A diffusion approximation for limit order book models

# A diffusion approximation for limit order book models

Ulrich Horst Humboldt-Universität zu Berlin, Germany  and  Dörte Kreher Humboldt-Universität zu Berlin, Germany
###### Abstract.

This paper derives a diffusion approximation for a sequence of discrete-time one-sided limit order book models with non-linear state dependent order arrival and cancellation dynamics. The discrete time sequences are specified in terms of an -valued best bid price process and an -valued volume process. It is shown that under suitable assumptions the sequence of interpolated discrete time models is relatively compact in a localized sense and that any limit point satisfies a certain infinite dimensional SDE. Under additional assumptions on the dependence structure we construct two classes of models, which fit in the general framework, such that the limiting SDE admits a unique solution and thus the discrete dynamics converge to a diffusion limit in a localized sense.

###### Key words and phrases:
Functional limit theorem, diffusion limit, scaling limit, convergence of stochastic differential equations, limit order book
###### 2010 Mathematics Subject Classification:
60F17, 91G80
This research was partially supported by CRC 649: Economic Risk. Moreover, part of this research was performed while the second author was visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. A previous version of this paper was entitled “A functional convergence theorem for interpolated Markov chains to an infinite dimensional diffusion with application to limit order books.”
\setitemize

leftmargin=20pt \setenumerateleftmargin=20pt

## 1. Motivation and setup

In modern financial markets almost all transactions are settled through limit oder books (LOBs). A LOB is a record of unexecuted orders awaiting execution. Stochastic analysis provides powerful tools for understanding the complex system of order aggregation and execution in limit order markets via the description of suitable scaling (“high-frequency”) limits. Scaling limits allow for a tractable description of the macroscopic LOB dynamics (prices and standing volumes) from the underlying microscopic dynamics (individual order arrivals and cancellations). In this paper we prove a novel functional convergence result for a class of Markov chains arising in microstructure models of LOBs to an infinite dimensional diffusion.

Scaling limits for LOBs have recently attracted considerable attention in the probability and finnacial mathematics literature. Depending on the scaling assumptions either fluid limits (cf. [6, 7, 8, 9]) or diffusion limits (cf. [1, 4, 18]) can be derived. Fluid limits for the full order book were first studied in  and afterwards in , where it was shown that under certain assumptions on the scaling parameters the sequence of discrete-time LOB models converges in probability to the solution of a deterministic differential equation. Although there is some work on probabilistic LOB models that assumes an SPDE or measure-valued dynamics for the volume process (cf. [10, 14]), there is little work on the derivation of a measure valued diffusion limit starting from a microscopic (“event-by-event”) description of the limit order book. Two exceptions are the particular models considered in  and . The work  extends the models in  and  by introducing additional noise terms in the pre-limit in which case the dynamics can then be approximated by an SPDE in the scaling limit. The papers [1, 8, 9] rely on the same scaling assumptions. Our work is motivated by the question whether under different scaling assumptions the same event-by-event dynamics can be approximated by a diffusion process in the high frequency regime without adding additional noise terms in the pre-limit.

### 1.1. The LOB dynamics

The one-sided LOB models considered in this paper are specified by a sequence of discrete time -valued processes , where for each , the non-negative one dimensional process specifies the dynamics of the best bid price, and the -valued process specifies the dynamics of the bid-side volume density function.

We fix some and introduce the scaling parameters and . They denote the tick-size, the impact of an individual order on the state of the book, and the time between two consecutive order arrivals, respectively. We put , and for all and . For all and we define the interval as

 I(n)(x):=[x(n)j,x(n)j+1)forx(n)j≤x

The initial best bid price is given by for some . The initial volume density function is given by a non-negative deterministic step function on the -grid. Following the modelling framework of  we assume that there are three events that change the state of the book: price increases (event ), price decreases (event ) and limit order placements, respectively cancellations (event ). In terms of the placement operator

 (1) M(n)k(⋅):=\mathbbm1C(ϕ(n)k)ω(n)kΔx(n)\mathbbm1I(n)(π(n)k)(⋅)

the dynamics of the one-sided LOB models can then be described by the following point process: for each and all ,

 (2) B(n)k=B(n)k−1+Δx(n)[\mathbbm1B(ϕ(n)k)−\mathbbm1A(ϕ(n)k)]v(n)k=v(n)k−1+Δv(n)M(n)k

where the event indicator function is a random variable taking values in the set , the -valued random variable specifies the size of a placement or cancellation , and the non-negative random variable specifies the location of a placement or cancellation.

