A Scaling Limit for Limit Order Books Driven by Hawkes Processes

A Scaling Limit for Limit Order Books Driven by Hawkes Processes Ulrich Horst∗ and Wei Xu†

arXiv:1709.01292v1 [q-fin.MF] 5 Sep 2017

September 6, 2017

Abstract In this paper we derive a scaling limit for an infinite dimensional limit order book model driven by Hawkes random measures. The dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator. With our choice of scaling the dynamics converges to a coupled SDE-ODE system where limiting best bid and ask price processes follows a diffusion dynamics, the limiting volume density functions follows an ODE in a Hilbert space and the limiting order arrival and cancellation intensities follow a Volterra-Fredholm integral equation.

1

Introduction

A significant part of financial transactions is nowadays carried out through electronic limit order books (LOBs). A LOB is a record of unexecuted orders awaiting execution. From a mathematical perspective, LOBs are infinite-dimensional complex interactive stochastic processes. Incoming limit orders can be placed at infinitely many different price levels, and incoming market orders are matched against standing limit orders according to a set of priority rules. Stochastic analysis provides powerful tools to describe the complex dynamics and interdependencies of the order arrivals and their impact on the state of the order book through suitable scaling limits. In this paper we prove a novel scaling result for LOBs that are driven by Hawkes random measures. Hawkes random measures capture the empirically well documented clustering and cross-dependencies between different order arrivals and cancellations. With our choice of scaling the limiting dynamics of the LOB can be described by a fully coupled SDE-ODE system. The dynamics of the best bid and ask prices follows an SDE, the dynamics of the volume density functions follows an ODE on a Hilbert space, and the dynamics of the order arrival and cancellation intensities follows a Volterra-Fredholm integral equation. Scaling limits for limit order books have received considerable attention in the financial mathematics literature in recent years. When the analysis of the order book is limited to prices or prices and aggregate volumes (e.g. at the top of the book) as in [1, 6, 10], then the limiting dynamics can naturally be described by ordinary differential equations or real-valued diffusion processes, depending on the choice of scaling. The analysis of the full book including the distribution of standing volume across different price levels is much more complex. Horst and Paulsen [20] and Horst and Kreher [19] where the first to obtain fluid limits for the full LOB dynamics. Starting from a microscopic event-by-event description of the LOB, they proved convergence of the price-volume process to coupled ODE-PDE systems. Their scaling limits required two time scales: a fast time scale for cancelations and limit order placements outside the spread, and a comparably slow time scale for market order arrivals and limit order placements in the spread. The different times scales had at least two drawbacks: first, they imply that the proportion of market orders and spread placements is negligible in the limit; second, as shown in the recent paper [18] they make it impossible to obtain a non-degenerate second-order approximation for the full LOB dynamics. Our scaling limit does not require different time scales. A model similar to [19, 20] has been studied by Gao and Deng [13]. They derived a deterministic ODE limit using weak convergence in the space of positive measures on a compact interval. Lakner et al. [27] derived a ∗ Department of Mathematics, and School of Business and Economics, Humboldt-Universit¨ at zu Berlin Unter den Linden 6, 10099 Berlin, Germany; email: [email protected] † Department of Mathematics, Humboldt-Universit¨ at zu Berlin Unter den Linden 6, 10099 Berlin, Germany; email: [email protected]

1

high frequency limit for a one-sided order book model under the assumption that on average investors place their limit orders above the current best ask price. The opposite case when orders are placed in the spread with higher probability is analyzed in [26], where the authors use a coupling between a simple one-sided limit order book model and a branching random walk to characterize the diffusion limit. Bayer et al. [5] extends the models in [19, 20] 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. With a different choice of scaling an SPDE limit for LOB models has recently been established in [17]. Macroscopic SPDE models of limit order markets were studied in [21, 25]. These models describe the volume dynamics by exogenous SPDEs while [17] endogenously derived a semi-martingale random measure driving volumes from a microscopic approach. There is considerable empirical evidence that the state of the order book, especially order imbalance at the top of the book, has a noticeable impact on order dynamics; see [7, 8] and references therein. Many of the aforementioned LOB models therefore allow for a dependence of the order arrival dynamics on the current state of the book. There is also empirical evidence of clustering of and cross-dependencies between order arrivals; see [9, 16] and references therein. Hawkes processes provide a powerful tool to model clustering and crossdependencies of events. They were first introduced in [14, 15] and have since been applied in many areas, ranging from earthquake modelling [28] to financial analysis [12]. Recently, they have been extensively used to model the dynamics of bid and ask prices in limit order markets [2, 3, 4, 23, 29, 32]. In this paper, we derive a limit result for LOBs driven by Hawkes random measures. Our framework allows us to model the clustering of many small market orders (cancellations at the top of the book) and the arrival of lager market orders that exceed the liquidity at the top of the book and are hence sliced into smaller orders by the exchange that are consecutively matched against the liquidity standing at less competitive price levels. This is not possible within the Markovian models [19, 20]. Our framework also allows for a dependence of the probability of order placements and/or cancellations at different price levels on past price changes. This is important to model peg orders. Peg orders follow the best bid, when buying a stock, and the best offer, when selling a stock. As such they are typically cancelled and resubmitted after a price change occurred. Our main theorem sates that, with our choice of scaling, the dynamics of the LOB converges in law to a fully coupled SDE-ODE system. The SDEs describe the limiting dynamics of the best bid and ask prices, respectively. The ODEs describe the joint limiting dynamics of the volume density functions, and the order arrival and cancellation intensities. Unlike [5, 19, 20] our limiting results does not require different time scales, i.e. we allow for a non-trivial limiting proportions of market orders and spread placements. We introduce a class of Hawkes random measures that can be viewed as an extension of the standard Hawkes processes, and a sequence of LOB models driven by these measures. Under standard assumptions on the model parameters we prove that the sequence of prices, volumes and order arrival and cancellation intensities is tight as a sequence of processes taking values in a suitable Skorohood space, and that any weak accumulation point solves a certain dynamic stochastic system. Uniqueness of solutions to this system can not be expected in general and requires additional assumptions. Under additional conditions on the model parameters we prove that the limiting LOB model always has a strictly positive spread. From this, we deduce that the limiting stochastic system is non-degenerate and that it hence has a unique solution. In order to characterize weak accumulation points as solutions to the stochastic systems we prove that any accumulation point solves the martingale problem associated with the generalized generator of the limit stochastic system. The remainder of this paper is organized as follows. The sequence of LOB models driven by Hawkes random measures is introduced in Section 2. Section 3 states the main result of this paper, namely the characterisation of weak accumulation points as solutions to a certain stochastic system. In Section 4 we state additional conditions under which the limiting spread is strictly positive from which we then deduce uniqueness of solution to the limiting system. Section 5 establishes tightness of the state sequences and hence the existence of a weak accumulation point. In Section 6 we prove our result on the characterization of accumulation points. Appendix A introduces the class of Hawkes random measures that we use for our modelling of LOBs.

2

LOB models driven by Hawkes random measures

In this section, we introduce a class of LOB models driven by Hawkes random measures on a fixed time interval [0, T ]. The microscopic dynamics follows [19, 20] to which we refer for any modelling details. For each n ∈ N, the dynamics of the n-th LOB model will be described by a continuous-time stochastic (n) (n) (n) (n) process S(n) := (Pa , Pb , Va , Vb ). All the processes are defined on the common filtered probability space 2

(Ω, F , (Ft )t∈[0,T ] , P). They take values in the space S := R2 × (L2 (R; R+ ))2 . The state space is a Hilbert space when endowed with the norm kSk2S 2 := |pa |2 + |pb |2 + kva k2L2 + kvb k2L2 . (n)

(n)

The R-valued processes Pa and Pb denote the best ask and the best bid price process, respectively, and (n) (n) the L2 -valued step functions Va and Vb denote the standing ask and bid side volumes, respectively. More (n) (n) (n) (n) precisely, for a given tick size δx > 0, the set of price levels is {xj }j∈Z where xj := j · δx , and for (n)

(n)

x ∈ [xj , xj+1 ),

(n)

Va/b (t, x) :=

#{sell/buy orders at price xj at time t } (n)

δx

.

Following [20], we assume that there are four active events and four passive events that change the state of the book. Active events (market order arrivals and limit order placements in the spread at the bid/ask side) change prices; passive events (limit order placements outside of the spread and cancellations at the bid/ask side) do not change prices. We put I = {a, b}, J = {M, L}, and K = {L, C} and assume that the arrivals of active and passive events are driven by the following random measures: (A) Active events: (n)

(n)

(n)

(A1) NaM (dt): a (Ft )-random point measure on R+ with intensity ρaM (S(n) (t))µaM (t)dt, which represents the arrival of sell market orders; (n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(A2) NaL (dt): a (Ft )-random point measure on R+ with intensity ρaL (S(n) (t))µaL (t)dt, which represents the arrival of sell limit orders placed in the spread; (A3) NbM (dt): a (Ft )-random point measure on R+ with intensity ρbM (S(n) (t))µbM (t)dt, which represents the arrival of buy market orders; (A4) NbL (dt): a (Ft )-random point measure on R+ with intensity ρbL (S(n) (t))µbL (t)dt, which represents the arrival of buy limit orders placed in the spread. (n)

Here t ∈ [0, T ] represents the event arrival time, {ρij (S)}i∈I,j∈J are mappings from S to [0, C] for some (n)

constant C > 0, and {µij (t)}i∈I,j∈J are nonnegative and (Ft )-progressive processes.

(P) Passive events: (n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(P1) MaL (dt, dx, dz): a (Ft )-random point measure on R+ × R× R+ with intensity λaL (t, x)dtdxνaL (dz), which represents the arrival of sell limit orders at the distance x from the best bid price; (P2) MaC (dt, dx, dz): a (Ft )-random point measure on R+ × R× R− with intensity λaC (t, x)dtdxνaC (dz), which represents the arrival of cancellation of sell volume at the distance x from the best bid price; (P3) MbL (dt, dx, dz): a (Ft )-random point measure on R+ × R× R+ with intensity λbL (t, x)dtdxνbL (dz), which represents the arrival of buy limit orders at the distance x from the best ask price; (P4) MbC (dt, dx, dz): a (Ft )-random point measure on R+ × R× R− with intensity λbC (t, x)dtdxνbC (dz), which represents the arrival of cancellations of buy volume at the distance x from the best ask price; Here (t, x, z) represents the event arrival time, the distance from the top of the book where a placement (n) or cancellation takes place, and the size of a cancellation or placement, respectively, {νik (dz)}i∈I,k∈K are (n) finite measures on R and {λik (t, ·)}i∈I,k∈K are (Ft )-progressive, nonnegative function-valued processes. (n)

We denote by δv book, and by

a scaling parameter that measures the impact of an individual order on the state of the (n)

(n)

∆(n) (x) := [xj , xj+1 ) for 3

(n)

xj

(n)

≤ x < xj+1

the price interval that contains x. Assuming that order placements are additive and that cancellations are proportional, the dynamics of the LOB models can then be described in terms of the previously introduced random measures as follows: Z t Z t (n) (n) Pa(n) (t) = Pa(n) (0) + δx(n) NbM (ds) − δx(n) NaL (ds), 0 0 Z t Z t (n) (n) (n) (n) (n) Pb (t) = Pb (0) + δx NbL (ds) − δx(n) NaM (ds), 0

Va(n) (t, x) = Va(n) (0, x) + +

Z tZ 0

(n)

Vb

0

(n)

∆(n) (x−Pa

(n)

(t, x) = Vb +

0

Z tZ

(0, x) +

(n)

∆(n) (x−Pa

(s−))

Z tZ 0

0

(n)

R−

(n) ∆(n) (Pb (s−)−x)

(n)

δx

(n)

(ez − 1)MaL (ds, dy, dz)

R−

(n)

(2.1)

V (n) (s−, y + Pa(n) (s−))(ez − 1)MaC (ds, dy, dz), (n) a

δx

(n)

Z

R+

(n)

δv

δv

Z

∆(n) (Pb

Z tZ

(s−))

Z

(s−)−x) (n)

δv

(n) δx

Z

(n)

Vb

R+

(n)

δv

(n)

δx

(n)

(ez − 1)MbL (ds, dy, dz)

(n)

(n)

(s−, Pb (s−) − y)(ez − 1)MbC (ds, dy, dz).

In order to guarantee that the best ask price is never smaller than the best bid price, we assume throughout that the following condition is satisfied. Condition 2.1 For any I ∈ I, J ∈ J and S = (pa , pb , va , vb ) ∈ S have (n)

(n)

if pa − pb < δx(n) .

ρaL (S) = ρbL (S) = 0,

In order to capture clustering and cross-dependencies between order arrivals we assume that the event arrival intensities depend on the past placements and cancellations as follows: for any I ∈ I, J ∈ J and K ∈ K, X Z t (n) (n) (n) (n) (n) ˆIJ (t, S (t−)) + µIJ (t) = µ φIJ,ij (t − s)Nij (ds) i∈I,j∈J

+

X

i∈I,k∈K (n) λIK (t, x)

Z tZ Z 0

R

R

0

(n)

ˆ(n) (t, S(n) (t−), x) + = λ IK

X

i∈I,j∈J

+

X

i∈I,k∈K

Z tZ Z 0

R

R

(n)

ΦIJ,ik (y, t − s)Mik (ds, dy, dz), Z

0

t

(2.2)

(n)

(n)

ψIK,ij (x, t − s)Nij (ds)

(n)

(n)

ΨIK,ik (x, y, t − s)Mik (ds, dy, dz).

(2.3)

ˆ (n) are exogenous densities that depend on the current state of the book only. The kernels φ(n) Here, µ ˆ (n) and λ and Φ(n) measure the impact of past active/passive events of the price dynamics while the kernels ψ (n) and Ψ(n) measure the impact of past passive events on placements/cancellations. Tables 1 and 2 summarize the notation. Remark 2.2 The assumption that the kernels do not depend on the jump size and the state of the book is made for notational simplicity only.

