Shizan Fang

Andrey Pilipenko

Additive functionals and push forward measures under Veretennikov's flow

Fukushima with

Introduction

Let (X t ) t≥0 be a Brownian flow on R m , that is, W t = X t -X 0 is a standard Brownian motion; then for a function u ∈ C 2 (R m ), Itô formula says that

u(X t ) -u(X 0 ) = t 0 ∇u(X s ) • dW s + t 0 1 2 ∆u(X s ) ds.
(1.1)

In a celebrated paper [START_REF] Fukushima | A decomposition of additive functionals of finite energy[END_REF] M. Fukushima extended a C 2 function u in (1.1) to a function u in the Sobolev space H 1 (R m ); in order to reach this end, he used an additive functional N

[u] t of X • to express the last term in (1.1), moreover he showed that N

[u] t /t tends to [START_REF] Pilipenko | Transport equation and Cauchy problem for BV vector fields[END_REF] 2 ∆u in distribution sense. In this work, we will be concerned with the stochastic differential equation (SDE) on R m dϕ t (x) = dW t + a(ϕ t (x))dt, ϕ 0 (x) = x ∈ R m , (

where a : R m → R m is a measurable map.

The SDE (1.2), due to the non-degenerated noise W t , makes illuminating difference with ordinary differential equations (ODE). In the context of ODE, the existence of a flow of quasi-invariant measurable maps associated to a vector field a on R m belonging to Sobolev space, having a bounded divergence div(a), was established in a seminal paper by Di Perna and Lions in Ref. [START_REF] Perna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF] ; their result was extended later in Ref. [START_REF] Pilipenko | Transport equation and Cauchy problem for BV vector fields[END_REF] by L. Ambrosio to a vector field having only bounded variation regularity and bounded divergence (see also Ref. [START_REF] Cipriano | Flows associated with irregular R d -vector fields[END_REF] ).

There are various considerations to SDE (1.2). When a is bounded, it was proved by Veretennikov in Ref. [START_REF] Veretennikov | On strong solutions and explicit formulas for solutions of stochastic integral equations[END_REF] that there exists a unique strong solution ϕ t (x) to SDE (1.2). Moreover if a is Hölderian, it was proved in Ref. [START_REF] Flandoli | Well-posedness of the transport equation by stochastic perturbation[END_REF] as well as in Ref. [START_REF] Zhang | Stochastic flows of SDEs with irregular coefficients and stochastic transport equations[END_REF] that x → ϕ t (x) is a flow of diffeomorphisms. Recently, it was proved in Ref. [START_REF] Aryasova | On differentiability of stochastic flow for multidimensional SDE with discontinuous drift[END_REF] that if a is of bounded variation, and µ k,j = ∂a k ∂xj are signed measures satisfying (1.19) for all k, j,, then the solution ϕ t to SDE (1.2) is in Sobolev space:

ϕ t (•) ∈ ∩ p≥1 W 1 p,loc (R m , R m ), t ≥ 0.
Moreover, the Sobolev derivative ∇ϕ t is a solution to the equation

∇ϕ t = I + t 0 Āϕ (ds)∇ϕ s (x), t ≥ 0,
where Āϕ is the additive functional associated to ∇a.

In Ref. [START_REF] Zhang | Strong solutions of SDES with singular drift and Sobolev diffusion coefficients[END_REF] , X. Zhang allowed a to be time-dependent, and established the existence of strong solutions under integrability conditions on the drift a, while in Ref, [START_REF] Krylov | Strong solutions of stochastic equations with singular time dependent drift[END_REF] Krylov and Röckner considered such a SDE on a domain of R m and established the existence of strong solutions. In another direction, in Ref. [START_REF] Bass | Brownian motion with singular drift[END_REF] Bass and Chen took the point of view of additive functionals

A i t = t 0 a i (ϕ s (x)) ds,
where a i denotes the ith-component of a, to generalize the drift a; a i (x)dx seen as the Revuz measure associated to A i t , was extended to the Kato class K α for some α > 0, where K α is the class of signed measures on R m defined by

K α = π(dx); lim ε→0 sup x∈R m B(x,ε) |x -y| α |π|(dy) = 0 (1.3)
where |π| denotes the total variation of π. More precisely, they proved that if the Revuz measures π 1 , . . . , π m are in K m-1 with m ≥ 3, then with help of associated additive functionals A t = (A 1 t , . . . , A m t ), the SDE

X t = x + W t + A t
admits a unique weak solution. The interest of considering π 1 , . . . , π m in Kato class is they are not necessarily absolutely continuous with respect to the Lebesgue measures. In the case where a = ∇ log ρ, by considering

E(u, v) = R m ∇u • ∇v ρ(x)dx,
the theory of Dirichlet forms provides a powerful tool, which allows ρ to be only locally integrable (see Refs. [START_REF] Fukushima | Dirichlet forms and Markov processes[END_REF][START_REF] Fukushima | On a stochastice calculus related to Dirichlet forms and distorted Brownian motion[END_REF] ).

In this work, we are interested in push-forward measures under the map x → ϕ t (x) defined by SDE (1.2). It is well-known that if a is smooth and bounded, then x → ϕ t (x) is a diffeomorphism of R m and the inverse flow ϕ -1 t can be expressed by SDE with reversed Brownian motion. More precisely, for t > 0 given, let W t s = W (ts) -W (t), and ψ t s solve the SDE

dψ t s (x) = -a(ϕ t s (x))dt + dW t s , s ∈ [0, t], ψ t 0 (x) = x; (1.4) then ϕ -1 t = ψ t t .
Let γ be the standard Gaussian measure on R m . By Kunita 21 the push forward measure (ϕ -1 t ) * γ admits the density Kt with respect to γ given by

Kt (x) = exp - t 0 ϕ s (x), •dW s - t 0 δ(a)(ϕ s (x)) ds (1.5)
where •dW s means the stochastic integral in Stratanovich's sense, and δ(a) is the divergence with respect to γ, that is,

R m ∇f, a dγ = R f δ(a) dγ for all f ∈ C 1 0 (R m ).
We have δ(a)(x) = a, xdiv(a) so that

t 0 δ(a)(ϕ s (x)) ds = t 0 a(ϕ s (x)), ϕ s (x) ds - t 0 div(a)(ϕ s (x)) ds. (1.6)
Here is the main result of this paper Theorem 1.1. Let a : R m → R m be a bounded measurable map. Assume that the divergence div(a) in generalized sense is a signed measure µ satisfying the condition

lim t→0 sup x∈R m R m t 0 s -m/2 e -|x-y| 2 /(2s) ds |µ|(dy) = 0, (1.7)
where |µ| denotes the total variation of µ. Let ϕ t be given by SDE (1.2); then almost surely the push forward measure (ϕ t ) * γ is equivalent to γ, and the density Kt of the push forward measure (ϕ -1 t ) * γ with respect to γ has the expression

Kt (x) = exp -A t + t 0 a(ϕ s (x)), ϕ s (x) ds - t 0 ϕ s (x), dW s - mt 2 , (1.8)
where A t is the additive functional associated to div(a). Shizan Fang and Andrey Pilipenko

Notice that if f is a positive function in the Kato class K m-2 , then (see Ref. [START_REF] Bass | Brownian motion with singular drift[END_REF] ),

lim t→0 sup x∈R m E t 0 f (W s + x) ds = 0,
that is nothing but (1.7) for µ(dy) = f (y)dy.

