Limit order books, diffusion approximations and reflected SPDEs: from microscopic to macroscopic models
LLimit order books, diffusion approximations and reflected SPDEs:from microscopic to macroscopic models
Ben Hambly ∗ , Jasdeep Kalsi † and James Newbury ‡ Mathematical Institute, University of OxfordJune 27, 2019
Abstract
Motivated by a zero-intelligence approach, the aim of this paper is to connect the microscopic(discrete price and volume), mesoscopic (discrete price and continuous volume) and macroscopic(continuous price and volume) frameworks for the modelling of limit order books, with a view toproviding a natural probabilistic description of their behaviour in a high to ultra high-frequencysetting. Starting with a microscopic framework, we first examine the limiting behaviour ofthe order book process when order arrival and cancellation rates are sent to infinity and whenvolumes are considered to be of infinitesimal size. We then consider the transition betweenthis mesoscopic model and a macroscopic model for the limit order book, obtained by lettingthe tick size tend to zero. The macroscopic limit can then be described using reflected SPDEswhich typically arise in stochastic interface models. We then use financial data to discuss apossible calibration procedure for the model and illustrate numerically how it can reproduceobserved behaviour of prices. This could then be used as a market simulator for short-termprice prediction or for testing optimal execution strategies.
The rising prevalence of order-driven markets in recent years has generated a significant interestin the modelling of limit order books, for an overview see the survey paper [14]. In such markets,three specific types of orders can be submitted. Firstly, limit orders are orders to buy or sell adesignated number of shares at a specified price or better. Secondly, market orders are orders toimmediately buy or sell a certain number of shares at the best available price. Finally, cancellationorders enable a market participant to cancel an existing limit order. Whilst market orders areinstantly matched against the best available limit orders of the opposite quote, the collection ofunexecuted and uncancelled limit orders is recorded in the limit order book (LOB), according toprice and time priority. In limit order book terminology, the bid refers to the price of the bestlimit buy order, whereas the ask designates the price of the best sell order. The average of thebid and the ask is referred to as the mid. Two other quantities of interest are the spread, whichcorresponds to the difference between the ask and the bid, and the tick size, which is the smallestprice increment in the market. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ q -f i n . M F ] J un here is now a large amount of high-quality financial time series of orders enabling one toconduct statistical analyses of various limit order book features and to guide the development ofmodels. A good model for the order book should describe the evolution of prices and capture someof the stylised facts that are observed in financial time series. It can then be used for examiningstrategies for order placement and for the optimal execution of large orders.Limit order book models have been developed by two independent schools of thought. The firstone, initiated by economists, has been based on a ‘perfect-rationality’ approach, where market par-ticipants all employ optimal strategies to place limit orders. The second one, led by econophysicistsand mathematicians, has been associated with a ‘zero-intelligence’ framework, that is limit orderarrivals can be viewed as a purely random process and no strategic order placement is taken intoaccount.Within the realm of perfect-rationality, where order flow is considered as static, the centralissue concerns the strategic trading decisions of agents in which they maximise their individualutility. Notable models in the perfect-rationality literature include those due to Mendelson [22], whoanalysed the statistical behaviour of the market from a clearing house perspective, Kyle [20], wherethe question of insider trading with sequential auctions is addressed, Ro¸su [25], who introducedthe notion of optimal choice between market orders and limit orders, and Almgren and Chriss [2],where a model for the optimal execution of large orders is developed.In the zero-intelligence approach the order flow is treated as dynamic and the focus is shiftedto the random nature of order arrivals. One of the first models dealing with this was developedby Kruk [19], where he established a functional limit theorem for the order flow in a continuousdouble-auction setting. More recently, there has been a significant interest in modelling the bookas a multiclass queueing system, [1], [5], [9], [24], [16]. In order to deal with a market where ordersare submitted at high frequency, Cont and Larrard [8] considered a heavy traffic approximation ofthe order book process from a queueing theory perspective. The order book is reduced to the bestbid and ask queues: once the bid (or ask) queue has been depleted, it takes a new value drawnfrom a stationary distribution representing the depth of the order book after a price change.Over the years, there have also been a number of attempts to establish scaling limits for the fullorder book. One of the first papers to explore this direction is [6], where the order book is modelledas a two-species interacting particle system and a hydrodynamic limit is obtained for the associatedempirical process. This particle system approach is also used by [10], where their limit is actuallyan ODE with a constant price. The work of [21] obtains a limit for one side of the order book as ameasure-valued process. In [4] and [15] Horst et al. establish functional limit theorems for two-sidedorder books and obtain PDE or SPDE limits depending on the initial scaling procedure. Finally,Zheng [29] (using the results of Kim et al. [18] on stochastic Stefan problems) and M¨uller [23] and[17] developed models for the order book as a stochastic free boundary problem. Motivated by a zero-intelligence approach, the aim of this paper is to bridge the gap between microscopic (discrete volume and price), mesoscopic (continuous volume and discrete price) and macroscopic (continuous volume and price) models of limit order books. The financial context ofour study is the following: we consider an order-driven market where orders and cancellations aresubmitted at very high frequency. Starting with a discrete-space model describing the microscopicevolution of the order book, we prove that by sending order arrival and cancellation rates to infinityand by rescaling order volumes, the behaviour of the book can be described in terms of a systemof coupled stochastic differential equations. This is what we call the mesoscopic limiting process.Next, by sending the tick size to zero, we derive a macroscopic SPDE limit from the mesoscopic2rocess.Even though we send the tick size to zero we wish to capture the fact that in a high frequencytrading environment the price changes are comparatively rare in the evolution of the order book.Thus we will consider our model for the book as generating price changes at a much lower fre-quency, so there is a natural separation of time scales. Our price changes will be a macroscopictick movement occurring as a result of imbalances created in the book by the order flow. Ourmain mathematical result is Theorem 3.6 showing that the queueing system converges weakly to areflected SPDE for the dynamics of the order book along with a discrete price process evolving ina realistic way.An outline of the paper is as follows. In the next section we will develop our microscopic model.We allow quite a degree of flexibility in the arrival rates and cancellation rates of orders and show inTheorem 2.3, that by letting the volume size of orders go to 0 as their rate of arrival goes to infinitythat the microscopic system has a scaling limit which is a coupled system of stochastic differentialequations. The initial result is for the static order book and we then show how to incorporate pricechanges which are functions of the two sides of the order book. In Section 3 we let the tick sizego to zero and show that this system converges weakly to a reflected SPDE. In our main result(Theorem 3.6) we also incorporate price changes to get a full model for the order book and thediscrete price dynamics it generates. In Section 4 we illustrate with some examples how our generalframework can incorporate some natural models. In Section 5 we develop the numerical applicationfor our framework. By considering data from LOBSTER we show how to determine some of theparameters in a simple version of the model and how it will produce realistic order book profilesand prices series. The proofs of the Theorems are then given in the Appendix.
We begin this section by introducing a simple microscopic order book model, where order arrivalsand cancellations are driven by Poisson processes. One could also use more complicated pointprocesses as the basis for the model, for instance Hawkes processes. We choose Poisson processesboth to simplify the analysis, and since such dynamics can be considered an approximation to alarge class of point processes which are natural in this context. Other papers which assume Poissondriven order book dynamics include [1], [4], [9], [10], [21] and [24]. The rates for order arrivals,market orders and cancellations will be allowed to depend on the current mid price, the pricerelative to the mid and the number of orders currently on the book at the point in question.We will first consider a model for the evolution of the book in between price changes, and willthen add price changes by introducing stopping times which depend on how the book has evolved.This will allow us to easily maintain a separation of time-scales between the evolution of the orderbook profiles and the corresponding price process when passing to mesoscopic and macroscopiclimits, where we will scale time and space for our models. Maintaining this separation is sensiblesince, typically, price movements occur on a significantly slower time-scale to order book events.For example, examining order book data for the SPDR Trust Series I from June 21 2012 between11:00am and 12:00pm, we see that there were 840549 order book events and approximately 2453price changes.Once our microscopic model has been described, our goal is then to establish a diffusion ap-proximation for the model. This is done initially for the static model, where price changes are notincluded, and then for the dynamic case. 3 .1 The Discrete Order Book Process in a Static Setting
In this section we describe the dynamics of our microscopic order book model when it is statici.e. we describe its behaviour in between price changes. The index n here will be used later inorder to take a diffusive scaling of the model.We will work on a relative price grid in that, at any given time, the i th price point of thebid/ask side of the book refers to the price which is i ticks away from the best bid/ask respectively.The grid is given by { , , , ..., N } for some N ∈ N . For every n ∈ N , we consider two N − Z bn = ( Z b, n , ...Z b,N − n ) and Z an = ( Z a, n , ...Z a,N − n ), each taking values in Z N − and representing the limit order volumes currently on the bid and ask sides respectively of the staticdiscrete order book process. The mid price m ∈ R is taken to be fixed here, and by conventionwe think of the spread as being constantly equal to two ticks. For each i ∈ { , , ..., N − } , Z b,in then represents the number of outstanding orders to buy at price m − i , whilst Z a,in represents thenumber of outstanding limit orders to sell at price m + i . Order and cancellation sizes are assumedto be 1 in the static setting, although the results here can easily be adapted to the case where ordersizes are assumed only to be bounded. We choose the rates at which different orders arrive in ourmodel such that they possess the following three features.1. At each price level, there is a common high frequency rate for limit orders and cancella-tion/market orders. We allow for these rates to be dependent on the relative price (withrespect to the mid), the current position of the mid and the number of offers currently atthat price. These terms are intended to capture the effects of high frequency trading.2. Residual imbalance between limit orders and cancellations/market orders at different pricelevels gives lower frequency terms. These are once again allowed to be dependent on price,the current midprice and the number of offers at that price.3. Orders undergo a lower frequency random walk. This is intended to capture the effects oftraders repositioning their offers in the book, as many of the cancellations that occur will bequickly followed by a limit order at an adjacent queue. We assume that each order moves toa neighbouring queue at a certain rate. This will have a smoothing effect on the profile of theorder book.Altogether, this motivates the following description for the dynamics of the bid side of the orderbook in our model. To simplify the notation here, we define e i to be the usual basis functions for R N − for i = 1 , , ..., N −
1, and we use the convention e = e N = 0.(i) For i ∈ { , ..., N − } , Z bn → Z bn + e i at exponential rate12 σ b,m,n ( i, Z b,in ) (cid:32) (cid:110) Z b,in =0 (cid:111) (cid:33) + f b,m,n (cid:16) i, Z b,in (cid:17) . (ii) For i ∈ { , ..., N − } , Z bn → Z bn − e i at exponential rate12 σ b,m,n ( i, Z b,in ) (cid:110) Z b,in ≥ (cid:111) + g b,m,n (cid:16) i, Z b,in (cid:17) (cid:110) Z b,in ≥ (cid:111) . (iii) For i ∈ { , ..., N − } , Z bn → Z bn + e i − − e i at exponential rate α b,n Z b,in i ∈ { , ..., N − } , Z bn → Z bn + e i +1 − e i at exponential rate α b,n Z b,in Similarly, the dynamics of the ask side of the book are given by:(i) For i ∈ { , ..., N − } , Z an → Z an + e i at exponential rate12 σ a,m,n ( i, Z a,in ) (cid:32) (cid:110) Z a,in =0 (cid:111) (cid:33) + f a,m,n (cid:16) i, Z a,in (cid:17) . (ii) For i ∈ { , ..., N − } , Z an → Z an − e i at exponential rate12 σ a,m,n ( i, Z a,in ) (cid:110) Z a,in ≥ (cid:111) + g a,m,n (cid:16) i, Z a,in (cid:17) (cid:110) Z a,in ≥ (cid:111) . (iii) For i ∈ { , ..., N − } , Z an → Z an + e i − − e i at exponential rate α a,n Z a,in (iv) For i ∈ { , ..., N − } , Z an → Z an + e i +1 − e i at exponential rate α a,n Z a,in In the above we have that:(a) For k ∈ { b, a } , every n and every m ∈ R , σ k,m,n is a map from { , , ..., N − } × N → R + .(b) For k ∈ { b, a } , every n and every m ∈ R , f k,m,n and g k,m,n are maps from { , , ..., N − }× N → R + . Remark 2.1.
