Convergence of Jump Processes with Stochastic Intensity to Brownian Motion with Inert Drift

Submitted to Bernoulli

Convergence of jump processes with stochastic intensity to Brownian motion with inert drift

CLAYTON BARNES1 1E-mail: [email protected]

Consider a random walker on the nonnegative lattice, moving in continuous time, whose positive transition intensity is proportional to the time the walker spends at the origin. In this way, the walker is a jump process with a stochastic and adapted jump intensity. We show that, upon Brownian scaling, the sequence of such processes converges to Brownian motion with inert drift (BMID). BMID was introduced by Frank Knight in 2001 and generalized by White in 2007. This conﬁrms a conjecture of Burdzy and White in 2008 in the one-dimensional setting.

Keywords: Brownian motion; Discrete Approximation; Random Walk; Local Time

1. Introduction

Brownian motion with inert drift (BMID) is a process X that satisﬁes the SDE

dX = dB + dL + KLdt, (1)

where K ≥ 0 is a constant and L is the local time of X at zero. This process behaves as a Brownian motion away from the origin but has a drift proportional to its local time at zero. Note that such a process is not Markovian because its drift depends on the past history. See Figure1 for sample path comparisons between reﬂected Brownian motion and BMID. BMID can be constructed path-by-path from a standard Brownian motion via the employment of a Skorohod map. This is discussed in more −n detail in Section2. We consider continuous time processes (Xn,Vn) on 2 N×R such that for K ≥ 0, n (i) Vn(t) := K2 · Leb(0 < s < t : Xn(s) = 0) is the scaled time Xn spends at the origin.

2n n (ii) Xn is a jump process with positive jump intensity 2 + 2 Vn(t) and downward jump intensity 2n 2 , modiﬁed appropriately so Xn does not transition below zero.

The existence of such a process and its rigorous deﬁnition is presented in Section3. Intuitively, Xn is −n a random walker on the lattice 2 N whose transition rates depend linearly on the amount of time the walker spends at zero. In other words, the positive jump rate of Xn increases each time Xn reaches arXiv:1809.04428v3 [math.PR] 17 Feb 2021 zero. We show that as the lattice size shrinks to zero, i.e. as n → ∞, (Xn,Vn) converges in distribution to (X,V ), where X is BMID and V = KL is its velocity. See Theorem 4.4 and Corollary 4.5 for pre- cise statements. By setting K = 0 we recover the classical result that random walk on the nonnegative lattice converges to reﬂected Brownian motion.

1.1. Outline

In Section 2 we introduce BMID and its construction using Skorohod maps. We also give an equivalent formulation of the process (X,V ). In Section 3 we introduce the necessary background on jump

1 imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 2 Clayton Barnes

2.5

1.5

0.5

0 0 500 1000 1500 2000 2500

Figure 1. The figure shows a path of reflected Brownian motion and a path of BMID, each coming from the same Brownian sample path. This drift of BMID becomes evident as its domination over reflected Brownian motion grows with time. The drift of BMID comes from its contact with zero, which, in this figure, occurs in the beginning of the process. process with stochastic intensity and introduce the setting used by Burdzy and White in [6]. Section 4 contains the statement and proof of the main results, Theorem 4.4 and Corollary 4.5. We conclude by briefly discussing BMID in a multidimensional setting in Section5.

1.2. Background

The study of BMID began in 2001 when Knight [11] described a Brownian particle reflecting above a particle with Newtonian dynamics. This two-particle system of Knight is equivalent in some sense to BMID in that the gap between the Brownian particle and the Newtonian particle is BMID. See Section 2. For more background on BMID see [13], where White constructs a multidimensional analog to BMID; [3], where Bass, Burdzy, Chen and Hairer study the stationary distribution; [1], where Barnes describes the hydrodynamic behavior of systems of Brownian motions with inert drift. Burdzy and White studied similar processes from a discrete state point of view [6]. They consider d a pair of processes (X,L) with state space L × R , where L is a finite set, and where the transition rate of X depends on L, the scaled time X has spent on previous states. See subsection 3.2 for definitions. The authors find necessary and sufficient conditions for such a process (X,L) to have stationary distribution µ × γ, where µ is uniform on space and γ is Gaussian. Burdzy and White make many conjectures involving approximating BMID, and its variants, and suggest the results of Bass, Burdzy, Chen, and Hairer [3] concerning a multidimensional analog of BMID stem from a discrete approximation scheme where the continuous process of BMID is a limit of these processes whose values take place in discretized space. The main result of this article confirms the discrete approximation scheme converges to the continuous model in the one dimensional setting. BMID is just one example where a process with memory has a Gaussian stationary distribution. Gauthier [8] studies diffusions whose drift is also dependent on the process history through a linear combination of sine and cosine functions. He shows the average displacement across time obtains a