### 1.2. Preview of the main results

In deriving a diffusion limit for the sequence of LOB models (2), the first challenge is to define a suitable convergence concept. While for any ,

 ∥∥\mathbbm1I(n)(π)∥∥L2(R+)=(Δx(n))1/2,

we have for any bounded ,

 ⟨\mathbbm1I(n)(π),f⟩L2(R+)=∫I(n)(π)f(x)dx=O(Δx(n)).

Hence, it seems impossible to formulate a scaling assumption with and that allows to prove convergence of the volume density functions to an -valued diffusion process. However, observe that for any we have

 ∥∥ ∥∥Δx(n)⌊⋅/Δx(n)⌋∑j=0\mathbbm1I(n)(π)(x(n)j)\mathbbm1[0,m](⋅)∥∥ ∥∥L2=Δx(n)(m−Δx(n)⌊πΔx(n)⌋)1/2=O(Δx(n))

and for any bounded also

 ⟨Δx(n)⌊⋅/Δx(n)⌋∑j=0\mathbbm1I(n)(π)(x(n)j),f\mathbbm1[0,m]⟩L2=Δx(n)∫mΔx(n)⌊πΔx(n)⌋f(x)dx=O(Δx(n)).

This suggests to study the convergence of the cumulated volume processes with

 (3) V(n)k(x):=Δx(n)⌊x/Δx(n)⌋∑j=0v(n)k(x(n)j),x∈R+,

instead of analyzing directly the convergence of the volume density functions. To do this we will choose a localized convergence concept, since the functions are not square integrable on the whole line.

Our main contribution is to establish a convergence concept and a convergence result for the sequence , . In particular, we state sufficient conditions that guarantee that (i) this sequence is relatively compact; (ii) any limit point solves an infinite dimensional SDE driven by a standard Brownian and a cylindrical Brownian motion; (iii) the limiting SDE has a unique solution.

Having established a convergence concept, the second major challenge is that the dynamics of the process , , is not given in standard SDE form, due to the event-by-event dynamics, and that the system can only be controlled by specifying the conditional distribution of the random variables and . Much of our work is, therefore, devoted to the identification of suitable integrands and semimartingale random measures such that can be represented as

 (4) S(n)(t)=S(n)0+∫t0G(n)(S(n)(u))dY(n)(u),t∈[0,T]

after continuous time-interpolation. Once the dynamics of the sequence , , has been brought into standard SDE form, it remains to study its convergence. The convergence of infinite dimensional stochastic integrals has been studied by several authors. Chao  and Walsh  consider semimartingale random measures as distribution valued processes in some nuclear space. Kallianpur and Xiong  prove diffusion approximations of nuclear space-valued SDEs. Their approach requires a dependence structure that is incompatible with our spatial pointwise dynamics, and is hence not applicable to our modelling framework. Jakubowski  provides convergence results for Hilbert space valued semimartingales under a uniform tightness condition. Kurtz and Protter  work with the same uniform tightness condition, but allow for a more general setting. Especially, they also study the convergence of solutions of stochastic differential equations in infinite dimension. The results are further extended by Ganguly  to study the convergence of infinite dimensional stochastic differential equations when the approximating sequence of integrators is not uniformly tight anymore.