The organization of the paper is as follows. In section 1.2, we will recall and collect some facts concerning continuous additive functionals. Section 1.3 is devoted to the proof of Theorem 1.1. In section 1.4, we will consider some examples of vector fields a whose divergence div(a) is a signed measure, but is not absolutely to Lebesgue measure. In section 1.5, we will discuss briefly generalizations of Theorem 1.1.

Continuous additive functionals

In this section, we recall some definitions and facts about continuous additive functionals of Markov processes. There are a lot of publications in the litterature on this topic, see for example Refs. [START_REF] Dynkin | Markov Processes. Fizmatlit[END_REF][START_REF] Fukushima | Dirichlet forms and Markov processes[END_REF][START_REF] Gihman | The theory of stochastic processes[END_REF][START_REF] Gikhman | The theory of stochastic processes[END_REF][START_REF] Revuz | Continuous martingales and Brownian motion[END_REF][START_REF] Uemura | Positive continuous additive functionals of mutidimensional Brownian motion and the Brownian local time[END_REF] . Here we will follow Chapters 6 -8 in Ref. [START_REF] Dynkin | Markov Processes. Fizmatlit[END_REF] Chapter II, section 6 in Ref. [START_REF] Gikhman | The theory of stochastic processes[END_REF] . We don't need the theory on the whole generality; so some assumptions, statements or definitions are simplified in our exposition.

Let {X t , t ≥ 0} be a continuous R m -valued homogeneous Markov processes adapted to a filtration {F t , t ≥ 0} with infinite life-time, P x be the distribution of

X • given X 0 = x. Denote N t = σ(X s , s ∈ [0, t]). Definition 1.1. A non negative additive functional of X is a R + -valued, N t - adapted process A = {A t (X), t ≥ 0} such that 1) it is almost surely continuous in t and A 0 (X) = 0; 2) it is additive, i.e. ∀t ≥ 0 ∀s ≥ 0 ∀x ∈ R m : A t+s (X) = A s (X) + A t (θ s X), P x -a.s.,
where θ s is the shift operator.

Following the terminology of Dynkin 9 , we introduce the notion of W -functional.

Definition 1.2. A non negative continuous additive functional

A t (X) is called W - functional if ∀t ≥ 0 : sup x E x (A t (X)) < ∞.
(1.9)

The function

f t (x) = E x (A t (X)) is called the characteristics of A t (X).
Here is an obvious example

Example 1.1. Let b : R m → [0, ∞) be a bounded measurable function. Then A t (X) := t 0 b(X s )ds (1.10) is a W -functional of X.
Assume that for any t > 0, X t has a transition density p(t, x, y). Then the characteristics of A t (X) defined in (1.10) is equal to

f t (x) = E x t 0 b(X s )ds = t 0 R m b(y)p(s, x, y)dyds = R m t 0 p(s, x, y)ds b(y)dy.
(

1.11)

There are a close relations between convergence of W -functionals and their characteristics. The first one is the following Proposition 1.1. (see Ref. [START_REF] Dynkin | Markov Processes. Fizmatlit[END_REF] , Theorem 6.3) A W -functional is defined by its characteristics uniquely up to the equivalence.

The second one concerns the convergence, that is,

Theorem 1.2. (Ref. 9 , Theorem 6.4, Lemma 6.1 ′ ) Let {A (n) t (X)} be a sequence of W -functionals of X and f (n) t (x) = E x A (n) t (X) be their characteristics. Assume that a function f t (x) is such that for each t > 0 lim n→∞ sup 0≤s≤t sup x∈ R m |f (n) s (x) -f s (x)| = 0.
(1.12)

Then f t (x) is the characteristics of a W -functional A t (X). Moreover, for each t > 0, lim n→+∞ E x (|A t (X) -A (n) t (X)| 2 ) = 0,
and in probability,

lim n→+∞ sup s∈[0,t] |A (n) s (X) -A s (X)| = 0. Example 1.2. Let {X t = B t , t ≥ 0} be a one-dimensional Brownian motion; set A (n) t := t 0 2n1 B(s)∈ -1 n , 1 n ds.
Then a function b n in expression (1.10) is equal to 2n1 {|x|≤1/n} and converges to the Dirac mass δ 0 at 0. It is easy to verify that (1.12) holds with

f t (x) = R t 0 p(s, x, y)dsδ 0 (dy) = t 0 p(s, x, 0)ds, where p(s, x, y) = 1 √ 2πs exp |x -y| 2 2s
is the transition density of a Brownian motion. The limiting additive functional is the local time of a Brownian motion at 0. Now let us write (1.11) as

f t (x) = R d t 0 p(s, x, y)ds µ(dy), (1.13)
with µ(dy) = a(y)dy. Note that representation (1.13) makes a sense even µ is not absolutely continuous with respect to Lebesgue measure.

Similarly to Example 1.2, sometimes it is possible to assign a W -functional to a measure. For example, assume that there exists a sequence of non-negative bounded, measurable functions {b n , n ≥ 1} such that for t > 0

lim n→0 sup x R m t 0 p(s, x, y)ds (µ n (dy) -µ(dy)) = 0,
where µ n (dy) = b n (y)dy. Then a function f t (x) defined in (1.13) is the characteristic of a W -functional. We will formally denote it by

A t := t 0 dµ dy (X s )ds.
If there are some a priori estimates on the transition density of X t , then using the described approach it is possible to characterize a class of measures corresponding to its W -functionals. See for example Ref. [START_REF] Dynkin | Markov Processes. Fizmatlit[END_REF] , Ch.8 for W -functionals of a Brownian motion.