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 3

Gaussian stationary distribution as time approaches infinity. In [2], Barnes, Burdzy, and Gauthier use this discrete approximation scheme, taking limits of Markov processes in the same class considered here, to demonstrate billiards with certain Markovian reflection laws have µ × γ as the stationary measure for space and velocity, where as above µ is the uniform stationary measure in the spatial component and γ the Gaussian stationary measure in the velocity component. In Section5 we briefly discuss a multidimensional analog of BMID that inspired conjectures of Burdzy and White.

2. An equivalent formulation of BMID

In this section we describe the process (X,Y,V ), where X is Brownian motion reﬂecting from the inert particle Y and where Y has velocity V . We begin with a probability space (Ω, P, (Ft)t≥0), with the ﬁltration (Ft)t≥0 satisfying the usual conditions, supporting a Brownian motion B.

Theorem 2.1 (Existence and Uniqueness, Knight [11], White [13]). Choose K ≥ 0 and v ∈ R. There exists a unique strong solution of continuous Ft-adapted processes (X,Y,V ) satisfying: X(t) = B(t) + L(t), for all t ≥ 0, almost surely, X(t) ≥ Y (t), for all t ≥ 0, almost surely, (2) dY = V (t)dt := (v − KL(t))dt, for all t ≥ 0, almost surely, L is nondecreasing, and is ﬂat away from the set {s : X(s) = Y (s)}.

R ∞ Remark 2.2. Flatness of L off {s : X(s) = Y (s)} means 0 1(X(s) 6= Y (s)) dL(s) = 0. One can use the Ito-Tanaka formula to show that L is the local time of X − Y at zero; see [13, Th. 2.7]. That is,

1 Z t L(t) = lim 1(|X(s) − Y (s)| < ) ds, →0 2 0 where the right hand side is the local time of X − Y at zero.

BMID together with its velocity is equivalent, in a certain sense, to the process (X,Y,V ) whose existence is given in Theorem 2.1. We will refer to the following result by Skorohod.

Lemma 2.3 (Skorohod, see [10]). Let f ∈ C([0,T ], R) with f(0) ≥ 0. There is a unique, continuous, nondecreasing function mf (t) such that

xf (t) = f(t) + mf (t) ≥ 0,

mf (0) = 0, mf (t) is ﬂat off {s : xf (s) = 0}, that is given by

mf (t) = sup [−f(s)] ∨ 0. 0

Remark 2.4. The classical Lévy’s theorem states that for a Brownian motion B, xB is distributed as |B|. See [10, Section 3.6C].

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 4 Clayton Barnes

To see the equivalence between BMID and the process (X,Y,V ) from Theorem 2.1, consider the gap process G(t) = X(t)−Y (t). Obviously G ≥ 0, almost surely, and from (2) it follows that (when v = 0)

dG = dB + dL + KLdt, (3) where L is continuous, nondecreasing, and flat off U −1(0). From the comment in Remark 2.2 on local time, G is a reflected diffusion whose drift is proportional to its local time at zero. Consequently, the gap process G is BMID as it satisfies (1). Assume we have a pair of processes (U, V ) adapted to a continuous filtration Ft that supports a Brownian motion B, and that for fixed K ≥ 0, v ∈ R Z t U(t) = B(t) + V (x) dx, 0 V (t) = −v + KM U (t), (4)

M U (t) = sup [−U(s)] ∨ 0. 0

In the system (4), it is clear from the deﬁnition of M U that U + M U ≥ 0 and it follows from the Skorohod Lemma 2.3 that M U is ﬂat off the set {s : U(s) + M U (s) = 0}. Therefore

Z t B(t) + M U ≥ − V (s) ds, 0

U U R t and M is flat off of {s : B(t) + M = − 0 V (s) ds}. Consequently, Z t B(t) + M U (t), − V (s) ds, −V (t) 0 satisfies the original equation (2) with respect to the filtration Ft. Similarly, one can use the uniqueness statement in Skorohod’s Lemma to go from a solution of (4) to a solution of (2). This demonstrates the equivalence of the two systems (2) and (4) in the sense that if one solution exists for a given probability space (Ω, P, Ft), where Ft supports a given Brownian motion, then the other solution can be given by a path-by-path transformation. Existence of a solution to (2) was first shown by Knight in [11]. A strong solution to a more general process was attained via the employment of a Skorohod map by David White [13] in a more general version of Theorem 2.5 given below.