3

Characterization of accumulation points

We are now going to state conditions on the model parameters (intensities, kernels, tick sizes and order sizes) that guarantee that any weak accumulation point of the state process solves a coupled SDE-ODE system. Before stating our main result, we need to introduce some notation. In what follows we write (Ai1 ,...,in ) for a matrix A with entries Ai1 ,...,in if there is no risk of confusion about the index sets. We denote by l(dz) = 1R (z)dz + 1∞ (dz) 4

A1 (n) NaM (dt) R+ (n) (n) ρaM (S(n) (t))µaM (t)

Type Notation Space Intensity Exogenous density

A2 (n) NaL (dt) R+ (n) (n) ρaL (S(n) (t))µaL (t)

(n)

(n)

(n)

(n)

µ ˆbM (t, S(n) (t−))

(n)

φaM,ij (t) (n) ΦaM,ik (x, t)

A4 (n) NbL (dt) R+ (n) (n) ρbL (S(n) (t))µbL (t)

(n)

µ ˆaL (t, S(n) (t−))

µ ˆaM (t, S(t−))

Kernel

A3 (n) NbM (dt) R+ (n) (n) ρbM (S(n) (t))µbM (t)

µ ˆbL (t, S(n) (t−))

(n)

φaL,ij (t) (n) ΦaL,ik (x, t)

(n)

φbM,ij (t) (n) ΦbM,ik (x, t)

φbL,ij (t) (n) ΦbL,ik (x, t)

Table 1: Active events Type Notation Space Intensity Exogenous density

P1 (n) MaL (dt, dx, dz) R+ × R × R+ (n) λaL (t, x)dtdx (n) ×νaL (dz)

P2 (n) MaC (dt, dx, dz) R+ × R × R− (n) λaC (t, x)dtdx (n) ×νaC (dz)

P3 (n) MbL (dt, dx, dz) R+ × R × R+ (n) λbL (t, x)dtdx (n) ×νbL (dz)

P4 (n) MbC (dt, dx, dz) R+ × R × R− (n) λbC (t, x)dtdx (n) ×νbC (dz)

ˆ (n) (t, S(n) (t−), x) λ aL

ˆ (n) (t, S(n) (t−), x) λ aC

ˆ (n) (t, S(n) (t−), x) λ bL

ˆ (n) (t, S(n) (t−), x) λ bC

(n)

(n)

ψaL,ij (x, t) (n) ΨaL,ik (x, y, t)

Kernel

(n)

ψaC,ij (x, t) (n) ΨaC,ik (x, y, t)

(n)

ψbL,ij (x, t) (n) ΨbL,ik (x, y, t)

ψbC,ij (x, t) (n) ΨbC,ik (x, y, t)

Table 2: Passive events ¯ := R ∪ {∞} and put a measure on R Z n o ¯ R) := f : R ¯ 7→ R : kf kp p := Lpl (R, |f (z)|p l(dz) < ∞ . L ¯ R

l

Furthermore, we introduce the space

D := R4 × (L1 (R; R+ ) ∩ L2 (R; R+ ))4 . 2 This space is a Banach space endowed with the norm k · kD1,2 := k · kD12 + k · kD22 , where k · kDqp (p, q ∈ Z+ ) is defined for any D := (D1 , · · · , D8 ) ∈ D by

kDkpDqp =

4 X

k=1

|Dk |p +

8 X

k=5

kDk kpLq .

Finally, in order to simplify the statements of our conditions, we put (n) ˆ (n) (t, S, ·))i∈I,j∈J ,k∈K ∈ D ˆ (n) (t, S) := (|δ (n) |2 µ D ˆij (t, S), δv(n) λ x ik

and 

A(n) (x, y, t) :=  



(n)

φIJ,ij (t)1(x,y)=(∞,∞)

δv(n)

(n) |δx |2



ψ (n) (x, t)1(x,y)∈(R,∞)

(n)





(n) 2 |δx |



(n) δv

Φ(n) (y, t)1(x,y)∈(∞,R)

(n)

ΨIK,ik (x, y, t)1(x,y)∈(R,R)

(n)

 

 .

(n)

We assume throughout that ρaL and ρbL satisfy Condition 2.1 and w.l.o.g that νIK (R) = 1, for any n ∈ N and I ∈ I, K ∈ K. Furthermore, we assume that the following four conditions are satisfied. Condition 3.1 There exists a constant C > 0 such that for any n > 1 and I ∈ I, J ∈ J , K ∈ K i i h h (n) E kS(n) (0)k2S 2 + E kVI (0, ·)k4L4 ≤ C

(3.1)

and

(n)

νIK (|ez − 1|4 ) +

(n)

(n)

|ρbM (S) − ρaL (S)| (n)

δx

5

(n)

+

(n)

|ρbL (S) − ρaM (S)| (n)

δx

≤ C.

(3.2)

The second condition concerns the following rescaled active event arrival intensities. For any n > 1, t ∈ [0, T ] and S ∈ S, (n)

(n)

(n)

(n)

(n)

βb (t) = δx(n) [µbL (t) − µaM (t)];

βa(n) (t) = δx(n) [µbM (t) − µaL (t)],

(n) (n) (n) βˆb (t, S) = δx(n) [ˆ µbL (t, S) − µ ˆaM (t, S)];

(n) (n) βˆa(n) (t, S) = δx(n) [ˆ µbM (t, S) − µ ˆaL (t, S)],

¯ and for any y ∈ R (n)

B

(n)

(n)

(φML,ij (t)1y=∞ ) (ΦML,ik (y, t)1y∈R ) (n) (n) (φLM,ij (t)1y=∞ ) (ΦLM,ik (y, t)1y∈R )

(y, t) :=

!

,

where (n)

(n)

φML,ij (t) =

(n)

φbM,ij (t) − φaL,ij (t) (n)

δx (n)

(n)

ΦML,ik (y, t) =

(n)

(n)

,

φLM,ij (t) =

(n)

ΦbM,ik (y, t) − ΦaL,ik (y, t) (n)

(n)

δv /δx

(n)

φbL,ij (t) − φaM,ij (t) (n)

δx (n)

(n)

,

ΦLM,ik (y, t) =

;

(n)

ΦbL,ik (y, t) − ΦaM,ik (y, t) (n)

(n)

δv /δx

(n)

(n)

.

(n)

The process βa denotes the rescaled drift of the best ask price process, while φML,ij and ΦLM,ik denote the net impact of ask side market order arrivals and spread placements, respectively the net impact of ask side placements and cancellations on the dynamics of the best ask price process. We assume that the rescaled drifts are finite, that the intensities of market order arrivals and spread placements (n) (n) are of the order O(|δx |−2 ), that the intensities of placements and cancellations are of the order O(|δv |−1 ), and that the rescaled (net) impact of past passive events on the order arrival intensities is finite. Condition 3.2 There exists a constant C > 0 such that for any t ∈ [0, T ], n > 0, I ∈ I, S ∈ S and p ∈ {1, 2, 4}, ˆ (n) (t, S)kD1 + |βˆ(n) (t, S)| ≤ C. kD I p

(3.3)

¯ Moreover, for any y ∈ R, |B(n) (y, t)| + kA(n) (·, y, t)kLpl ≤ C.

(3.4)

The next two conditions assert the convergence of the rescaled model parameters to sufficient regular functions. Condition 3.3 There exist a S-valued random variable S(0) and some constants {αIK : I ∈ I, K ∈ K} such that as n → ∞ h i (n) (3.5) E kS(n) (0) − S(0)k2S 2 + |νIK (ez − 1) − αIK | → 0.

Moreover, there exist continuous functions {ρIJ (S), ̺I (S) : I ∈ I, J ∈ J } such that as n → ∞ the following locally uniform convergence holds: ρ(n) (S) − ρ(n) (S) ρ(n) (S) − ρ(n) (S) bL (n) aL aM |ρIJ (S) − ρIJ (S)| + bM − ̺ (S) − ̺ (S) + → 0. a b (n) (n) δx δx

(3.6)

ˆIK (t, S, ·))I∈I,J∈J ,K∈K ∈ D and (βˆI (t, S))I∈I that ˆ S) := (ˆ Condition 3.4 There exist functions D(t, µIJ (t, S), λ satisfy for t, t′ ∈ [0, T ], S, S ′ ∈ S, δ > 0 and p ∈ {1, 2} ˆ IK (t, S, · + δ) − λ ˆ IK (t′ , S ′ , ·)kLp |ˆ µIJ (t, S) − µ ˆIJ (t′ , S ′ )| + kλ +|βˆI (t, S) − βˆI (t′ , S ′ )| ≤ C(δ + |t − t′ | + kS − S ′ kS 2 ). Moreover, there exist functions B(y, t) :=



(φML,ij (t)1y=∞ ) (φLM,ij (t)1y=∞ ) 6

(ΦML,ik (y, t)1y∈R ) (ΦLM,ik (y, t)1y∈R )



,

(3.7)

and
φIJ,ij (t)1(x,y)=(∞,∞)
ψIK,ij (x, t)1(x,y)∈(R,∞)

A(x, y, t) :=

¯ and p ∈ {1, 2} that satisfy for any 0 ≤ t ≤ t + s ≤ T , y ∈ R


! ΦIJ,ik (y, t)1(x,y)∈(∞,R)
. ΨIK,ik (x, y, t)1(x,y)∈(R,R)

kA(· + δ, y, t + s) − A(·, y, t)kLpl + |B(y, t + s) − B(y, t)| ≤ C(γ(s) + δ)

(3.8)

where γ(·) is a continuous function that vanishes at zero. Moreover, there exists a sequence {γn }n≥1 that converges to zero as n → ∞ such that for any t ∈ [0, T ] and S ∈ S: ˆ (n) (t, S) − D(t, ˆ S)kD2 + βˆ(n) (t, S) − βˆI (t, S) ≤ γn , (3.9) kD I 1,2 ¯ and p ∈ {1, 2} and for any y ∈ R

kA(n) (·, y, t) − A(·, y, t)kLpl + |B(n) (y, t) − B(y, t)| ≤ γn .

(3.10)

From Condition 3.4 and the definition of the function β, it is easy to see that ρa := ρbM = ρaL ,

ρb := ρbL = ρaM ,

µ ˆa := µ ˆbM = µ ˆaL ,

µ ˆb := µ ˆ bL = µ ˆaM ,

and for any i ∈ I, j ∈ J , k ∈ K, φa,ij := φbM,ij = φaL,ij ,

φb,ij := φbL,ij = φaM,ij ,

Φa,ik := ΦbM,ik = ΦaL,ik ,

Φb,ik := ΦbL,ik = ΦaM,ik .

We are now ready to state the main result in this paper. Its proof is given in Section 6 below. (n)

(n)

(n) (n)

Theorem 3.5 Suppose Condition 3.1-3.4 hold. Let D(n) := (|δx |2 µij , δv λik )i∈I,j∈J ,k∈K ∈ D. Then, (n)

ˆ(n) , D(n) , βa(n) , β ) ⇒ (S, D, βa , βb ) (S b weakly in D(R+ , S × D × R2 ), where S = (Pa , Pb , Va , Vb ) and D = (µij , λik )i∈I,j∈J ,k∈K with µa := µbM = µaL , µb := µbL = µaM . Moreover, the limit is a solution to the following stochastic dynamic system: Z tp Z th i 2ρa (S(s))µa (s)dBa (s), ρa (S(s))βa (s) + ̺a (S(s))µa (s) ds + Pa (t) = Pa (0) + 0 0 Z tp Z th i ρb (S(s))βb (s) + ̺b (S(s))µb (s) ds + Pb (t) = Pb (0) + 2ρb (S(s))µb (s)dBb (s), 0 0 (3.11) Z th i αaL λaL (s, x − Pa (s)) + αaC λaC (s, x − Pa (s))Va (s, x) ds, Va (t, x) = Va (0, x) + 0 Z th i αbL λbL (s, Pb (s) − x) + αbC λbC (s, Pb (s) − x)Vb (s, x) ds, Vb (t, x) = Vb (0, x) + 0

where (Ba , Bb ) is a standard two-dimensional Brownian motion, and XZ t µI (t) = µ ˆI (t, S(t)) + φˆIi (t − s)ρi (S(s))µi (s)ds +

X

i∈I,k∈K

Z t i∈I Z 0

R

0

ΦI,ik (y, t − s)λik (s, y)dsdy,

ˆIK (t, S(t), x) + λIK (t, x) = λ +

X

i∈I,k∈K

Z tZ 0

ˆ S(t)) + βI (t) = β(t,

R

XZ i∈I

ψˆIK,i (x, t − s)ρi (S(s))µi (s)ds

ΨIK,ik (x, y, t − s)λik (s, y)dsdy,

XZ i∈I

0

t

0

t

φ˜Ii (t − s)ρi (S(s))µi (s)ds 7

(3.12)

(3.13)

+

X

i∈I,k∈K

Z tZ 0

R

˜ I,ik (y, t − s)λik (s, y)dsdy, Φ

(3.14)

˜ a,ik = ΦML,ik , Φ ˜ b,ik = ΦLM,ik , where Φ φˆIa = φI,bM + φI,aL ,

φˆIb = φI,bL + φI,aM ,

φ˜Ib = φLM,bL + φLM,aM ,

φ˜Ia = φML,bM + φML,aL ,

ψˆIK,a = ψIK,bM + ψIK,aL ,

ψˆIK,b = ψIK,bL + ψIK,aM .

