Scaling limit of a limit order book model via the regenerative characterization of Lévy trees
Abstract.
We consider the following Markovian dynamic on point processes: at constant rate and with equal probability, either the rightmost atom of the current configuration is removed, or a new atom is added at a random distance from the rightmost atom. Interpreting atoms as limit buy orders, this process was introduced by Lakner et al. [23] to model a onesided limit order book. We consider this model in the regime where the total number of orders converges to a reflected Brownian motion, and complement the results of Lakner et al. [23] by showing that, in the case where the mean displacement at which a new order is added is positive, the measurevalued process describing the whole limit order book converges to a simple functional of this reflected Brownian motion.
The cornerstone of our approach is the regenerative characterization of Lévy trees proved in Weill [34], which provides an elegant and intuitive proof strategy which we unfold. Moreover, the proofs rely on new results of independent interest on branching random walks with a barrier.
Contents
1. Introduction
Context
The limit order book is a financial trading mechanism that facilitates the buying and selling of securities by market participants. It keeps track of orders made by traders, which makes it possible to fulfill them in the future. For instance, a trader may place an order to buy a security at a certain level . If the price of the security is larger than when the order is placed, then the order is kept in the book and will be fulfilled if the price of the security falls below . Due to its growing importance in modern electronic financial markets, the limit order book has attracted a significant amount of attention in the applied probability literature recently. One may consult, for instance, the survey paper by Gould et al. [15] for a list of references. Several mathematical models of the limit order book have been proposed in recent years, ranging from stylized models such as Yudovina [35] to more complex models such as those proposed by Cont et al. [10] and Garèche et al. [13]. Broadly speaking, these models may be categorized as being either discrete and closely adhering to the inherent quantized nature of the limit order book, or as being continuous in order to better capture the high frequency regime in which the order book typically evolves.
In the present paper, we attempt to bridge the gap between the discrete and continuous points of view by establishing the weak convergence of a discrete limit order book model to a continuous one in an appropriately defined high frequency regime where the speed at which orders arrive grows large. Similar weak convergence results have recently been considered in various works. However, most of the time, only finitedimensional statistics of the limit order book are tracked, such as the bid price (the highest price associated with a buy order on the book), the ask price (defined in a symmetric way from the limit sell orders) or the spread (equal to the difference between these two quantities), see for instance [1, 6, 8, 9, 22]. In contrast, in the present paper we establish the convergence of the full limit order book which we model by a measurevalued process. This approach has also been taken in [30]. In [16] the authors also model the entire book, but with a different approach, namely, they track the density of orders which they see as random elements of an appropriate Banach space.
Relation with previous work
The particular discrete model that we study is a variant of the limit order book model proposed by Lakner et al. [23]. We are interested in a onesided limit order book with only limit buy orders, which are therefore fulfilled by market sell orders. In this model, limit buy orders and market sell orders arrive according to two independent Poisson processes. Each time a limit buy order arrives, it places an order on the book at a random distance from the current existing highest buy price. The sequence of differences between the arriving limit buy orders and the current highest buy price forms an i.i.d. sequence of random variables with common distribution identical to the distribution of a random variable . This sequence of differences is also assumed to be independent of the two Poisson processes according to which limit and market orders arrive (cf. next section for a precise definition).
Under the assumption that traders on average place their limit buy orders at a price lower than the bid price, it was shown in [23] that, under some appropriate rescaling in the high frequency regime, the entire limit order book is asymptotically concentrated at the bid price, and that the latter converges to a monotonically decreasing process.
In the present paper we complement this result by considering the case . In stark contrast to the case , our main result (Theorem 2.1 below) shows that when , the price process converges to a reflected Brownian motion and that at any point in time the measure describing the book puts mass on a nonempty interval.
It is worthwhile to note that in the high frequency regime that we consider, the total number of orders in the book converges to a reflected Brownian motion and that in both cases and , the limiting measure is a deterministic function of this reflected Brownian motion. However, this deterministic function changes completely depending on the sign of , and this interesting dichotomy reflects the asymmetric nature of the discrete limit order book model itself. Another interpretation of this dichotomy is discussed at the end of Section 6.