Let's come back to SDE (1.2). It is known in Ref. [START_REF] Aronson | Bounds for the fundamental solution of a parabolic equation[END_REF] that the transition density of ϕ t (x) exists and there are constants c 1 , c 2 > 0 depending only on sup x |a(x)| such that ∀t ∈ (0, T ],

c -1 1 t -m/2 exp - |x -y| 2 c 2 t ≤ p(t, x, y) ≤ c 1 t -m/2 exp - c 2 |x -y| 2 t . (1.14)
Observe that ( Ref. [START_REF] Dynkin | Markov Processes. Fizmatlit[END_REF] Ch.8)

t 0 p(s, x, y)ds ≍ ω(|x -y|),
where

ω(r) =        1, m = 1, ln(r -1) ∨ 1, m = 2, r 2-m , m ≥ 3,
More precisely, for each t > 0, there exists a positive constant C such that for all x = y ∈ R m with m > 1,

C -1 ω(|x -y|) ≤ t 0 p(s, x, y)ds ≤ Cω(|x -y|). ( 1
(X) = A + t (X) -A - t (X)
, where {A ± t (X), t ≥ 0} are continuous non negative additive functionals of X.

For a signed measure µ = µ + -µ -such that lim t→0 sup x R m t 0 s -m/2 exp - |x -y| 2 2s ds |µ|(dy) = 0, (1.19)
where |µ| is the total variation of µ, we can construct a signed W-functional A t = A + t -A - t , where functionals A + t , A - t correspond to µ + , µ -respectively (see Ref. [START_REF] Dynkin | Markov Processes. Fizmatlit[END_REF] ).

Proof of Theorem 1.1

Let a be a bounded measurable vector field on R m .

Definition 1.4. We say that a signed measure µ on R m is the divergence in a generalized sense of a if for any test function

g ∈ C ∞ 0 (R m ) : R m a(x), ∇g(x) dx = - R m g(x)µ(dx),
where dx on the left hand side denotes the Lebesgue measure; we denote µ = div(a).

In what follows, we will assume that div(a) exists and satisfies condition (1.7).

Let {g n , n ≥ 1} ⊂ C ∞ 0 (R m ) be a sequence of non-negative smooth functions with compact support such that R m g n (x)dx = 1, and g n (x) = 0 for |x| > 1 n .

Put

a n (x) := a * g n (x) = R m a(x -y)g n (y)dy. (1.20) Note that a n ∈ C ∞ (R m , R m ), Shizan Fang and Andrey Pilipenko a n ∞ = sup x |a n (x)| ≤ sup x |a(x)| = a ∞ , (1.21)
and a n converges to a in all L p loc (R m , R m ). Without loss of generality we may assume that as n → +∞, a n (x) → a(x) for almost everywhere x ∈ R m .

(1.22)

Let ϕ n t (x) be the stochastic flow of diffeomorphisms defined by

dϕ n t (x) = dW t + a n (ϕ n t (x))dt, ϕ n 0 (x) = x ∈ R m . (1.23)
Let γ be the standard Gaussian measure on R m . We set

K n t (x) = d(ϕ n t ) * γ dγ , Kn t (x) = d(ϕ n t ) -1 * γ dγ .
It is well-known (see Ref. [START_REF] Kunita | Stochastic flows and stochastic differential equations[END_REF] ) that

K n t (ϕ n t (x)) = 1 Kn t (x) , (1.24)
and

Kn t (x) = exp - t 0 (δa n )(ϕ n s (x))ds - t 0 ϕ n s (x), •dW s = exp - t 0 (δa n )(ϕ n s (x))ds - t 0 ϕ n s (x), dW s - mt 2 , (1.25)
where δa n (x) = (div a n )(x)a n (x), x . In, 10 the L p estimates on densities were established and used to prove the absolute continuity for a limit of push-forward measures. Here we will use the following result of Gikhman and Skorokhod Ref. [START_REF] Gihman | Densities of probability measures in function spaces. (Russian) Uspehi Mat. Nauk[END_REF] .

Theorem 1.3. (see Ref. [START_REF] Gihman | Densities of probability measures in function spaces. (Russian) Uspehi Mat. Nauk[END_REF] ) Let (X 1 .F, µ 1 ) be a probability space, X 2 be a complete separable metric space, µ 2 be a probability measure on the Borel σ-algebra B(X 2 ).

Assume that a sequence of measurable mappings {F n :

X 1 → X 2 , n ≥ 0} is such that 1) as n → +∞, F n converges to F 0 in measure µ 1 ;
2) for all n ≥ 1, the push forward measure (F n ) * µ 1 is absolutely continuous with respect to µ 2 ;

3) the sequence of the densities ρ n := d(Fn) * µ1 dµ2

, n ≥ 1 is uniformly integrable with respect to µ 2 . Then the push forward measure (F 0 ) * µ 1 is absolutely continuous with respect to µ 2 . Moreover, if ρ n converges to ρ in measure µ 2 , then ρ = d(F0) * µ1 dµ2 .

Let us apply Theorem 1.3 to the sequence {ϕ n t , n ≥ 1}. First of all, we remark that although in Ref. [START_REF] Luo | Absolute continuity under flows generated by SDE with measurable drift coefficients[END_REF] D. Luo assumed that the drift admits the divergence as a function satisfying some integrability condition, but in the proof of Theorem 3.4 in Ref., [START_REF] Luo | Absolute continuity under flows generated by SDE with measurable drift coefficients[END_REF] he only used Krylov estimate for non-degenerated diffusions, without involving the divergence. Since a n converges to a in all L p loc , we can use directly Theorem 3.4 in Ref. [START_REF] Luo | Absolute continuity under flows generated by SDE with measurable drift coefficients[END_REF] to get that for each x ∈ R m , we have

lim n→+∞ E sup t∈[0,T ] |ϕ n t (x) -ϕ t (x)| = 0. ( 1
|ϕ n t (x) -ϕ t (x)| = 0 = 1. (1.27)
It follows that for almost surely ω, for all t ∈ [0, T ], ϕ n t converges to ϕ t in measure with respect to γ.

Next, we will establish the uniform integrability of {K n t ; n ≥ 1}.