Remark 3.6 The uniqueness of solutions to stochastic systems of the form (3.11)-(3.14) is an open problem in general. In the next section we prove the uniqueness of solutions under the additional assumption that the spread always strictly positive for which we shall also give sufficient conditions. The system (3.12)-(3.14) can be viewed as the solution to a linear Volterra-Fredholm integral equation. In ¯ R) to order to see this, we now define a class of linear operators {T(x, y, S, t, s) : 0 ≤ s ≤ t < ∞} from L2l (R, 2 ¯ Ll (R, R) by   
 ΦI,ik (y, t − s)1(x,y)∈(∞,R) φˆIi (t − s)ρi (S(s))1(x,y)=(∞,∞)  T(x, y, S, t, s) :=  
, ΨIK,ik (x, y, t − s)1(x,y)∈(R,R) ψˆIK,i (x, t − s)ρi (S(s))1(x,y)∈(R,∞) then (3.12)-(3.14) can be written in the following form: Z t Z ˆ S(t), x) + D(t, x) = D(t, ds T(x, y, S, t, s)D(s, y)l(dy). ¯ R

0

(3.15)

The solution to this linear Volterra-Fredholm integral equation is given by Z t Z ˆ S(s), y)l(dy), ˆ S(t), x) + D(t, x) = D(t, ds T∞ (x, y, S, t, s)D(s, ¯ R

0

where T∞ (x, y, S, s, t) =

P∞

Tn (x, y, S, s, t) and Z t Z ds Tn−1 (x, z, S, t, r)T(z, y, S, r, s)l(dz). Tn (x, y, S, s, t) = k=1

¯ R

0

Example 3.1 (Exponential kernel) Consider the special case ρa (S) = ρb (S) = 1 when Pa > Pb and φˆIi (t) = e−βt ,

ΦI,ik (y, t) = ΦI,ik (y)e−βt

ψIK,i (x, t) = ψIK,i (x)e−βt ,

ΨIK,ik (x, y, t) = ΨIK,ik (x, y)e−βt

¯ R) to L2 (R, ¯ R) as follows for some β > 0. Define an linear operator from L2l (R, l T(x, y) := If T(x, y) is bounded, then


1(x,y)=(∞,∞)
ψI,ik (x)1(x,y)∈(R,∞)

ˆ S(t), x) + D(t, x) = D(t,

Z

0

4

t

ds

Z

¯ R


! ΦI,ik (y)1(x,y)∈(∞,R)
. ΨI,ik (x, y)1(x,y)∈(R,R)

ˆ S(s), z)l(dz). (T2 e(T−β)(t−s) )(x, z)D(s,

Uniqueness of accumulation points

In this section, we prove the pathwise uniqueness of solutions to the stochastic dynamic system (3.11)-(3.14) under mild additional assumptions on the model parameters. We first prove the positivity of the spread P¯ (t) := Pa (t) − Pb (t). This result is then used to prove the uniqueness of solutions. In what follows we assume µ ˆI (0, S) > 0 for any S ∈ S and P¯ (0) > 0. 8

4.1

Positivity of the spread

We start with the following simple result on the non-negativity of degenerate diffusion processes. The proof follows immediately from the continuity of the sample paths. Lemma 4.1 Let x0 > 0 be a F0 -measurable random variable and bt (x) ∈ R and σt (x) ≥ 0 be (Ft )-progressive processes such that the diffusion process Z t Z t xt = x0 + bs (xs )ds + σs (xs )dBs , (4.1) 0

0

is well defined and continuous. If bt ≥ 0 on R− and σt = 0 on R− , then P{xt ≥ 0, t ≥ 0} = 1. Corollary 4.2 (Non price-crossing) Suppose that Conditions 2.1 and 3.3 hold. Then, for any solution (S, D, βa , βb ) to the stochastic dynamic system (3.11)-(3.14), we have P{Pa (t) ≥ Pb (t) : t ≥ 0} = 1. Moreover, if ̺a (S) − ̺b (S) > 0 for any S ∈ S with pa = pb , then {t : Pa (t) = Pb (t)} is sparse on R+ almost surely. Proof. The first statement follows from Lemma 4.1 along with Condition 2.1 and 3.3. For the second statement, define τ− = inf{t ≥ 0 : P¯ (t) = 0} and τ+ = inf{t > τ− : P¯ (t) > 0}. It suffices to prove τ+ = τ− almost surely. This follows from the fact that µa , µb > 0 almost everywhere and ̺a (S) − ̺b (S) > 0 for any S ∈ S with Pa = Pb , implies for any t > 0 P¯ (t ∧ τ+ ) =

Z

t∧τ+

τ−

h i ̺a (S(s))µa (s) − ̺b (S(s))µb (s) ds > 0. ✷

We proceed with the following lemma from which we shall deduce the strict positivity of the spread. Lemma 4.3 Let x0 > 0 be a F0 -measurable random variable and at ≥ ct ≥ 0, bt ∈ R are (Ft )-progressive processes. If (xt , Bt ) is a weak solution to the following stochastic equation: xt = x0 +

Z

t 0

(as − bs xs )ds +

Z

t

√ 2cs xs dBs ,

0

then P{xt > 0, t ≥ 0} = 1. Rt

Proof. Applying Itˆ o’s formula to zˆt := e 0 bs ds xt , we have Z t Z t Rs Rs √ 2cs xs e 0 br dr dBs zˆt = x0 + as e 0 br dr ds + Z 0t p Z 0t Rs Rs 1 b dr r as e 0 ds + = zˆ0 + 2cs zˆs e 2 0 br dr dBs 0

0

Let τt be a strictly increasing process defined as follows: Z t Rs [cs + 1{cs =0} ]e 0 br dr ds, τt := 0

and it inverse process σt := τt−1 . It is easy to see zt := zˆσt satisfies the following equation: Z σt p Z σt Rs Rs 1 br dr 0 2cs zˆs e 2 0 br dr dBs as e ds + zt = z0 + 0

0

9

(4.2)

Z tp R σs 1 = z0 + 2cσs zˆσs e 2 0 br dr dBσs aσs e dσs + 0 Z0 t h i aσs = z0 + 1{cσs >0} + aσs 1{cσs =0} ds cσs 0 Z tq q R σs 21{cσs >0} zs e 0 br dr [cσs + 1{cσs =0} ]dBσs . + Z

R σs

t

0

br dr

0

R t q R σs Obviously, Wt := 0 e 0 br dr [cσs + 1{cσs =0} ]dBσs is a standard Brownian motion and (zt , Wt ) is a weak solution to Z tq Z th i aσs 21{cσs >0} zs dWs . (4.3) 1{cσs >0} + aσs 1{cσs =0} ds + zt = z0 + cσs 0 0 Since as ≥ cs , a standard comparison theorem [31, Theorem 1.1] yields P{zt ≥ z˜t , t ≥ 0} = 1, where z˜t is the unique solution to Z t Z tq z˜t = z0 + 1{cσs >0} ds + 21{cσs >0} z˜s dWs . 0

0

From [30, p.442], P{˜ zt > 0, t ≥ 0} = 1. Hence the desired result follows from the definition of z˜.

✷

Condition 4.4 There exists ǫ > 0 such that for any for any S ∈ S with pa − pb ∈ (0, ǫ) have ρa (S) ≤ ̺a (S)(pa − pb )

and

ρb (S) ≤ −̺b (S)(pa − pb ).

The following result shows that the spread is strictly positive under the preceding condition. Proposition 4.5 Suppose that Conditions 2.1, 3.3 and 4.4 hold. Then any solution (S, D, βa , βb ) to the stochastic dynamic system (3.11)-(3.14) satisfies P{Pa (t) > Pb (t), t ≥ 0} = 1. Proof. For any S ∈ S with pa − pb > 0, Condition 4.4 yields, ρˆa (S) :=

ρa (S) ρb (S) ≤ ̺a (S) and ρˆb (S) := ≤ −̺b (S). pa − pb pa − pb

(4.4)

Let B ′ (s) be another Brownian motion independent to Ba/b and put W (t) :=

t

Z

0

+

Z

p p ρa (S(s))µa (s)dBa (s) + ρb (S(s))µb (s)dBb (s) p 1{ρa (S(s))µa (s)+ρb (S(s))µb (s)>0} ρa (S(s))µa (s) + ρb (S(s))µb (s) t

1{ρa (S(s))µa (s)+ρb (S(s))µb (s)=0} dB ′ (s).

0

Then, W (t) is a standard Brownian motion and P¯ (t) satisfies, Z th i P¯ (t) = P¯ (0) + ̺a (S(s))µa (s) − ̺b (S(s))µb (s) + (ˆ ρa (S(s))βa (s) − ρˆb (S(s))βb (s))P¯ (s) ds Z tq 0 + 2[ˆ ρa (S(s))µa (s) + ρˆb (S(s))µb (s)]P¯ (s)dW (s). 0

Hence, the desired result follows from (4.4) and Lemma 4.3.

10

✷

4.2

Pathwise uniqueness

We are now going to state sufficient conditions on the model parameters that guarantee uniqueness of solutions to the stochastic dynamic system (3.11)-(3.14). From Condition 3.1-3.4 we see that there exists a constant C > 0 ¯ and p ∈ {1, 2} such that for any t ∈ [0, T ], S ∈ S, y ∈ R ˆ S)kD2 + kA(·, y, t)kLp + |B(y, t)| ≤ C. |ρI (S)| + |βˆI (t, S)| + kD(t, 1,2

(4.5)

ˆ IL . The following condition introduces additional bounds on ψIL,ik , ΨIL,ik and λ Condition 4.6 There exists a constant C > 0 such that for any y ∈ R, t ∈ [0, T ] and S ∈ S, ˆ IL (t, S, x)} ≤ C. sup{VI (0, x) + ψIL,ij (x, t) + ΨIL,ik (x, y, t) + λ x∈R

Lemma 4.7 Suppose Condition 4.6 holds. Then, for any solution (S, D, βa , βb ) to the stochastic dynamic system (3.11)-(3.14) there exists a constant C > 0 such that,   E kS(t)k2S 2 + kD(t)k2D2 + |βa (t)|2 + |βb (t)|2 + kVI (t, ·)k∞ + kλIK (t, ·)k∞ ≤ C. 1,2

Proof. From (4.5) and the Gr¨onwall inequality, we have

kD(t)kD11 ≤ C + C

Z

t 0

kD(s)kD11 ds

and kD(t)kD11 ≤ C. By the H¨ older inequality, Z Z tZ 2 ΨIK,ik (x, y, t − s)λik (s, y)dyds dx R 0 Z R Z Z Z tZ t ≤ |ΨIK,ik (x, y, t − s)|2 dxλik (s, y)dyds λik (s, y)dyds 0 R Z0 t ZR R 2 ≤ λik (s, y)dyds ≤ C 0

R

and

kλIK (t, ·)k2L2

X Z t t |µi (s)|2 ds ≤ C+C i∈I

+C

X

i∈I,k∈K

0

Z Z R

r

Z

0

R

2 ΨIK,ik (x, y, t − s)λik (s, y)dyds dx ≤ C.

The upper boundary of βI can be proved similarly. Since VI (t, x) ≥ 0 and αIC ≤ 0, Z t 2 i h Z t p 2 2 2 2 2ρI (S(s))µI (s)dBI (s) E[|PI (t)| ] ≤ C|PI (0)| + Ct E[|βI (s)| + |µI (s)| ]ds + CE 0 Z0 t 2 2 2 E[|βI (s)| + |µI (s)| + 2ρI (S(s))µI (s)]ds ≤ C = C|PI (0)| + Ct 0

and kVI (t, ·)k2L2 ≤ CkVI (0, ·)k2L2 + Ct Furthermore, from Condition 4.6, for any x ∈ R, XZ t λIK (t, x) ≤ C + C µ ˆi (s)ds + C i∈I

0

Z

0

t

kλIL (s, ·)k2L2 ds ≤ C.

X

i∈I,k∈K

Z tZ 0

R

λik (s, y)dsdy ≤ C

and Va (t, x) ≤ Va (0, x) + All other terms can be estimated similarly.

Z

0

t

αaL λaL (s, x − Pa (s))ds ≤ C. ✷

11

Theorem 4.8 Suppose Condition 4.4 and 4.6 hold in addition to the conditions of Theorem 3.5. If the functions {ρI (S), ̺I (S) : I ∈ I} are Lipschitz continuous and ρI (S) > 0 for any S ∈ S with pa > pb , then the pathwise uniqueness of solutions holds for the stochastic dynamic system (3.11)-(3.14). ˜ IK )I∈I,J∈J ,K∈K and β˜I := βI − βˆI , where µ ˜IK = λIK − λ ˆ IK . Then ˜ := (˜ Proof. Define D µI , λ ˜ I = µI − µ ˆI and λ (3.12)-(3.14) can be written as XZ t µ ˜I (t) = φˆIi (t − s)ρi (S(s))(˜ µi (s) + µ ˆi (s, S(s)))ds 0

i∈I

X

+

i∈I,k∈K

˜ IK (t, x) = λ

t

XZ

X

+

i∈I,k∈K

β˜I (t) =

t

XZ

X

+

i∈I,k∈K

Let

˜ (1) , βa(1) , β (1) ) (S(1) , D b

R

˜ ik (s, y) + λ ˆ ik (s, S(s), y))dsdy, ΦI,ik (y, t − s)(λ

Z tZ 0

R

˜ ik (s, y) + λ ˆik (s, S(s), y))dsdy, ΨIK,ik (x, y, t − s)(λ

φ˜Ii (t − s)ρi (S(s))(˜ µi (s) + µ ˆi (s, S(s)))ds

0

i∈I

0

ψˆIK,i (x, t − s)ρi (S(s))(˜ µi (s) + µ ˆi (s, S(s)))ds

0

i∈I

Z tZ

Z tZ 0

R

˜ ik (s, y) + λ ˆ ik (s, S(s), y))dsdy. ˜ I,ik (y, t − s)(λ Φ

˜ (2) , βa(2) , β (2) ) be two solutions and put and (S(2) , D b

¯ D, ¯ β¯a , β¯b ) := (S(1) , D ˜ (1) , β˜a(1) , β˜(1) ) − (S(2) , D ˜ (2) , β˜a(2) , β˜(2) ). (S, b b

From Lemma 4.7 and Lipschitz continuity of ρI , we deduce that Z t 2 2 2 2 ¯ ¯ ¯ ¯ kD(t)kD2 + |βI (t)| ≤ C [kS(s)k S 2 + kD(s)kD 2 ]ds. 1

1

0

(4.6)

By the H¨ older inequality, ¯ IK (t, x)|2 ≤ C |λ

XZ

t

XZ i∈I

+C

ˆ|ψIK,i (x, t − s)|2 |ρi (S(1) (s)) − ρi (S(2) (s))|2 ds

0

i∈I

+C

t

0

X

|ψˆIK,i (x, t − s)|2 [|¯ µi (s)|2 + |ˆ µi (s, S(1) (s)) − µ ˆi (s, S(2) (s))|2 ]ds

i∈I,k∈K

+C

X

t

ds

0

Z

i∈I,k∈K Z

×

and

Z

0

R

¯ ik (s, y)|dy |λ

Z

ˆ ik (s, S(1) (s), y) − λ ˆik (s, S(1) (s), y)|dy |ΨIK,ik (x, y, t − s)|2 |λ

R

t

ds

Z

Z

R

R

¯ ik (s, y)|dy |ΨIK,ik (x, y, t − s)|2 |λ

ˆ ik (s, S(1) (s), y) − λ ˆ ik (s, S(1) (s), y)|dy |λ

¯ IK (t, ·)k2 2 ≤ C kλ L

Z

0

t

2 2 ¯ ¯ [|S(s)| S 2 + kD(s)kD 2 ]ds. 1

In order to estimate the square of the norm of the price difference, we denote, for any ε > 0,   q (l) τε = inf t ≥ 0 : 2ρI (S(l) (s))µI (s) ≤ ε, I ∈ I, l = 1, 2 .