Link with Lévy trees
The most demanding part of the proof of our main result is to show that the price process converges to a reflected Brownian motion. To prove this, we exploit an unexpected connection between our model and Lévy trees. Informally speaking, Lévy trees are the continuous scaling limits of Galton Watson trees, and both Lévy trees and Galton Watson trees are coded by socalled contour functions. The definition of a Galton Watson tree translates immediately to some regenerative property, say (R’), satisfied by its contour function: informally, successive excursions above a certain level are i.i.d., see Section 2 for more details. In the discrete setting this regenerative property is easily seen to actually characterize contour functions of Galton Watson trees. Weill [34] extended this characterization to the continuous setting, i.e., showed that any continuous stochastic process satisfying some regenerative property (R) (the analog of (R’) for continuous processes) must be the contour function of some Lévy tree.
The above mentioned result lies at the heart of the proof of our main result. Indeed, it follows by simple inspection that, in our model, the price process satisfies a regenerative property very close to (R’) and that, in the asymptotic regime that we are interested in, this difference should be washed out in the limit. This suggests an elegant way to study the asymptotic behavior of the price process, namely by showing that any accumulation point of the price process must satisfy the continuous regenerative property (R). We will then know thanks to [34] that any accumulation point must be the contour function of a Lévy tree, thereby drastically reducing the possible accumulation points. A few additional arguments will then make it possible to deduce that, among this class of stochastic processes, the limit process must actually be a reflected Brownian motion and that the whole measurevalued process converges to a simple functional thereof.
Although this proof strategy is very attractive at a conceptual level, many technical details need to be taken care of along the way and Section 5 of the paper is dedicated to working these details out. A key argument which is used repeatedly in the proofs is the coupling established in Simatos [33] between the model studied here and a branching random walk. Leveraging this coupling leads us to derive new results of independent interest on branching random walks with a barrier, which are gathered in Appendix A.
As an aside, it is interesting to note that the discrete regenerative property (R’) is satisfied by a classical queueing system, namely the LastInFirstOut (LIFO) queue, also called LastComeFirstServed (LCFS). This property was for instance exploited in NúñezQueija [29] to study its stationary behavior. Combined with the above reasoning it provides a new interpretation for the results by Limic [27, 28], where it is proved that the scaling limit of the LIFO queue is the height process of a Lévy tree.
Organization of the paper
Section 2 introduces basic notation, presents our main result (Theorem 2.1) and discusses more formally the connection with Lévy trees. Before proceeding to the proof of Theorem 2.1 in Section 5, we introduce in Section 3 the coupling of Simatos [33], and additional notation together with preliminary results in Section 4. We conclude the paper by discussing in Section 6 possible extensions of our results. Finally, as mentioned above, some results of independent interest on branching random walks, which we need along the way, are gathered and proved in Appendix A.
Acknowledgements
F. Simatos would like to thank N. Broutin for useful discussions about branching random walks that lead to the results of Appendix A.
2. Model and main result
Model and main result
Let be the set of positive measures on . We equip with the weak topology and consider the class of càdlàg mappings from to , which we endow with the Skorohod topology. Let be the zero measure, be the Dirac mass at and be the set of finite point measures, i.e., measures with finite support and of the form for some integers . For a measure let be the supremum of its support:
with the convention ; will be called the price of the measure , and an atom of will be referred to as an order.
We use the canonical notation and denote by the canonical valued process. Let be the law of the valued (strong) Markov process started at and with generator given by
where for , and where and , a realvalued random variable, are the only two parameters of the model under consideration. In words, the dynamic is as follows. We are given two independent Poisson processes, each of intensity . When the first one rings, a new order is added to the process and is located at a distance distributed like to the current price, independently from everything else ( will sometimes be referred to as the displacement of the newly added order). Note however that an order cannot be placed in the negative halfline, and so an order with displacement is placed at (this boundary condition will be discussed in Section 6). When the second Poisson process rings and provided that at least one order is present, an order currently sitting at the price is removed (it does not matter which one).
Let be the law of under , where acts on measures as follows:
(2.1) 
In the sequel we will omit the subscript when the initial state is the empty measure , i.e., we will write and for and , respectively, with their corresponding expectations and . For convenience we will also use and to denote the probability and expectation of other generic random variables (such as when we write , or when we consider random trees).