Theorem 2.5 (White, [13]). For every f ∈ C([0,T ], R),K ≥ 0, v ∈ R there is a unique pair of continuous functions (I,V ) such that

x(t) := f(t) + I(t), V (t) = v + Km(t), Z t (5) I(t) = V (s) ds, 0 m(t) = sup [−x(s) ∨ 0]. 0

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 5

Remark 2.6. In Remark 2.4 it is mentioned that replacing the function f with a Brownian motion in the formulation of Skorohod’s Lemma 2.3 gives rise to a representation of reﬂected Brownian motion. Similarly, when replacing f in Theorem 2.5 pathwise by Brownian motion, the corresponding process (x, −I,V ) is a solution to (4). Note that Skorohod’s Lemma 2.3 implies that m(t) in Theorem 2.5 is the unique monotonically increasing, continuous, function which is ﬂat off of the level set {s : x(s) + m(s) = 0} such that x + m is nonnegative.

Remark 2.7. One can also see from the above arguments that (U, V ) of (4) solves

dU(t) = dB(t) + dL(t) + V (t)dt, dV (t) = KL(t)dt, V (0) = −v, where L is the local time of U at zero.

3. Markov Processes with Memory

The title of this section seems contradictory because Markov processes lose their memory when con- ditioning on their current location. The processes considered are pairs of processes, one process taking values in “space," and the other process storing the history of the space-valued process. The transition rate of the space-valued process depends on this stored history. We let C denote the class of such processes which we introduce more formally in this section. We will later construct a sequence of processes in C that will approximate BMID. First, we review well known facts of Poisson processes and point process with stochastic intensity. For reference, see Brémaud’s description of a doubly-stochastic point process in [5, Chapter 2].

3.1. Non-homogeneous Poisson processes

A non-homogeneous Poisson process with a nonnegative locally integrable rate (or intensity) function λ(t) is a process N such that (i) N(0) = 0, a.s. (ii) N has independent increments, (iii) N is RCLL, a.s. d R b (iv) N(a, b] = N(b) − N(a) = Poisson( a λ(s)). If we let T = inf{t : N(t) > 0} be the ﬁrst jump time of N, then

− R t λ(s) ds P(T > t) = P(N(t) = 0) = e 0 .

Lemma 3.1. Let λ1, λ2 be two rate functions such that λ1(t) ≤ λ2(t) for all t ≥ 0 and let T1,T2 be the ﬁrst jump time of their corresponding Poisson process. Then T1 stochastically dominates T2.

In [5, Chapter II], Brémaud discusses the notion of point processes adapted to a ﬁltration Ft whose intensity λ(s) is not a deterministic function but rather a process adapted to Ft with certain conditions.

Deﬁnition 3.2. [5, II] Let Nt be a point process adapted to the ﬁltration Ft and let λt be a nonneg- R t ative Ft-progressive process such that 0 λs ds < ∞ almost surely for each t ∈ [0,T ]. If Z ∞ Z ∞ E Cs dNs = E Csλs ds , (6) 0 0

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 6 Clayton Barnes for all nonnegative Ft-predictable processes Ct then we say Nt has stochastic intensity λt.

Remark 3.3. In the proofs of later results we will refer to point processes with a given intensity or jump/step size. By a point process of jump/step size a > 0 and (stochastic) intensity λ we mean a process aNt where Nt is a point process with (stochastic) intensity λ. By the positive (resp. negative) jump process for a process we mean the process aNt (resp. −aNt). For example, a process with jump −n −n size 2 with positive jump rate λ1(t), and negative jump rate λ2(t), is 2 (N1 − N2) where Ni is a point process with (stochastic) rate λi.

Some well known facts of Poisson processes have analogous results for Poisson processes with stochastic intensities, which we list below.

1 2 Lemma 3.4. Let N1,N2 be two independent point processes with stochastic intensities λ , λ 1 2 adapted to ﬁltrations F , F respectively. Then N1 + N2 is a point process with stochastic intensity 1 2 1 2 λ + λ , adapted to Ft := σ Ft , Ft .

Sketch. The fact that N1,N2 are independent implies the two processes do not have common jumps, so that (N1,N2) is a multivariate point process. The result follows from [5, T15, Chapter II.2].

