows from the identities thatcos θ = σ b (cid:113) σ a + σ b , sin θ = σ a (cid:113) σ a + σ b . This proves (i). For (ii), we see that for t ∈ [0 , T δ,n +1 − T δ,n ) , n = 1 , , . . . ,˜ X a,n ( t ) ≡ ln A δ ( t + T δ,n ) − ln A δ ( T δ,n − ) = X a ( t + T δ,n ) − X a ( T δ,n ) + δ, ˜ X b,n ( t ) ≡ ln B δ ( t + T δ,n ) − ln B δ ( T δ,n − ) = X b ( t + T δ,n ) − X b ( T δ,n ) − δ. Thus from the strong Markov property of Brownian motions, { ˜ X a,n ( t ) , t ≥ } and { ˜ X b,n ( t ) , t ≥ } are independent Brownian motions with the initial values δ and − δ , and the same drifts and19ariances as X a and X b . Furthermore, they are independent of F T δ,n , where F t = σ { ( X a ( s ) , X b ( s )) , ≤ s ≤ t } . (6.9)Thus if we let ˜ T n and ˜ L n denote the first meeting time and meeting point of ˜ X a,n and ˜ X b,n , then { ( ˜ L n , ˜ T n ) , n = 1 , , . . . } is an i.i.d. sequence, which is independent of ( U δ, , V δ, ), and has the samedistribution as ( U δ, , V δ, ) with α = δ and β = − δ. Finally, noting that X a ( T δ,n ) − X b ( T δ,n ) = − n − δ , we have that the first meeting time of ˜ X a,n and ˜ X b,n is given by˜ T n = inf { t ≥ X a,n ( t ) = ˜ X b,n ( t ) } = inf { t ≥ X a ( t + T δ,n ) − X a ( T δ,n ) + δ = X b ( t + T δ,n ) − X b ( T δ,n ) − δ } = inf { t ≥ X a ( t + T δ,n ) − X b ( t + T δ,n ) = − nδ } = T δ,n +1 − T δ,n = V n , and the first meeting point of ˜ X a,n and ˜ X b,n is given by˜ L n = ˜ X a,n ( ˜ T n ) = ln A δ ( T δ,n +1 − ) − ln A δ ( T δ,n − ) = ln P n +1 − ln P n = U n . To summarize, we have shown that { ( U n , V n ) , n = 2 , , . . . } is an i.i.d. sequence, which is indepen-dent of ( V δ, , U δ, ), and has the same distribution as ( V δ, , U δ, ) with α = δ and β = − δ. Proof of Lemma 4.4.
Assume A (0) = e δ and B (0) = e − δ . Then ( U δ, , V δ, ) has the same distribu-tion as ( U δ , V δ ) . Let Y a ( t ) = exp (cid:26) θ X a ( t ) − ( θ µ a + 12 θ σ a ) t (cid:27) ,Y b ( t ) = exp (cid:26) θ X b ( t ) − ( θ µ b + 12 θ σ b ) t (cid:27) , where θ and θ are arbitrary real numbers. Then { Y a ( t ) , t ≥ } and { Y b ( t ) , t ≥ } are independent,and { ( Y a ( t ) , Y b ( t )) , t ≥ } is a {F t } t ≥ martingale (see the beginning of Section 5 of Chapter 7 in[21]), where F t is defined in (6.9). We also note that V δ, is an {F t } t ≥ stopping time with finitemean and variance ( V δ, follows IG distribution from Lemma 4.1). Hence optional stopping theoremyields ( E ( Y a ( V δ, )) , E ( Y b ( V δ, ))) = ( E ( Y a (0)) , E ( Y b (0))) , and so E [ Y a ( V δ, ) Y b ( V δ, )] = E [ Y a (0) Y b (0)]. More precisely, we have E (cid:26) exp { ( θ + θ ) U δ, − ( θ µ a + 12 θ σ a + θ µ b + 12 θ σ b ) V δ, } (cid:27) = exp { [ θ − θ ] δ } . Let θ + θ = s,θ µ a + 12 θ σ a + θ µ b + 12 θ σ b = − t. θ and θ in terms of s and t , we obtain θ ( s, t ) = ( µ b − µ a + sσ b ) ± (cid:113) ( µ b − µ a + sσ b ) − ( σ a + σ b )( s σ b + 2 t + 2 sµ b ) σ a + σ b ,θ ( s, t ) = s − θ ( s, t ) . Letting s = 0, and noting that V δ, follows inverse Gaussian distribution (see (4.2)), the momentgenerating function of V δ, is E (exp( tV )) = ( µ b − µ a ) − (cid:113) ( µ b − µ a ) − t ( σ a + σ b ) σ a + σ b . Thus the solutions of θ ( s, t ) should be as in (4.4), and the moment generating function φ ( s, t ) of( U δ , V δ ) is given by (4.3) with θ ( s, t ) instead of θ ( s, t ) as in (4.4). To compute the moments, wefirst need some simple results about θ ( s, t ) as follows. θ (0 ,
0) = 0 ,∂θ ( s, t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) s = t =0 = 1 µ b − µ a , ∂θ ( s, t ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s = t =0 = µ b µ b − µ a ,∂ θ ( s, t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) s = t =0 = σ a + σ b ( µ b − µ a ) , ∂ θ ( s, t ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s = t =0 = µ b σ a + µ a σ b ( µ b − µ a ) ,∂ θ ( s, t ) ∂s∂t (cid:12)(cid:12)(cid:12)(cid:12) s = t =0 = µ b σ a + µ a σ b ( µ b − µ a ) , ∂ θ ( s, t ) ∂t ∂s (cid:12)(cid:12)(cid:12)(cid:12) s = t =0 = 3( σ a µ b + σ b µ a )( σ a + σ b )( µ b − µ a ) . Therefore, E ( V δ ) = ∂φ ( s, t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) s = t =0 = ∂∂t exp { [2 θ ( s, t ) − s ] δ }| s =0 ,t =0 = 2 δµ b − µ a . Similarly, we obtain E ( U δ ) = δ ( µ b + µ a ) µ b − µ a E ( V δ ) = 4 δ ( µ b − µ a ) + 2( σ a + σ b ) δ ( µ b − µ a ) E ( U δ ) = δ ( µ a + µ b ) ( µ b − µ a ) + 2( µ b σ a + µ a σ b ) δ ( µ b − µ a ) E ( U δ V δ ) = 2 δ ( µ b + µ a )( µ b − µ a ) + 2( µ b σ a + µ a σ b ) δ ( µ b − µ a ) . k, l ∈ N ∪ { } and k + l ≥
1, (4.5) follows by noting that E ( U kδ V lδ ) = ∂ k + l φ ( s, t ) ∂s k ∂t l (cid:12)(cid:12)(cid:12)(cid:12) s = t =0 = δ (cid:18) φ ( s, t ) ∂ k + l ∂s k ∂t l (2 θ ( s, t ) − s ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) s = t =0 + o ( δ ) , where o ( δ ) → δ → . Proof of Theorem 4.6.
From Brown and Solomon [9], we have the following results for a renewalreward process generated by { ( U δ,n , V δ,n ) , n ≥ } : E ( Z δ ( t )) = mt + O (1) , where m = E ( U δ ) E ( V δ ) . Using the results of Lemma 4.4 in the above equation, we get Equation (4.7). The same paper alsostates that Var( Z δ ( t )) = st + O (1) , where s = E ( V δ ) E ( U δ ) E ( V δ ) − E ( U δ V δ ) E ( U δ ) E ( V δ ) + E ( U δ ) E ( V δ ) , Substituting the moments of ( U δ , V δ ) from Lemma 4.4 into the above equation and simplifying, weget Equation (4.9). Proof of Theorem 4.7.