We remark here that market orders have the same impact on the profile of the bookas cancellations at the best price levels. Market orders are therefore accounted for in these staticdynamics.
Remark 2.2.
Our model here only accounts for placement of small orders, which we have takenwithout loss of generality to be of size one. In Section 4 we give an example of how to includelarger orders on a longer timescale in our dynamic model.
We now switch our attention to the heavy traffic approximation of the suitably rescaled staticmicroscopic order book process. Time is accelerated by a factor of n and volumes are divided by √ n . Therefore, we are considering the limits of the processes˜ Z bn ( t ) := Z bn ( nt ) √ n and ˜ Z an ( t ) := Z an ( nt ) √ n . These processes therefore take values in √ n N N − and the limiting process will take values in[0 , ∞ ) N − . In order to obtain convergence, we need that various quantities in our microscopicmodel converge suitably. We will assume that 5i) For k ∈ { b, a } , m ∈ R , i ∈ { , , ..., N − } , u ∈ N and n ≥ σ k,m,n ( i, u ) = σ k,m (cid:18) i, u √ n (cid:19) . (ii) For k ∈ { b, a } , m ∈ R , i ∈ { , , ..., N − } , u ∈ N and n ≥ f k,m,n ( i, u ) = 1 √ n f k,m (cid:18) i, u √ n (cid:19) ,g k,m,n ( i, u ) = 1 √ n g k,m (cid:18) i, u √ n (cid:19) . (iii) For k ∈ { b, a } and n ≥ α k,n = n α k > k ∈ { b, a } , Z kn (0) √ n = ⇒ X k (0) in law in [0 , ∞ ) N − as n → ∞ .Here, the functions σ k,m , f k,m and g k,m are all measurable from { , , ..., N − } × R + → R + . Fortechnical reasons, we further assume that these functions are Lipschitz continuous in the secondargument. Note that this implies boundedness on compact sets. Theorem 2.3.
The √ n N × √ n N - valued process ( ˜ Z bn , ˜ Z an ) converges weakly in M (cid:16) D ([0 , ∞ ); R N − ) × D ([0 , ∞ ); R N − ) (cid:17) as n → ∞ to the unique [0 , ∞ ) N − × [0 , ∞ ) N − -valuedstrong Markov diffusion process ( X b , X a ) which satisfies the following system of reflected SDEs:d X b,it = α b ( X b,i +1 t + X b,i − t − X b,it ) dt + h b,m ( i, X b,it ) dt + σ b,m ( i, X b,it ) d W b,it + d η b,it , d X a,it = α a ( X a,i +1 t + X a,i − t − X a,it ) dt + h a,m ( i, X a,it ) dt + σ a,m ( i, X a,it ) d W a,it + d η a,it , for i = 1 , ..., N − with the pinning conditions that X k, = X k,N = 0 , where W k,i are independentBrownian motions. The η k,i are reflection measures which maintain positivity of the X k,i . We now describe the mechanism for price movements in the model to give our microscopicdynamic model. Price changes in both directions will be assumed to occur at positive rates whichdepend on the state of the book at any given time (including the current position of the mid). Ourmotivating example is the case where these rates are dependent on the imbalance of the number ofbid limit orders compared to ask limit orders currently on the book near the mid. Here, relativelymore offers to buy near the mid make a price increase more likely and relatively more offers to sellnear the mid make a price decrease more likely.In order to formalise this, we introduce functions θ nu,m and θ nd,m for n ≥ m ∈ R , which map N N − × N N − → R > . These will determine the rate of upward and downward price movementsrespectively as a function of the profiles of the bid and ask sides of the book. We also define for n ≥ R n . These map N N − × N N − × { u, d } → M ( N N − × N N − ) and will determinethe distribution of the new profiles of the bid and ask sides of the book following price changes as afunction of the profiles at the time of the price change and the direction of the price change. We fixsome (cid:15) >
0, which determines the size of price changes. We additionally introduce i.i.d. rate oneexponential random variables ( Y in,u ) ∞ i =1 and ( Y in,d ) ∞ i =1 which will be used in the construction. Theseare independent of each other and of the other driving Poisson processes in the model. With this6s place, we can start to construct our dynamic process. Let Z bn, (0) , Z an, (0) ∈ N N − be the initialprofiles for the bid and ask sides of the book respectively, and let m n be the initial mid for the n th microscopic order book. We denote by Z bn, , Z an, the processes evolving according to our dynamicsfor the static microscopic order book with mid m n and initial profiles Z bn, (0) , Z an, (0). We definethe following stopping times. τ n,u := inf (cid:40) t ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t θ nu,m n ( Z bn, ( s ) , Z an, ( s )) d s ≥ Y n,u (cid:41) ,τ n,d := inf (cid:40) t ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t θ nd,m n ( Z bn, ( s ) , Z an, ( s )) d s ≥ Y n,d (cid:41) . Therefore, τ n,u and τ n,d are exponential waiting times, whose arrival rates at time t are given by θ nu,m n ( Z bn, ( s ) , Z an, ( s )) and θ nd,m n ( Z bn, ( s ) , Z an, ( s )) respectively. We define τ n := τ n,u ∧ τ n,d . Thetime τ n triggers a price change. If τ n = τ n,u , we have an upward price change, and set m n = m n + (cid:15) . Similarly, if τ n = τ n,d , we have an downward price change, and set m n = m n − (cid:15) . We thenlet Z bn, , Z an, be new processes which follow the dynamics of the bid and ask sides of the staticmicroscopic order book with mid m n . The initial profile has the law of R n ( Z bn, ( τ n ) , Z an, ( τ n ) , u ) ifthe price change was upward and R n ( Z bn, ( τ n ) , Z an, ( τ n ) , d ) if the price change was downward. Theprocesses Z bn, , Z an, are taken to be conditionally independent of the past of the dynamic order bookgiven ( Z bn, ( τ n ) , Z an, ( τ n )), the direction of the price change, and m n . We can iterate this procedureto define further stopping times, price points and processes Z bn,i , Z an,i , describing the dynamics afterthe i th price change. Having defined ( Z bn,i ) Mi =1 , ( Z an,i ) Mi =1 , ( τ in ) M − i =1 and ( m in ) Mi =1 , we set τ Mn,u := inf (cid:40) t ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t θ nu,m Mn ( Z bn,M ( s ) , Z an,M ( s )) d s ≥ Y Mn,u (cid:41) ,τ Mn,d := inf (cid:40) t ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t θ nd,m Mn ( Z bn,M ( s ) , Z an,M ( s )) d s ≥ Y Mn,d (cid:41) . As before, we define τ Mn := τ Mn,u ∧ τ Mn,d , with the time τ Mn triggering a price change. If τ Mn = τ Mn,u , weset m M +1 n = m Mn + (cid:15) and if τ Mn = τ Mn,d , we set m M +1 n = m Mn − (cid:15) . We then let Z bn,M +1 , Z an,M +1 be newprocesses which follows the dynamics of the bid and ask sides of the static microscopic order bookwith the mid now given by m M +1 n . The initial profile has the law of R n ( Z bn,M ( τ Mn ) , Z an,M ( τ Mn ) , u )if the price change was upward and R n ( Z bn,M ( τ Mn ) , Z an,M ( τ Mn ) , d ) if the price change was down-ward. Once again, Z bn,M +1 , Z an,M +1 are taken to be conditionally independent of the past of thedynamic order book given ( Z bn,M ( τ Mn ) , Z an,M ( τ Mn )), the direction of the previous price change, andthe new mid position. Our dynamic microscopic order book is then described by the processes( ˆ Z bn ( t ) , ˆ Z an ( t ) , m n ( t )) describing the evolution of the two sides of the book and the mid throughtime, where ˆ Z bn ( t ) := ∞ (cid:88) i =1 Z bn,i t − i − (cid:88) j =1 τ jn (cid:40) i − (cid:80) j =1 τ jn ≤ t< i (cid:80) j =1 τ jn (cid:41) , ˆ Z an ( t ) := ∞ (cid:88) i =1 Z an,i t − i − (cid:88) j =1 τ jn (cid:40) i − (cid:80) j =1 τ jn ≤ t< i (cid:80) j =1 τ jn (cid:41) , n ( t ) := ∞ (cid:88) i =1 m in (cid:40) i − (cid:80) j =1 τ jn ≤ t< i (cid:80) j =1 τ jn (cid:41) . We now present the convergence of our dynamic microscopic model to a dynamic mesoscopicmodel. We should first, of course, define the dynamic mesoscopic model. This is done in essentiallythe same way as in the microscopic case. We therefore only give an overview here and refer tothe previous section for the precise details. The static mesoscopic model determines the behaviourof the dynamic model in between price changes. The functions determining rates of upward pricechanges and downward price changes as functions of the book profile are now denoted by θ u,m and θ d,m respectively. They are maps from ( R + ) N − × ( R + ) N − → R > , which we assume to beuniformly bounded over m and such that there exists c > θ u,m , θ d,m ≥ c . We further assumethat these maps are continuous. Our regenerative function which determines the distribution of thenew profile following a price change is denoted by R and now maps ( R + ) N − × ( R + ) N − × { u, d } → M (( R + ) N − × ( R + ) N − ). This function is also assumed to be continuous, where M (( R + ) N − × ( R + ) N − ) is equipped with the topology of weak convergence. For ease of notation, we introducethe maps P n : N N − × N N − → ( R + ) N − × ( R + ) N − , which simply divide each coordinate by √ n .The functions here are then approximated by the corresponding functions in the microscopic modelin the following ways.(i) For k ∈ { u, d } , v , v ∈ N N − and u , u ∈ ( R + ) N − , (cid:12)(cid:12)(cid:12) nθ nk,m ( v , v ) − θ k,m ( u , u ) (cid:12)(cid:12)(cid:12) ≤ r ( (cid:107) P n (( v , v )) − ( u , u ) (cid:107) ) , where lim x → r ( x ) = 0.(ii) For k ∈ { u, d } , if ( v n , v n ) is a sequence in N N − × N N − such that P n (( v n , v n )) → ( u , u ),then R n ( v n , v n , k ) ◦ P − n = ⇒ R ( u , u , k )in law in M (( R + ) N − × ( R + ) N − ).With this in place, we define the processes X bi , X ai stopping times τ iu , τ id , τ i and price sequence m i analogously to the previous section, with price jumps once again of size (cid:15) . The underlying collectionsof exponential random variables which are used in the construction of the stopping times τ iu and τ id are now denoted Y iu and Y id . Hence, X i is the sequence of static mesoscopic models with suitablemids, ( τ iu , τ id ) are stopping times determining the times in between price changes as well as thedirection of these price changes, and m i is the sequence of mids. Our dynamic mesoscopic modelis then given by ( ˆ X b ( t ) , ˆ X a ( t ) , m ( t )), whereˆ X b ( t ) := ∞ (cid:88) i =1 X bi t − i − (cid:88) j =1 τ j (cid:40) i − (cid:80) j =1 τ j ≤ t< i (cid:80) j =1 τ j (cid:41) , ˆ X a ( t ) := ∞ (cid:88) i =1 X ai t − i − (cid:88) j =1 τ j (cid:40) i − (cid:80) j =1 τ j ≤ t< i (cid:80) j =1 τ j (cid:41) , ( t ) := ∞ (cid:88) i =1 m i (cid:40) i − (cid:80) j =1 τ j ≤ t< i (cid:80) j =1 τ j (cid:41) . The following is our convergence result for approximating the dynamic mesoscopic model withour sequence of rescaled dynamic microscopic models. The mesoscopic model has the benefit ofbeing more computationally efficient for large scale systems than the microscopic model, whilstmaintaining a discrete spatial structure.
Theorem 2.4.