From Proposition 4.5, we can see that τε → ∞ a.s. as ε → 0. Hence, it is enough to consider t ∈ [0, τε ]. In particular, Z t q q 2 (1) (2) (1) 2ρa (S (s))µa (s) − 2ρa (S(2) (s))µa (s) ds 0 Z th i 1 (1) (2) 2 2 (1) (2) 2 ds. |ρ (S (s)) − ρ (S (s))| + |¯ µ (s)| + |ˆ µ (s, S (s)) − µ ˆ (s, S (s))| ≤ a a a a a 2ǫ2 0 12

Thus, an application of Itˆ o’s formula to |P¯a (t)|2 yields, Z t Z t 2 |P¯a (s)|2 ds + C |P¯a (t)|2 ≤ C ρa (S(1) (s)) − ρa (S(2) (s)) ds 0 0Z i th 2 ¯ ˆ |βa (s)| + |βa (s, S(1) (s)) − βˆa (s, S(2) (s))|2 ds +C Z0 t h i |̺a (S(1) (s)) − ̺a (S(2) (s))|2 + |¯ µa (s)|2 + |ˆ µa (s, S(1) (s)) − µ ˆa (s, S(2) (s))|2 ds +C Z0 i 2 th + 2 |ρa (S(1) (s)) − ρa (S(2) (s))|2 + |¯ µa (s)|2 + |ˆ µa (s, S(1) (s)) − µ ˆa (s, S(2) (s))|2 ds ε 0 Z t q hq i (1) (2) (1) ¯ + 2PI (s) 2ρa (S (s))µa (s) − 2ρa (S(2) (s))µa (s) dBa (s), 0

and hence E[|P¯a (t)|2 ] ≤ C(1 + 2/ε2 )

Z th i 2 2 2 ¯ ¯ ¯ E[kS(s)k S 2 ] + E[|βa (s)| ] + E[kD(s)kD 2 ] ds. 2

0

The following estimate allows us to estimate the norms of the volume density functions. For any δ > 0, ˜ (l) (s, · + δ) − λ ˜(l) (s, ·k2 2 kλ L IK IK Z X Z t Z ≤C ds |λik (s, y)|dy kΨIK,ik (· + δ, y, t − s) − ΨIK,ik (·, y, t − s)k2L2 |λik (s, y)|dy 0

i∈I,k∈K

+C

XZ i∈I

t

0

≤ Cδ 2 + Cδ 2

R

R

kψˆIK,i (· + δ, t − s) − ψˆIK,i (·, t − s)k2L2 ds X

i∈I,k∈K

Z

0

(4.7)

t

kλik (s, ·)k2L1 ds ≤ Cδ 2 .

Thus, by direct computation we verify that kV¯a (t, ·)k2L2 ≤ C

Z

0

t

2 2 ¯ ¯ [kS(s)k S 2 + kD(s)kD 2 ]ds. 2

As a result, 2 2 ¯ ¯ E[kS(t)k S 2 + kD(t)kD 2 ] ≤ C 1,2

Z

t

0

2 2 ¯ ¯ E[kS(s)k S 2 + kD(s)kD 2 ]ds. 1,2

By the Gr¨onwall inequality, this yields 2 2 ¯ ¯ E[kS(t)k S 2 + kD(t)kD 2 ] = 0. 1,2

Along with the continuity of the solutions this yields the desired pathwise uniqueness.

✷

Proposition 4.9 Suppose conditions in Theorem 4.8 hold. Then there exists a unique strong solution to (3.11)(3.14). Proof. By [22, Theorem 1.1, p.163-166], distributional existence and pathwise uniqueness imply strong existence. Hence the result follows from the Theorems 3.5 and 4.8. ✷ Example 4.1 Let νa/bL (dz) and νa/bC (dz) be probability measure on R+ and R− respectively, with finite forth moments. We consider a sequence of LOBs defined as follows: for any n ≥ 1, (1) (Initial state) S(n) (0) = S(0); (n)

(2) (Jump size) νIK (dz) = νIK (dz) and νIL (ez − 1) = −νIC (ez − 1) = 1; 13

ˆIK (t, S, x)/δv(n) and ˆ(n) (t, S, x) = λ (3) (Exogenous intensities) λ IK (n)

µ ˆa (t, S) + δx

(n)

µ ˆbM (t, S) =

(n)

(n) 2 |δx | (n)

δv

µ ˆaL (t, S) =

,

µ ˆaM (t, S) =

(n)

µ ˆb (t, S) + δx

(n)

µ ˆbL (t, S) = (4) (Kernels) ψIK,ij (x, t) =

(n)

,

(n) |δx |2

(n) |δx |2

(n)

µ ˆa (t, S) (n)

|δx |2 µ ˆb (t, S) (n)

|δx |2

,

;

(n)

ψIK,ij (x, t), ΨIK,ik (x, y, t) = ΨIK,ik (x, y, t) and

(n)

φaL,ij (t) = φa,ij (t),

(n)

(n)

φaM,ij (t) = φb,ij (t),

φbM,ij (t) = φa,ij (t) + δx(n) ,

(n)

φbL,ij (t) = φb,ij (t) + δx(n) , (n)

(n)

δv

(n)

ΦbM,ij (t) =

(n) |δx |2

Φa,ij (t) +

Φb,ij (t) +

(n)

|δx |2

(n)

(n)

(n)

,

ΦaL,ij (t) =

, (n)

ΦaM,ij (t) =

(n) δx (n)

(n)

δv

(n)

ΦbL,ij (t) =

δv

δv

δx

(n) |δx |2

Φa,ij (t),

(n)

(n)

(n)

δv

δv

(n)

|δx |2

Φb,ij (t);

(n)

(5) (Non-crossing conditions) ρaL (S) = ρbL (S) = [pa − pb − δx ]+ ∧ n and (n)

(n)

ρbM (S) = ρaM (S) = [pa − pb − δx(n) ]+ ∧ n + δx(n) . Then all conditions above are satisfied and with P¯ (t) = Pa (t) − Pb (t), Pa (t) = Pa (0) +

Z th 0

Z tq i ¯ 2µa (s)P¯ (s)dBa (s), P (t) + µa (s) ds + 0

Z th i λaL (s, x − Pa (s)) − λaC (s, x − Pa (s))Va (s, x) ds, Va (t, x) = Va (0, x) + 0

with Pb and Vb being defined analogously, and µI (t) = µ ˆI (t, S(t)) +

XZ

0

i∈I

X

+

i∈I,k∈K

Z tZ 0

R

+

i∈I,k∈K

βI (t) = 1 +

XZ i∈I

5

Z tZ 0

t

φˆIi (t − s)µi (s)P¯ (s)ds

ΦI,ik (y, t − s)λik (s, y)dsdy,

ˆ IK (t, S(t), x) + λIK (t, x) = λ X

t

XZ i∈I

R

0

t

ψˆIK,i (x, t − s)µi (s)P¯ (s)ds

ΨIK,ik (x, y, t − s)λik (s, y)dsdy,

µi (s)P¯ (s)ds +

0

X

i∈I,k∈K

Z tZ 0

λik (s, y)dsdy.

R

Tightness of the LOB models (n)

(n)

In this section, we prove the tightness of the processes (S(n) , D(n) , βa , βb ) by showing that the pointwise moment conditions on the state sequence and the moment conditions on the increments of the state sequence in Kurtz’s tightness criterion hold. In what follows we assume without loss of generality that the constants C in Condition 3.1-3.4 equal to 1.

14

5.1

Pointwise norm estimates

Our norm estimates use the following quantity: for any t ≥ 0, X Z t X Z tZ Z (n) (n) (n) 2 (n) J (t) = 1 + |δx | Nij (ds) + δv(n) Mik (ds, dy, dz). 0

i∈I,j∈J

0

i∈I,k∈K

R

R

Lemma 5.1 There exists a constant C > 0 such that for any 0 ≤ r ≤ t ≤ T and p ∈ {1, 2, 4} h i h i EFr kD(n) (t)kpDp ≤ CT |J (n) (r)|p and E sup kD(n) (t)kpDp ≤ C. 1

1

t∈[0,T ]

Proof. From Condition 3.2, (n)

(n)

|δx(n) |2 µIJ (t) = |δx(n) |2 µ ˆIJ (t, S(n) (t−)) + +

Z tZ Z

X

0

i∈I,k∈K

R

X

i∈I,j∈J

(n)

|δx |2 (n)

δv

R

t

Z

0

(n)

(n)

φIJ,ij (t − s)|δx(n) |2 Nij (ds)

(n)

(n)

ΦIJ,ik (y, t − s)δv(n) Mik (ds, dy, dz) ≤ J (n) (t)

(5.1)

and h i (n) EFr |δx(n) |2 µIJ (t) ≤ 1 + +

0

i∈I,j∈J

t

Z

X

r

i∈I,j∈J

≤ J (n) (r) + Similarly, we also have

r

Z

X

Z

r

t

(n) |δx(n) |2 Nij (ds)

X

+



(n) EFr |δx(n) |2 µij (s)

i∈I,k∈K

+

r

Z

0

X

i∈I,k∈K

h i EFr kD(n) (s)kD11 ds.

Z Z R

Z

R

(n)

R

δv(n) Mik (ds, dy, dz) 

(n) δv(n) λik (s, y)dy  ds

(n)

kδv(n) λIK (t, ·)kL1 ≤ J (n) (t)

(5.2)

and Z t i h i h (n) (n) (n) EFr kD(n) (s)kD11 ds. EFr kδv λIK (t, ·)kL1 ≤ J (r) + r

Hence,

Z t i h i h EFr kD(n) (s)kD11 ds. EFr kD(n) (t)kD11 ≤ 8J (n) (r) + 8 r

From the Gr¨onwall inequality, we have

i h EFr kD(n) (t)kD11 ≤ 8e8(t−r) J (n) (r).

(5.3)

For p = 4 (the case p = 2 is similar),

Z t (n) 2 (n) 4 (n) 4 3 kD(n) (s)k4D4 ds + C |δx | µIJ (t) ≤ C|J (r)| + C(t − r) 1 r

+C

X

i∈I,k∈K

and (n)

R

Z t 4 ˜ (n) (ds) |δx(n) |2 N ij

X

Z t 4 ˜ (n) (ds) |δx(n) |2 N ij

i∈I,j∈J

Z tZ Z 4 ˜ (n) (ds, dy, dz) δv(n) M ik r

X

r

R

kδv(n) λIK (t, ·)k4L1 ≤ C|J (n) (r)|4 + C(t − r)3

Z

r

t

kD(n) (s)k4D4 ds + C 1

15

i∈I,j∈J

r

+C

X

i∈I,k∈K

Z tZ Z 4 ˜ (n) (ds, dy, dz) . δv(n) M ik r

R

R

By the Burkholder-Davis-Gundy inequality, the Cauchy inequality and the H¨ older inequality,  Z t    Z t 4 2 (n) 4 (n) ˜ (n) (ds) ≤ CEFr EFr |δx(n) |2 N (ds) |δ | N x ij ij r r  Z t  Z t 2  2  (n) ˜ (n) (ds) ≤ CEFr |δx(n) |4 µij (s)ds + CEFr |δx(n) |4 N ij r r Z t i h (n) (n) ≤ C EFr |δx(n) |8 |µij (s)|2 + |δx(n) |8 µij (s) ds r

and  Z t Z Z 4  ˜ (n) (ds, dy, dz) EFr δv(n) M ik r R R  Z t Z Z 2  (n) (n) 2 ≤ CEFr |δv | Mik (ds, dy, dz) r R R Z t i h (n) (n) ≤ C EFr k|δv(n) |2 λik (s, ·)k2L1 + k|δv(n) |4 λik (s, ·)kL1 ds. r

Hence, Z t i i h h (n) 4 3 (n) 4 EFr kD(n) (s)k4D4 ds EFr kD (t)kD4 ≤ C|J (r)| + C(t − r) 1 1 r Z t i h (n) (n) EFr |δx(n) |8 |µij (s)|2 + |δx(n) |8 µij (s) ds +C Zr t i h (n) (n) EFr k|δv(n) |2 λik (s, ·)k2L1 + k|δv(n) |4 λik (s, ·)kL1 ds +C r Z t i h (n) ≤ C|J (r)|4 + C EFr kD(n) (s)k4D4 ds. 1

r

The first result now follows from the Gr¨onwall inequality. The second result follows from (5.1)-(5.2) together with the first result with r = 0. ✷ The following estimate of the conditional moments of the increment of J (n) (t) follows directly from the above proof and is hence omitted. Proposition 5.2 There exists a constant C > 0 such that for any 0 ≤ r ≤ t ≤ T and p ∈ {1, 2, 4} i i h h EFr |J (n) (t) − J (n) (r)|p ≤ C|J (n) (r)|p |t − r| and E |J (n) (T )|p ≤ C. Using the same arguments as in the proof of Lemma 5.1, the following moment estimates of the drift follow from Condition 3.2. Proposition 5.3 There exists a constant C > 0 such that for any 0 ≤ r ≤ t ≤ T and I ∈ I have i i h h (n) (n) EFr |βI (t)|2 ≤ C|J (n) (r)|2 and E sup |βI (t)|2 ≤ C. t∈[0,T ]