1 Lemma 3.5. Let N1 be a point process with stochastic intensity λ (t) ≥ λ, almost surely, for some λ ∈ R+. Then we can enlarge the probability space to support a Poisson point process N2 with constant 3 1 intensity λ and a point process N3 of stochastic intensity λ = λ − λ such that N3 is independent of 1 N2 and N2 + N3 has stochastic intensity λ .

Remark 3.6. It is clear that one can generate a Poisson point process N2 with constant intensity λ which is independent of N1. Lemma 3.5 could be generalized to include more general lower bounds than a constant, however, we don’t require this and sketch the proof only in the case when N2 has constant intensity.

0 0 0 Sketch. Enlarge the probability space to support two independent processes N2,N3 where N2 is a 0 3 1 Poisson point process of rate λ and N3 is a point process with stochastic rate λ = λ − λ. By Lemma d 3.4, N1 = N2 + N3, and the processes are adapted to the ﬁltration generated by (N1,N2,N3).

3.2. Class C of Markov Processes with Memory

As mentioned, Burdzy and White [6] study continuous time Markov processes (X,L) on L × R where L = {1, 2,...,N} is a finite set. For each j ∈ L we associate a vector vj ∈ R and define Lj(t) = Leb(0 < s < t : X(s) = j) as the time X has spent at location j until time t. We also define X L(t) = vjLj(t) j∈L as the accumulated time X spends at each location, weighting the time spent at location j by the factor vj. The transition rates of X will depend on L(t). More precisely, we are given RCLL functions aij : R → R where aij is the rate function for the Poisson process defining the transition of X from i to j. Conditional on X(t0) = i, L(t0) = l, the jump rate of X transitioning from i to j is aij(l +[t−t0]vi)

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 7 with t ≥ t0. To construct such a process, for each i we create independent random variables (Ti,j)j∈L which represents the jump time from i to j. Since this jump has intensity aij(l + [t − t0]vi) with t ≥ t0, Z t P(Ti,j > t + t0|X(t0) = i, L(t0) = l) = exp − aij(l + svi) ds , (7) 0 0 for all t > 0. Pick j such that Ti,j0 = minj6=i Ti,j, and deﬁne the ﬁrst transition of X after time t0 to 0 be location j and occur at time Ti,j0 .

These dynamics can be produced from a collection of independent exponential random variables

(Ei,j)i,j∈N of rate one. Set T0 = 0 and recursively deﬁne Z t T j = inf t > T : a (L(T ) + v )ds > E i+1 i X(Ti)j i X(Ti)(s−Ti) i,j (8) 0 j Ti+1 = min T . (9) j i+1

We use the convention inf ∅ = ∞. Deﬁne

L(s) = L(Ti) + vXn(Ti)(s − Ti), for s ∈ [Ti,Ti+1] (10)

X(s) = X(Ti), for s ∈ [Ti,Ti+1) (11) j X(Ti+1) = argminjTi+1. (12) The pair (X,L) is a strong Markov process with generator X Af(j, l) = vj · ∇lf(j, l) + aji(l)[f(i, l) − f(j, l)], j = 1, . . . , N, l ∈ R. i6=j

Burdzy and White assume (X,L) is irreducible in the sense that there is some {j0} × U ⊂ L × R such that

P((X(t),L(t)) ∈ {j0} × U|X(0) = i, L(0) = l) > 0, for all (i, l) ∈ L × R.

It should be noted that although they consider L to be a ﬁnite set, their main results hold assuming that supij aij(l) is bounded on compact sets of l and supi |vi| < ∞. We denote C as the class of such processes with these conditions, allowing L = N.

4. Discrete Approximation

4.1. Deﬁnition of Processes

The reﬂected diffusion (1) describing BMID is a process whose drift depends on the local time of the −n diffusion at zero. Intuitively, to approximate this diffusion with a Markov process on the lattice 2 N one would want the “velocity” to depend on the accumulated time spent at zero. This is modeled as a jump process whose intensity function is stochastic and depends linearly on the accumulated time the process spends at zero. These jump processes need to converge, as n → ∞, to a process whose drift is the appropriate local time.