Suppose that (cid:32) Z an, (0) √ n , Z bn, (0) √ n (cid:33) = ⇒ ( X a (0) , X b (0)) weakly in M (cid:16) ( R + ) N − × ( R + ) N − (cid:17) . Let ( ˆ Z an ( t ) , ˆ Z bn ( t ) , m n ( t )) be dynamic microscopic models withinitial data (cid:32) Z an, (0) √ n , Z bn, (0) √ n , m (cid:33) , and let ( ˆ X a ( t ) , ˆ X b ( t ) , m ( t )) be the dynamic mesoscopic model with initial data ( X a (0) , X b (0) , m ) .Then (cid:32) ˆ Z an ( nt ) √ n , ˆ Z bn ( nt ) √ n , m n ( nt ) (cid:33) = ⇒ ( ˆ X a ( t ) , ˆ X b ( t ) , m ( t )) weakly in M (cid:16) D ([0 , ∞ ) , R N − ) × D ([0 , ∞ ) , R N − ) × D ([0 , ∞ ) , R ) (cid:17) .Proof. See Section A.2 in the appendix.
In this section we rescale our previously obtained dynamic mesoscopic system and bridge thegap between mesoscopic and macroscopic models of limit order books. We show convergence to adynamic macroscopic model limit by letting the tick size tend to zero and rescaling suitably. Thisconnects our description of the order book to SPDE systems, which have been considered as orderbook models in the literature. See, for example, Zheng [29], M¨uller [23] and [17]. As with thedynamic result in the previous section, the proof of this relies heavily on convergence of the staticmodels, which we present first. We are able to obtain convergence in the static setting by applyingTheorem 2.1 in T. Zhang [28].
Remark 3.1.
Note that, although we take the tick size to zero for the bid and ask sides of thebook, the price process moves according to macroscopic price jumps. This both simplifies theanalysis, avoiding the need to consider a model with a continuously moving boundary, and allowsus to maintain a natural separation of time-scales for the order book evolution and price changes.
Remark 3.2.
When simulating the model, any numerical scheme for the SPDE limit requires aprojection which would essentially return us to an SDE framework. Nonetheless, it is interestingto note that the Poisson driven framework introduced in Section 2 has a natural SPDE analogue.The work in this section also completes our objective of connecting existing orderbook models inthe literature at the particle, SDE and SPDE level, by noting that these correspond to one anotherif we rescale time and space appropriately. 9 .1 SPDE limit in a static setting
We begin this section by describing the sequence of rescaled static mesoscopic models which wewill consider. For every N ≥ X bN ( t ) , X aN ( t ) satisfy the dynamics of the bid and ask sidesof the static mesoscopic model on the price grid { , , , ..., N } with mid m ∈ R . These modelsare now indexed by N , which was suppressed in the previous section. We wish to emphasise thishere since we will be taking N to infinity. We will rescale space and time appropriately and mapthe coordinates of X bN ( t ) and X aN ( t ) to equally spaced points on [0 , Q N : R N − → C ((0 , i = 1 , , ..., N − Q N ( x )( iN ) = x i √ N .(ii) Q N ( x )(0) = Q N ( x )(1) = 0.(iii) For i = 0 , , , ..., N − Q N ( x ) is linear between the points [ iN , i +1 N ].The aim will be to take the limit of ( Q N ( X bN ( N t )) , Q N ( X aN ( N t ))). We should have that theparameters for our SDE systems are consistent in some way. Given the rescaling, they are chosensuch that for k ∈ { b, a } , m ∈ R , N ≥ i = 0 , , ..., N − u ∈ ( R + ) N − :(i) h Nk,m ( i, u ) := N − h k,m ( iN , u √ N ),(ii) σ Nk,m ( i, u ) := σ k,m ( iN , u √ N ),where h k,m , σ k,m are measurable maps from [0 , × [0 , ∞ ) → R . We further assume the Lipschitzcondition: | h k,m ( x, u ) − h k,m ( y, v ) | + | σ k,m ( x, u ) − σ k,m ( y, v ) | ≤ C ( | x − y | + | u − v | ) . We will show that our rescaled mesoscopic models converge to a reflected SPDE. Before doingso, we first give the definition of a solution to a reflected SPDE. This is the same as in T. Zhang[28].
Definition 3.3.
We say that the pair ( u, η ) is a solution the SPDE with reflection ∂u∂t = α ∆ u + h ( x, u ( t, x )) + σ ( x, u ( t, x )) ∂ W∂x∂t + η ( t, x ) (3.1) with Dirichlet conditions u ( t,
0) = u ( t,
1) = 0 and initial data u (0 , x ) = u ∈ C ((0 , + if(i) u is a continuous adapted random field on R + × [0 , such that u ≥ almost surely.(ii) η is a random measure on R + × (0 , such that:(a) For every t ≥ , η ( { t } × (0 , ,(b) For every t ≥ , (cid:82) t (cid:82) x (1 − x ) η ( ds,dx ) < ∞ ,(c) η is adapted in the sense that for any measurable mapping ψ : (cid:90) t (cid:90) ψ ( s, x ) η ( ds,dx ) is F t − measurable . iii) For every t ≥ and every φ ∈ C ([0 , with φ (0) = φ (1) = 0 , (cid:90) u ( t, x ) φ ( x ) d x = (cid:90) u (0 , x ) φ ( x ) d x + α (cid:90) t (cid:90) u ( s, x ) φ (cid:48)(cid:48) ( x ) d x d s + (cid:90) t (cid:90) h ( x, u ( s, x )) φ ( x ) d x d s + (cid:90) t (cid:90) φ ( x ) σ ( x, u ( s, x )) W ( d s, d x )+ (cid:90) t (cid:90) φ ( x ) η ( d s, d x ) almost surely.(iv) (cid:82) ∞ (cid:82) u ( t, x ) η ( d t, d x ) = 0 . The intuition for this equation is that the reflection measure is analogous to the local timefor a one dimensional diffusion. The solution follows the dynamics of a standard SPDE (withoutreflection), except when the solution meets the x -axis, where the profile is minimally pushed up bythe reflection measure and kept positive.Existence of strong solutions of to equations of the form (3.1), under our conditions on thecoefficients, is proved by C.Donati-Martin and E. Pardoux in [12]. This means that, given a whitenoise process and the filtration that it generates, we can construct a solution which is adapted.Uniqueness was then proved by T. Xu and T. Zhang in [27].We now pass to the macroscopic limit in the static setting. The following result can be shown byadapting Theorem 2.1 in T.Zhang [28]. We equip the space C ([0 , ∞ ); C ((0 , f n → f in C ([0 , ∞ ); C ((0 , f n → f in C ([0 , T ]; C ((0 , T > Theorem 3.4.
Suppose that ( Q N ( X bN (0)) , Q N ( X aN (0))) = ⇒ ( v b , v a ) in law in C ((0 , + × C ((0 , + . Then ( Q N ( X bN ( N t )) , Q N ( X aN ( N t )) = ⇒ ( v b , v a ) in law in C ([0 , ∞ ); C ((0 , ,where ( v b , v a ) is the unique solution to the pair of reflected stochastic heat equations ∂v b ∂t = α b ∆ v b + h b,m ( x, v b ) + σ b,m ( x, v b ) ∂ W b ∂x∂t + η b ( d t, d x ) ,∂v a ∂t = α a ∆ v a + h a,m ( x, v a ) + σ a,m ( x, v a ) ∂ W a ∂x∂t + η a ( d t, d x ) , with pinning conditions v b ( t,
0) = v b ( t,
1) = v a ( t,
0) = v a ( t,
1) = 0 and initial data v b (0 , x ) = v b ( x ) and v a (0 , x ) = v a ( x ) , where the space-time white noises W b and W a are independent.Proof. See Section A.3 in the appendixWe call the limiting process ( v b , v a ) our static macroscopic order book process with mid m . Let ( ˆ X bN ( t ) , ˆ X aN ( t ) , m N ( t )) be our N th dynamic mesoscopic model, with the parameters definedas in the previous section. The price change rates are now denoted by θ Nb,m and θ Na,m , and theregenerative functions by R N . We aim to prove convergence to a dynamic macroscopic model bytaking the limit of the sequence ( Q N ( ˆ X bN ( N t )) , Q N ( ˆ X aN ( N t )) , m N ( N t )).11e describe our dynamic macroscopic order book model. The principles are the same as forthe dynamic microscopic and mesoscopic models, in that we assume the dynamics follow the staticmodel in between price changes, with these occurring at state driven rates. As in previous sections,the stopping times determining the time in between the ( i − th and i th price change, as well as thedirection of the price change, are denoted τ iu and τ id , with τ i := τ iu ∧ τ id . When an upward/downwardprice change is triggered, the price process m ( t ) increases/decreases by (cid:15) . The bid/ask profiles whichgive the evolution of the profile in between the ( i − th and i th price change are denoted by u bi and u ai respectively. We introduce θ u,m and θ d,m for m ∈ R , which are uniformly bounded over m and continuous from C ((0 , → R > , with there existing c > θ u,m , θ d,m ≥ c . Thesefunctions determine the rates of upward and downward price changes respectively as functions ofthe profile of the book when the mid is at m . We also introduce the regenerative function for ourmacroscopic model, R , which maps C ((0 , × C ((0 , × { u, d } → M ( C ((0 , × C ((0 , u b ( t ) , ˆ u a ( t ) , m ( t )), whereˆ u b ( t ) := ∞ (cid:88) i =1 u bi t − i − (cid:88) j =1 τ j (cid:40) i − (cid:80) j =1 τ j ≤ t< i (cid:80) j =1 τ j (cid:41) , ˆ u a ( t ) := ∞ (cid:88) i =1 u ai t − i − (cid:88) j =1 τ j (cid:40) i − (cid:80) j =1 τ j ≤ t< i (cid:80) j =1 τ j (cid:41) ,m ( t ) := ∞ (cid:88) i =1 m i (cid:40) i − (cid:80) j =1 τ j ≤ t< i (cid:80) j =1 τ j (cid:41) . For ease of notation, we introduce here the map ˜ Q N : R N − × R N − → C ((0 , × C ((0 , Q N to each coordinate. We connect the dynamic aspects of the mesoscopicand macroscopic models by assuming that(i) For k ∈ { b, a } , m ∈ R , ( X , X ) ∈ R N − × R N − and ( u , u ) ∈ C ((0 , × C ((0 , (cid:12)(cid:12)(cid:12) N θ Nk,m (( X , X )) − θ k,m (( u , u )) (cid:12)(cid:12)(cid:12) ≤ r ( (cid:107) ˜ Q N (( X , X )) − ( u , u ) (cid:107) ) , where lim x → r ( x ) = 0.(ii) For m ∈ R , if ( X , X ) ∈ N N − × N N − and ( u , u ) ∈ C ((0 , × C ((0 , (cid:107) ˜ Q N (( X , X )) − ( u , u ) (cid:107) → , then, for k ∈ { b, a } , R N ( X , X , k ) ◦ ˜ Q − N = ⇒ R ( u , u , k )in law in C ((0 , × C ((0 , Remark 3.5.
For X ∈ N N − and u ∈ C ((0 , (cid:107) Q N ( X ) − u (cid:107) → i =1 , ,...,N − (cid:12)(cid:12)(cid:12) X i − u (cid:0) i/N (cid:1)(cid:12)(cid:12)(cid:12) → . heorem 3.6. Suppose that ( Q N ( X bN (0)) , Q N ( X aN (0))) = ⇒ ( u b (0) , u a (0)) in law in C ((0 , × C ((0 , . Let ( ˆ X bN ( t ) , ˆ X aN ( t ) , m N ( t )) be our dynamic mesoscopic model with initial data ( X bN (0) , X aN (0) , m (0)) , and let (ˆ u b ( t ) , ˆ u a ( t ) , m ( t )) be our dynamic macroscopic model with initial data ( u b (0) , u a (0) , m (0)) . Then we have that ( Q N ( ˆ X bN ( N t )) , Q N ( ˆ X aN ( N t )) , m N ( N t )) = ⇒ (ˆ u b ( t ) , ˆ u a ( t ) , m ( t )) in law in the space D ([0 , ∞ ); C ((0 , × D ([0 , ∞ ); C ((0 , × D (cid:0) [0 , ∞ ); R (cid:1) . In this section we illustrate the flexibility of our set-up by discussing some ideas for differentaspects of the model.
Example 4.1.