Proposition 1.2. We have

sup n≥1 E sup t∈[0,T ] R m K n t (x)| ln K n t (x)|γ(dx) < +∞.
(1.28)

Proof. We have

R m K n t (x)| ln(K n t (x))| dγ(x) = R m | ln(K n t (ϕ n t (x)))| dγ(x).
But by (1.24) and (1.25), we have

ln(K n t (ϕ n t (x))) = t 0 (div(a n )(ϕ n s (x)) ds - t 0 a n (ϕ n s (x)), ϕ n s (x) ds + t 0 ϕ n s (x), dW s + mt 2 .
Let T > 0 be fixed; then

E sup t∈[0,T ] R m K n t (x)| ln K n t (x)|γ(dx) ≤ R m E sup t∈[0,T ] | t 0 (div a n )(ϕ n s (x))ds| + E sup t∈[0,T ] | t 0 ϕ n s (x), dW s | + E T 0 |a n (ϕ n s (x))| |ϕ n s (x)|ds + mT dγ(x).
(1.29) Shizan Fang and Andrey Pilipenko By (1.21), it is well-known that there exists a constant c 0 > 0 independent of n such that

sup n E sup t∈[0,T ] |ϕ n t (x)| ≤ c 0 (1 + |x|), (1.30)
Using Burkholder's inequality and (1.30), we also have

sup n E sup t∈[0,T ] | t 0 ϕ n s (x), dW s | ≤ c 0 (1 + |x|). (1.31) Let us estimate E sup t∈[0,T ] | t 0 (div a n )(ϕ n s (x))ds| . Denote µ(dy) = (div a)(dy), µ n (dy) = div a n (y)dy. We have div(a n ) = div(a * g n ) = div a * g n = µ * g n . Let A n (t) = t 0 (div a n )(ϕ n s (x))ds;
then A n is a signed additive functional of ϕ n . Let p n (t, x, y) be the transition density of ω → ϕ n t (x, ω). By (1.14), there are two constants c 1 , c 2 > 0 independent of n such that

e -|x-y| 2 /c22t c 1 t m/2 ≤ p n (t, x, y) ≤ c 1 e -c2|x-y| 2 /2t t m/2 .
So there exists a constant β > 0 independent of n such that

t 0 p n (s, x, y) ds ≤ β t/c2 0 e -|x-y| 2 /2s s m/2 ds. Let k t (r) = t/c2 0 e -r 2 /2s s m/2 ds.
It follows that

E(|A n (t)|) ≤ R m t 0 p n (s, x, y)| div a n (y)| dsdy ≤ β R m k t (|x -y|)|µ n |(dy).
(1.32)

We have

| div(a n )(y)| ≤ R m g n (y -z) |µ|(dz), so that R m k t (|x -y|)|µ n (y)| dy ≤ R m ×R m k t (|x -y|)g n (y -z) |µ|(dz)dy = R m R m k t (|x + z -y|)g n (y) |µ|(dz) dy ≤ σ(t),
where

σ(t) = sup x,y R m k t (|x + z -y|) |µ|(dz).
Then by condition (1.7), C T := sup t∈[0,T ] σ(t) < +∞. Now combining this with (1.30) and (1.31), and by (1.29), we finally obtained (1.28).

Now by Fatou's lemma,

E lim n→∞ sup t∈[0,T ] R m K n t (x)| ln K n t (x)|ν(dx) ≤ lim n→∞ E sup t∈[0,T ] R m K n t (x)| ln K n t (x)|ν(dx) < ∞.
So, for almost surely ω :

lim n→∞ sup t∈[0,T ] R m K n t (x)| ln K n t (x)|ν(dx) < ∞.
Hence for almost surely ω, there exists a random subsequence {n k } such that

sup k sup t∈[0,T ] R m K n k t (x)| ln K n k t (x)|ν(dx) < ∞. (1.33)
Now we can apply Theorem 1.3 to conclude that for almost surely ω, and for all t ∈ [0, T ], the push-forward measure (ϕ t ) * γ is absolutely continuous with respect to γ. Actually it remains to prove that (ϕ t ) * γ is equivalent to γ.

Proposition 1.3. The map x → ϕ t (x) admits an inverse map x → ψ t (x), which is given by the reserved SDE

dψ s (x) = dW t s -a(ψ s (x)) ds, ψ 0 (x) = x, s ∈ [0, t]. (1.34)
Proof. For each n, the inverse map of x → ϕ n t (x) is given by ψ n t where ψ n s solves

dψ n s (x) = dW t s -a(ψ n s (x)) ds, ψ n 0 (x) = x, s ∈ [0, t].
In the same way, we have

lim n→+∞ E sup s∈[0,t] |ψ n s (x) -ψ s (x)| = 0.
In order to prove that ψ t is the inverse map of ϕ t , we will use the following result Lemma 1.1. Let X, Y be complete, separable metric spaces, ν be a finite measure on X.

Assume that a sequence of X-valued random elements {ξ n , n ≥ 0} and a sequence of measurable functions f n : X → Y are such that 1) ξ n → ξ 0 in probability, as n → ∞;

2) f n → f 0 in measure with respect to ν, as n → ∞;

3) the push forward measure (ξ n ) * P is absolutely continuous with respect to ν;

4) the sequence of densities d(ξ n ) * P dν , n ≥ 1 is uniformly integrable with respect to ν. Then as n → +∞

f n (ξ n ) → f 0 (ξ 0 ) in probability.
We refer to Corollary 9.9.11 in Ref., [START_REF] Bogachev | Measure Theory[END_REF] as well as to Lemma 2 in Ref. [START_REF] Kulik | Nonlinear transformations of smooth measures on infinite-dimensional spaces[END_REF] for a proof.

Proof (continued) of Proposition 1.3. For almost surely ω, up to a subsequence, the family of densities {K n t ; n ≥ 1} is uniformly integrable. In Lemma 1.1, we take

X = Y = R m , ξ n = ϕ n t , f n = ψ n t .
Then ϕ n t (ψ n t ) converges to ϕ t (ψ t ) in probability. So that ϕ t • ψ t = Id. In the same way, we prove that ψ t • ϕ t = Id.

End of the proof of Theorem 1.1. Let A t be a signed additive functional of ϕ t that corresponds to the measure µ = div a. Then using Theorem 1.2, similarly to the proof of Lemma 3 in Ref.

A(t) = t 0 dµ(X(s)) dx ds is well-defined.
Remark that for m = 1, any finite measure µ satisfies condition (1.18). Indeed,

sup x R m t 0 s -1/2 exp - |x -y| 2 s dsµ(dy) ≤ µ(R) t 0 s -1/2 ds → 0, t → 0 + .
b) Functions of bounded variation. Assume that derivatives ∂ai ∂x k considered in a generalized sense are measures. Such function a = (a 1 , . . . , a m ) are called functions of bounded variation (BV). If this measures are of the form (1.36) with bounded b j , then a satisfies condition (1.7). Let now g ∈ C 1 (R m , R m ), D be a bounded domain with C 1 boundary. Then a(x) = g(x)1 {x∈D} also satisfies condition (1.7) since the generalized divergence div a equals to (div g(x))1 {x∈D} dx + g(x), n(x) σ ∂D (dx), where n(x) is the normal vector at x ∈ ∂D (see Ref. [START_REF] Federer | Geometric measure theory[END_REF] ). Linear combinations of the form

b 0 (x) + k j=1 g j (x)1 x∈Dj also satisfy condition (1.7), if b 0 ∈ Lip, g j ∈ C 1 , D j are bounded with C 1 boundary.
It should be noted that if a = (a 1 , . . . , a m ) is a vector field of bounded variation and µ k,j = ∂a k ∂xj satisfies (1.19) for all k, j, it has been proved in Ref.