Our proof relies on the results in . We first establish sufficient conditions that guarantee that the sequence , converges to some -semimartingale . Subsequently we prove that the sequence , satisfies a compactness property and converges in a localised sense to some function . Finally, we show that the sequence of stochastic differential equations in (4) converges in law in a localised sense to a solution to an SDE of the form

 (5) S(t)=S0+∫t0G(S(u))dY(u),t∈[0,T].

The challenge in proving the converges of the SDEs is the verification of the conditions in  on the integrators and coefficient functions of the approximating sequence, and the fact that our convergence concept localises in space, not time. Finally, we give sufficient conditions for the uniqueness of solutions to the above SDE. For instance, we show that uniqueness holds if only the drift but not the volatility is state-dependent.

### 1.3. Structure of the paper

The rest of the paper is structured as follows. In Section 2 we state conditions on the dynamics of the price processes that guarantee the converge of their normalized fluctuations to a standard Brownian motion. In Section 3 we state conditions on the dynamics of the order arrivals and cancelations that guarantee convergence of the standardized fluctuations of the volume processes to a cylindrical Brownian motion. While the analysis of the price is quite standard, deriving similar results for the volumes is much more tedious. First we show in Subsections 3.2 the convergence of the drift, volatility and correlation functions. Using an orthogonal decomposition of the covariance matrix we then establish in Subsection 3.3 a representation of the volume process as a discrete stochastic differential equation driven by “infinitely many discretised Brownian motions”. In Subsection 3.5 we prove the convergence in law of the “infinitely many discretised Brownian motions” to a cylindrical Brownian motion. In Section 4 we define the stochastic integrals and stochastic differential equations that describe the LOB dynamics and verify that the conditions from  are satisfied. This allows us to derive our results on the characterisation of the limiting LOB dynamics as solutions to an infinite dimensional SDE in Section 5. We conclude with two specific examples in which the LOB dynamics converges weakly to the unique solution of an infinite dimensional SDE.

### 1.4. Notation

For each we fix a probability space 111For ease of notation we will simply write and in the following instead of and , since it is clear from the context on which probability space we work. with filtration

 {∅,Ω(n)}=F(n)0⊂F(n)1⊂⋯⊂F(n)k⊂⋯⊂F(n)Tn⊂F(n).

We assume that the random vector is -measurable for all and . We define the Hilbert space

 E:=R×L2(R+;R),∥(X1,X2)∥E:=|X1|+∥X2∥L2

and its localized version

 Eloc:=R×L2loc(R+;R)

with

 L2loc(R+):={f:R+→R ∣∣∣ ∫m0f2(x)dx<∞ ∀ m∈N}.

Moreover, we define for all the -valued stochastic process via

 S(n)k:=(B(n)k,V(n)k),

where and were defined in equations (2) and (3). For all and we set

 δV(n)k :=V(n)k−V(n)k−1, δB(n)k :=B(n)k−B(n)k−1, δ^v(n)k(x) :=E(δV(n)k(x)∣∣F(n)k−1),  δ^B(n)k :=E(δB(n)k∣∣F(n)k−1), δ¯¯¯v(n)k(x) :=δV(n)k(x)−δ^v(n)k(x),δ¯¯¯¯B(n)k :=δB(n)k−δ^B(n)k.

W.l.o.g. we will assume that for all .

## 2. Fluctuations of the price process

In this section we analyse the fluctuations of the best bid price process . To this end, we introduce a fourth scaling parameter that controls the proportion of price changes among all events. The scaling limits in [1, 8, 9] require two time scales, a fast time scale for limit order placements and cancellations and a comparably slow time scale for price changes. The scaling parameter introduces the “slow” time scale.

###### Assumption 2.1.

For each there exist two functions and satisfying the boundary condition

 (6) (r(n)(s))2=Δx(n)p(n)(s)∀ s=(0,v)∈Eloc,

such that for all ,

 (7) P(ϕ(n)k∈{A,B} ∣∣F(n)k−1)=Δp(n)(r(n)(S(n)k−1))2a.s.

and

 (8) P(ϕ(n)k=B ∣∣F(n)k−1)−P(ϕ(n)k=A ∣∣F(n)k−1)=Δp(n)Δx(n)p(n)(S(n)k−1)a.s.