Rate functions can be chosen such that θ u,m ( u , u ) = γF (cid:18)(cid:90) (cid:15) (cid:16) u ( x ) − u ( x ) (cid:17) dx (cid:19) + δ and θ d,m ( u , u ) = γF (cid:18)(cid:90) (cid:15) (cid:16) u ( x ) − u ( x ) (cid:17) dx (cid:19) + δ where F is a non-negative continuous function, and γ, δ >
0. The rate at which price movementsoccur then has two components which are natural from a modelling standpoint. The first is afunction of the local imbalance (the difference between the number of offers to buy and the numberof offers to sell in a region close to the mid). The second is a fixed rate, intended to represent pricemovements due to exogenous factors. We will see in Section 5 how one might fit these parametersto data.
Example 4.2.
We can incorporate large orders into our model. So far we have only directlyconsidered small order sizes in our models, taking these wlog to be of size 1 in our microscopicmodels. We haven’t, however, mentioned larger order sizes which would appear as “jumps” in themacroscopic model profile in the limit. These can easily be incorporated by being assumed to appear(as one would expect) on a slower time scale to small orders. Large market orders which cause pricechanges are already accounted for in the existing set-up. We can also easily include extra stoppingtimes τ ic into our models, which allow us to create a jump in the profile of the book without changingthe price. This can model large cancellations or large limit orders for the book which do not triggerprice changes. The rate at which these occur can then be given by some third rate function θ c,m , withthe new profile again given by R via the definition of R ( u , u , c ), where we extend the definitionof R so that it now maps from C ((0 , × C ((0 , × { u, d, c } → M ( C ((0 , × C ((0 , Example 4.3.
As a particular case of the drift and volatility functions, h k,m and σ k,m we can take13. h k,m ( x, u ) = h ( x ) + h ( x ) u , and2. σ k,m ( x, u ) = σ ( x ) + σ ( x ) u .The multiplicative terms h ( x ) u and σ ( x ) u can be thought of as self-exciting components forthe order rates, whereby more orders on the book leads to faster trading. The remaining termsrepresent orders placed independently of the current order book profile. Example 4.4.
We note that our regeneration functions, which determine the profiles of the twosides of the order book following price changes, allow us to choose profiles which both depend on thestate of the book when the price change occurs, and are random. Therefore, natural deterministicchoices, such as suitably removing orders from the previous profiles when the price changes, arepermitted. We are also able to choose random profiles, such as sampling from invariant measures,or some combination of these two mechanisms.
The aim of this final section is to demonstrate that even simple versions of the model canreproduce features of the price series and order books seen in financial data. We describe theparticular parameter choices for the model which will be used in this section, before giving anoverview of the numerical scheme and introducing the LOBSTER dataset. Following this, webriefly explain the parameter estimation procedure used, before presenting numerical illustrationsand results by plugging in the estimated parameters into the simulation algorithms. We would liketo emphasise here that the intention behind this section is to demonstrate that sensible order booksimulations can readily be obtained from the model - we do not claim that the numerical schemeimplemented, or the methods used to fit the parameters, are optimal.
We will work with a special case of the models which fall within the framework described inthe earlier sections. The two sides of the book, u b and u a , will evolve in between price changesaccording to the reflected SPDEs ∂u k ∂x = α ∆ u k + f ( x ) + σ ( x ) ∂ W k ∂x∂t + η k , for k ∈ { b, a } . The implicit assumption here is that the order arrival rates, which give rise to thedrift and volatility terms f and σ in our SPDE limit, depend on the distance from the mid only,and there is no dependence on the number of orders which are currently on the book at that price.We have also imposed that the coefficients f and σ here do not depend on k . This is due to thesymmetry in the order arrival rates for the bid and ask sides of the book, which can be seen in thedata.We will now describe the form of the rate functions which will be used in this section, θ u ( u )and θ d ( u ), which determine the rates at which the price moves up and down respectively. In aparticular case of Example 1 of Section 4, the rate at time t for an upward price jump will be givenby θ u ( u b ( t, · ) , u a ( t, · )) = γ max (cid:18)(cid:90) (cid:15) (cid:16) u b ( t, x ) − u a ( t, x ) (cid:17) d x, (cid:19) + δ (5.1)14nd the rate of a downward price movement will similarly be given by θ d ( u b ( t, · ) , u a ( t, · )) = γ max (cid:18)(cid:90) (cid:15) (cid:16) u a ( t, x ) − u b ( t, x ) (cid:17) d x, (cid:19) + δ. (5.2)The first terms in these rates represents the contribution of order imbalance close to the mid as adriving factor for price movement, whilst the second terms here represent price movements due toadditional exogenous factors.Our regeneration functions, which determine the profiles of the two sides of the book followingprice changes, will simply shift the profiles of the two sides of the book by the size of a price jump,in the relevant direction. That is, R is given by R (cid:16) u b , u a , u (cid:17) = (0 , u a ( x + (cid:15) )) for x ∈ [0 , (cid:15) ] , ( u b ( x − (cid:15) ) , u a ( x + (cid:15) )) for x ∈ [ (cid:15), − (cid:15) ] , ( u b ( x − (cid:15) ) ,
0) for x ∈ [1 − (cid:15), ,R (cid:16) u b , u a , d (cid:17) = ( u b ( x + (cid:15) ) ,
0) for x ∈ [0 , (cid:15) ] , ( u b ( x + (cid:15) ) , u a ( x − (cid:15) )) for x ∈ [ (cid:15), − (cid:15) ] , (0 , u a ( x − (cid:15) )) for x ∈ [1 − (cid:15), , (5.3)We note here that, strictly speaking, some of these new profiles do not satisfy the Dirichletconditions imposed earlier in the analysis. However, the equations can be shown to have solutionswhen started from a general positive continuous initial profile, with the solution in C ((0 , + at allpositive times. In addition, the regeneration function as described here has a natural interpretationsimply as the best bid/ask queue having been depleted to trigger the price change. We use a forward time-stepping scheme in order to simulate our equations in between pricemovements. The equation is discretised into M time steps and N space steps, and we define t i := iT /M , x i := i/N . We denote the simulated height of the bid/ask sides of the book at time t j and position x i by u b ( t j , x i ) and u a ( t j , x i ) respectively. The simulated price process at time t j isdenoted by p ( t j ). Given our simulated solution up to the j th time step, we begin our approximationof the following time-step by first determining whether there should be a price movement at thistime-step. Note that we should take the price jump size, (cid:15) , to be equal to a multiple of the spacestep in that (cid:15) = k/N for an integer k . In our simulations, we will take k = 1. We start by defining π + [ t j ] and π − [ t j ] which give the probabilities of upward of downward price jumps in that time-step.These are approximations of the probabilities of price jumps in that time period given by the rates θ u ( u b ( t, · ) , u a ( t, · )) and θ d ( u b ( t, · ) , u a ( t, · )) as in (5.1) and (5.2). Writing these explicitly, we havethat π + [ t j ] = (cid:18) max( γ N (cid:16) u b ( t j , x ) − u a ( t j , x ) (cid:17) ,
0) + δ (cid:19) × T /M. and π − [ t j ] = (cid:18) max( γ N (cid:16) u a ( t j , x ) − u b ( t j , x ) (cid:17) ,
0) + δ (cid:19) × T /M.
We simulate a uniform random variable on [0 , Y [ t j ]. If Y [ t j ] < π + [ t j ] wetake this as an indication that the price has moved up in that time-step. Similarly, if π + [ t j ] ≤ Y [ t j ] < π + [ t j ] + π − [ t j ], we take this as an indication that there has been a downward price change15t this time-step. If we are in neither of these cases, the simulated order book does not change priceduring this time-step. If there has been a price increase, we update our price process by setting p [ t j +1 ] = p [ t j ] + (cid:15) , and adjust the profiles u b and u a by discrete approximations to (5.3). That is,we simply set u b ( t j , x ) = u a ( t j , x N ) = 0 , and define u b ( t j , x i ) = u b ( t j , x i − )for i ∈ { , .., N } , and u a ( t j , x i ) = u a ( t j , x i +1 )for i ∈ { , ..., N − } . We analogously update u b , u a and p in the event of a downward pricemovement. If there is no price change at this time-step, we do not update u b , u a at this point, andset p ( t j +1 ) = p ( t j ). At the end of this price updating procedure, we then simulate the profiles of u b and u a at the next time-step by setting, for i ∈ { , , ..., N − } u b ( t j +1 , x i ) := max (cid:40) u b ( t j , x i ) + T N M (cid:16) u b ( t j , x i +1 ) + u b ( t j , x i − ) − u b ( t j , x i ) (cid:17) + TM f ( x i ) + √ T N √ M σ ( x i ) Z bi,j , (cid:41) , and similarly u a ( t j +1 , x i ) := max (cid:40) u a ( t j , x i ) + T N M (cid:0) u a ( t j , x i +1 ) + u a ( t j , x i − ) − u a ( t j , x i ) (cid:1) + TM f ( x i ) + √ T N √ M σ ( x i ) Z ai,j , (cid:41) . The Z ki,j here are simply simulated unit normal random variables, and appear due to the discreti-sation of the space-time white noise component of our equations. We take the maximum with zeroat each time-step in order to capture the influence of the reflection measures in the equations. Thisstep completes our forward time-step, and we return the profiles u b ( t j +1 , · ), u a ( t j +1 , · ) and the price p ( t j +1 ) as our simulated order book at time t j +1 . Remark 5.1.
By simulating the problem in this way, we can also think of our numerical schemeas a simulation of a mesoscopic model consisting of 50 coupled SDEs. Our work from Section 3shows that, for sufficiently many queues, the SDE model is essentially an SPDE model. We chooseto present our numerics here as a discretisation of the macroscopic SPDE model.
The data we have at our disposal originates from the LOBSTER (Limit Order Book System,The Efficient Reconstructor) database project initiated by the Humboldt University of Berlin, whichgives access to reconstructed limit order book data for all NASDAQ traded stocks between June2007 up to the present day. For each trading day of a given ticker, LOBSTER generates two distinctfiles. On the one hand, a message file, which lists indicators for the different kinds of events whichcause an update of the book (limit order arrivals and cancellations, executions or market orders,trading halts) within a prespecified price range. On the other hand, an order book file, which16isplays the evolution of the book up to a chosen number of occupied price levels (which can go upto 200, depending on the selected ticker). Order book events are timestamped according to secondsafter midnight, and the decimal precision available ranges from milliseconds to nanoseconds. Oursample consists of data from the SPDR Trust Series I, and covers the 50 best levels on each side ofthe book on June 21 2012 between 11:00:00.000 and 12:00:00.000 EST, with tick size size being 1cent for the dataset.
In this section, we aim to obtain input parameters for our model based on the data described inthe previous section. We would like to emphasise here that we do not claim that our fitting methodsin this section are particularly robust - we simply aim to fit the parameters in a reasonable wayso that we can demonstrate that our model can provide sensible simulations of the evolution ofthe order book. In addition, we do not fit the smoothing parameter, α and instead choose thisvalue such that the profiles obtained are on the correct scale. In order to account for the reflectionmeasure, we will only use the data of limit orders placed in price levels which are currently occupied.This is done since we should only take into consideration the order arrivals when the order volumesare away from zero, so as not to bias our drift estimates with components which could be attributedto the reflection measure.Throughout this section, we will denote by X bj,i the order volume in the i th price point belowthe best bid at the j th time-step for our dataset. We similarly denote by X aj,i the order volume inthe i th price point above the best ask at the j th time-step for our dataset.The scaling used will measure order volumes in units of 10 orders, and the 50 queues of theorder book will map to positions i/
51. This simply results in the SPDEs for each side of the bookeach being on a spatial interval of length 1. Since each queue of our dataset represents a pricechange of size 1 cent, a spatial interval of size 1 /
51 corresponds to 1 cent under our scaling. Timeunits for the SPDEs will be measured in minutes. In this scale, our choice of smoothing parameter, α will be given by α = 0 . We begin this section by fitting the volatility and drift parameters for our SPDE. Let ˆ σ ( i/ σ ( i/ i/
51, corresponding to i pricepoints away from the best bid/ask. We calculate ˆ σ by equating51 × × ˆ σ ( i/
51) = 12 (cid:88) (cid:104)
Order Size × − (cid:105) , where the sum is taken over all orders on the bid/ask side of the book of all types which are i pricepoints away from the best bid/ask price respectively, with the omission of limit orders placed inunoccupied queues. This simply matches the quadratic variation of different spatial intervals inthe model with the corresponding value from the dataset. The factors of 51 and 3600 appear heredue to our space and time scaling respectively, whilst the 1 / − appears as we areworking with units of 10 orders.We now fit the drift term. Let d bi be the total number of bid limit orders placed in the hourlong period in queues which are i price points away from the best bid, once again disregardingthose orders placed in unoccupied queues. We similarly define d ai , with c bi and c ai the correspondingvalues for market/cancellation orders at the different relative price points for the bid and ask sides17f the book respectively. The net order flow over the hour long period at a price point which is i ticks away from the best bid/ask queue is then obtained from the dataset as12 (cid:16) d ai + d bi − c ai − c bi (cid:17) × − . Denote by ˜ f ( i/
51) our estimate for the drift term at spatial position i and recall that we are workingin time units of minutes. We estimate the drift parameter by equating60 × ( ˆ f ( i/
51) + 0 .