3 P ϕ t (•) ∈ ∩ p≥1 W 1 p,loc (R m , R m ), t ≥ 0 = 1.
Moreover, the Sobolev derivative is a solution of the equation Hence it follows from Ch. 9.2 in Ref. [START_REF] Bogachev | Differentiable measures and the Malliavin calculus[END_REF] the absolute continuity (ϕ t ) * γ with respect to γ.

∇ϕ t = I + t 0 Āϕ (ds)∇ϕ s (x), t ≥ 0, ( 1
c) Example of a / ∈ BV with div a = 0. Functions of bounded variation is not unique example satisfying condition (1.7). For m = 2, let

a(x 1 , x 2 ) = (g(x 1 -x 2 ), g(x 2 -x 1 )),
where g is only measurable, bounded function. Then div a = 0 in the generalized sense, but partial derivatives ∂a ∂x k may not be measures.

Generalizations and localization.

In this section, we give briefly some generalization of Theorem 1.1.

Assume now that the vector field a is locally bounded and for any x ∈ R m . Assume that SDE (1.2) is conservative in the sense of Kunita 21 , that is, if τ (x) is the life-time of ϕ t (x), then P ({ω; τ (x) = +∞}) = 1.

For example, this is the case if a has a linear growth.

Let {f n ; n ≥ 1} be a sequence of functions in C ∞ 0 (R m ) such that sup Let ϕ n t be the solution to SDE (1.2) with a n instead of a. Observe that a n is a bounded vector field on R m . By uniqueness of solutions, almost surely, for t ≤ τ n (x), ϕ t (x) = ϕ n t (x). So for any bounded Borel function h : R m → R, {τn(x)≥t} h(ϕ t (x)) dγ(x) = {τn(x)≥t} h(ϕ n t (x)) dγ(x).

(1.39)

Observe that (ϕ n t ) * (1 {τn(x)≥t} γ) is absolutely continuous with respect to (ϕ n t ) * γ and for almost surely ω, (ϕ t ) * (1 {τn(x)≥t} γ) converges to (ϕ t ) * γ weakly as n → +∞ since τ n (x) → +∞, n → ∞.

Assume that for each n ≥ 1 where µ = div a. Then for any n, the vector field a n satisfies condition (1.7) in Theorem 1.1; therefore the push forward measure (ϕ n t ) * γ is absolutely continuous with respect to γ. Now let E be a Borel subset of R m such that γ(E) = 0; then by (1.39), then {τn(x)≥t} 1 E (ϕ t (x)) dγ(x) = {τn(x)≥t} 1 E (ϕ n t (x)) dγ(x) ≤ [(ϕ n t ) * γ](E) = 0.

Letting n → +∞ yields [(ϕ t ) * γ](E) = 0. in other words, (ϕ t ) * γ is absolutely continuous with respect to γ. Note also that in this case A ϕ n n (t) = A ϕ m m (t), t ∈ [0, τ n (x)] a.s.

for all m ≥ n. Therefore we can define A ϕ (t) = lim n→∞ A ϕ n n (t) and expression (1.8) also holds true if the reverse SDE (1.34) is conservative.

(

  |f n (x)| + |∇f n (x)|) < ∞; f n (x) = 1, for |x| ≤ n. Denote a n (x) = a(x)f n (x),andτ n (x) = inf{t ≥ 0 : |ϕ t (x)| ≥ n}.

  [START_REF] Aryasova | On differentiability of stochastic flow for multidimensional SDE with discontinuous drift[END_REF] , we get A t in L 2 , as n → ∞.Using again Lemma 1.1, we have for s fixed, a n • ϕ s converges to a • ϕ s in measure. 0} be a Markov process in R m with transition density satisfying condition(1.14). Let D 1 , . . . , D k be bounded domains of R m with C 1 boundary, and σ ∂Dj be the surface measure on ∂D j . , . . . , b k are bounded measurable functions. Then conditions (1.16) and (1.18) are satisfied. So the additive functional

	a) Examples of W -functionals.
	Let {X(t), t ≥ Let µ be a signed measure defined by
							k
		µ(dx) = b 0 (x)dx +	b k (x)σ ∂Dj (dx),	(1.36)
							j=1
	where b 0
		t
	0 s (x))ds → Therefore by expression div a n (ϕ n
	Kn t (ω, x) = exp -	0	t	(div a n )(ϕ n s (x)) ds +	0	t	a n (ϕ s (x)), ϕ n s (x) ds
				-	0	t	ϕ n s (x), dW s -	mt 2	,
	when n → +∞, K n t converges in measure P ⊗ γ to
	exp -A t +	0	t	a(ϕ s (x)), ϕ s (x) ds -		0	t	ϕ s (x), dW s -	mt 2	.	(1.35)
	The proof of Theorem 1.1 is completed.
	1.4. Examples
	In this section, we will construct examples of vector fields a satisfying the condition
	in Theorem 1.1.

Additive functionals and push forward measures under Veretennikov's flow


Kaj Nyström

Mohamed Ould Sidi

Aly

Kaj Nystr

Sidi Mohamed

Ould Aly
email: souldaly@math.uu.se

A framework

Keywords: 2000 Mathematics Subject Classification high frequency trading, modeling, order book, price dynamics, timeseries, random iterative function system, ergodicity

come   L'archive ouverte pluridisciplinaire

Introduction

In recent years, numerous electronic trading systems have been designed to replace or complement traditional markets. In electronic order-driven markets, participants may submit limit order (or cancel an existing limit order), by specifying whether they wish to buy or sell, the amount (volume) and the desired price. Limit orders wait in a queue to be canceled or executed (when a sell order is matched against one or more buy limit orders). All outstanding limit orders are aggregated in a limit order book which is available for market participant. The order book at a given instant of time is the list of all buy and sell limit orders (that have not been executed/canceled) with their corresponding price and volume. The price dynamics is therefore the result of the interplay between anonymous traders who place orders in the limit order book. Nowadays, more than half of the stock exchanges in the world are order driven and the submissions of limit orders have massively increased. The latter makes it extremely hard to build a model for the order book and its dynamics which is realistic and still amenable to rigorous quantitative analysis.

There is now a growing literature on limits order books and attempts to modeling their dynamics. Several authors have proposed models based on self-exciting point processes (see e.g. [START_REF] At-Sahalia | Modeling financial contagion using mutually exciting jump processes[END_REF], [START_REF] Bauwens | Modelling financial high frequency data using point processes[END_REF]) focusing on the modeling of the frequency at which the different types of orders arrive.