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 8 Clayton Barnes

In Section2 we introduced an equivalent formulation for BMID given by (U, V ) in (4). In this subsection we will describe two equivalent discrete processes that mirror the equivalence of the continuous processes described earlier; see Proposition 4.3. We do this because in order to prove the convergence result described in the introduction we actually prove the convergence result for the equivalent formulation. Jump processes whose intensity depends linearly on the accumulated time at zero are described by X −n the class C in subsection 3.2. Consider a process (Xn,Vn ) on the state space 2 N × R where vj = 0 n for all j 6= 0, v0 = K2 as given in the notation in that subsection. (We may hide the dependence on n for convenience.) For an initial “velocity” v ∈ R, we deﬁne X n n Vn (t) = −v + K2 Ln(t) = −v + K2 · Leb(0 < s < t : Xn(s) = 0).

The rate functions aij : R → R are

2n n 2n n X ai(i+sign(l)2−n)(l) = 2 + 2 |l| = 2 + 2 |Vn (t)|, (13) 2n ai(i−sign(l)2−n)(l) = 2 ,

X where l = Vn (t), except when i = 0 where we do not allow a downward transition. By Lemma 3.4 the jump process Xn can be decomposed into a sum of independent processes, Sn and Zn, whose rate functions sum to that of Xn. The following deﬁnition will be used throughout the paper.

Deﬁnition 4.1. For a process Q(t) we deﬁne M Q(t) as the signed running minimum below zero of Q. That is, M Q(t) = sup [−Q(s)] ∨ 0. 0

−n 2 Deﬁnition 4.2. Consider the processes (Sn,Zn,Vn) on 2 Z × R where −n (i) Sn is a continuous time simple random walk on 2 Z with positive (and negative) jumps of size 2−n and rate 22n. −n (ii) Zn is a point process with jump size 2 and with positive (resp. negative) jumps having stochas- n tic and adapted rate 2 |Vn| when Vn > 0 (resp. Vn < 0). (iii) We have

n Vn(t) = −v + K2 · Leb(0 < s < t : Un(s) = −Mn(s)),

Un = Sn + Zn,

Un Mn(t) := M (t).

That is, Sn and Zn are point processes with adapted intensity functions as discussed in [5, Chapter 2].

Note the similarity to the equivalent formulation of BMID given by (U, V ) in (4) to (Un,Vn) given above. Existence of (Sn,Zn,Vn) follows from the fact that it is of class C, or, equivalently, one can construct the processes via the dynamics given in (8) by using the intensity functions (13).

X Proposition 4.3. The processes (Un + Mn,Vn) and (Xn,Vn ) have the same law.

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 9

X Proof. With these definitions (Un + Mn,Vn) has the same law as (Xn,Vn ) because it is of class C and satisfies (13). To see this, note that Vn is adapted to the right continuous filtration Ft generated by −n the pair (Sn,Zn). Also note that Un + Mn = Sn + Zn + Mn is a nonnegative process on 2 N. By 2n n Lemma 3.4, Sn + Zn has a jump rate function of 2 + 2 Vn(t) where

n Vn(t) = −v + K2 · Leb{0 < s < t : Un(s) = −Mn(s)} n = −v + K2 · Leb{0 < s < t : Un(s) + Mn(s) = 0}.

X Consequently, Vn = Vn if we deﬁne Xn := Un + Mn. Therefore (Un + Mn,Vn) is one realization of X the process (Xn,Vn ) given by (13).

X We will work with (Sn,Zn,Vn) as an equivalent formulation of (Xn,Vn ) deﬁned by (13).

4.2. Theorem Statement

R · The main result of this article is that (Sn,Vn,Zn,Un) converges in an appropriate sense to (B,V, 0 V,U).

Theorem 4.4. For K ≥ 0 and v ∈ R, let (Sn,Zn,Vn,Un) be given as in Definition 4.2 in subsection 4.1. Then · d Z (Sn,Vn,Zn,Un) −→ (B,V, V,U), 0 4 R · in the Skorohod topology on D([0,T ], R ), where (B,V, 0 V,U) is a quadruple of continuous processes adapted to the Brownian filtration of the first coordinate B with the following holding for all t ∈ [0,T ], almost surely:

Z t U(t) = B(t) + V (x) dx, 0 V (t) = −v + KM U (t), and where M U is the running minimum given in Deﬁnition 4.1.

Theorem 4.4 has the following corollary.

X Corollary 4.5. Let (Xn,Vn ) be the process deﬁned by (13) in subsection 4.1. Then

X d (Xn,Vn ) −→ (X,V ), in the Skorohod topology on D([0,T ], R). Here (X,V ) is BMID together with its velocity. That is, dX(t) = dB(t) + dL(t) + V (t)dt, dV (t) = KL(t)dt, V (0) = −v, (14) where L is the local time of X at zero.