Let . In the sequel we will denote by for the only measure in such that . Let in the sequel be a standard Brownian motion reflected at and . The following result, which is the main result of the paper, shows that converges weakly to a measurevalued process which can simply be expressed in terms of .
Theorem 2.1.
Assume that and that for some . Then as , converges weakly to the probability measure under which is equal in distribution to and for each is absolutely continuous with respect to Lebesgue measure with density , i.e.,
(2.2) 
Remark.
We will prove more than is stated, namely, we will show that converges jointly with its mass and price processes, and also with their associated local time processes at (see Lemma 5.2).
Link with Lévy trees: detailed discussion
The following lemma is at the heart of our approach to prove Theorem 2.1. Let in the sequel be the set of realvalued càdlàg functions with domain and for . We call excursion, or excursion away from , a function with and for all (note that we only consider excursions with finite length). We call height of an excursion its supremum, and denote by the set of excursions. For and we say that the function is an excursion of above level if , for every and .
Lemma 2.2.
Under , the sequence of successive excursions of above level , for any integer , are i.i.d., with common distribution the first excursion of away from under .
Proof.
Consider under for any with that only puts mass on integers. Then when the first excursion (of ) above begins, the price is at and an order is added at . Thus if is the left endpoint of the first excursion above , must be of the form with . This excursion lasts as long as at least one order sits at , and if is the right endpoint of the first excursion above , then what happens during the time interval above is independent from and is the same as what happens above during the first excursion of away from under . Moreover, only puts mass on integers and satisfies , so that thanks to the strong Markov property we can iterate this argument. The result therefore follows by induction. ∎
Remark.
For let be defined by , and call an excursion of above level if is an excursion of above level . Then the above proof actually shows that the successive excursions above level of are i.i.d., with common distribution the first excursion above of under .
Lemma 2.2 is at the heart of our proof of Theorem 2.1. Indeed, this regenerative property is strongly reminiscent of Galton Watson branching processes. More precisely, consider a stochastic process with finite length and continuous sample paths, that starts at , increases or decreases with slope and only changes direction at integer times.
For integers and and conditionally on having excursions above level , let be these excursions. Then is the contour function of a Galton Watson tree if and only if for each and , the are i.i.d. with common distribution . Indeed, can always be seen as the contour function of some discrete tree. With this interpretation, the successive excursions above of code the subtrees rooted at nodes at depth in the tree. The being i.i.d. therefore means that the subtrees rooted at a node at depth in the tree are i.i.d.: this is precisely the definition of a Galton Watson tree.
The difference between this regenerative property and the regenerative property satisfied by under and described in Lemma 2.2 is that, when conditioned to belong to the same excursion away from , consecutive excursions of above some level are neither independent, nor identically distributed. If for instance we condition some excursion above level to be followed by another such excursion within the same excursion away from , this biases the number of orders put in during the first excursion above . Typically, one may think that more orders are put in in order to increase the chance of the next excursion above to start soon, i.e., before the end of the current excursion away from .
However, this bias is weak and will be washed out in the asymptotic regime that we consider. Thus it is natural to expect that under , properly renormalized, will converge to a process satisfying a continuous version of the discrete regenerative property satisfied by the contour function of Galton Watson trees.
Such a regenerative property has been studied in Weill [34], who has showed that it characterizes the contour process of Lévy trees (see for instance Duquesne and Le Gall [11] for a background on this topic). Thus upon showing that this regenerative property passes to the limit, we will have drastically reduced the possible limit points, and it will remain to show that, among the contour processes of Lévy trees, the limit that we have is actually a reflected Brownian motion. From there, a simple argument based on local time considerations allows us to conclude that Theorem 2.1 holds.
In summary, our proof of Theorem 2.1 will be divided into four main steps: showing tightness of ; showing, based on Lemma 2.2, that for any accumulation point , under satisfies the regenerative property studied in Weill [34] (most of the proof is devoted to this point); arguing that among the contour processes of Lévy trees, under must actually be a reflected Brownian motion; showing that under has density with respect to Lebesgue measure.