In other works (e.g. [START_REF] Parlour | Price dynamics in limit order markets[END_REF], [START_REF] Foucault | Order flow composition and trading costs in a dynamic limit order market[END_REF]) equilibrium models for limit order markets are proposed, while another category of models is based on some dynamic expected utility maximization models (e.g. [START_REF] Parlour | Price dynamics in limit order markets[END_REF], [START_REF] Rous | A dynamic model of the limit order book[END_REF]). More recently, Cont et al [START_REF] Cont | A stochastic model for order book dynamics[END_REF] considered the limit order book as a queuing system where the frequencies at which orders are submitted and canceled, all orders are assumed to be of unit size, are modeled as independent Poisson processes. Although all these models are theoretically attractive they have some difficulties describing the reality of the order book dynamics and main drawbacks of the models include Markov type assumption concerning the order flow for some and the difficulty of calibrating the parameters of the model for others. However the main reason that these models produce order book dynamics which deviates from what is observed in real data stems from the fact that it is difficult, or even impossible, to predict the arrival times of orders due to the inhomogeneity of these arrival times. This is a problem even for self-exciting Hawkes processes [START_REF] Hawkes | Point spectra of some mutually exciting point processes[END_REF] where the main difficulty is to accurately and properly calibrate the kernel defining the range of influence of previous events on the current intensity of events.

Based on the main drawback of the above models an idea is to model order book dynamics using a clock different from chronological time and in this paper we propose a modeling framework for the order book and its dynamics in event-based time, i.e. we are only interested in modeling the order book as it is changed by the occurrence of an event and by definition an event is the submission of an order, market, limit or cancelation, which changes the order book. In particular, in the proposed framework we make no reference to chronological time and the specific times at which the events occur. Instead, for us event time is simply an index, set to 0 at midnight, which represents an enumeration of the events and which increases by one every time a new order is submitted to the order book. The idea to use a different clock has already appeared in the literature, e.g. see Mandelbrot and Taylor [START_REF] Mandelbrot | On the Distribution of Stock Price Differences[END_REF], Clark [START_REF] Clark | A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices[END_REF], and more recently in Ané and Geman [START_REF] Ané | Order Flow, Transaction Clock and Normality of Asset Returns[END_REF] where it is showed that the normality of asset return can be recovered through a stochastic time change.

Our framework for the order book is influenced by Cont and De Larrard [START_REF] Cont | Order book dynamics in liquid markets: limit theorems and diffusion approximations[END_REF] and as in [START_REF] Cont | Order book dynamics in liquid markets: limit theorems and diffusion approximations[END_REF] we represent a reduced form of the state of the order book, right after event j, by (S b j , S a j , Q b j , Q a j ) where the best bid price S b j right after event j, the best ask price S a j right after event j, the size of the best bid queue Q b j right after event j, the size of the best ask queue Q a j right after event j.

(1.1)
The state of this reduced form of the order book changes when an order (market order, limit order or cancelation) effects the queues at the best bid/ask. When the best bid queue (or best ask queue) is depleted the price moves one (or more) step(s) down (or up) and the best bid ask spread widens immediately to more than one tick. In [START_REF] Cont | Order book dynamics in liquid markets: limit theorems and diffusion approximations[END_REF] the authors assume that after such an event the flow of limit buy/sell orders quickly fills the gap and that the spread instantly reduces to one tick again. In particular, we emphasize that in [START_REF] Cont | Order book dynamics in liquid markets: limit theorems and diffusion approximations[END_REF] it is assumed that the second step occurs instantly and hence that the spread, i.e. distance between the best bid and the best ask, always is one tick. Based on this assumption the process Q j = (Q b j , Q a j ) can be constructed from the order flow process, O F , until the next price move, as

(1.2) Q j = (Q b j , Q a j ) = O F j 0 ,j := Q j 0 +   j i=j 0 +1 V b i , j i=j 0 +1 V a i   , j > j 0 .
Here V b i is the change of the size of the bid queue occurring at event number i, V a i is the change of the size of the ask queue occurring at event number i.

(1.3) Note that V b i (V a i
) is positive if event i is a limit order and negative if it is a market order or cancelation and that V b i and V a i can not simultaneously be different from zero as one event can not change the queues on both sides of the order book. Whenever a component of O F j 0 ,. hits zero there is a price change: the price moves one step (tick) up or one step down, the spread widens to more than one tick and Q deviates temporarily from the order flow process (as Q b and Q a are both strictly positive, by definition of the best bid and best ask price). In [START_REF] Cont | Order book dynamics in liquid markets: limit theorems and diffusion approximations[END_REF] it is assumed, when a component of O F j 0 ,. hits zero, that the order book immediately jumps to the next step where the spread is again one tick and in [START_REF] Cont | Order book dynamics in liquid markets: limit theorems and diffusion approximations[END_REF] the queue sizes are updated according to some distribution functions.

In this paper we consider the reduced form of the order book briefly detailed above, but compared to [START_REF] Cont | Order book dynamics in liquid markets: limit theorems and diffusion approximations[END_REF] we allow the best bid ask spread to widen to more than one tick. (1.4) Our main modeling assumption is that the best bid and ask prices can only change by one tick at each event. (1.5) We emphasize that this is a very week assumption which may even be considered as a fact in liquid markets. In particular, for our data this assumption is satisfied in 100% of all cases (see Remark ?? below). Based on the assumption in (1.5) we show that the dynamics of this simplified order book can be fully described by only two processes (X, Y ). The first of these processes, X, is the (signed) size of the new order and the second processes Y is the size of the queue on the side of the order book which is changed by X. X and Y are a priori dependent and autocorrelated and the challenge is then to model these processes in a consistent way. A contribution of the paper is that apart from the assumption in (1.5), we make no additional assumptions concerning the order book dynamics. (1.6) Instead we propose a modeling framework for the order book dynamics, through (X, Y ), which is completely data driven and contains no adhoc assumptions.

The rest of the paper is organized as follows. In section 2 we show how the modeling of (Q b j , Q a j , V b j , V a j ) is reduced to the modeling of processes (X, Y ). We also show how (S b , S a ) is derived given (Q b , Q a , V b , V a ). In section 3, which is of empirical nature, we empirically describe the spread process found in high frequency data, the time to queue depletion as well as the statistical properties of the empirically observed (X, Y ). Section 3 serves as an empirical basis for our modeling framework proposed and outlined in section 4 and section 5. While we in section 4 give a short background on ARMA and AR models we in section 5 outline a semi linear model which is the core of our framework. An interesting aspect here is that we use results from the theory of random iterative function systems to understand issues concerning stationarity and ergodicity in our models. Section 6 is devoted to an application of our framework to the predication of the order flow and price moves in a the context of high frequency data. In section 7 we give a brief discussion and/of future research. Finally, in an appendix the inversion of Toeplitz matrices is discussed.