Consider an arbitrary nonnegative sequence { δ m } m ≥ such that δ m → m → ∞ . Define for m, n ≥
1, ˜ U m,n = (cid:112) δ m ( U δ m ,n − E ( U δ m ,n )) , ˜ V m,n = (cid:112) δ m ( V δ m ,n − E ( V δ m ,n )) . We note that for each m , { ( ˜ U m,n , ˜ V m,n ) , n ≥ } is an i.i.d. sequence. Furthermore, (cid:98) tδ m (cid:99) (cid:88) n =1 Var( ˜ U m,n ) → µ b σ a + µ a σ b ) t ( µ b − µ a ) , and (cid:98) tδ m (cid:99) (cid:88) n =1 Var( ˜ V m,n ) → σ a + σ b ) t ( µ b − µ a ) , as m → ∞ . We claim that { ( ˜ U m,n , ˜ V m,n ) , m ≥ , ≤ n ≤ (cid:98) tδ m (cid:99)} satisfies Lindeberg condition, i.e., for any (cid:15) > (cid:98) tδ m (cid:99) (cid:88) n =1 E (cid:16) ˜ U m,n {| ˜ U m,n |≥ (cid:15) } (cid:17) → , and (cid:98) tδ m (cid:99) (cid:88) n =1 E (cid:16) ˜ V m,n {| ˜ V m,n |≥ (cid:15) } (cid:17) → , as m → ∞ . (6.10)22e will prove (6.10) at the end of this proof. Thus from [4, Theorem 18.2], letting u m ( t ) = (cid:98) tδ m (cid:99) (cid:88) n =1 ˜ U m,n , and v m ( t ) = (cid:98) tδ m (cid:99) (cid:88) n =1 ˜ V m,n , then ( u m , v m ) ⇒ W, as m → ∞ . where W is a two dimensional Brownian motion with drift 0 and covariance matrix2( µ b − µ a ) (cid:18) µ b σ a + µ a σ b µ b σ a + µ a σ b µ b σ a + µ a σ b σ a + σ b (cid:19) . Next from [18, Theorem 1] and [19, Corollary 3.33], if˜ N m ( t ) = ( E ( V δ m , )) / (cid:18) N δ m ( t/δ m ) − tδ m E ( V δ m , ) (cid:19) = (cid:18) δ m µ b − µ a (cid:19) / (cid:18) N δ m ( t/δ m ) − ( µ b − µ a ) t δ m (cid:19) , then ( u m , v m , ˜ N m ) ⇒ ( W , W , − W ) as m → ∞ , where W and W are the first and secondcomponents of the Brownian motion W . Finally, we note thatˆ Z δ m ( t ) = 1 √ s (cid:34) u m (cid:0) δ m N δ m ( t/δ m ) (cid:1) + µ b + µ a µ b − µ a (cid:18) µ b − µ a (cid:19) / ˜ N m ( t ) (cid:35) . Furthermore, observing that δ m N δ m ( t/δ m ) = δ m (cid:34) ˜ N m ( t )( E ( V δ m , )) / + ( µ b − µ a ) t δ m (cid:35) → ( µ b − µ a ) t , as m → ∞ , we have that ˆ Z δ m ( · ) ⇒ W ( µ b − µ a · ) + µ b + µ a µ b − µ a (cid:0) µ b − µ a (cid:1) / W ( · ) √ s , and it is easy to check that the weak limit on the right hand side is a standard Brownian motion.Consequently, ˆ Z δ converges weakly to a standard Brownian motion as δ →
0. At last, we give theproof of the claim given in (6.10). The proofs for ˜ V m,n and ˜ U m,n are similar, and we only consider˜ V m,n . We first note that from Lemma 4.4, E ( V δ m , | A (0) , B (0)) = ln A (0) − ln B (0) µ b − µ a , Var( V δ m , | A (0) , B (0)) = (ln A (0) − ln B (0))( σ a + σ b )( µ b − µ a ) , b ∈ (0 , ∞ ),Var( V δ m , ) = E (Var( V δ m , | A (0) , B (0))) + Var( E ( V δ m , | A (0) , B (0))) ≤ b (cid:0) E [ln( A (0) /B (0))] + E [ln ( A (0) /B (0))] (cid:1) < ∞ . Next using Markov inequality, Holder’s inequality and (4.5), we have for some c ∈ (0 , ∞ ) , (cid:98) tδ m (cid:99) (cid:88) n =1 E (cid:16) ˜ V m,n {| ˜ V m,n |≥ (cid:15) } (cid:17) ≤ E ( ˜ V m, ) + (cid:100) tδ m (cid:101) (cid:113) E ( ˜ V m, ) P ( | ˜ V m, | ≥ (cid:15) ) ≤ δ m Var( V δ m , ) + (cid:100) tδ m (cid:101) (cid:113) E ( ˜ V m, ) (cid:15) − E ( ˜ V m, ) ≤ δ m Var( V δ m , ) + (cid:15) − (cid:100) tδ m (cid:101) δ / m (cid:113) E [( V δ m , − E ( V δ m , )) ]Var( V δ m , ) ≤ δ m Var( V δ m , ) + (cid:15) − (cid:100) tδ m (cid:101) δ / m (cid:112) c δ m → , as m → ∞ . Proof of Lemma 5.1.