Next, we establish moment estimates for the intensities of passive order arrivals. Lemma 5.4 There exists a constant C > 0 such that for any 0 ≤ r ≤ t ≤ T and p ∈ {2, 4} i i h h (n) (n) EFr kδv(n) λIK (t, ·)kpLp ≤ C|J (n) (r)|p and E sup kδv(n) λIK (t, ·)kpLp ≤ C. t∈[0,T ]

16

Proof. Here we just prove this result with p = 4. From (3.4) and the H¨ older inequality, we have Z Z

t

(n)

4 (n) (n) ψIK,ij (x, t − s)|δx(n) |2 Nij (ds) dx

|δx |2 Z t Z (n) 4 Z t 3 δv (n) (n) (n) 2 (n) ≤ |δx(n) |2 Nij (ds) (n) ψIK,ij (x, t − s) dx|δx | Nij (ds) 2 0 R |δx | 0 Z t 4 (n) ≤ |δx(n) |2 Nij (ds)

0

R

(n)

δv

0

and

Z Z tZ Z 4 (n) (n) ΨIK,ik (x, y, t − s)δv(n) Mik (ds, dy, dz) dx R 0 Z R RZ Z Z Z tZ Z 3 t (n) (n) (n) ≤ |ΨIK,ik (x, y, t − s)|4 dxδv(n) Mik (ds, dy, dz) δv(n) Mik (ds, dy, dz) 0 R R R 0 R R Z t Z Z 4 (n) ≤ δv(n) Mik (ds, dy, dz) . 0

R

R

By (3.3) and the Cauchy inequality, (n)

kδv(n) λIK (t, x)k4L4 ≤ C + C +C

X

i∈I,j∈J

X

i∈I,k∈K

Z t 4 (n) |δx(n) |2 Nij (ds) 0

Z t Z Z 4 (n) δv(n) Mik (ds, dy, dz) ≤ C|J (n) (t)|4 . 0

R

R

The desired result follows directly from the proof of Lemma 5.1.

✷

The conditional moment estimates on the volume density functions use the following oservation. Since 1∆(n) (x−P (n) (s)) (y) = 1∆(n) (y+P (n) (s)) (x), a

a

x, y ∈ R

it follows from Fubini’s theorem, for any integrable function g(y), Z Z Z Z g(y)1∆(n) (x−P (n) (s)) (y)dydx g(y)dydx = a (n) R ∆(n) (x−Pa (s)) ZR ZR g(y)1∆(n) (y+P (n) (s)) (x)dydx = a Z Z ZR R g(y)dy 1∆(n) (y+P (n) (s)) (x)dx = δx(n) g(y)dy. = R

R

a

(5.4)

R

Lemma 5.5 There exists a constant C > 0 such that for any 0 ≤ r ≤ t ≤ T and I ∈ I, # "   (n) (n) 4 4 4 4 EFr kVI (t, ·)kL4 ≤ C[kVI (r, x)kL4 + |J (r)| ] and E sup kVI (t, ·)kL4 ≤ C. t∈[0,T ]

(n)

Proof. Since the third term on the right side of the third equations in (3.11) is non-positive and Va (t, x) is always nonnegative, it follows from the H¨ older inequality that (n) 4 (n) δv n λ (s, y)dsdy aL (n) (n) (n) r ∆ (x−Pa (s)) δx Z Z Z (n) t 4 δv z ˜ (n) (ds, dy, dz) +C (e − 1) M aL (n) (n) r ∆(n) (x−Pa (s)) R+ δx Z tZ (n) 4 4 |α | (n) n 3 |t − r| C|Va(n) (r, x)|4 + C aL δ λ (s, y) dyds v aL (n) (n) r ∆(n) (x−Pa (s)) δx

Z tZ |Va(n) (t, x)|4 ≤ C|Va(n) (r, x)|4 + C ≤

αaL

17

Z t Z +C r

(n)

∆(n) (x−Pa

(s))

(n)

4 z ˜ (n) (ds, dy, dz) . (e − 1) M aL (n)

δv

Z

R+

δx

From the Burkholder-Davis-Gundy inequality, there exists a constant C > 0 such that # " Z Z Z (n) 4 t δv (n) z ˜ (ds, dy, dz) (e − 1)M EFr aL (n) (n) r ∆(n) (x−Pa (s)) R+ δx " Z Z # Z (n) t 2 2 δv (n) z ≤ CEFr (n) (e − 1) MaL (ds, dy, dz) (n) r ∆(n) (x−Pa (s)) R+ δx # " Z Z 2 t δ (n) 2 v (n) ≤ CEFr (n) λaL (s, y)dyds (n) r ∆(n) (x−Pa (s)) δx "Z Z # (n) t |δv |2 (n) ≤ C (n) |t − r|EFr |δv(n) λaL (s, y)|2 dyds (n) r ∆(n) (x−Pa (s)) |δx |3 " # Z tZ δ (n) 4 v (n) +CEFr (n) λaL (s, y)dyds . (n) r ∆(n) (x−Pa (s)) δx From (5.4) and Fubini’s theorem, EFr



kVa (t, ·)k4L4



≤

CkVa(n) (r, x)k4L4

+ Z

(n) C|αaL |4 |t

3

− r|

Z

t r

i h EFr kδv(n) λnaL (s, ·)k4L4 ds

δ (n) 2 t i h v (n) +C (n) |t − r| EFr kδv(n) λaL (s, ·)k2L2 ds r δx δ (n) 3 Z t i h v (n) +C (n) EFr kδv(n) λaL (s, ·)kL1 ds. r δx

The first result now follows from Lemma 5.1 and Lemma 5.4. The second result follows from: # " EFr

sup kVa (t, ·)k4L4

t∈[0,T ]

≤

CkVa(n) (0, x)k4L4

"Z Z + CEFr

≤ CkVa(n) (0, x)k4L4 + C

0 T

R

X

p∈{1,2,4}

Z

0

T

Z

(n)

∆(n) (x−Pa

(s))

Z

R+

(n)

δv

(n)

δx i

z

(e −

#

4 (n) 1)MaL (ds, dy, dz) dx

h EFr kδv(n) λnaL (s, ·)kpLp ds ≤ C.

✷

5.2

Moment estimates for the increments

We are now going to prove moment estimates for the increments of the state processes. To this end, we put (n) (n) αIK = µIK (ez − 1) for any I ∈ I and K ∈ K. Then, Z t Z t Z t (n) (n) (n) (n) (n) (n) ˜ (n) ˜ (n) (ds) Pa (t) = Pa (0) + ρbM (S (s))βa (s)ds + δx NbM (ds) − δx(n) N aL 0 0 Z t (n) 0(n) (n) ρbM (S (s)) − ρaL (S(n) (s)) (n) 2 (n) + |δx | µaL (s)ds, (n) 0 δx Z tZ Z (n) δv (n) (n) ˜ (n) (ds, dy, dz) (ez − 1)M Va (t, x) = Va (0, x) + aL (n) (n) (n) 0 ∆ (x−Pa (s−)) R+ δx Z tZ Z (n) δv ˜ (n) (ds, dy, dz) + V (n) (s−, y + Pa(n) (t))(ez − 1)M aC (n) a (n) 0 ∆(n) (x−Pa (s−)) R− δx Z tZ (n) h i δv (n) n (n) (n) (n) n α λ (s, y) + α V (s, y + P (t))λ (s, y) dsdy, + a aC aL aL aC a (n) (n) 0 ∆(n) (x−Pa (s)) δx (n)

(n)

with Pb (t) and Vb

(t, x) being defined analogously. 18

Lemma 5.6 There exists a constant C > 0 such that for any 0 ≤ r ≤ t ≤ T i h X (n)
kVI (r, x)k4L4 + |J (n) (r)|4 (|t − r|2 + |t − r|) EFr kS(n) (t) − S(n) (r)k2S 2 ≤ C I∈I

and

E

h

i sup kS(n) (t)k2S 2 ≤ C.

t∈[0,T ]

Proof. It suffices to prove the first inequality, since the second one follows directly from the first one with r = 0 and (3.1). From Lemma 5.1 and Proposition 5.3, we have  Z t  i h (n) 2 (n) 2 (n) 2 (n) (n) 2 EFr |βa (s)| + |δx | µaL (s) ds EFr |Pa (t) − Pa (r)| ≤ C(t − r)  rZ t  Z t 2  2  (n) ˜ (n) ˜ (n) (ds) + CEFr +CEFr δx(n) N δ N (ds) x aL bM r Z tr i h (n) ≤ C(t − r) EFr |βa(n) (s)|2 + ||δx(n) |2 µaL (s)|2 ds Z t r h i (n) (n) +C EFr |δx(n) |2 µbM (s) + |δx(n) |2 µaL (s) ds r

≤ C|J

(n)

(r)|2 (|t − r|2 + |t − r|).

Moreover, by the Markov inequality,

Hence

Z Z t−r t |δv(n) λnaL (s, y)|2 dyds |Va(n) (t, x) − Va(n) (r, x)|2 ≤ C (n) (n) (x−P (n) (s)) r ∆ δ a x Z Z t−r t +C (n) |Va(n) (s−, y + Pa(n) (t))|2 |δv(n) λnaC (s, y)|2 dsdy (n) r ∆(n) (x−Pa (s)) δx Z (n) Z t Z 2 δv z ˜ (n) (ds, dy, dz) +C (e − 1) M aL (n) (n) r ∆(n) (x−Pa (s−)) R+ δx Z (n) 2 Z t Z δv (n) (n) (n) z ˜ (ds, dy, dz) V (s−, y + P (t))(e − 1) M +C . a aC (n) a (n) r ∆(n) (x−Pa (s−)) R− δx i h EFr kVa(n) (t, ·) − Va(n) (r, ·)k2L2 Z t i h EFr kδv(n) λnaL (s, ·)k2L2 + kVa(n) (s, · + Pa(n) (t))δv(n) λnaC (s, ·)k2L2 ds ≤ C|t − r| r (n) Z t i h δv (n) (n) +C (n) EFr kδv(n) λaL (s, y)kL1 + k|Va(n) (s, · + Pa(n) (t))|2 δv(n) λaC (s, ·)kL1 ds δx Z r h i t ≤ C|t − r| EFr kδv(n) λnaL (s, ·)k2L2 + kδv(n) λnaC (s, ·)k4L4 + kVa(n) (s, ·)k4L4 ds r (n) Z t i h δv (n) (n) +C (n) EFr kδv(n) λaL (s, y)kL1 + kδv(n) λaC (s, ·)k2L2 + kVa(n) (s, ·)k4L4 ds r δx ≤ C[kVa(n) (r, x)k4L4 + |J (n) (r)|4 ](|t − r|2 + |t − r|).

We can get the similar results for the other terms. In conclusion, i hX i h (n) EFr kS(n) (t) − S(n) (r)k2S 2 ≤ C kVI (r, x)k4L4 + |J (n) (r)|4 (|t − r|2 + |t − r|). I∈I

The second result can be proved using similar arguments as in the proof of Lemma 5.5. Lemma 5.7 There exists a constant C > 0 such that for any 0 ≤ r ≤ t ≤ T i h EFr kD(n) (t) − D(n) (r)k2D2 1,2 hX i (n) 4 ≤ C kVI (r, ·)kL4 + |J (n) (r)|4 [γn + γ(t − r) + |t − r|2 + |t − r|]. I∈I

19

✷

older inequality, Proof. Here we just deal with EFr [kD(n) (t) − D(n) (r)k2D2 ]. From (3.8), (3.10) and the H¨ 2

r

2 (n) (n) (n) |φIJ,ij (t − s) − φIJ,ij (r − s)||δx(n) |2 Nij (ds) 0 Z Z r r (n) (n) (n) 2 (n) 2 (n) |δx(n) |2 Nij (ds) |φIJ,ij (t − s) − φIJ,ij (r − s)| |δx | Nij (ds) ≤ 0 0 Z r 2 (n) ≤ C[γn + γ(t − r)] |δx(n) |2 Nij (ds)

Z

0

and

(n) 2 |δx |2 (n) (n) (n) (n) (y, t − s) − Φ (y, r − s) M (ds, dy, dz) Φ δ v IJ,ik IJ,ik ik (n) 0 R R δv Z r Z Z (n) 2 (n) 2 |δx |2 (n) |δx | (n) (n) ≤ (n) ΦIJ,ik (y, t − s) − (n) ΦIJ,ik (y, r − s) δv(n) Mik (ds, dy, dz) 0Z RZ RZ δv δv

Z

r

Z Z

r

(n)

δv(n) Mik (ds, dy, dz) 0 R R Z r Z Z 2 (n) ≤ C[γn + γ(t − r)] δv(n) Mik (ds, dy, dz) . ×

0

R

R

From the above inequalities, (3.7), (3.9) and the Cauchy inequality, 2 (n) 2 (n) (n) |δx | µIJ (t) − |δx(n) |2 µIJ (r) ≤ CkS(n) (t) − S(n) (r)k2S 2 + C[γn + γ(t − r)]|J (n) (r)|2 +C[J (n) (t) − J (n) (r)]2 .