3. Coupling with a branching random walk
In this section we introduce the coupling of [33] between our model and a particular random walk with a barrier. As mentioned in the Introduction, this coupling plays a crucial role in the proof of Theorem 2.1. Let be the set of colored, labelled, rooted and oriented trees. Trees in are endowed with the lexicographic order. Thus in addition to its genealogical structure, each edge of a tree has a realvalued label and each node has one of three colors: either white, green or red.
In the sequel we write to mean that is a node of T, and we denote by the root of T, by its size (the total number of nodes) and by its height. Nodes inherit labels in the usual way, i.e., the root has some label and the label of a node that is not the root is obtained recursively by adding to the label of its parent the label on the edge between them. If we write for the label of (in T), for the depth of (so that, by our convention, and ) and for for the node at depth on the path from the root to (so that and ). Also, is the largest label in T, is the green node in T with largest label, with if T has no green node and in case several nodes have the largest label, is the last one, and is the point measure that records the labels of green nodes:
We say that a node is killed if the label of is than the label of the root, and if the label of every other node on the path from the root to has a label to the one of the root. Let be the set of killed nodes:
and consider the tree obtained from T by removing all the descendants of the killed nodes (but keeping the killed nodes themselves), and the tree obtained from by applying the map to the label of every node in . Note that since is a subtree of T, we always have .
Let be the operator acting on a tree as follows. If T has no green node then . Else, changes the color of one node in T according to the following rule:

if has at least one white child, then its first white child becomes green;

if has no white child, then becomes red.
Let be the th iterate of , i.e., is the identity map and , and let also . We will sometimes refer to the process as the exploration of the tree T.
Consider a tree such that all the nodes are white, except for the root which is green. For such a tree, the dynamic of is such that is the smallest at which the nodes of are red, the nodes of are green and the other nodes are still white. It has taken one iteration of to make the nodes of green, and two to make the nodes of red (first each of them had to be made green), except for the root which was already green to start with. Thus for such a tree we have .
Let finally for be the following random tree:

its genealogical structure is a (critical) Galton Watson tree with geometric offspring distribution with parameter ;

and labels on the edges are i.i.d., independent from the genealogical structure, and with common distribution ;

all nodes are white, except for the root which is green.
Because of the last property and the preceding remark, we have
(3.1) 
Note that since , we have , and in particular . The following result is a slight variation of Theorem in Simatos [33], where the same model in discretetime and without the boundary condition (i.e., an order may be added in the negative halfline) was studied. The intuition behind this coupling is to create a genealogy between orders in the book, a newly added order being declared the child of the order corresponding to the current price, see Section in Simatos [33] for more details.
Theorem 3.1.
[Theorem in [33]] Let be any integer and be the endpoints of the first excursion of above level . Then the process under and embedded at jump epochs is equal in distribution to the process .
Ambient tree
Thanks to this coupling, we can see any piece of path of corresponding to an excursion of the price process above some level as the exploration of some random tree : we will sometimes refer to this tree as the ambient tree. Note that the ambient tree of an excursion above , say , is a subtree of the ambient tree of the excursion above containing . Moreover, the remark following Lemma 2.2 implies that the ambient trees corresponding to successive excursions above some given level are i.i.d..
Exploration time
Theorem 3.1 gives, via (3.1), the number of steps needed to explore the ambient tree, say . However, we are interested in in continuous time. Since jumps in under occur at rate , independent from everything else, the length of the corresponding excursion is given by , where, here and in the sequel, is a random walk with step distribution the exponential random variable with parameter , independent from the ambient tree .
More generally, we will need to control the time needed to explore certain regions of , which will translate to controlling for some random times defined in terms of , and thus independent from . As it turns out, the random variables that need be considered have a heavy tail distribution. Since on the other hand jumps of are lighttailed, the approximation will accurately describe the situation. Let us make this approximation rigorous: for the upper bound, we write
Then, a large deviations bound shows that with . Carrying out a similar reasoning for the lower bound, we get
(3.2) 
with .
4. Notation and preliminary remarks
4.1. Additional notation and preliminary remarks
We will write in the sequel , , , for , , and , respectively, and denote by , etc, the corresponding expectations. Remember that we will also use and to denote the probability and expectation of other generic random variables (such as when we write ). In the sequel it will be convenient to consider some arbitrary probability measure on and to write to mean that the law of under converges weakly to the law of under ( and are measurable functions of the canonical process). When we will have proved the tightness of , then we will fix to be one of its accumulation points, but until then remains arbitrary. Let for be the mass of , i.e., . If is continuous, we will denote by the function defined for by .