For convenience, we omit the superscript n for the estimators of µ a , µ b , σ a , σ b and δ . Using the moments in Lemma 4.4, we consider the following equations. x = 2ˆ δ ˆ µ b − ˆ µ a (6.11) x = ˆ δ (ˆ µ b + ˆ µ a )ˆ µ b − ˆ µ a (6.12) x = 4ˆ δ (ˆ µ b − ˆ µ a ) + 2(ˆ σ a + ˆ σ b )ˆ δ (ˆ µ b − ˆ µ a ) (6.13) x = ˆ δ (ˆ µ a + ˆ µ b ) (ˆ µ b − ˆ µ a ) + 2(ˆ µ b ˆ σ a + ˆ µ a ˆ σ b )ˆ δ (ˆ µ b − ˆ µ a ) (6.14) x = 2ˆ δ (ˆ µ b + ˆ µ a )2(ˆ µ b − ˆ µ a ) + 2(ˆ µ b ˆ σ a + ˆ µ a ˆ σ b )ˆ δ (ˆ µ b − ˆ µ a ) . (6.15)Next we solve the above equations for ˆ µ a , ˆ µ b , ˆ σ a , ˆ σ b , ˆ δ in terms of x k , k = 1 , , . . . ,
5. Let y = 2 x x = ˆ µ b + ˆ µ a y = x − x x = ˆ σ a + ˆ σ b (ˆ µ b − ˆ µ a ) = x − x x = ˆ µ b ˆ σ a + ˆ µ a ˆ σ b (ˆ µ b − ˆ µ a ) y = x − x x x = ˆ µ b ˆ σ a + ˆ µ a ˆ σ b (ˆ µ b − ˆ µ a ) . We then note that y − y y − y y = (ˆ µ b − ˆ µ a ) . Letting ˆ µ b > ˆ µ a , we obtain ˆ µ a = y − (cid:113) y − y y − y y µ b = y + (cid:113) y − y y − y y σ a = (cid:112) ( y − ˆ µ a y )(ˆ µ b − ˆ µ a ) , ˆ σ b = (cid:112) (ˆ µ b y − y )(ˆ µ b − ˆ µ a ) , ˆ δ = (ˆ µ b − ˆ µ a ) x . To see the above estimators are well-defined, we only need to show (5.2). We first note that y − y y − y ) y = 4 x ( x − x ) (cid:2) x ( x − x ) − x x ( x − x x ) + x ( x − x ) (cid:3) . It is clear that x = (cid:32) n (cid:88) i =1 v i n (cid:33) > ,x − x = n n (cid:80) i =1 v i − n (cid:80) i =1 v i n > . We next note that x ( x − x ) − x x ( x − x x ) + x ( x − x ) ≥ x x (cid:113) ( x − x )( x − x ) − x x ( x − x x )= 2 x x ( (cid:113) ( x − x )( x − x ) − ( x − x x ))= 2 (cid:80) v i n (cid:80) u i n (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) (cid:80) v i n − (cid:18) (cid:80) v i n (cid:19) (cid:33) (cid:32) (cid:80) u i n − (cid:18) (cid:80) u i n (cid:19) (cid:33) − (cid:18) (cid:80) v i ∆ p i n − (cid:80) v i n (cid:80) u i n (cid:19)
25 2 (cid:80) v i n (cid:80) u i n (cid:115) (cid:80) ( v i − (cid:80) v i /n ) n (cid:80) ( u i − (cid:80) u i /n ) n − (cid:18) (cid:80) v i u i n − (cid:80) v i n (cid:80) u i n (cid:19) ≥ (cid:80) v i n (cid:80) u i n (cid:18) (cid:80) ( v i − (cid:80) v i /n ) ( u i − (cid:80) u i /n ) n − (cid:18) (cid:80) v i u i n − (cid:80) v i n (cid:80) u i n (cid:19)(cid:19) = 0 . This shows the first inequality in (5.2). To show the last two inequalities in (5.2), we observe that y − ˆ µ a y = y (cid:113) y − y y − y ) y (cid:16) y − y y (cid:17) , ˆ µ b y − y = y (cid:113) y − y y − y ) y − (cid:16) y − y y (cid:17) . Hence it suffices to show y (cid:16) y − y y − y ) y (cid:17) ≥ (cid:16) y − y y (cid:17) . After simplifying above inequality, it suffices to show that y y ≥ y . Note that y y ≥ y is equivalent to ( x − x )( x − x ) ≥ ( x − x x ) , and the latter one is proved above. This completes the proof of (5.2). Next from the constructionof the estimators, we see that they are the unique solutions of (6.11) – (6.15). Using the strong lawof large numbers and the continuous mapping theorem, we have (5.3). Finally, the central limittheorem for ˆΘ follows immediatly from Delta method (see [11]) and the central limit theorem for( x , x , . . . , x ), i.e., √ n [( x , x , x , x , x ) − E ( x , x , x , x , x )] ⇒ N (0 , Σ) , where Σ is the covariance matrix of ( V δ , U δ , V δ , U δ , U δ V δ ) . References [1] J. Abate and W. Whitt. Transient behavior of regulated brownian motion, i: Starting at theorigin.
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