There exists such that for all and ,

 (9) r(n)(s)+(p(n)(s))+>η.

Note that the conditional distribution of the event variables is uniquely determined by equations (7) and (8). Moreover, equation (6) guarantees that the price process will always stay positive.

The next assumption controls the relative speed at which the different scaling parameters converge to zero. Since the discrete system dynamics are the same as in , we must use a different scaling to get a diffusion limit instead of a fluid limit. Intuitively, the average impact of all individual events must be of larger size to generate volatility. By comparing the scaling assumption from  with Assumption 2.2 below, we see that this is indeed the case.

###### Assumption 2.2.

For all ,

 Δt(n)=Δp(n)(Δx(n))2=(Δv(n))2=o(1).
###### Remark 2.3.

The fact that the conditional distribution of the event variables is uniquely determined by equations (8) and (7) is different from the corresponding assumption made in  to derive a large of large numbers in the high frequency regime. Indeed, while (7) can also be found in , (8) is the only important additional assumption - apart from the different scaling - which is needed to derive a diffusion dynamic for the price process in the high frequency limit. A similar assumption can also be found in .

The equations (7) and (8) of Assumption 2.1 yield together with Assumption 2.2 that for all and almost surely

 Δt(n)[r(n)(S(n)k−1)]2 = E[(δB(n)k)2∣∣∣F(n)k−1], Δt(n)p(n)(S(n)k−1) = E[δB(n)k∣∣F(n)k−1]=δ^B(n)k.

Let us define the process of the (nearly) normalized increments of as

 (10) δZ(n)k:=δ¯¯¯¯B(n)kr(n)(S(n)k−1),Z(n)k:=k∑j=1δZ(n)jfor% all k=1,…,Tn.

Then we may write for all ,

 (11) B(n)(t)=B(n)0+⌊t/Δt(n)⌋∑k=1δB(n)k=B(n)0+⌊t/Δt(n)⌋∑k=1[p(n)(S(n)k−1)Δt(n)+r(n)(S(n)k−1)δZ(n)k]

Through linear interpolation of the , we obtain the continuous time process

 Z(n)(t):=Tn∑k=0Z(n)k\mathbbm1[t(n)k,t(n)k+1)(t),t∈[0,T].
###### Theorem 2.4.

Under Assumptions 2.1 and 2.2, converges weakly in to a standard Brownian motion as .

###### Proof.

First note that equations (7) and (8) imply that for all and ,

 (12) −1≤Δx(n)p(n)(S(n)k−1)(r(n)(S(n)k−1))2≤1a.s.

Moreover, by definition

 (13) Δt(n)∣∣p(n)(S(n)k−1)∣∣≤E(∣∣δB(n)k∣∣∣∣F(n)k−1)≤Δx(n)\lx@stackreln→∞⟶0a.s.

Hence, for any

Second, (9) and (12) imply that for all and ,

 [r(n)(S(n)k−1)]−2 ≤ [r(n)(S(n)k−1)]−2\mathbbm1{r(n)(S(n)k−1)>η2}+[r(n)(S(n)k−1)]−2\mathbbm1{(p(n)(S(n)k−1))+>η2} ≤ 4η2+[r(n)(S(n)k−1)]−2\mathbbm1{[r(n)(S(n)k−1)]2>Δx(n)η2} ≤ 4η2+2Δx(n)ηa.s.

Therefore, there exists a deterministic sequence converging to zero such that for all ,

 (14) ∣∣δZ(n)k∣∣2=[Δx(n)(\mathbbm1B(ϕ(n)k)−\mathbbm1A(ϕ(n)k))−Δt(n)p(n)(S(n)k−1)]2[r(n)(S(n)k−1)]2≤2[(Δx(n))2+(Δt(n)p(n)(S(n)k−1))2]2η(2η+1Δx(n))≤cn a.s.