Order Flow and Price Dynamics

In the following we outline the reduced form of the order book considered in this paper. Recall the notation (S b j , S a j , Q b j , Q a j , V b j , V a j ) introduced in the introduction. In the following we use the notation that 1 event = 1 if event is true and 1 event = 0 if event is not true. Note that the processes V a j and V b j are given in terms of S b , S a , Q b , Q a , through

V b j = Q b j -Q b j-1 1 S b j =S b j-1 -Q b j-1 1 S b j <S b j-1 + Q b j 1 S b j >S b j-1 , V a j = Q a j -Q a j-1 1 S a j =S a j-1 -Q a j-1 1 S a j >S a j-1 + Q a j 1 S a j <S a j-1 . (2.1)
We will describe the dynamics of the simplified order book using the process (X, Y ) where

X j = V a j 1 j=ask-event -V b j 1 j=bid-event , Y j = Q a j 1 j=ask-event -Q b j 1 j=bid-event . (2.2)
The modeling in this paper is based on the assumption in (1.5) stated above, i.e. we assume that the best bid as well as the best ask price can only move by at most one tick (up or down) when an event occurs and we first discuss two consequences of (1.5). Firstly, assume the difference between the best bid and the best ask prices is exactly one tick. Then these prices can only move if either the best bid queue Q b or the best ask queue Q a is depleted (i.e. one component of the order flow (1.2) hits 0) and when this occurs the spread widens to two or more ticks. However, the assumption in (1.5) now implies that at this event the spread is only allowed to widen to exactly two ticks. In particular, this means, for example, that if best bid queue Q b is depleted then it is assumed that the second best bid queue, one tick away, is non-empty. Secondly, assume the difference between the best bid price and the best ask price is at least two ticks. Then two things can happen: either the best bid queue Q b or the best ask queue Q a is depleted or a new limit order is placed between the best bid and ask. In the first case, based on (1.5) and as above, the spread widens with exactly one tick. In the second case (1.5) implies that the spread narrows by exactly one tick since the new limit order is assumed to be submitted either one tick to the right of the best bid or one tick to the left of the best ask. In particular, an other way of formulating (1.5) is to state that the bid/ask spread can only change by exactly one tick at each event.

(2.3) However, we emphasis that we put no restrictions on the absolute size of the bid/ask spread.

Based on (1.5), (2.3), the dynamics of (Q b , Q a , V b , V a ) can be fully recovered from the dynamics of the process (X, Y ). Indeed, assume first the spread after event j -1 is exactly one tick. Then either the spread remains the same or it widens by one tick. If event j occurs on the ask side of the order book then Y j = Q a j and if event j occur on the bid side of the order book then Y j = -Q b j . Hence Y j > 0 if and only if event j occurs on the ask side of the order book and Y j < 0 if event j occurs on the bid side of the order book. Note that the event Y j = 0 can not occur as, by definition, neither Q a nor Q b can hit zero. Furthermore, using this notation we see that V a j = X j 1 Y j >0 and V b j = -X j 1 Y j <0 , while Q b and Q b are given in terms of V a and V b through the order flow equation (1.2) until the next price move. In this case, the next price move occurs when the spread widens by one tick and then Q a (Q b ) is re-initialized by Y (-Y ) if the price moves up (down). Next, assume that the spread after event j -1 is at least two ticks. Then either the spread remains the same or it widens by one tick or it narrows by one tick. In the first two cases we have as above that V a j = X j 1 Y j >0 and V b j = -X j 1 Y j <0 while again Q b and Q b are given in terms of V a and V b through the order flow equation (1.2), until the next price move, and when this occurs Q a (Q b ) is re-initialized by Y (-Y ) as above. In the third case a new limit order is submitted either one tick to the right of the best bid or one tick to the left of the best ask. Assume that the new limit order is an ask order, i.e. an ask event occurs. Then V a j > 0 and X j = V a j = Q a j = Y j since V a j becomes the new best ask queue. Similarly, if the new limit order is a buy order, i.e. a buy event occurs, then V b j > 0 and

X j = -V b j = -Q b j = Y j
since V b j becomes the new best bid queue. We can conclude that (V b , V a ) can be fully recovered from the process (X, Y ) and hence the dynamics of (Q b , Q a ) is determined by the dynamics of (V b , V a ) and hence by (X, Y ). Finally, the dynamics of (S b , S a ) follows from (X, Y ) through (Q b , Q a , V b , V a ) and (1.5). In particular, the dynamics of (S b , S a ) can be described as

[S b , S a ] -→                        (S b , S a + δ) -→        (S b , S a + 2δ) (S b + δ, S a + δ) (S b -δ, S a + δ) (S b , S a ) -→ . . . (S b -δ, S a ) -→        (S b -δ, S a + δ) (S b , S a ) (S b -δ, S a -δ) (S b -2δ, S a ) -→ . . .
where δ is the tick. In conclusion, based on (1.5) the dynamics of (S b , S a , Q b , Q a , V b , V a ) is completely determined by (X, Y ). In particular, the complexity in our set up is reduced to the process (X, Y ).

2.1. Price dynamics. In the following we let 0 denote the index of the last event up to now and we let F 0 be the history (σ-algebra) of all events that have occurred so far. We here discuss the dynamics of the best bid and ask prices and when we simply refer to the price we mean the mid price. The mid price can only change at an event as a consequence of that either the best bid and ask price change by the event.