Similarly, Z Z

r

(n)

δv

(n)

2 (n) (n) (n) |ψIK,ij (x, t − s) − ψIK,ij (x, r − s)||δx(n) |2 Nij (ds) dx

|δx |2 Z r (n) 2 rZ δv (n) (n) (n) (n) 2 (n) |δx(n) |2 Nij (ds) ≤ (n) |ψIK,ij (x, t − s) − ψIK,ij (x, r − s)| dx|δx | Nij (ds) 2 0 0 R |δx | Z r 2 (n) 2 (n) ≤ [2γn + γ(t − r)] |δx | Nij (ds) R

0

Z

0

and

Z Z r Z Z 2 (n) (n) (n) |ΨIK,ik (x, y, t − s) − ΨIK,ik (x, y, r − s)|δv(n) Mik (ds, dy, dz) dx R Z0 ZR ZR Z r (n) (n) (n) ≤ |ΨIK,ik (x, y, t − s) − ΨIK,ik (x, y, r − s)|2 dxδv(n) Mik (ds, dy, dz) 0 R ZR Z R Z r (n) × δv(n) Mik (ds, dy, dz) 0 R R Z Z Z r 2 (n) ≤ [2γn + γ(t − r)] δv(n) Mik (ds, dy, dz) . 0

R

R

From the above inequalities and the Cauchy inequality, (n)

(n)

kδv(n) λIK (t, x) − δv(n) λIK (r, x)k2L2

≤ CkS(n) (t) − S(n) (r)k2S 2 + C[γn + γ(t − r)]|J (n) (r)|2 + C[J (n) (t) − J (n) (r)]2 . In conclusion, from Proposition 5.2 and Lemma 5.6, i h i h EFr kD(n) (t) − D(n) (r)k2D2 ≤ C[γn + γ(t − r)]|J (n) (r)|2 + CEFr kS(n) (t) − S(n) (r)k2S 2 2 i h +CEFr |J (n) (t) − J (n) (r)|2 ≤ C[γhn + γ(t − r) + |t − r|]|J (n) (r)|2 i X (n) +C kVI (r, x)k4L4 + |J (n) (r)|4 (|t − r|2 + |t − r|). I∈I

20

✷ Using (3.6), (3.8) and (3.10) the following result can be proved similarly to the previous one. Lemma 5.8 There exists a constant C > 0 such that for any 0 ≤ r ≤ t ≤ T and I ∈ I have i h (n) (n) EFr |βI (t) − βI (r)|2 hX i (n) kVI (r, x)k4L4 + |J (n) (r)|4 [γn + γ(t − r) + |t − r|2 + |t − r|]. ≤ C I∈I

(n)

(n)

We are ready to prove the tightness of (S(n) , D(n) , βa , βb ) and the continuity of the cluster points. (n)

(n)

Proposition 5.9 Suppose Condition 3.1-3.4 hold. Then the sequence (S(n) , D(n) , βa , βb ) is tight as a sequence in D([0, ∞), S × D × R2 ). Moreover, any cluster point (S∗ , D∗ , βa∗ , βb∗ ) is continuous, i.e.  P (S∗ , D∗ , βa∗ , βb∗ ) ∈ C([0, ∞), S × D × R2 ) = 1. (n)

(n)

Proof. The tightness of (S(n) , D(n) , βa , βb ) follows from Lemma 5.1, 5.3, 5.6, 5.7 and 5.8 using Kurtz’s criterion. It remains to prove the continuity of the cluster points. From the Skorokhod representation theorem [22, Theorem 2.7], we can construct a new sequence of processes defined on a common space and with the same law to the initial sequence such that they converge almost surely. Continuity of the price processes follows from standard arguments. In order to prove the continuity of Va∗ , let f ∈ Cb2 (R). Then X E[ hVa(n) (s, ·) − Va(n) (s−, ·), f i2 ] s≤t

≤

Cδv(n) E

R hZ tZ 

(n)

∆(n) (y−Pa

0

(s)) (n) δx

RR

f (x)dx 2

(n)

δv(n) λaL (s, y)dyds

i

i f (x)dx  Z (n) ∆(n) (y−Pa (s)) (n) (n) 2 (n) (n) +δv(n) E V (s, y + P (t)) δ λ (s, y)dyds a a v aC (n) R δx Z t 0 (n) (n) ≤ Cδv(n) E[kδv(n) λaL (s, ·)kL1 + kVa(n) (s, ·)k4L4 + kδv(n) λaC (s, ·)k2L2 ]ds ≤ Cδv(n) . hZ t

0

i ∗ ∗ 2 = 0. Hence continuity follows from standard arguments. By Fatou’s lemma, E s≤t hVa (s) − Va (s−), f i Likewise, 2 (n) 2 (n) (n) (n) (n) ˆIJ (t−, S(n) (t−))] ˆIJ (t, S(n) (t))] − |δx(n) |2 [µIJ (t−) − µ |δx | [µIJ (t) − µ Z Z 2 X X (n) (n) (n) (n) = ||δx(n) |2 ΦIJ,ik (y, 0)|2 Mik (dt, dy, dz) |δx(n) |2 φIJ,ij (0) Nij (dt) + i∈I,j∈J i∈I,k∈K R R X X Z Z (n) (n) ≤ |δx(n) |4 Nij (dt) + |δv(n) |2 Mik (dt, dy, dz). hP

i∈I,j∈J

and as n → ∞

i∈I,k∈K

R

R



 2 X (n) (n) (n) (n) E ˆIJ (t, S(n) (t))] − |δx(n) |2 [µIJ (t−) − µ ˆIJ (r, S(n) (r))]  |δx(n) |2 [µIJ (t) − µ t≤T



≤ CE 

X

i∈I,j∈J

≤

C(|δx(n) |2

∨

Z

0

T

(n)

|δx(n) |4 Nij (dt) +

δv(n) )E

h i J (n) (T ) → 0.

X

i∈I,k∈K

Z

0

T

Z Z R

R



(n) |δv(n) |2 Mik (dt, dy, dz)

Hence we educe the continuity of µ∗IJ (t) − µ ˆIJ (t, S(t)). The continuity of µ∗IJ (t) follows from (3.7) and the continuity of S(t). The continuity of other terms can be proved similarly. ✷

21

6

Proof of the characterization result In this section, characterize the weak accumulation points of the sequence of LOB models.

S Definition 6.1 Let Gt = σ{X(s) : s ∈ [0, t], X ∈ C([0, ∞), S × H × R2 )} and G = t≥0 Gt . We say that a mapping Z : [0, ∞) × C([0, ∞), S × D × R2 ) 7→ D × R2 is a progressively measurable functional, if for any t ≥ 0 fixed, Z restricted to [0, t] × C([0, ∞), S × D × R2 ) is B([0, t]) × Gt /B(D × R2 ) measurable. The process D(t) depends on the whole trajectory {S(s) : s ∈ [0, t]}. We rewrite D(t) as D(t, S). For any S = (pa , pb , va , vb ) ∈ C([0, ∞), S), define θI (t, S) := ρI (S(t))βI (t, S) + ̺I (S(t))µI (t, S),

σI (t, S) := ρI (S(t))µI (t, S),

ηa (t, x, S) := αaL λaL (t, x − pa (t), S) + αaC λaC (t, x − pa (t), S)va (s, x), ηb (t, x, S) := αbL λbL (t, pb (t) − x, S) + αbC λbC (t, pb (t) − x, S)vb (t, x),

which are progressively measurable functionals. Then (3.11) turns to be Z tp Z t θa (s, S)ds + Pa (t) = Pa (0) + 2σa (s, S)dBa (s), 0 0 Z t Va (t, x) = Va (0, x) + ηa (s, x, S)ds

(6.1)

0

with Pb and Vb being represented analogously. For any fa (x), fb (x) ∈ L2 (R), define S f := (pa , pb , vaf , vbf ), where Z Z f f va (t, x)fa (x)dx, vb (t) := vb (t, x)fb (x)dx. va (t) := R

R

For any t ≥ 0, we introduce a second-order differential operator: for any G ∈ C 2 (R4 ), (Atf G)(S) =

X I∈I

σI (t, S)

∂ 2 G(S f ) X ∂G(S f ) ∂G(S f ) X f , + + ηI (t, S) θI (t, S) 2 ∂pI ∂pI ∂vIf I∈I

I∈I

where ηIf (t, S) =

Z

ηI (t, x, S)fI (x)dx.

R

Definition 6.2 A probability measure P on C([0, ∞), S) is called a solution to the martingale problem associated to Atf , if for any fa (x), fb (x) ∈ L2 (R) and G ∈ C 2 (R4 ), under P Z t MfG (t) := G(S f (t)) − (Asf G)(S)ds; t ≥ 0, 0

is a continuous, local martingale. If P is induced by a C([0, ∞), S)-valued random variable S, we also say S is a solution to the martingale problem associated to Atf . Theorem 6.3 For some two-dimensional Brownian motion (Ba , Bb ), if (S, Ba , Bb ) is a weak solution to the equation (6.1), then S is a solution to the martingale problem associated to Atf . Conversely, if S is a solution to the martingale problem associated to Atf , then there exists a two-dimensional Brownian motion (Ba , Bb ) defined on an enlarged probability space, such that (S, Ba , Bb ) is a weak solution to (6.1). Proof. The result can be considered as an extension of Problem 4.3 and Proposition 4.6 in [24, p.313-315]. We just give a brief outline. If S is a solution to the stochastic dynamic system, by Itˆ o’s formula it is easy to identify that MfG (t) is a continuous, local martingale. Conversely, suppose that MfG (t) is a continuous, local martingale for every {fi (x) ∈ L2 (R) : i ∈ I} and G(x) ∈ C 2 (R4 ). By the standard stopping time argument, we have S(t) = S(0) + AS (t) + MS (t), 22

where AS (t) := (APa , APb , AVa , AVb ) is a predictable, S-valued process with locally bounded variations and MS (t) := (MPa , MPb , MVa , MVb ) is a continuous, S-valued, local martingale. Moreover, for any {fi (x) ∈ L2 (R) : i ∈ I}, like the definition of Sf let Af := (APa , APb , AfVa , AfVb )

and M f := (MPa , MPb , MVfa , MVfb ).

By Itˆ o’s formula, for any G(x) ∈ C 2 (R4 ) Z 1 t ∂ 2 G(Sf (s)) ∂G(Sf (s)) f dA (s) + dhM f , M f is + Local Mart. f 2 0 ∂|S f |2 0 Z ∂S X t ∂G(Sf (s)) X Z t ∂G(Sf (s)) f f = G(S (0)) + dAPI (s) + dAVII (s) fI ∂PI ∂V 0 0 I I∈I I∈I X 1 Z t ∂ 2 G(Sf (s)) X 1 Z t ∂ 2 G(Sf (s)) f dhMVfII , MVII′′ is dhMPI , MPI ′ is + + fI ′ fI ′ 2 ∂P ∂P 2 I I ∂V ∂V 0 0 I I′ I,I ′ ∈I I,I ′ ∈I X 1 Z t ∂ 2 G(Xf (s)) fI ′ dhMPI , MVI ′ is + Local Mart. + 2 0 ∂PI ∂V f′I ′ ′

G(Sf (t)) = G(Sf (0)) +

Z

t

I,I ∈I

I

By the uniqueness of canonical decompositions of special semi-martingales [11, p.213], for any I, I ′ ∈ I and I 6= I ′ Z t Z t θa (s, S)ds, AfVI (t) = APa (t) = ηIf (s, S)ds 0

0

and f

f

hMPa , MPb it = hMVfII , MVII′′ it = hMPI , MVII′′ it ≡ 0.

hMPI , MPI it = 2ρI (S(t))σI2 (t),

By the representation theorem for semi-martingales, the existence of weak solutions to (6.1) follows from, e.g. [22, p.90]. ✷ We are now ready to prove the main result of this paper. Proof for Theorem 3.5: By the Skorokhod representation theorem, we may without loss of generality assume (n) (n) that (S(n) , D(n) , βa , βb ) → (S, D, βa , βb ) almost surely. We now proceed in three steps. (i) we prove that µbM (t) = µaL (t) and that µbL (t) = µaM (t); (ii) we show that µI (t) satisfies equations (3.12); (iii) we show that λIK (t, x) can be given by (3.13); (iv) we show that βI (t) can be described by (3.14); (v) we identify that S(t) is a solution to the martingale problem associated with Atf . Step 1. Since (n) (n) |µbM (t) − µaL (t)| = lim |δx(n) |2 µbM (t) − |δx(n) |2 µaL (t) = lim |δx(n) βa(n) (t)| = 0 n→∞

and

n→∞

(n) (n) (n) |µbL (t) − µaM (t)| = lim |δx(n) |2 µbL (t) − |δx(n) |2 µaM (t) = lim |δx(n) βb (t)| = 0, n→∞

n→∞

we have µbM (t) = µaL (t) := µa (t) and µbL (t) = µaM (t) := µb (t). n o (n) (n) . For any I ∈ I, J ∈ J , define Step 2. Convergence of |δx |2 µIJ (t) n≥1

(n)

MIJ (t) :=

X

i∈I,j∈J

+

X

Z

t

0

i∈I,k∈K

(n)

(n)

˜ (ds) φIJ,ij (t − s)|δx(n) |2 N ij

Z tZ Z

(n)

0

R

R

(n)

|δx |2 (n) δv

(n) ˜ (n) (ds, dy, dz) ΦIJ,ik (y, t − s)δv(n) M ik

(n)

ˆIJ (t, S(n) (t−)) − = |δx(n) |2 µIJ (t) − |δx(n) |2 µ 23

X

i∈I,j∈J

Z

0

t

(n)

(n)

φIJ,ij (t − s)|δx(n) |2 µij (s)ds

X

−

i∈I,k∈K

Z tZ 0

(n)

|δx |2 (n) δv

R

(n)

(n)

ΦIJ,ik (y, t − s)δv(n) λik (s, y)dyds.

(6.2)

(n)

We are going to show the a.s. convergence of {MIJ (t)}n≥1 . To this end, we put (n)

2 τK := inf{t ∈ [0, T ] : kD(n) (t)kD1,2 ≥ K} (K > 0).