We conclude that for all ,

 ≤ cnε2⌊t/Δt(n)⌋∑k=1E∣∣δZ(n)k∣∣2≤tε2⋅cn→0,

i.e. the Lindeberg condition is satisfied. Therefore, the functional central limit theorem for martingale difference arrays (cf. Theorem 18.2 in ) implies that converges weakly to a standard Brownian motion. ∎

In order to obtain the convergence of the full price process in Section 5 below we also have to assume that the drift and volatility functions and satisfy a continuity condition and that they converge to some functions and as .

###### Assumption 2.5.
1. There exist functions and such that for all ,

 |p(s)|+r(s)≤C(1+|b|)

and for all ,

 sups=(b,v)∈Eloc∣∣p(n)((b∧m,v))−p((b∧m,v))∣∣+∣∣r(n)((b∧m,v))−r((b∧m,v))∣∣→0.
2. There exists such that for all and ,

 max{∣∣p(n)(s)−p(n)(˜s)∣∣,∣∣r(n)(s)−r(n)(˜s)∣∣}≤L(1+|b|+|˜b|)(1+∥∥v\mathbbm1[0,b∨˜b]∥∥L2+∥∥˜v\mathbbm1[0,b∨˜b]∥∥L2){∣∣b−˜b∣∣+∥∥(v−˜v)\mathbbm1[0,b∨˜b]∥∥L2}.

Assumption 2 is similar to a local Lipschitz assumption. It will play a key role in the proof of the main theorem later on. The following example illustrates the assumed dependence structure.

###### Example 2.6.

In order to model dependence on standing volumes we can integrate a Lipschitz continuous function against cumulated volumes standing to the left of the price process. If we suppose that has compact support in , then for all ,

 ∣∣⟨v(⋅+b)\mathbbm1[−b,0],h⟩−⟨˜v(⋅+˜b)\mathbbm1[−˜b,0],h⟩∣∣=∣∣⟨v,h(⋅−b)\mathbbm1[0,b]⟩−⟨˜v,h(⋅−˜b)\mathbbm1[0,˜b]⟩∣∣ ≤ ∣∣⟨v−˜v,h(⋅−˜b)\mathbbm1[0,˜b]⟩∣∣+∣∣⟨v\mathbbm1[0,b∨˜b],h(⋅−b)−h(⋅−˜b)⟩∣∣ ≤ ∥h∥L2⋅∥∥(v−˜v)\mathbbm1[0,b∨˜b]∥∥L2+∥∥v\mathbbm1[0,b∨˜b]∥∥L2⋅L∥∥\mathbbm1[0,b∨˜b](b−˜b)∥∥L2 ≤ ∥h∥L2⋅∥∥(v−˜v)\mathbbm1[0,b∨˜b]∥∥L2+L∥∥v\mathbbm1[0,b∨˜b]∥∥L2(1+|b|+|˜b|)∣∣b−˜b∣∣.

Now if are Lipschitz continous functions, we may define for all ,

 p(n)(s):=P(⟨v(⋅+b)\mathbbm1[−b,0],h⟩),r(n)(s):=R(⟨v(⋅+b)\mathbbm1[−b,0],h⟩)

and the so defined functions and satisfy Assumption 2.

## 3. Fluctuations of the volume process

In this section we analyze the fluctuation of the infinite dimensional volume process . In a first step we compute its conditional moments and prove their convergence as . Subsequently, we represent it as the solution to a stochastic differential equations driven by infinite dimensional martingale that converges in distribution to a cylindrical Brownian motion as . Since is not an -valued process, but only -valued, we need to localize the analysis.

We make the following assumption on the joint distribution of the random variables and .

###### Assumption 3.1.

There exists an such that for all and ,

 (15) P(ω(n)k∈[−M,M], π(n)k∈[0,∞))=1.