Assume first that the bid-ask spread is exactly one tick, i.e. S a 0 = S b 0 + δ. Define l b 0 and l a 0 through (2.4)

l b 0 ≡ l b 0 (Q b 0 ) := min    l ≥ 1 : l j=1 V b j ≤ -Q b 0    , and (2.5)
l a 0 ≡ l a 0 (Q a 0 ) := min    l ≥ 1 : l j=1 V a j ≤ -Q a 0    .
The next change in the mid price, best bid price or best ask price, occurs at the event index τ 0 = min(l b 0 , l a 0 ), i.e. when the best bid queue or the best ask queue is depleted, and if l b 0 > l a 0 , i.e. τ 0 = l a 0 , then the next price move will be up and if l b 0 < l a 0 , i.e. τ 0 = l b 0 , then the next price move will be down. Naturally one can also ask what the probabilities for these events are. When Q b,0 0 ≪ Q a,0 0 , the bid queue will most likely be depleted before the ask queue, which means that the next price move will most likely be down. However, the outcome is less obvious when Q b 0 is close to Q a 0 . Let p + 0 (q b , q a ) := P l a 0 (q a ) < l b 0 (q b )| F 0 , p - 0 (q b , q a ) := P l a 0 (q a ) > l b 0 (q b )| F 0 , (2.6) whenever (q b , q a ) ∈ R 2 . Then p + 0 (q b , q a ) is the probability that ask queue, currently having size q a , will be depleted before that bid queue, currently having size q b . In particular, conditioned on (q b , q a ), p + 0 (q b , q a ) is the probability that the next prove move will be up. p - 0 (q b , q a ) have a similar interpretation. By (2.4), we see that q a -→ l a 0 (q a ) and q b -→ l b 0 (q b ) are non decreasing functions. Furthermore, for q b > 0 fixed the function q a -→ p + 0 (q b , q a ) is non increasing and the function q a -→ p - 0 (q b , q a ) is non decreasing. Also lim q a →0 + p + 0 (q b , q a ) = 1, lim q a →0 + p - 0 (q b , q a ) = 0, lim q a →∞ p + 0 (q b , q a ) = 0, lim q a →∞ p - 0 (q b , q a ) = 1. (2.7)

In particular, given q b > 0, there exists a unique point q * ≡ q * (q b ) such that p + 0 (q b , q a ) ≤ p - 0 (q b , q a ) for every q a ≥ q * (q b ) and p + 0 (q b , q a ) > p - 0 (q b , q a ) for every q a < q * (q b ). Note that q * (q b ) can be written as (2.8)

q * (q b ) = inf q a ∈ R : p + 0 (q b , q a ) ≥ p - 0 (q b , q a ) ∨ 0.

Furthermore, the function q b -→ q * (q b ) is non decreasing. The model predicts a price increase if Q a 0 < q * (Q b 0 ) and a price decrease if Q a 0 > q * (Q b 0 ). Next, using τ 0 as introduced above, we see that if S a 0 = S b 0 + δ, i.e. the spread is exactly one tick, then the state of (S a(b) , Q a(b) ) after event k ∈ {0, . . . , τ 0 -1} is given by (2.9)

S a k = S a 0 , S b k = S b 0 , Q a k = Q a 0 + k j=1 V a j , Q b k = Q b 0 + k j=1 V b j .
When either the best bid or ask queue is depleted, i.e at τ 0 , the spread widens immediately by one tick and S a(b)

0 , Q a(b) 0 are re-initialized according to S a τ 0 = S a 0 + δ 1 l a 0 <l b 0 , S b τ 0 = S b 0 -δ 1 l a 0 >l b 0 , Q a τ 0 =   Q a j + τ 0 j=1 V a j   1 l a 0 >l b 0 + Y τ 0 1 l a 0 <l b 0 , Q b τ 0 =   Q b 0 + τ 0 j=1 V b j   1 l a 0 <l b 0 -Y τ 0 1 l a 0 >l b 0 . (2.10)
By the above the spread is now two ticks and again are now two scenarios for the next change in the spread. Either it widens again by one tick, in analogy with the outline above, or it narrows again if a new limit order in placed between the best bid and ask. Let

:= min    l ≥ τ 0 + 1 : l j=τ 0 +1 V a(b) j ≤ -Q a(b) τ 0   
and la 1 = min {l ≥ τ 0 + 1 : a new limit ask order is submitted between} , lb 1 = min {l ≥ τ 0 + 1 : a new limit bid order is submitted between} . (2.12)

The time l = la 1 ∧ lb 1 is also the first event time, after τ 0 , after which X l and Y l have the same sign and |Y l | ≤ |X l |. Indeed, assume first the price does not change after event l, then

Y l = -Q b l-1 +X l if Y l < 0 and Y l = Q a l-1 + X l if Y l > 0.
Hence either Y l and X l do not have the same sign or |X l | < |Y l |. Next assume the spread widens after event l, then either Y l > 0, in which case the best ask queue is depleted and

X l = V a l = -Q a l-1 < 0, or Y l < 0 and X l = -V b l = Q b l-1 > 0.
Hence X l and Y l do not have the same sign. Finally, assume the spread narrows after event l, then a new limit buy (or sell) order is submitted between the best bid price and the best ask price and X l = V a l = Q a l = Y l , if Y l > 0 (new sell order), and X l = -V b l = -Q b l = Y l , if Y l < 0 (new buy order). We can therefore write la 1 and lb 1 as lb 1 = min {l ≥ τ 0 + 1 : X l ≤ Y l < 0} , la 1 = min {l ≥ τ 0 + 1 : 0 < Y l ≤ X l } . (2.13) Finally, let (2.14) τ 1 = l a 1 ∧ l b 1 ∧ lb 1 ∧ la 1 . Using this notation, we see that the next price move, after τ 0 , occurs at event τ 1 . In particular, if spread widens again by one tick then τ 1 = l a 1 ∧l b 1 and we see that we can describe the dynamics of (S b , S a , Q b , Q a ) as above. Similarly, if the spread next narrows by one tick then τ 1 = lb 1 ∧ la 1 and once the order is submitted the size of the new queue is either V a ),

S a τ 1 = S a τ 0 + δ(1 τ 1 =l a 1 -1 τ 1 = la 1
), (2.15) and the queues Q a τ 1 and Q b τ 1 are re-initialized to

Q b τ 1 = V b τ 1 1 τ 1 = lb 1 -Y τ 1 1 τ 1 =l b 1 +   Q b τ 0 + τ 1 l=τ 0 +1 V b l   1 τ 1 < lb 1 ∧l b 1 , Q a τ 1 = V a τ 1 1 τ 1 = la 1 + Y τ 1 1 τ 1 =l a 1 +   Q a τ 0 + τ 1 l=τ 0 +1 V a l   1 τ 1 < la 1 ∧l a 1 . (2.16)
We emphasize that while we have here used a mix of the notation (S b , S a , Q b , Q a , V b , V a ) and (X, Y ), each component of (S b , S a , Q b , Q a , V b , V a ) can in the above displays be expressed solely using (X, Y ) as discussed in the previous subsection.

la 1 or V b lb 1 . 1 -1 τ 1 =l b 1

 1111 Under this scenario, at τ 1 , the best bid and ask prices move from (S b τ 0 , S a τ 0 ) to (S b τ 1 , S a τ 1 ), whereS b τ 1 = S b τ 0 + δ(1 τ 1 = lb

This research has been supported by a grant from Riksbankens Jubileumsfond (P10-0113:1). The empirical analysis in this paper is carried out using high frequency data supplied by NasdaqOMX.