We will need various local time processes at . First of all, denotes the Lebesgue measure of the time spent by the price process at . For discrete processes, i.e., under , we will also need the following local time processes at of and :
For the continuous processes that will arise as the limit of and , we consider the operator acting on continuous functions as follows:
We will only consider applied at random processes equal in distribution to for some , in which case this definition makes sense and indeed leads to a local time process at . Note that for any and any for which is welldefined, we have . Moreover, according to Tanaka’s formula the canonical semimartingale decomposition of is given by
(4.1) 
where is a standard Brownian motion.
In the sequel we will repeatedly use the fact that the process under (or ) is regenerative at , in the sense that successive excursions away from are i.i.d.. Note also that the time durations between successive excursions away from are also i.i.d., independent from the excursions, with common distribution the exponential random variable (with parameter under , and under ).
Moreover, jumps of under have size , and so if under converges weakly, then the limit must be almost surely continuous (see for instance Theorem in Billingsley [5]).
Let and for be the shift and stopping operators associated to , i.e., and . Since by the previous remark, accumulation points of under are continuous, these operators are continuous in the following sense (see for instance Lambert et al. [25, Lemma ]).
Lemma 4.1 (Continuity of the shift and stopping operators).
Consider some arbitrary random times . If , then .
We will finally need various random times. For and let
Note that and are the endpoints of the excursion of straddling , where we say that an excursion straddles if its endpoints satisfy . For we also define and
so that are the endpoints of the first excursion of above level with height and is its length. Note that, in terms of trees, the interval corresponds to the exploration of a tree distributed like conditioned on , since the height of the excursion corresponds to the largest label in the ambient tree. Also, it follows from the discussion at the end of Section 3 that is equal in distribution to under the same conditioning.
4.2. An aside on the convergence of hitting times
At several places in the proof of Theorem 2.1 it will be crucial to control the convergence of hitting times. For instance, we will need to show in the fourth step of the proof that if , then for any . Let us explain why, in order to show that , it is enough to show that for any ,
(4.2) 
Let us say that goes across if for every , and let . Then, the following property holds (see for instance Proposition VI.. in Jacod and Shiryaev [17] or Lemma in Lambert and Simatos [24]): if , then .
On the other hand, the complement of is precisely the set of discontinuities of the process . Since is càglàd, as the leftcontinuous inverse of the process , the set is at most countable, see for instance Billingsley [5, Section ]. Gathering these two observations, we see that for all outside a countable set. Then, writing for any
gives
Since for all outside a countable set, and since for those we have for all ’s outside a countable set, we obtain for all outside a countable set
Next, observe that as , almost surely. Indeed, decreases as , and its limit must satisfy , since , and also , since and is almost surely continuous. Thus letting first and then in the previous display, we obtain by (4.2)
which shows that by the Portmanteau theorem. This reasoning, detailed for and used in the proof of Lemma 5.6, will also be used in Section 5.4 to control the asymptotic behavior of the hitting times , and .
We will also use the following useful property: if and converge weakly, then the convergence actually holds jointly. The reasoning goes as follows. If and under converge to and under , then under is tight (we always consider the product topology). Let be any accumulation point.
Since projections are continuous, is equal in distribution to under , in particular it is almost surely continuous, and is equal in distribution to under , in particular it is almost surely . Further, assume using Skorohod’s representation theorem that is a version of under which converges almost surely to . Since and is continuous, we get and thus, since , . Since these two random variables are both equal in distribution to under , they must be (almost surely) equal. This shows that is equal in distribution to under , which uniquely identifies accumulation points.
This reasoning applies to all the hitting times considered in this paper, in particular to , and . Thus, once we will have shown the convergence of and, say, , then we will typically be in position to use Lemma 4.1 and deduce the convergence of and .
4.3. Convention
In the sequel we will need to derive numerous upper and lower bounds, where only the asymptotic behavior up to a multiplicative constant matters. It will therefore be convenient to denote by a strictly positive and finite constant that may change from line to line, and even within the same line, but which is only allowed to depend on and the law of .