For every there exist two measurable functions such that for all and all ,

 E((ω(n)k)2\mathbbm1C(ϕ(n)k)\mathbbm1D(π(n)k) ∣∣∣ F(n)k−1)=∫Dg(n)(S(n)k−1;y)dya.s.

and

 E(ω(n)k\mathbbm1C(ϕ(n)k)\mathbbm1D(π(n)k) ∣∣ F(n)k−1)=Δv(n)∫Dh(n)(S(n)k−1;y)dya.s.

According to Assumptions 2.1 and 3.1 the process is a homogeneous Markov chain for each . Furthermore, (15) and Assumption 2.2 imply that for all , , and ,

 ∥∥δV(n)k\mathbbm1[0,m]∥∥2L2 ≤ (Δv(n))2M2∥∥ ∥∥⌊⋅/Δx(n)⌋∑j=0\mathbbm1I(n)(π(n)k)(x(n)j)\mathbbm1[0,m]∥∥ ∥∥2L2 ≤ Δt(n)M2ma.s.

and therefore for all also

 (16) ∥∥δ¯¯¯v(n)k\mathbbm1[0,m]∥∥2L2≤∥∥δ^v(n)k\mathbbm1[0,m]∥∥2L2+∥∥δV(n)k\mathbbm1[0,m]∥∥2L2≤E(∥∥δV(n)k\mathbbm1[0,m]∥∥2L2∣∣∣F(n)k−1)+∥∥δV(n)k\mathbbm1[0,m]∥∥2L2≤2M2mΔt(n)a.s.

The next two assumptions deal with the convergence and continuity of and .

###### Assumption 3.2.
1. There exists a measurable function satisfying

 infs∈Elocg(s;y)>0∀ y∈R+

such that

 sups∈Eloc∫∞0∣∣g(n)(s;y)−g(s;y)∣∣dy→0.
2. There exists an such that for all and ,

 ∫∞0∣∣g(n)(s;y)−g(n)(˜s;y)∣∣dy≤L(1+|b|+]˜b])(1+∥∥v\mathbbm1[0,b∨˜b]∥∥L2+∥∥˜v\mathbbm1[0,b∨˜b]∥∥L2){∣∣b−˜b∣∣+∥∥(v−˜v)\mathbbm1[0,b∨˜b]∥∥L2}.

The next assumption is key to the derivation of a diffusion limit for the -valued functions . It states that order placements and cancellations are expected to be approximately of the same size and that the expected disbalance between both also scales in . This guarantees that the cumulated volume process will not explode when passing to the scaling limit.

###### Assumption 3.3.
1. There exists a measurable function satisfying

 sups∈Eloc∫∞0|h(s;y)|2dy<∞

such that

 sups∈Eloc∫∞0∣∣h(n)(s;y)−h(s;y)∣∣2dy→0.
2. There exists an such that for all and ,

 (∫∞0∣∣h(n)(s;y)−h(n)(˜s;y)∣∣2dy)1/2≤L(1+|b|+|˜b|)(1+∥∥v\mathbbm1[0,b∨˜b]∥∥L2+∥∥˜v\mathbbm1[0,b∨˜b]∥∥L2){∣∣b−˜b∣∣+∥∥(v−˜v)\mathbbm1[0,b∨˜b]∥∥L2}.

### 3.1. Basis functions

Our goal is to represent the volume function as a stochastic differential equation driven by an infinite dimensional martingale whose increments are orthogonal across different basis functions of . We choose the Haar basis, i.e. we specify the basis functions as follows: for each we set . Moreover, we set for all ,

 gkl(x):=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩2l/2: x∈[k2−l,(k+12)2−l)−2l/2: x∈[(k+12)2−l,(k+1)2−l)0: else.

To define the we now reorder the in a diagonal procedure:

 f1:=g0−1, f2:=g1−1, f3:=g00, f4:=g2−1, f5:=g10, f6:=g01, …

In the following we denote by and the indeces such that .

Let us define for each the functions and via

 Fi(y):=∫∞yfi(x)dx,F(n)i(y):=∫∞Δx(n)