It is easy to see that as n → ∞ (n)

2 τK → τK := inf{t ∈ [0, T ] : kD(t)kD1,2 ≥ K},

(n)

a.s.

(n)

It is enough to prove the almost sure convergence of {MIJ (t ∧ τK )}n≥1 . We analyse the different term separately. • The first term on the right side of the last equality in (6.2) converges a.s. by assumption (D(n) → D a.s.). Convergence of the second term follows from (3.7): as n → ∞, (n)

(n)

(n)

||δx(n) |2 µ ˆIJ (t ∧ τK , S(n) ((t ∧ τK )−)) − µ ˆIJ (t ∧ τK , S(t ∧ τK ))| (n) (n) (n) ≤ C(|τK − τK | + kS ((t ∧ τK )−) − S(t ∧ τK )kS 2 → 0, a.s. (n)

• For the third term, we assume without loss of generality that τK ≤ τK a.s. Then, Z

(n)

t∧τK 0

t∧τK

Z

≤

(n) φIJ,ij (t

0

−

(n) s)|δx(n) |2 µij (s)ds

(n) |φIJ,ij (t

(n) s)|δx(n) |2 µij (s)

−

−

Z

0

t∧τK

φIJ,ij (t − s)µij (s)ds

− φIJ,ij (t − s)µij (s)|ds

Z t∧τK(n) (n) (n) φIJ,ij (t − s)|δx(n) |µij (s)ds + t∧τK Z t∧τK (n) (n) (n) |φIJ,ij (t − s)|δx(n) |2 µij (s) − φIJ,ij (t − s)µij (s)|ds + CK|τK − τK |. ≤ 0

The second term above tends to 0 as n → ∞. For the first term, by (3.4), (3.10) and the dominated convergence theorem, Z t∧τK (n) (n) |φIJ,ij (t − s)|δx(n) |2 µij (s) − φIJ,ij (t − s)µij (s)|ds 0 Z t∧τK (n) (n) |φIJ,ij (t − s)|2 |δx(n) |2 µij (s) − µij (s)|ds ≤ 0Z t∧τK (n) + |φIJ,ij (t − s) − φIJ,ij (t − s)|2 |µij (s) ds Z0 t∧τK (n) ||δx(n) |2 µij (s) − µij (s)|ds ≤ C 0Z t (n) +K |φIJ,ij (t − s) − φIJ,ij (t − s)|ds → 0, a.s. 0

(n)

• For the fourth term, we assume again without loss of generality that τK ≤ τK a.s. Then, Z

(n)

t∧τK

0

Z

R

Z ≤

(n)

|δx |2

(n) t∧τK

t∧τK t∧τK

Z +

0

(n)

δv Z

(n)

(n)

ΦIJ,ik (y, t − s)δv(n) λik (s, y)dyds −

Z

t∧τK

Z

ΦIJ,ik (y, t − s)λik (s, y)dyds

0 R (n) 2 |δx | (n) (n) ΦIJ,ik (y, t − s)δv(n) λik (s, y)dyds (n) R δ Z h v (n) 2 |δx | (n) (n) ΦIJ,ik (y, t − s)δv(n) λik (s, y) − ΦIJ,ik (y, t (n) R δv

24

i − s)λik (s, y) dyds

≤

Z (n) 2 |δx | (n) (n) (n) ΦIJ,ik (y, t − s)δv(n) λik (s, y) − ΦIJ,ik (y, t − s)λik (s, y) dyds 0 R δv (n) +CK|τK − τK |.

Z

t∧τK

The second term vanishes as n → ∞. For the first term, by (3.10) Z (n) 2 |δx | (n) (n) (n) ΦIJ,ik (y, t − s)δv(n) λik (s, y) − ΦIJ,ik (y, t − s)λik (s, y) dyds 0 R δv Z t∧τK |δ (n) | x (n) (n) Φ (y, s) − Φ (y, s) kδv(n) λik (s, ·)kL1 ds ≤ sup (n) IJ,ik IJ,ik y∈R,s∈[0,T ] δv 0 Z t∧τK (n) (n) kδv λik (s, ·) − λik (s, ·)kL1 ds +C

Z

t∧τK

0

≤ K

|δ (n) |2 x (n) (n) (n) ΦIJ,ik (x, s) − ΦIJ,ik (x, s) + Ckδv(n) λik (s, ·) − λik (s, ·)kL1 → 0, x∈R,s∈[0,T ] δv sup

a.s.

(n)

Since K > 0 is arbitrary the limit MIJ (t) := limn→∞ MIJ (t) exists and equals X Z t MIJ (t) = µIJ (t) − µ ˆIJ (t, S(t)) − φIJ,ij (t − s)µij (s)ds i∈I,j∈J

−

X

i∈I,k∈K

Z tZ 0

R

0

ΦIJ,ik (y, t − s)λik (s, y)dyds,

a.s.

We are now going to prove that MIJ (t) = 0. By Cauchy’s inequality and Condition 3.2, for any t ∈ [0, T ], X Z T (n) (n) 2 2 (n) 2 (n) |MIJ (t)| ≤ C + C sup ||δx | µIJ (t)| + CT ||δx(n) |2 µij (s)|2 ds t∈[0,T ]

X

+C

i∈I,k∈K

i∈I,j∈J

Z

0

T

0

Z 2 h i (n) δv(n) λik (s, y)dy ds ≤ C 1 + sup kD(n) (t)k2D12 . t∈[0,T ]

R

(n)

Hence, it follows from Lemma 5.1 that supn≥1 E[|MIJ (t)|2 ] < ∞. In particular, (n)

{|MIJ (t)|2 }n≥1

is uniformly integrable. (n)

Thus, almost sure convergence implies convergence of {MIJ (t)}n≥1 to MIJ (t) in L1 (P). Hence, by Fatou’s lemma, for any K > 0 large enough, i i h h   (n) (n) E |MIJ (t)|2 1{|MIJ (t)|2 ≤K} = E lim |MIJ (t)|2 1{|M(n) (t)|2 ≤K} ≤ lim inf E |MIJ (t)|2 n→∞ n→∞ IJ  Z t 2  X (n) ˜ (n) (ds) ≤ lim inf C E φIJ,ij (t − s)|δx(n) |2 N ij n→∞

0

i∈I,j∈J

# " Z Z Z (n) 2 t X |δx |2 (n) (n) ˜ (n) ΦIJ,ik (y, t − s)δv Mik (ds, dy, dz) + lim inf C E (n) n→∞ 0 R R δv i∈I,k∈K i X Z t h (n) (n) = lim inf C E |φIJ,ij (t − s)|2 |δx(n) |4 µij (s) ds n→∞

i∈I,j∈J

0

# "Z 2 |δ (n) |2 t X (n) 2 (n) x (n) + lim inf C E (n) ΦIJ,ik (y, t − s) |δv | λik (s, y)dyds n→∞ R δv i∈I,k∈K 0 Z t i h X (n) ≤ lim C|δx(n) |2 E |δx(n) |2 µij (s) ds Z

n→∞

+ lim Cδv(n) n→∞

i∈I,j∈J

X

i∈I,k∈K

0

Z

0

t

i h (n) E kδv(n) λik (s, ·)kL1 ds = 0. 25

(6.3)

The continuity of MIJ (t) now implies P{MIJ (t) = 0; t ∈ [0, T ]} = 1, and hence X Z t µIJ (t) = µ ˆIJ (t, S(t)) + φIJ,ij (t − s)µij (s)ds 0

i∈I,j∈J

X

+

i∈I,k∈K

Step 3. Convergence of

n

o (n) (n) δv λIK (t)

(n)f δv(n) λIK (t)

:=

Z

R

n≥1

Z tZ 0

R

ΦIJ,ik (y, t − s)λik (s, y)dyds

a.s.

. For any f ∈ L2 (R) and I ∈ I, K ∈ K, let

(n) δv(n) λIK (t, x)f (x)dx,

λfIK (t)

:=

Z

λIK (t, x)f (x)dx

R

and Z (n)f (n)f ˆ(n) (t, S(n) (t−), x)f (x)dx MIK (t) := δv(n) λIK (t) − δv(n) λ IK R Z Z (n) t X δv (n) (n) − (x, t − s)f (x)dx|δx(n) |2 µij (s)dx ψ (n) 2 IK,ij 0 R |δ | x i∈I,j∈J X Z tZ Z (n) (n) − ΨIK,ik (x, y, t − s)f (x)dxδv(n) λik (s, y)dyds 0

i∈I,k∈K

=

R

R

(n) δv (n) ˜ (n) (ds) (x, t − s)|δx(n) |2 N ψ ij (n) 2 IK,ij i∈I,j∈J 0 |δx | X Z tZ Z (n) ˜ (n) (ds, dy, dz), + ΨIK,ik (x, y, t − s)δv(n) M ik R R i∈I,k∈K 0

X

Z

t

Using the same arguments as in the previous step, Z (n)f f f ˆIK (t, S(t), x)f (x)dx MIK (t) → MIK (t) = λIK (t) − λ R X Z tZ − ψIK,ij (x, t − s)f (x)dxµij (s)ds i∈I,j∈J

−

X

i∈I,k∈K

0

R

Z tZ Z 0

R

R

ΨIK,ik (x, y, t − s)f (x)dxλik (s, y)dyds

a.s.

as well as Z (n)f (n)f ˆ(n) (t, S(n) (t−), x)f (x)dx |MIK (t)| ≤ C|δv(n) λIK (t)| + C δv(n) λ IK R Z Z (n) t X δv (n) (n) 2 (n) (s)ds (x, t − s)f (x)dx|δ | µ ψ +C x ij (n) 2 IK,ij 0 R | |δ x i∈I,j∈J X Z t Z Z (n) (n) +C ΨIK,ik (x, y, t − s)f (x)dxδv(n) λik (s, y)dyds ≤

≤

0 R R i∈I,k∈K (n) (n) ˆ(n) (t, S(n) (t−), ·)kL2 Ckf kL2 kδv λIK (t, ·)kL2 + Ckf kL2 kδv(n) λ IK Z (n)

t X

δv

(n) (n) +Ckf kL2

(n) ψIK,ij (·, t − s) 2 |δx(n) |2 µij (s)ds 2 L |δx | i∈I,j∈J 0 X Z tZ (n) (n) +CT kf kL2 kΨIK,ik (·, y, t − s)kL2 δv(n) λik (s, y)dyds 0 R i∈I,k∈K X Z t (n) (n) (n) Ckf kL2 kδv λIK (t, ·)kL2 + Ckf kL2 + Ckf kL2 |δx(n) |2 µij (s)ds i∈I,j∈J 0 i h X Z t (n) +Ckf kL2 kδv(n) λik (s, ·)kL1 ds ≤ C 1 + sup kD(n) (t)kD11 . t∈[0,T ] i∈I,k∈K 0

26

As in (6.3), it follows from Lemma 5.1, 5.4 and Fatou’s lemma that n o P MfIK (t) = 0, t ∈ [0, T ] = 1.  Step 4. Convergence of β (n) n≥1 . As in Step 2, we can define X

M(n) a (t) :=

i∈I,j∈J

Z

0

X

+

t

i∈I,k∈K

(n) ˜ (n) (ds) φML,ij (t − s)|δx(n) |2 N ij

Z tZ Z 0

R

(n) ˜ (n) (ds, dy, dz) ΦML,ik (y, t − s)δv(n) M ik

R

= βa(n) (t) − βˆa(n) (t, S(n) (t−)) − −

X

i∈I,k∈K

Z tZ 0

R

t

Z

X

0

i∈I,j∈J

(n)

(n)

(n)

φML,ij (t − s)|δx(n) |2 µij (s)ds

(n)

ΦML,ik (y, t − s)δv(n) λik (s, y)dyds

(n)

and prove the convergence of {Ma (t)}n≥1 to Ma (t) = βa (t) − βˆa (t, S(t)) − −

X

i∈I,k∈K

Z tZ 0

R

X

i∈I,j∈J

Z

t

φML,ij (t − s)µij (s)ds

0

ΦML,ik (y, t − s)λik (s, y)dyds

and E[|Ma (t)|2 1{|Ma (t)|2 ≤K} ] ≤ lim E[|Ma(n) (t)|2 ] = 0, n→∞

for any K > 0. From this, we can again deduce that P{Ma (t) = 0} = 1 and hence that (3.14) holds. Step 5. S is a weak solution to (3.11). In view of Theorem 6.3 it is enough to prove that S solves the martingale problem associated with Atf . Since C(R) is dense in L2 (R), we may chose C(R) as our set of test functions. (n) (n) (n)f (n)f For fa , fb ∈ C(R), let Sf (n) := (Pa , Pb , Va , Vb ), where Z tZ Z (n) (n) JaL (s, y + Pa(n) (s−), z)δv(n) MaL (ds, dy, dz) Va(n) (0, x)fa (x)dx + R 0 R R+ Z tZ Z (n) (n) JaC (s, y + Pa(n) (s−), z)δv(n) MaC (ds, dy, dz), + 0 R R− Z tZ Z Z (n) (n) (n) (n) (n)f JbL (s, Pb (s−) − y, z)δv(n) MbL (ds, dy, dz) Vb (0, x)fb (x)dx − Vb (t) = R 0 R R+ Z tZ Z (n) (n) (n) JbC (s, Pb (s−) − y, z)δv(n) MbC (ds, dy, dz), − Va(n)f (t) =

Z

0

R

R−

and for any I ∈ I (n) JIL (s, y, z)

z

= (e − 1)

(n)

Z

fI (x)

∆(n) (y) (n)

JIC (s, y, z) = (ez − 1)VI

(n)

δx Z

dx, fI (x)

(s−, y)

∆(n) (y)

(n)

δx

dx.