5. Proof of Theorem 2.1
We decompose the proof of Theorem 2.1 into several steps. The coupling of Theorem 3.1 makes it possible to translate many questions on to questions on , and in order to keep the focus of the proof on , we postpone to the Appendix A the proofs of the various results on which we need along the way.
At a high level, it is useful to keep in mind that, since , the law of large numbers prevails and the approximation describes accurately enough (for our purposes) the labels in the tree . In some sense, most of the randomness of lies in its genealogical structure, and the results of the Appendix A aim at justifying this approximation.
Note that similar results than the ones we need here are known in a more general setting, but for the tree without the barrier, i.e., for instead of , see, e.g., Durrett et al. [12] and Kesten [21].
We begin with a preliminary lemma: recall that is a reflected Brownian motion, that and that for any .
Lemma 5.1.
As , under converges weakly to
(5.1) 
Moreover,
(5.2) 
Proof.
By definition, under is a critical queue with input rate , which is wellknown to converge under to . Further, is the finite variation process that appears in its canonical (semimartingale) decomposition, and standard arguments show that it converges, jointly with , to the finite variation process that appears in the canonical decomposition of , equal to by (4.1). Dividing by we see that under converges to . This shows that under converges weakly to .
We now show that under converges weakly to jointly with and . Since under is equal to under , it is enough to show that as , almost surely. Indeed, this would imply that under converges in the sense of finitedimensional distributions to (jointly with and ), and so, since and are continuous and increasing, Theorem VI.. in Jacod and Shiryaev [17] would imply the desired functional convergence result.
Let , where stands for the rightcontinuous inverse of . The composition with makes evolve only when the price is at . Under and while the price is at , the dynamic of is as follows:

increases by one at rate (which corresponds to an order with a displacement being added) and decreases by one at rate , provided (which corresponds to an order being removed);

when an order with displacement is added, which happens at rate , the price makes an excursion away from . When it comes back to , resumes evolving and, by the coupling, a random number of orders distributed like and independent from everything else have been added at .
Thus we see that under is stochastically equivalent to a singleserver queue, with two independent Poisson flows of arrivals: customers arrive either one by one at rate , or by batch of size distributed according to at rate . Then, customers have i.i.d. service requirements following an exponential distribution with parameter . In particular, the load of this queue is which by (A.4) is equal to . Since , is positive recurrent and in particular, the longterm average idle time is equal to one minus the load, i.e.,
(5.3) 
Fix on the other hand some : then
which, combined with (5.3), proves that and achieves the proof of the convergence of .
It remains to prove (5.2): since spends more time at when started empty (this can be easily seen with a coupling argument), we have which gives the uniformity in . Further, since as mentioned previously is a martingale, we have . Since is a reflected critical random walk with jump size and jump rates , one easily proves that which gives the desired result by CauchySchwarz inequality. ∎
5.1. I don’t know what to do with this! : tightness of
To show the tightness of , it is enough to show that under is tight, and that for each continuous which is infinitely differentiable with a compact support, under is tight (recall that ), see for instance Theorem in RoellyCoppoletta [32]. The tightness of is a direct consequence of Lemma 5.1, and so it remains to show the tightness of . First of all, note that jumps of under are upper bounded by , and so we only need to control the oscillations of this process (see for instance the Corollary on page in Billingsley [5]). Using standard arguments, we see that the process is a special semimartingale with canonical decomposition
where is the generator of , given for any and any function by
and is a local martingale with predictable quadratic variation process given by , with
(see, e.g., Lemma VIII. in Jacod and Shiryaev [17]). In particular, we have
and
from which it follows that
Thus there exists a finite constant , that only depends on , the law of and , such that for any finite stopping time and any we have
where the last inequality follows from (5.2) combined with the strong Markov property at time . Similarly, and these upper bounds imply the tightness of by standard arguments for the tightness of a sequence of semimartingales, see for instance Theorem VI.. in Jacod and Shiryaev [17], or Theorem in RoellyCoppoletta [32].
We now know that is tight: it remains to identify accumulation points. As planned in Section 4.1, we now let be an arbitrary accumulation point of