01 ˆ∆ i u ) = 12 (cid:16) d ai + d bi − c ai − c bi (cid:17) × − . (5.4)The term 0 .
01 ˆ∆ i u here represents the contribution of the laplacian term in our equation to the netorder flow per minute at the i th level. ˆ∆ i u is obtained by calculating the average Dirichlet laplacianat the bid/ask sides of the book at price points which are i ticks away from the best bid/ask prices.By Dirichlet here, we mean that the laplacian is calculated at the first and last queues using theconvention that the 0 th and 51 th queues are empty.Figure 1 displays the estimated drift and volatility functions obtained from the data by thetechniques described above. We note that the volatility is at its largest close to the mid, representingthat there was significantly more activity at these price points over the trading period, as one wouldexpect. (a) (b) Figure 1:
Estimated Drift (a), and volatility (b).
It is only left to produce estimates for γ , the rate at which the model changes price due toimbalance of the bid and ask queues, and δ , the rate at which the model changes price due toexogenous factors. Let P ( t ) denote the price process of our dataset, measured in cents (recall thatthe tick size for the data is one cent). Our estimate of γ , ˆ γ , is chosen such that it satisfies theequation P (1) − P (0) = ˆ γ × × I. (5.5)The value I here is the average local imbalance of the data over the entire period, given by I = 1 J J (cid:88) j =1 (cid:34)(cid:90) / (cid:16) ˜ X bj ( x ) − ˜ X aj ( x ) (cid:17) d x (cid:35) , J is the number of timesteps for our dataset, and ˜ X bj ( x ) and ˜ X aj ( x ) are obtained from X bj,i and X aj,i by setting ˜ X bj ( i/
51) = 10 − X bj,i and ˜ X aj ( i/
51) = 10 − X aj,i , and then linearly interpolatingin between these points. Writing I in terms of X b and X a , we have I = 12 × × × J J (cid:88) j =1 (cid:16) X bj, − X aj, (cid:17) . The factor of 60 appears in (5.5) since we are working in units of minutes, and we have used an hourof data. The rate for price movements in either direction due to exogenous factors, ˆ δ , is then fittedso that the quadratic variation of the price process from the dataset matches with the expectedquadratic variation had the price moved due to our price changing mechanism with parameters ˆ γ and ˆ δ . The expected number of price changes due to the local imbalance component is given by60ˆ γ ˜ I , where ˜ I = 12 × × × J J (cid:88) j =1 (cid:12)(cid:12)(cid:12) X bj, − X aj, (cid:12)(cid:12)(cid:12) . For a given δ we would expect an extra 2 × × δ price changes over the time period. Our parameterˆ δ is therefore chosen such that it solves120ˆ δ = J (cid:88) j =1 (cid:0) P ( j/J ) − P (( j − /J ) (cid:1) − γ ˜ I. Implementing the procedures described, we obtain the following values for ˆ δ and ˆ γ .ˆ γ ˆ δ In this section, we will present the results of our numerical simulation of the order book underour model. In addition to the parameters which were fitted in the previous section, we fix thefollowing additional parameters for the simulation. α T N M α refers to the smoothing parameter, T the time period in minutes for which werun the simulation and N , M the number of space and time steps used respectively.We will now present graphs illustrating the outcome of our simulations. In order to emphasisethe strength of the fit, we present graphs of our simulations next to the corresponding graphs fromthe dataset. 19 a) (b) Figure 2:
Price process of SPDR1 (a), simulated price process from fitted model (b). (a) (b)
Figure 3:
Price process of SPDR1 (a), simulated price process from fitted model (b).
We note that the quadratic variation of the price process from the dataset over the hour longtime period is 0 . . . . . . a) (b) Figure 4:
Time averaged order book profiles for the ask side of SPDR1 (a) and our simulation (b). (a) (b)
Figure 5:
Static snapshots of order book profiles at 12 minute intervals for the ask side of SPDR1 (a) and oursimulation (b).
We note that the average profile from our simulation is on the correct scale, and, like theaverage profile from the data, has most mass concentrated close to the mid. The snapshots fromthe simulation also demonstrate this.
A Appendix
A.1 Proof of Theorem 2.3
Proof.
Since the particle/SDE systems decouple into two independent systems, it is sufficient toprove convergence of ˜ Z bn to X b in M ( D ([0 , ∞ ); R N − )). It follows from the dynamics of Z bn thatthe rescaled process, ˜ Z bn ( t ), has dynamics given by:(i) For i ∈ { , ..., N − } , ˜ Z bn → ˜ Z bn + e i √ n at exponential rate n σ b,m ( i, ˜ Z b,in ) (cid:32) (cid:110) ˜ Z b,in =0 (cid:111) (cid:33) + √ nf b,m (cid:16) i, ˜ Z b,in (cid:17) . i ∈ { , ..., N − } , ˜ Z bn → ˜ Z bn − e i √ n at exponential rate n σ b,m ( i, ˜ Z b,in ) (cid:110) ˜ Z b,in ≥ √ n (cid:111) + √ ng b,m (cid:16) i, ˜ Z b,in (cid:17) (cid:110) ˜ Z b,in ≥ √ n (cid:111) . (iii) For i ∈ { , ..., N − } , ˜ Z bn → ˜ Z bn + e i − √ n − e i √ n at exponential rate α b √ n ˜ Z b,in (iv) For i ∈ { , ..., N − } , ˜ Z bn → ˜ Z bn + e i +1 √ n − e i √ n at exponential rate α b √ n ˜ Z b,in We note that the indicator functions above will ensure that our processes converge to reflected
SDEs. In order to prove convergence of these processes, we will argue that their infinitesimalgenerators converge (see Corollary 4.8.7, [13]). We start by calculating the infinitesimal generatorsfor the processes ˜ Z bn . Define:∆ ln,k F ( y ) := √ n (cid:34) F ( y ) − F (cid:18) y − e k √ n (cid:19)(cid:35) , ∆ rn,k F ( y ) := √ n (cid:34) F (cid:18) y + e k √ n (cid:19) − F ( y ) (cid:35) , ∆ n,k := n (cid:34) F (cid:18) y + e k √ n (cid:19) + F (cid:18) y − e k √ n (cid:19) − F ( y ) (cid:35) . For ease of notation, we use the convention that terms are zero whenever they refer to the 0 th or N th queues. Then we have that, for F ∈ C ([0 , ∞ ) N − )), the continuous functions on [0 , ∞ ) N − which vanish at ∞ ,1 t (cid:18) E (cid:104) F ( ˜ Z n ( t )) | ˜ Z n (0) = y (cid:105) − F ( y ) (cid:19) = N − (cid:88) k =1 ∆ rn,k F ( y ) (cid:32) √ n σ b,m ( k, y k ) (cid:16) { y k =0 } (cid:17)(cid:33) + N − (cid:88) k =1 ∆ rn,k F ( y ) f b,m ( k, y k )+ N − (cid:88) k =1 (cid:18) ∆ rn,k − F ( y − e k √ n ) − ∆ ln,k F ( y ) (cid:19) y k + N − (cid:88) k =1 ∆ ln,k F ( y ) (cid:32) − √ n σ b,m ( k, y k ) (cid:110) y k ≥ √ n (cid:111) (cid:33) + N − (cid:88) k =1 ∆ ln,k F ( y ) (cid:32) − g b,m ( k, y k ) (cid:110) y k ≥ √ n (cid:111) (cid:33) + N − (cid:88) k =1 (cid:18) ∆ rn,k +1 F ( y − e k √ n ) − ∆ ln,k F ( y ) (cid:19) y k + R y,t , R y,t → t → y . Note that this control on R y,t is aconsequence of the local boundedness of σ , f and g , since these conditions ensure that the jumprate of the process after the first jump from y can be bounded. If we further assumed that F weresmooth with compact support, so F ∈ C ∞ c ([0 , ∞ ) N − ), we obtain that the remainder term R y,t converges uniformly to zero for y ∈ ( √ n N ) N − . Therefore, writing A n for the generator of ˜ Z n andrearranging gives that for F ∈ C ∞ c ([0 , ∞ ) N − ), A n F ( y ) = N − (cid:88) k =1
12 ∆ n,k F ( y ) (cid:110) y k ≥ √ n (cid:111) σ b,m ( k, y k ) + N − (cid:88) k =1 ∆ rn,k F ( y ) √ nσ b,m ( k, y k ) { y k =0 } + N − (cid:88) k =1 ∆ rn,k F ( y ) f b,m ( k, y k ) − N − (cid:88) k =1 ∆ ln,k F ( y ) g b,m ( k, y k ) (cid:110) y k ≥ √ n (cid:111) + N − (cid:88) k =1 (cid:18) ∆ rn,k − F ( y − e k √ n ) + ∆ rn,k +1 F ( y − e k √ n ) − ln,k F ( y ) (cid:19) y k . Recall that our candidate limiting process, X b , satisfies the system of reflected SDEsd X b,it = α b ( X b,i +1 t + X b,i − t − X b,it )dt + h b,m ( i, X b,it )dt + σ b,m ( i, X b,it )d W b,it + d η b,it , for i = 1 , , ..., N −
1, with X = X N = 0 and h b,m := f b,m − g b,m . Note that by Theorem 4.1 in [26],we have existence of a strong solution and pathwise uniqueness for this system of SDEs. Inspectionof the proof also reveals that the solution here, where the diffusion and drift coefficients do nothave any explicit time-dependence, is continuous. We can calculate the corresponding generator tobe AF ( x ) = 12 N − (cid:88) k =1 σ b,m ( k, x k ) ∂ F∂x k + N − (cid:88) k =1 (cid:2) h b,m ( k, x k ) + α b ( x k +1 + x k − − x k ) (cid:3) ∂F∂x k , acting on the domain D ( A ) = (cid:40) F ∈ C ([0 , ∞ ) N − ) s.t. ∀ k, ∂F∂x k (cid:12)(cid:12)(cid:12)(cid:12) x k =0 = 0 (cid:41) . This has a core given by C ( A ) = (cid:40) F ∈ C ∞ c ([0 , ∞ ) N − ) s.t. ∀ k, ∂F∂x k (cid:12)(cid:12)(cid:12)(cid:12) x k =0 = 0 (cid:41) . In our setting it is enough to prove (see Corollary 4.8.7 in [13]) that for all F ∈ C ( A ),sup y ∈ √ n N N − | A n F ( y ) − AF ( y ) | → . First suppose that y k ≥ √ n . Then, by Taylor’s Theorem we have that12 σ b,m ( k, y k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ n,k F ( y ) − ∂ F∂y k ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ √ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ F∂y k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:107) σ b,m (cid:107) F, ∞ , (cid:107) σ b,m (cid:107) F, ∞ is the supremum of σ b,m over the support of F. Similarly (cid:12)(cid:12)(cid:12)(cid:12) ∆ rn,k F ( y ) − ∂F∂y k ( y ) (cid:12)(cid:12)(cid:12)(cid:12) f b,m ( k, y k ) ≤ √ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ F∂y k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:107) f b,m (cid:107) F, ∞ , and (cid:12)(cid:12)(cid:12)(cid:12) ∆ ln,k F ( y ) − ∂F∂y k ( y ) (cid:12)(cid:12)(cid:12)(cid:12) g b,m ( k, y k ) ≤ √ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ F∂y k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:107) g b,m (cid:107) F, ∞ . Now suppose that y k = 0. Then, again by Taylor’s theorem (extending F by reflection about theaxes and using that ∂F∂y k is zero when y k = 0) we have that σ b,m ( k, y k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ n ∆ rn,k F ( y ) − ∂ F∂y k ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ √ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ F∂y k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:107) σ b,m (cid:107) F, ∞ ,f b,m ( k, y k ) (cid:12)(cid:12)(cid:12)(cid:12) ∆ rn,k F ( y ) − ∂F∂y k ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = f b,m ( k, y k ) (cid:12)(cid:12)(cid:12) ∆ rn,k F ( y ) (cid:12)(cid:12)(cid:12) ≤ √ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ F∂y k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:107) f b,m (cid:107) F, ∞ ,g b,m ( k, y k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ ln,k F ( y ) (cid:110) y k ≥ √ n (cid:111) − ∂F∂y k ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . We can argue similarly for the sum N − (cid:88) k =1 (cid:18) ∆ rn,k − F ( y − e k √ n ) + ∆ rn,k +1 F ( y − e k √ n ) − ln,k F ( y ) (cid:19) y k , and find thatsup y ∈ √ n N N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) BF ( y ) − N − (cid:88) k =1 (cid:20) ∆ rn,k − F ( y − e k √ n ) + ∆ rn,k +1 F ( y − e k √ n ) − ln,k F ( y ) (cid:21) y k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) → , where BF ( y ) := N − (cid:88) k =1 (cid:20) ∂F∂y k − + ∂F∂y k +1 − ∂F∂y k (cid:21) y k . Recalling our notational convention that terms referring to the 0 th and N th queues are zero, wehave that BF ( y ) = N − (cid:88) k =1 ∂F∂y k ( y k +1 + y k − − y k ) . Putting these cases together gives that Theorem 2.3 holds.