For any x ∈ R, let 01 (x) = (x, 0, 0, 0) and let 0i (i = 2, 3, 4) be defined similarly. By the Itˆ o formula, for any G(x) ∈ Cb2 (R4 ), f (n)

MG

(t) := G(Sf (n) (t)) − G(Sf (n) (0)) −

Z

t

∆01 (δ(n) ) G(Sf (n) (s)) x

(n) δx

0

27

(n)

ρbM (S(n) (s))βa(n) (s)ds

t

Z

−

∆01 (δ(n) ) G(Sf (n) (s)) ρ(n) (S(n) (s)) − ρ(n) (S(n) (s)) bM

x

0 t

Z

aL

(n) δx

(n) δx

∆01 (δ(n) ) G(Sf (n) (s)) + ∆−01 (δ(n) ) G(Sf (n) (s))

(n)

|δx(n) |2 µaL (s)ds (n)

(n)

x |δx(n) |2 ρaL (S(n) (s))µaL (s)ds (n) |δx |2 Z t∆ f (n) (s)) (n) (n) G(S 02 (δx ) (n) − ρbL (S(n) (s))βb (s)ds (n) 0 δx Z t∆ f (n) (s)) ρ(n) (S(n) (s)) − ρ(n) (S(n) (s)) (n) G(S 02 (δx ) (n) aM bL δx(n) µaM (s)ds − (n) (n) 0 δx δx Z t∆ f (n) (s)) + ∆−02 (δ(n) ) G(Sf (n) (s)) (n) (n) G(S 02 (δx ) (n) x − ρaM (S(n) (s))|δx(n) |2 µaM (s)ds (n) 2 0 |δx | f (n) Z tZ Z ∆ (s)) (n) (n) G(S 03 (JaL (s,y,z)δv ) (n) (n) (n) − JaL (s, y, z)δv(n) λaL (s, y − Pa(n) (s))νaL (dz)dyds (n) (n) 0 R R+ JaL (s, y, z)δv f (n) Z tZ Z ∆ (s)) (n) (n) (n) G(S 03 (JaC (s,y,z)δv ) (n) (n) JaC (s, y, z)δv(n) λaC (s, y − Pa(n) (s))νaC (dz)dyds − (n) (n) 0 R R− JaC (s, y, z)δv f (n) Z tZ Z ∆ (s)) (n) (n) G(S 04 (−JbL (s,y,z)δv ) (n) (n) (n) (n) JbL (s, y, z)δv(n) λbL (s, Pb (s) − y)νbL (dz)dyds − (n) (n) 0 R R+ JbL (s, y, z)δv f (n) Z tZ Z ∆ (s)) (n) (n) G(S 04 (−JbC (s,y,z)δv ) (n) (n) (n) (n) JbC (s, y, z)δv(n) λbC (s, Pb (s) − y)νbC (dz)dyds − (n) 0 R R− JbC (s, y, z)

−

x

0

is a martingale, where ∆x F (y) := F (y + x) − F (y). f (n)

In proving that {MG (t)}n≥1 converges almost surely we may without loss of generality assume that G along with its first and second derivative is bounded by 1. From (3.6), (n)

(n)

|ρbM (S(n) (t)) − ρbM (S(t))| ≤ |ρbM (S(n) (t)) − ρbM (S(n) (t))| + |ρbM (S(n) (t)) − ρbM (S(t))| → 0. (n)

From the mean value theorem, there exists some ξs

(n)

∈ [Sf (n) (s), Sf (n) (s) + 01 (δx )] such that

f (n) (n) ∆ (s)) ∂G(ξs ) ∂G(Sf (s)) 01 (δx(n) ) G(S ′ f − G (S (s)) = | − | ≤ kξs(n) − Sf (s)kS 2 ≤ δx(n) . (n) ∂Pa ∂Pa δx

Using similar arguments as above, from these two results and the dominated convergence theorem, Z

t

∆01 (δ(n) ) G(Sf (n) (s)) x

(n)

δx

0

(n)

ρbM (S(n) (s))βa(n) (s)ds →

Z

0

t

∂G(Sf (s)) ρbM (S(s))βa (s)ds, ∂Pa

a.s.

Applying the mean value theorem again, we also have ∆01 (δ(n) ) G(Sf (n) (s)) + ∆−01 (δ(n) ) G(Sf (n) (s)) x

x

(n)

|δx |2

→

∂ 2 G(Sf (s))) ∂Pa2

and Z

0

t

∆01 (δ(n) ) G(Sf (n) (s)) + ∆−01 (δ(n) ) G(Sf (n) (s)) x

x

(n) |δx |2

→

Z

0

t

(n)

(n)

|δx(n) |2 ρaL (S(n) (s))µaL (s)ds

∂ 2 G(Sf (s)) ρaL (S(s))µaL (s)ds, ∂Pa2

a.s.

˜IK defined in the proof of Theorem 4.8, we have that In terms of the function λ ˜ aC (s, · − P (n) (s)) − λ ˜ aC (s, · − Pa (s))kL2 kλaC (s, · − Pa(n) (s)) − λaC (s, · − Pa (s))kL2 ≤ kλ a (n) ˆ ˆaC (s, · − Pa (s))kL2 . +kλaC (s, · − Pa (s)) − λ 28

From (3.7) and (4.7), kλaC (s, · − Pa(n) (s)) − λaC (s, · − Pa (s))kL2 ≤ C|Pa(n) (s) − Pa (s)| → 0,

a.s.

Similar to the argument above, we have ∆03 (J (n) (s,y,z)δ(n) ) G(Sf (n) (s)) v

aC

→

(n)

(n)

JaC (s, y, z)δv

∂G(Sf (s)) ∂Vaf

Like the time stopping argument in Step 2, we can prove Z Z tZ (n) (n) z [νaC (e − 1) − αaC ] VI (s−, y) 0

fI (x)

∆(n) (y)

R

and

Z Z tZ (n) [VI (s−, y) − VI (s, y)] 0

Thus

fI (x)

∆(n) (y)

R

(n)

δx

(n)

δx

,

a.s.

dx dyds → 0,

dx dyds → 0,

a.s.

a.s.

Z tZ Z

(n) (n) JaC (s, y, z)νaC (dz) − αaC VI (s, y)f (y) dyds 0 R R− Z tZ Z fI (x) (n) (n) z dx dyds ≤ [νaC (e − 1) − αaC ] VI (s−, y) (n) (n) 0 R Z tZ Z ∆ (y) δx fI (x) (n) dx dyds +αaC [VI (s−, y) − VI (s, y)] (n) ∆(n) (y) δx Z0 t ZR Z fI (x) +αaC VI (s, y) dx − f (y) dyds (n) (n) 0 R (y) δx Z ∆ Z Z t fI (x) (n) (n) ≤ [νaC (ez − 1) − αaC ] dx dyds VI (s−, y) (n) 0 R ∆(n) (y) δx Z tZ Z fI (x) (n) dx dyds +αaC [VI (s−, y) − VI (s, y)] (n) 0 R ∆(n) (y) δx Z tZ Z fI (x) +αaC VI (s, y) dx − f (y) dyds → 0, a.s. (n) 0 R ∆(n) (y) δx

Putting the last three results together, we have Z tZ Z 0

R

R−

∆03 (J (n) (s,y,z)δ(n) ) G(Sf (n) (s)) aC

v

(n) (n) JaC (s, y, z)δv Z

→ αaL

0

t

Z

R

(n)

(n)

(n)

JaC (s, y, z)δv(n) λaC (s, y − Pa(n) (s))νaC (dz)dyds

∂G(Sf (s)) ∂Vaf

fa (y)λaL (s, y − Pa (s))dyds,

a.s.

The other terms can be analyzed similarly. Thus, f (n)

MG

Z t ∂G(Sf (s)) a.s. ρbM (S(s))βa (s)ds (t) → MfG (t) = G(Sf (t)) − G(Sf (0)) − ∂Pa 0 Z t Z t 2 ∂G(Sf (s)) ∂ G(Sf (s)) − ̺a (S(s))µaL (s)ds − ρaL (S(s))µaL (s)ds ∂Pa ∂Pa2 0 Z Z0 t t ∂G(Sf (s)) ∂G(Sf (s)) ρbL (S(s))βb (s)ds − ̺b (S(s))µaM (s)ds − ∂Pb ∂Pb 0 Z0 t 2 ∂ G(Sf (s)) ρaM (S(s))µaM (s)ds − ∂Pb2 0 Z tZ ∂G(Sf (s)) − αaL fa (y)λaL (s, y − Pa (s))dyds ∂Vaf Z0 t ZR ∂G(Sf (s)) αaC fa (y)Va (s, y)λaL (s, y − Pa (s))dyds − ∂Vaf 0 R 29

− −

Z tZ 0

R

Z tZ 0

∂G(Sf (s)) ∂Vbf ∂G(Sf (s)) ∂Vbf

R

αbL fb (y)λbL (s, Pb (s) − y)dyds αbC fb (y)Va (s, y)λbC (s, Pb (s) − y)dyds.

Moreover, since G ∈ Cb2 (R2 ), by H¨ older inequality and Cauchy’s inequality, it is easy to see f (n)

|MG

Z th i (n) (n) (n) (t)|2 ≤ C + CT |βa(n) (s)|2 + ||δx(n) |2 µaL (s)|2 + |βb (s)|2 + ||δx(n) |2 µaM (s)|2 ds Z t Z t 0 (n) 2 (n) (n) kδv(n) λbL (s, y)k2L2 ds kδv λaL (s, y)kL2 ds + CT +CT 0 Z0 t (n) (n) (n) (n) 4 +CT [kVa (s, ·)kL4 + kVb (s, ·)k4L4 + kδv(n) λaC (s, ·)k4L4 + kδv(n) λaC (s, ·)k4L4 ]ds 0   (n) (n) (n) ≤ C 1 + sup kD(n) k2D2 + sup [|βI (t)|2 + kVI (s, ·)k4L4 + kδv(n) λIC (s, ·)k4L4 ] . 2

t∈[0,T ]

t∈[0,T ],I∈I

f (n)

This inequality along with Lemma 5.1, 5.4, 5.5, 5.6 and Proposition 5.3 proves that the sqeuence {MG (t)}n≥1 f (n) is uniformly integrable. Hence, since {MG (t)}n≥1 converges almost surely, it also converges to MfG (t) in f L1 (P). As a result, {MG (t) : t ≥ 0} is a martingale. A standard stopping argument shows that {MfG (t)} is a local martingale for any G ∈ C 2 (R4 ). Thus S solves the martingale problem associated to Atf . ✷

A

Hawkes random measures

In this appendix, we introduce a class of random point measures (“Hawkes random measures”) that can be viewed as an extension of the Hawkes processes introduced in [14, 15]. Let (Ω, F , P) be a complete probability space endowed with filtration {Ft }t≥0 that satisfies the usual hypotheses. Let (U, U ) be a measurable space endowed with a base measure m(du). Without loss of generality, we always assume that m(U ) = 1. A real-valued two-parameter process {h(t, x) : t ≥ 0, x ∈ U } is said to be (Ft )-progressive if for every t ≥ 0 the mapping (ω, s, x) 7→ h(ω, s, x) restricted to Ω × [0, t] × U is measurable relative to Ft × B([0, t]) × U . Let pt be a (Ft )-point process on U and N (dt, du) be a random point measure on [0, ∞) × U defined as follows: N (I, A) = #{s ∈ I : ps ∈ A},

I ∈ B(R+ ), A ∈ U .

Definition A.1 A nonnegative, (Ft )-progressive process λ(t, u) is called the intensity process of N (dt, du) with respect to the base measure m(du) if for any nonnegative (Ft )-predictable process H(t, u) on U have hZ tZ i hZ t Z i E H(s, u)N (ds, du) = E H(s, u)λ(s, u)m(du) . ds 0

U

0

U

For any nonnegative, (Ft )-progressive process λ(t, u) defined on U , we can construct a random point measure N (dt, du) on [0, ∞) × U with intensity process λ(t, u) as follows: Z tZ Z ∞ N ([0, t], A) = 1{z≤λ(s,u)} N0 (ds, du, dz), t ≥ 0, A ∈ U , 0

A

0

where N0 (ds, du, dz) is a Poisson random measure on [0, ∞) × U × [0, ∞) with intensity dsm(du)dz. Definition A.2 We say N (dt, du) is a Hawkes random measure on [0, ∞) × U if its intensity process λ(t, u) can be written as Z tZ λ(t, u) = µ(t, u) + φ(s, u, v, t − s)N (ds, dv), (1.1) 0

U

where µ(t, u) : [0, ∞) × U 7→ [0, ∞) and φ(t, u, v, r) : [0, ∞) × U 2 × [0, ∞) 7→ [0, ∞) are (Ft )-progressive. 30

The processes µ(t, u) and φ(t, u, v, r) are called the exogenous intensity and kernel of the Hawkes random measure N (dt, du), respectively. We always assume that there exists some C > 0 such that, for any t ∈ [0, T ], Z Z φ(t, u, v, s)m(du) ≤ C. (1.2) µ(t, u)m(du) + sup v∈U

U

U

Then, E

i hZ i hZ t Z Z i λ(t, u)m(du) = E µ(t, u)m(du) + E φ(s, u, v, t − s)m(du)N (ds, dv) U U 0 U U hZ t Z i ≤ C + CT E λ(s, v)m(dv)ds .

hZ

0

and by the Gr¨onwall inequality, E[

R

U

U

λ(t, u)m(du)] < ∞ and E[N ([0, T ], U )] < ∞.

The following lemma proves the existence of a Hawkes random measure for any (Ft )-progressive processes µ(t, u) and φ(t, u, v, r). Lemma A.3 For any nonnegative, (Ft )-progressive processes µ(t, u) and φ(t, u, v, r) satisfying (1.2), there exists a Hawkes random measure with intensity process defined by (1.1). Proof. Let N0 (ds, du, dz) be the Poisson random measure introduced above. For any t ≥ 0 and u ∈ U , define λ−1 (t, u) = 0, λ0 (t, u) = µ(t, u) and for any n ≥ 1, λn (t, u) = µ(t, u) +

n Z tZ X 0

m=1

φ(s, u, v, t − s)Nm (ds, dv),

U

where Nm (I, A) =

Z Z Z I

A

∞

1{λm−2 (s,u)≤z