A.2 Proof of Theorem 2.4
The proof of Theorem 2.3 will be used as a basis for the proof of Theorem 2.4. We startby proving some continuity type results for certain maps, connecting some of the features of themicroscopic models to their mesoscopic counterparts. These will be used in an inductive argumentwhich will allow us to prove that (cid:18)(cid:16) Z bn,i ( nt ) / √ n (cid:17) ∞ i =1 , (cid:16) Z an,i ( nt ) / √ n (cid:17) ∞ i =1 , ( τ in /n ) ∞ i =1 , ( m in ) ∞ i =1 (cid:19) = ⇒ (( X bi ) ∞ i =1 , ( X ai ) ∞ i =1 , ( τ i ) ∞ i =1 , ( m i ) ∞ i =1 )24n law in D ([0 , ∞ ) , R N − ) N × D ([0 , ∞ ) , R N − ) N × [0 , ∞ ] N × R N . Note that, for a metric space M ,we use the topology of pointwise convergence for M N , which is itself metrizable. We then prove acontinuity-type result for the map which sends these processes to their associated dynamic models.Finally, we conclude with an application of the Skorohod representation theorem. Proposition A.1.
Fix some m ∈ R . Let w n : [0 , ∞ ) → N N − × N N − and w : [0 , ∞ ) → R N − × R N − be such that, for every T > , sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) √ n ( w n ( nt ) , w n ( nt )) − ( w ( t ) , w ( t )) (cid:12)(cid:12)(cid:12)(cid:12) → . Then we have that, for every
T > , sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) nθ nu,m ( w n ( nt )) − θ u,m ( w ( t )) (cid:12)(cid:12)(cid:12) → , and sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) nθ nd,m ( w n ( nt )) − θ d,m ( w ( t )) (cid:12)(cid:12)(cid:12) → . Proof.
This is a direct consequence of assumption (i) in Section 2.4.
Proposition A.2.
Suppose ( Z n , Z n ) is a sequence in M ( N N − × N N − ) such that P n (( Z n , Z n )) = ⇒ ( X , X ) in M (( R + ) N − × ( R + ) N − ) . Let ˜ R n : M ( N N − × N N − ) × { u, d } → M ( N N − × N N − ) such thatfor ( µ, k ) ∈ M ( N N − × N N − ) × { u, d } , ˜ R n ( µ, k )( A ) := (cid:90) N N − × N N − R n ( x , x , k )( A ) µ ( d x , d x ) . Similarly, define ˜ R : M (( R + ) N − × ( R + ) N − ) × { u, d } → M (( R + ) N − × ( R + ) N − ) such that for ( ν, k ) ∈ M (( R + ) N − × ( R + ) N − ) × { u, d } , ˜ R ( ν, k )( B ) := (cid:90) ( R + ) N − × ( R + ) N − R ( x , x , k )( A ) ν ( d x , d x ) . Then, for k ∈ { u, d } , ˜ R n ( Z n , Z n , k ) ◦ P − n = ⇒ ˜ R ( X , X , k ) in M (( R + ) N − × ( R + ) N − ) .Proof. We apply the Skorohod representation theorem to the weak convergence of P n (( Z n , Z n )) to( X , X ), so we assume that P n (( Z n , Z n )) → ( X , X ) almost surely on some probability space(Ω , F , P ). We then have that R n ( Z n , Z n , k ) ◦ P − n → R ( X , X , k )in M (( R + ) N − ) × ( R + ) N − ) P - almost surely. Let A ∈ B (( R + ) N − × ( R + ) N − ) be a continuity setfor the measure ˜ R ( X , X , k ). We then have that A is a continuity set for R ( X , X , k ) P - almostsurely, from which it follows that R n ( Z n , Z n , k ) ◦ P − n ( A ) → R ( X , X , k )( A ) P - almost surely. Taking expectations then gives the result.25 roposition A.3. Suppose that f n : [0 , ∞ ) → R > and f : [0 , ∞ ) → R > are such that f iscontinuous, there exists a constant c > such that f ≥ c , and f n → f uniformly on compact sets.Suppose also that x n ∈ R , x ∈ R such that x n → x . Define τ n := inf (cid:40) t ≥ (cid:12)(cid:12) x n ≤ (cid:90) t f n ( s ) ds (cid:41) and τ := inf (cid:40) t ≥ (cid:12)(cid:12) x ≤ (cid:90) t f ( s ) ds (cid:41) . Then τ n → τ .Proof. Fix δ >
0. We argue that, for n large enough, τ n ≤ τ + δ. Note that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) x n − (cid:90) τ f n ( s )ds (cid:19) − (cid:18) x − (cid:90) τ f ( s )ds (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | x n − x | + τ × sup t ∈ [0 ,τ ] | f n ( t ) − f ( t ) | . Since f ≥ c >
0, we have that for n large enough inf t ∈ [0 ,τ ] f n ( t ) > c . Also, for n large enough, | x n − x | + τ × sup t ∈ [0 ,τ ] | f n ( t ) − f ( t ) | < δc . Therefore, x n − (cid:90) τ f n ( s )ds < x − (cid:90) τ f ( s )ds + δc δc x n − (cid:90) τ + δ f n ( s )ds ≤ x n − (cid:90) τ f n ( s )ds − δc < . Therefore τ n ≤ τ + δ for large enough n. Similarly, we have τ n ≥ τ − δ for large enough n, concludingthe proof.We can now prove the following result, whose proof constitutes the main step in proving Theorem2.4. Proposition A.4.
Suppose that (cid:32) Z bn, (0) √ n , Z an, (0) √ n (cid:33) = ⇒ ( X b (0) , X a (0)) in law in ( R + ) N − × ( R + ) N − . Let ( ˆ Z bn ( t ) , ˆ Z an ( t ) , m n ( t )) be dynamic microscopic models with ini-tial data ( Z bn, (0) , Z an, (0) , m ) , and let ( ˆ X b ( t ) , ˆ X a ( t ) , m ( t )) be the dynamic mesoscopic model withinitial data ( X b (0) , X a (0) , m ) . Then (cid:18)(cid:16) Z bn,i ( nt ) / √ n (cid:17) ∞ i =1 , (cid:16) Z an,i ( nt ) / √ n (cid:17) ∞ i =1 , ( τ in /n ) ∞ i =1 , ( m in ) ∞ i =1 (cid:19) = ⇒ (( X bi ) ∞ i =1 , ( X ai ) ∞ i =1 , ( τ i ) ∞ i =1 , ( m i ) ∞ i =1 ) in law in D ([0 , ∞ ) , R N − ) N × D ([0 , ∞ ) , R N − ) N × [0 , ∞ ) N × R N . roof. We begin the proof by introducing the following notation.(i) U b,Mn ( t ) := ( Z bn,i ( nt ) / √ n ) Mi =1 , U b,M ( t ) := ( X bi ( t )) Mi =1 . (ii) U a,Mn ( t ) := ( Z an,i ( nt ) / √ n ) Mi =1 , U a,M ( t ) := ( X ai ( t )) Mi =1 . (iii) ˜ τ Mn,u := ( τ in,u /n ) Mi =1 , ˜ τ Mu := ( τ iu ) Mi =1 . (iv) ˜ τ Mn,d := ( τ in,d /n ) Mi =1 , ˜ τ Md := ( τ id ) Mi =1 . (v) ˜ m Mn := ( m in ) Mi =1 , ˜ m M := ( m i ) Mi =1 . We will prove by induction that for every M ≥ U a,Mn , U b,Mn , ˜ τ M − n,a , ˜ τ M − n,b , ˜ m Mn ) = ⇒ ( U a,M , U b,M , ˜ τ M − a , ˜ τ M − b , ˜ m M ) (A.1)in law in D ([0 , ∞ ) , R N − ) M × D ([0 , ∞ ) , R N − ) M × [0 , ∞ ) M − × [0 , ∞ ) M − × R M , from which theresult follows. Note that by applying Theorem 2.3, we have( U a, n , U b, n , ˜ m n ) = ⇒ ( U a, , U b, , ˜ m )in D ([0 , ∞ ) , R N − ) × D ([0 , ∞ ) , R N − ) × R . Suppose we had for some M ≥ U a,Mn , U b,Mn , ˜ τ M − n,a , ˜ τ M − n,b , ˜ m Mn , Y Mn,a , Y
Mn,b ) = ⇒ ( U a,M , U b,M , ˜ τ M − a , ˜ τ M − b , ˜ m M , Y Ma , Y Mb ) . By the Skorohod convergence theorem, we can assume that this convergence holds almost surely.Recall that τ Mn,u := inf (cid:40) t ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t θ nu,m Mn ( Z bn,M ( s ) , Z an,M ( s )) d s ≥ Y Mn,u (cid:41) ,τ Mn,d := inf (cid:40) t ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t θ nd,m Mn ( Z bn,M ( s ) , Z an,M ( s )) d s ≥ Y Mn,d (cid:41) . By a change of variables, (cid:90) t θ nu,m Mn ( Z bn,M ( s ) , Z an,M ( s ))d s = (cid:90) t/n nθ nu,m Mn ( Z bn,M ( nr ) , Z an,M ( nr ))d r. Therefore, 1 n τ
Mn,u = 1 n inf (cid:40) t ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t/n nθ nu,m Mn ( Z bn,M ( nr ) , Z an,M ( nr ))d r ≥ Y Mn,u (cid:41) = inf (cid:40) t ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t nθ nu,m Mn ( Z bn,M ( nr ) , Z an,M ( nr ))d r ≥ Y Mn,u (cid:41) . As m Mn and m M both take values in m + (cid:15) Z and m Mn → m M almost surely, we have that m Mn = m M for large enough n almost surely. Note that since the limits X aM , X bM are continuous, convergenceof Z an,M ( nt ) / √ n to X aM ( t ) and Z bn,M ( nt ) / √ n to X bM ( t ) in D ([0 , ∞ ) , R N − ) implies that, for every T >
0, sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) √ n ( Z an,M ( nt ) , Z bn,M ( nt )) − ( X aM ( t ) , X bM ( t )) (cid:12)(cid:12)(cid:12)(cid:12) → . θ u,m was assumed to be continuous and X bM , X aM are continuous, θ u,m ( X bM ( t ) , X aM ( t )) iscontinuous almost surely. We can therefore apply Propositions A.1 and A.3 to deduce that1 n τ Mn,u → inf (cid:40) t ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t θ u,m ( X bM ( s ) , X aM ( s )) d s ≥ Y Mu (cid:41) = τ Mu almost surely. Similarly, n τ Mn,d → τ Md almost surely. Note that τ Mu (cid:54) = τ Md almost surely. Thisimplies that m M +1 n → m M +1 almost surely. Therefore, we have deduced that( U a,Mn , U b,Mn , ˜ τ Mn,a , ˜ τ Mn,b , ˜ m M +1 n ) = ⇒ ( U a,M , U b,M , ˜ τ Ma , ˜ τ Mb , ˜ m M +1 ) . (A.2)For i = 1 , ...M + 1, let A i ∈ B ( D ([0 , ∞ ); R N − )) be a continuity set for the D ([0 , ∞ ); R N − )- valuedrandom variable X ai . Similarly, let B i be continuity sets for the random variables X bi , and let p i ∈ ( m + (cid:15) Z ). For i = 1 , ...M , let C i , D i be continuity sets for the random variables τ iu and τ id respectively. Define:˜ A K := K (cid:89) i =1 A i , ˜ B K := K (cid:89) i =1 B i , ˜ C K := K (cid:89) i =1 C i , ˜ D K := K (cid:89) i =1 D i , P K := K (cid:89) i =1 { p i } . Suppose that P (cid:104) ( U b,M , U a,M , ˜ τ Ma , ˜ τ Mb , ˜ m M +1 ) ∈ ˜ A M × ˜ B M × ˜ C M × ˜ D M × P M +1 (cid:105) > . We then have that, for large enough n , P (cid:104) ( U b,Mn , U a,Mn , ˜ τ Mn,a , ˜ τ Mn,b , ˜ m M +1 n ) ∈ ˜ A M × ˜ B M × ˜ C M × ˜ D M × P M +1 (cid:105) > , from which it follows that P (cid:104) ( U b,M +1 n , U a,M +1 n , ˜ τ Mn,a , ˜ τ Mn,b , ˜ m M +1 n ) ∈ ˜ A M +1 × ˜ B M +1 × ˜ C M × ˜ D M × P M +1 (cid:105) = P (cid:104) ( U b,Mn , U a,Mn , ˜ τ Mn,a , ˜ τ Mn,b , ˜ m M +1 n ) ∈ ˜ A M × ˜ B M × ˜ C M × ˜ D M × P M +1 (cid:105) × P n (cid:20) √ n ( Z bn,M +1 ( nt ) , Z an,M +1 ( nt )) ∈ A M +1 × B M +1 (cid:21) , where P n denotes the conditional probability law of P given the event( U b,Mn , U a,Mn , ˜ τ Mn,a , ˜ τ Mn,b , ˜ m M +1 n ) ∈ ˜ A M × ˜ B M × ˜ C M × ˜ D M × P M +1 . By (A.2), we know that P (cid:104) ( U b,Mn , U a,Mn , ˜ τ Mn,a , ˜ τ Mn,b , ˜ m M +1 n ) ∈ ˜ A M × ˜ B M × ˜ C M × ˜ D M × P M +1 (cid:105) → P (cid:104) ( U b,M , U a,M , ˜ τ Ma , ˜ τ Mb , ˜ m M +1 ) ∈ ˜ A M × ˜ B M × ˜ C M × ˜ D M × P M +1 (cid:105) . Let Q n be the law on ( R + ) N − × ( R + ) N − induced by R n ( Z an,M ( τ Mn ) , Z bn,M ( τ Mn ) , k ) ◦ P − n and theprobability measure P n , where k is the direction of the last price change. That is Q n ( A ) = E P n (cid:104) ( R n ( Z an,M ( τ Mn ) , Z bn,M ( τ Mn ) , k ) ◦ P − n )( A ) (cid:105) Q on ( R + ) N − × ( R + ) N − by setting Q ( A ) = E ˜ P (cid:104) R ( X aM ( τ M ) , X bM ( τ M ) , k )( A ) (cid:105) , where ˜ P here is the conditional probability law given the event that( U b,M , U a,M , ˜ τ Ma , ˜ τ Mb , ˜ m M +1 ) ∈ ˜ A M × ˜ B M × ˜ C M × ˜ D M × P M +1 . It follows from (A.2) and the continuity of ( X bM , X aM ) that the law of √ n ( Z bn,M ( τ Mn ) , Z an,M ( τ Mn ) , k )under the measure P n converges to the law of ( X bM ( τ M ) , X aM ( τ M ) , k ) under ˜ P . It then follows byan application of Proposition A.2 that Q n = ⇒ Q . We can now apply our result for convergencein a static setting, Theorem 2.3, to deduce that P n (cid:20) √ n ( Z bn,M +1 ( n · ) , Z an,M +1 ( n · )) ∈ A M +1 × B M +1 (cid:21) → ˜ P (cid:104) ( X bM +1 , X aM +1 ) ∈ A M +1 × B M +1 (cid:105) . We have therefore deduced that P (cid:104) ( U b,M +1 n , U a,M +1 n , ˜ τ Mn,a , ˜ τ Mn,b , ˜ m M +1 n ) ∈ ˜ A M +1 × ˜ B M +1 × ˜ C M × ˜ D M × P M +1 (cid:105) → P (cid:104) ( U b,M +1 , U a,M +1 , ˜ τ Ma , ˜ τ Mb , ˜ m M +1 ) ∈ ˜ A M +1 × ˜ B M +1 × ˜ C M × ˜ D M × P M +1 (cid:105) . So we have that ( U b,M +1 n , U a,M +1 u , ˜ τ Mn,a , ˜ τ Mn,b , ˜ m M +1 n ) = ⇒ ( U b,M +1 , U a,M +1 , ˜ τ Ma , ˜ τ Mb , ˜ m M +1 ) whichconcludes our inductive argument. Proposition A.5.
Let g : D ([0 , ∞ ); R N − ) N × [0 , ∞ ) N → D ([0 , ∞ ); R N − ) such that g (( u i ) ∞ i =1 , ( t i ) ∞ i =1 )( t ) := ∞ (cid:88) j =1 u j t − j − (cid:88) i =1 t i (cid:40) j − (cid:80) i =1 t i ≤ t< j (cid:80) i =1 t i (cid:41) . Suppose that(i) t i > for every i .(ii) ∞ (cid:80) i =1 t i = ∞ . (iii) u i is continuous for every i Then g is continuous at the point (( u i ) ∞ i =1 , ( t i ) ∞ i =1 ) .Proof. Suppose that the sequence (( u ni ) ∞ i =1 , ( t ni ) ∞ i =1 ) converges to (( u i ) ∞ i =1 , ( t i ) ∞ i =1 ) in D ([0 , ∞ ); R N − ) N × [0 , ∞ ) N . It is sufficient to prove that g (( u ni ) ∞ i =1 , ( t ni ) ∞ i =1 ) → g (( u i ) ∞ i =1 , ( t i ) ∞ i =1 )in D ([0 , T ]; R N − ) for every T ≥ g (( u i ) ∞ i =1 , ( t i ) ∞ i =1 ) is continuous at T . That is, it isenough to prove convergence in D ([0 , T ]; R N − ) for every T ≥ L (cid:88) i =1 t i (cid:54) = T L . Let T ≥ K ≥ K (cid:88) i =1 t i > T. (A.3)The functions u i are continuous, so we have by convergence of u ni to u i in D ([0 , ∞ ); R N − ) thatsup t ∈ [0 ,T ] | u ni ( t ) − u i ( t ) | → i . Recall that f n → f in D ([0 , T ]; M ) for a metric space M iff there exist continuousincreasing bijections λ n : [0 , T ] → [0 , T ] such that(a) sup t ∈ [0 ,T ] | λ n ( t ) − t | → t ∈ [0 ,T ] | f n ( λ n ( t )) − f ( t ) | → λ n : [0 , T ] → [0 , T ] such that for m ≥ λ n (cid:32) m (cid:80) i =1 t i (cid:33) := m (cid:80) i =1 t ni , and linearly interpolate between these points, together with the endpoints λ n (0) = 0 and λ n ( T ) = T (this may define the function for values greater than T as well but weonly care about the restriction). Then, for large enough n so that the t ni are all strictly positive, λ n is a strictly increasing continuous bijection from [0 , T ] to [0 , T ], and the λ n satisfy (a) above.Note that this can only be done if T is a continuity point for g . We have that, for t ∈ [0 , T ], g (( u ni ) ∞ i =1 , ( t ni ) ∞ i =1 )( λ n ( t )) = ∞ (cid:88) j =1 u nj λ n ( t ) − j − (cid:88) i =1 t ni (cid:40) j − (cid:80) i =0 t ni ≤ λ n ( t ) < j (cid:80) i =1 t ni (cid:41) = ∞ (cid:88) j =1 u nj − λ n ( t ) − j − (cid:88) i =0 t ni (cid:40) j − (cid:80) i =0 t i ≤ t< j (cid:80) i =0 t i (cid:41) . It follows from (A.3) thatsup t ∈ [0 ,T ] | g (( u ni ) ∞ i =1 , ( t ni ) ∞ i =1 )( λ n ( t )) − g (( u i ) ∞ i =1 , ( t i ) ∞ i =1 )( t ) |≤ sup j ≤ K sup t ∈ [0 ,T ] | u nj − ( t ) − u j − ( t ) | → . Therefore (b) holds and we have the result.
Proposition A.6.
Let h : [0 , ∞ ) N × R N → D ([0 , ∞ ); R ) such that h (( t i ) ∞ i =1 , ( m i ) ∞ i =1 ) := ∞ (cid:88) i =1 m i (cid:40) i − (cid:80) j =1 t j ≤ t< i (cid:80) j =1 t j (cid:41) . Suppose that(i) t i > for every i . ii) ∞ (cid:80) i =1 t i = ∞ . Then h is continuous at the point (( t i ) ∞ i =1 , ( m i ) ∞ i =1 ) .Proof. This is essentially the same as the proof of Proposition A.5.We are now in a position to conclude the proof of Theorem 2.4.
Proof of Theorem 2.4.
We begin by applying the Skorohod representation theorem to the weakconvergence statement in Proposition A.4. This gives that (cid:18)(cid:16) Z bn,i ( nt ) / √ n (cid:17) ∞ i =1 , (cid:16) Z an,i ( nt ) / √ n (cid:17) ∞ i =1 , ( τ in /n ) ∞ i =1 , ( m in ) ∞ i =1 (cid:19) → (( X bi ) ∞ i =1 , ( X ai ) ∞ i =1 , ( τ i ) ∞ i =1 , ( m i ) ∞ i =1 )in D ([0 , ∞ ); R N − ) N × D ([0 , ∞ ); R N − ) N × [0 , ∞ ) N × R N almost surely, where the vectors representthe key objects for the dynamic microscopic and mesoscopic models respectively. By PropositionA.5, we clearly have that (( X ai ) ∞ i =1 , ( τ i ) ∞ i =1 ) and (( X bi ) ∞ i =1 , ( τ i ) ∞ i =1 ) are continuity points of the map g almost surely. Similarly, by Proposition A.6, (( τ i ) ∞ i =1 , ( m i ) ∞ i =1 ) is a continuity point for the map h almost surely. The result follows. A.3 Proof of Theorem 3.4
As in the case of the proof of Theorem 2.3 we notice that the static mesoscopic systems decoupleinto two independent problems on either side of the mid. We therefore focus once more on provingthe convergence on one side of the mid and, without loss of generality, choose to prove convergenceof the bid side. The proof relies on an adaptation of Theorem 2.1 in T.Zhang [28].
Theorem A.7 (Theorem 2.1, [28]) . Let ( u, η ) be a solution of the reflected stochastic heat equation(3.1) with respect to a given white noise W , with initial data u ∈ C ((0 , + . Suppose that σ and h are Lipschitz in both variables and have linear growth in the second variable. For n ≥ , let ( W n,k ) n − k =1 be the independent family of Brownian motions given by W n,kt := √ n (cid:34) W (cid:18) t, k + 1 n (cid:19) − W (cid:18) t, kn (cid:19)(cid:35) , and let u n be the solution of the system of reflected SDEsd u n,kt = αn (cid:16) u n,k +1 t + u n,k − t − u n,kt (cid:17) dt + h (cid:16) k/n, u n,kt (cid:17) dt + √ nσ (cid:16) k/n, u k,nt (cid:17) d W n,kt + d η n,kt , for k = 1 , ..., n − with u n, = u n,n = 0 and initial data u n ∈ ( R + ) n − . For each t ≥ , define thefunction u n ( t, x ) by setting u n ( t, kn ) := u n,kt and linearly interpolating between these points. Supposethat sup x ∈ [0 , | u n (0 , x ) − u ( x ) | → . Then for every p ≥ and every t ∈ [0 , T ]lim n →∞ E (cid:34) sup t ∈ [0 ,T ] sup x ∈ [0 , | u n ( t, x ) − u ( t, x ) | p (cid:35) = 0 . C ([0 , ∞ ); C ((0 , C ([0 , ∞ ); C ((0 , f n → f in C ([0 , ∞ ); C ((0 , f n → f in C ([0 , T ]; C ((0 , T > Corollary A.8.
Let u be the solution of the reflected stochastic heat equation (3.1) with initialdata given by the law µ ∈ M ( C ((0 , + ) . Suppose that σ and h are Lipschitz in both variablesand have linear growth in the second variable. For n ≥ , let ( W n,k ) n − k =1 be an independent familyof Brownian motions and let u n be the solution of the system of reflected SDEsd u n,kt = αn (cid:16) u n,k +1 t + u n,k − t − u n,kt (cid:17) dt + h (cid:16) k/n, u n,kt (cid:17) dt + √ nσ (cid:16) k/n, u k,nt (cid:17) d W n,kt + d η n,kt , for k = 1 , ..., n − with u n, = u n,n = 0 and initial data given by the law ν n ∈ M (( R + ) n − ) . Foreach t ≥ , define the function u n ( t, x ) by setting u n ( t,
0) = 0 , u n ( t,
1) = 0 and u n ( t, kn ) := u n,kt for k = 1 , ..., n − , and linearly interpolating between these points. Suppose that ( ν n ◦ ( √ nQ n ) − ) = ⇒ µ in law in M ( C ((0 , . Then u n = ⇒ u in law in M ( C ([0 , ∞ ); C ((0 , .Proof. In the case where the initial data are deterministic, convergence in M ( C ([0 , ∞ ); C ((0 , L p (Ω; C ([0 , T ] × [0 , T >
0. Turning to the case of randominitial data, let f ∈ C b ( C ([0 , ∞ ); C ((0 , µ n = ν n ◦ ( √ nQ n ) − . We want to prove that E µ n (cid:2) f ( u n ) (cid:3) → E µ (cid:2) f ( u ) (cid:3) . Let F n : C ((0 , → C ((0 , F n ( u )(0) = F n ( u )(1) = 0, F n ( u )( i/n ) = u ( i/n ) for i = 1 , ..., n −
1, and F n ( u ) is linear in the intervals [ i/n, ( i + 1) /n ] for i = 0 , , ..., n −
1. For n ≥ v ∈ C ((0 , g n ( v ) := E (cid:2) f ( u n ) | u n (0 , · ) = F n ( v ) (cid:3) . Similarly, for v ∈ C ((0 , g ( v ) := E (cid:2) f ( u ) | u (0 , · ) = v (cid:3) . Then we have that E µ n (cid:2) f ( u n ) (cid:3) = (cid:90) C ((0 , g n ( v ) µ n (d v ) , and E µ (cid:2) f ( u ) (cid:3) = (cid:90) C ((0 , g ( v ) µ (d v ) , By the Skorohod representation theorem, we can realise the convergence of µ n to µ with the randomvariables u n and u on a common probability space (Ω , F , P ), so that u n → u in C ((0 , P -almostsurely. By the deterministic initial data case, we then have g n ( u n ) → g ( u ) P - almost surely. The result then follows by the DCT. Proof of Theorem 3.4.
Recall that the dynamics for the bid side of the N th static mesoscopic modelare given byd X b,iN ( t ) = α b ( X b,i +1 N ( t )+ X b,i − N ( t ) − X b,iN ( t ))d t + h Nb,m ( i, X b,iN ( t ))d t + σ Nb,m ( i, X b,iN ( t ))d W N,b,it +d η N,b,it i = 1 , ..., N − X b, N = X b,NN = 0. It follows thatd v N,b,it = α b N (cid:16) v N,i +1 t + v N,i − t − v N,it (cid:17) d t + h b,m (cid:16) k/N, v N,it (cid:17) d t + √ N σ b,m (cid:16) i/N, v i,Nt (cid:17) d ˜ W N,b,it + d η N,b,it , (A.4)where we define v N,b,it := √ N X b,iN ( N t ) and ˜ W N,b,it = N W N,b,iN t . The result then follows by anapplication of Corollary A.8.
A.4 Proof of Theorem 3.6
As in the proof of Theorem 2.4, we build towards a proof by first presenting a series of smallerresults. The arguments here reflect those made when upgrading the proof of Theorem 2.3 to aproof of Theorem 2.4. We therefore refer to the proof of Theorem 2.4 to illustrate how to provethe key results in this section. In the same way as the proof of Theorem 2.4, the main part of theproof is showing that(( Q N (( X bN,i ( N t, · ))) ∞ i =1 , ( Q N (( X aN,i ( N t, · ))) ∞ i =1 , ( τ iN,u /N ) ∞ i =1 , ( τ iN,d /N ) ∞ i =1 , ( m iN ) ∞ i =1 )= ⇒ (( u bi ( t, · )) ∞ i =1 , ( u ai ( t, · )) ∞ i =1 , ( τ iu ) ∞ i =1 , ( τ id ) ∞ i =1 , ( m i ) ∞ i =1 )in law in D ([0 , ∞ ); C ((0 , N × D ([0 , ∞ ); C ((0 , N × [0 , ∞ ) N × [0 , ∞ ) N × R N . Once again, fora metric space M we equip M N with the topology of pointwise convergence, which is metrizable. Proposition A.9.
Fix some m ∈ R . Let w N : [0 , ∞ ) → R N − and w : [0 , ∞ ) → C ((0 , be suchthat, for every T > t ∈ [0 ,T ] | Q N ( w N ( t )) − w ( t ) | → . Then we have that, for every
T > , sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) N θ Nb,Nm ( w N ( t )) − θ b,m ( w ( t )) (cid:12)(cid:12)(cid:12) → , and sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) N θ Na,Nm ( w N ( t )) − θ a,m ( w ( t )) (cid:12)(cid:12)(cid:12) → . Proof.
This is a direct application of assumption (i) in Section 3.6.
Proposition A.10.
Suppose ( X N , X N ) is a sequence in M ( R N − × R N − ) such that ( Q N ( X N ) , Q N ( X N )) = ⇒ ( u , u ) in M ( C ((0 , × C ((0 , . Define ˜ R N : M ( R N − × R N − ) × { u, d } → M ( R N − × R N − ) suchthat, for ( µ, k ) ∈ M ( R N − × R N − ) × { u, d } , ˜ R N ( µ, k )( A ) := (cid:90) R N − × R N − R N ( x , x , k )( A ) µ ( d x , d x ) . Similarly, define ˜ R : M ( C ((0 , × C ((0 , × { u, d } → M ( C ((0 , × C ((0 , such that for ( ν, k ) ∈ M ( C ((0 , × C ((0 , × { u, d } , ˜ R ( ν, k )( B ) := (cid:90) C ((0 , × C ((0 , R ( u , u , k )( B ) ν ( d x , d x ) . hen for k ∈ { u, d } , ˜ R N (( X , X ) , k ) ◦ Q − N = ⇒ ˜ R (( u , u ) , k ) in M ( C ((0 , × C ((0 , .Proof. This is essentially the same as the proof of Proposition A.2.
Theorem A.11.
Suppose that ( Q N ( X bN, (0)) , Q N ( X aN, (0))) = ⇒ ( u b (0) , u a (0)) in law in C ((0 , + × C ((0 , + . Let ( X bN ( t ) , X aN ( t ) , m N ( t )) be dynamic microscopic models withinitial data ( X bN, (0) , X aN, (0) , m ) , and let ( u b ( t ) , u a ( t ) , m ( t )) be the dynamic macroscopic modelwith initial data ( u b (0) , u a (0) , m ) . Then (( Q N ( X bN,i ( N t ))) ∞ i =1 , ( Q N ( X aN,i ( N t ))) ∞ i =1 , ( τ iN,b /N ) ∞ i =1 , ( τ iN,a /N ) ∞ i =1 , ( m iN ) ∞ i =1 )= ⇒ (( u bi ) ∞ i =1 , ( u ai ) ∞ i =1 , ( τ ib ) ∞ i =1 , ( τ ia ) ∞ i =1 , ( m i ) ∞ i =1 (A.5) in law in D ([0 , ∞ ); C ((0 , N × D ([0 , ∞ ); C ((0 , N × [0 , ∞ ) N × [0 , ∞ ) N × R N .Proof. This follows the proof of Theorem A.4 and we refer to this for the details. We once againuse an inductive argument, and show that for every M ≥ Q N ( X bN,i ( N t ))) Mi =1 , ( Q N ( X aN,i ( N t ))) Mi =1 , ( τ iN,b /N ) M − i =1 , ( τ iN,a /N ) M − i =1 , ( m iN ) Mi =1 )= ⇒ (( u bi ) Mi =1 , ( u ai ) Mi =1 , ( τ ib ) M − i =1 , ( τ ia ) M − i =1 , ( m i ) Mi =1 ) (A.6)in law in D ([0 , ∞ ); C ((0 , N × D ([0 , ∞ ); C ((0 , N × [0 , ∞ ) N × [0 , ∞ ) N × R N , from which theresult follows. Given the inductive hypothesis, we use the Skorohod representation theorem andPropositions A.9 and A.3 to obtain convergence of the next rescaled stopping times in the sequence.It then follows that the next boundary position in the sequence also converges. By conditioningas in the proof of Theorem A.2 and making use of Proposition A.10, we can obtain convergence ofthe process up to the next stopping time, concluding the inductive argument. Proposition A.12.
Let g : D ([0 , ∞ ); C ((0 , N × [0 , ∞ ) N → D ([0 , ∞ ); C ((0 , such that g (( u i ) ∞ i =1 , ( t i ) ∞ i =1 )( t ) := ∞ (cid:88) j =1 u j t − j − (cid:88) i =1 t i (cid:40) j − (cid:80) i =1 t i ≤ t< j (cid:80) i =1 t i (cid:41) . Suppose that(i) t i > for every i .(ii) ∞ (cid:80) i =1 t i = ∞ . (iii) u i is continuous for every i Then g is continuous at the point (( u i ) ∞ i =1 , ( t i ) ∞ i =1 ) .Proof. This is essentially the same as the proof of Theorem A.5.34 roof of Theorem 3.6.
We can now conclude the proof of Theorem 3.6. We begin by Skorohodrepresenting the convergence in Theorem A.11. So we have that(( Q N ( X aN,i ( N t ))) ∞ i =1 , ( Q N ( X bN,i ( N t ))) ∞ i =1 , ( τ iN,a /N ) ∞ i =1 , ( τ iN,b /N ) ∞ i =1 , ( m iN ) ∞ i =1 ) → (( u ai ) ∞ i =1 , ( u bi ) ∞ i =1 , ( τ ia ) ∞ i =1 , ( τ ib ) ∞ i =1 , ( m i ) ∞ i =1 ) (A.7)in D ([0 , ∞ ); C ((0 , N × D ([0 , ∞ ); C ((0 , N × [0 , ∞ ) N × [0 , ∞ ) N × R N almost surely. By Propo-sition A.12, we clearly have that (( u ai ) ∞ i =1 , ( τ i ) ∞ i =1 ) and (( u bi ) ∞ i =1 , ( τ i ) ∞ i =1 ) are continuity points of themap g almost surely. Similarly, by Proposition A.6, (( τ i ) ∞ i =1 , ( m i ) ∞ i =1 ) is a continuity point for themap h almost surely. The result follows. References [1] F. Abergel and A. Jedidi,
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