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 10 Clayton Barnes

X Proof of Corollary. By Proposition 4.3 we set Xn := Un + Mn and Vn := Vn. Now Theorem 4.4 implies

X d U (Xn,Vn ) −→ (U, −v + KM ), and where (U, −v + KM U ) solves (14) as mentioned in Remark 2.7.

Remark 4.6. Let D([0,T ], R) denote the space of RCLL paths equipped with the Skorohod metric d [7, Chapter 3 Section 5]. If a process Wn with paths in D([0,T ], R) converges weakly to W , then according to the Skorohod representation, [7, Theorem 3.1.8], we can place Wn,W on the same probability space such that

d(Wn,W ) −→ 0, almost surely. If the limiting process W is continuous almost surely, then

kWn − W k[0,T ] := sup |Wn(s) − W (s)| −→ 0, 0

4.3. Proof of Theorem 4.4

In this section we prove Theorem 4.4 assuming the two lemmas below, one for tightness and the other for classifying the subsequential limits.

2 Lemma 4.7 (Tightness). The collection of processes {(Sn,Zn,Vn): n ∈ N} is tight in D([0,T ], R ) 3 ×C[0,T ] ⊂ D([0,T ], R ). Because Un = Sn + Zn, it follows that {(Sn,Vn,Zn,Un): n ∈ N} is also 2 tight in D([0,T ], R) × C[0,T ] × D([0,T ], R ). Furthermore, all limiting processes are continuous.

We prove Lemma 4.7 in Section 4.4. Assuming Lemma 4.7 holds, it remains to show there is a unique limit.

Lemma 4.8 (Classiﬁcation of Limits). Consider a subsequence nk with processes (Snk ,Znk ,Vnk , 4 Unk ) converging to (S, Z, V, U) in D([0,T ], R ) with the Skorohod topology. Then (S, Z, V, U) is continuous and satisﬁes the equivalent formulation for BMID given in (4). That is, (i) U(t) = S(t) + Z(t), (ii) S(t) is a Brownian motion, (iii) V (t) = KL(t) − v, where L(t) = max0

Proof of Theorem 4.4. Since the formulation of BMID described by (B,V,U) in the statement of The- orem 4.4 is unique in law, Lemmas 4.7 and 4.8 characterizes the subsequential limits of (Sn,Zn,Vn). See [13] where existence (and uniqueness) of such a system is proved. Consequently, we have convergence of the entire sequence to this equivalent formulation of BMID.

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 11

4.4. Lemma 4.7: Tightness of (Sn,Zn,Vn)

3 Recall that our process (Sn,Zn,Vn) is in D([0,T ], R ), the space of RCLL paths with the Skorohod topology deﬁned by the product metric d × d × d where d is the Skorohod metric, see Billingsley [4]. The following deﬁnition is taken from Jacod and Shiryaev [9].

Remark 4.9. In general, it is not true that if αn → α, βn → β in the Skorohod topology, then αn + βn → α + β. However, this does hold if either α or β is continuous. See Jacod and Shiryaev

[9, Proposition VI.1.23]. Similarly, by Remark 4.6 one can assume a sequence (Snk ,Znk ,Vnk ) that 3 converges in distribution on D([0,T ], R ) in fact converges almost surely to a continuous process, say (S0,Z0,V 0) in the uniform metric if the limit is continuous; at least on some probability space. This 0 0 immediately implies Unk = Snk + Znk converges almost surely to S + Z .

Deﬁnition 4.10. [9, Deﬁnition VI.3.25] A sequence of processes {Xi : i ∈ N} in D([0,T ], R) is said to be C-tight if {Xi : i ∈ N} is tight on D([0,T ], R) and all limiting processes are continuous.

See Remark 4.6, which mentions the Skorohod metric when the limit processes are continuous. The n proof that (Sn,Zn,Vn) is tight is broken into multiple lemmas. Recall that Ln(t) := 2 Leb(0 < s < U t : Un(s) = −Mn(s)) where Un := Sn + Zn, Mn := M n , and Vn := −v + KLn.

Deﬁnition 4.11. For f ∈ D([0,T ], R), let ω(f, δ) := sup |f(t) − f(s)| 0

Recall that kfk[a,b] = supa

Lemma 4.12. [9, Proposition VI.3.26] A sequence of processes Xn in D([0,T ], R) is C-tight if and only if for every > 0,

(i) limC→∞ lim supn→∞ P(kXnk[0,T ] ≥ C) = 0,

(ii) limδ→0 lim supn→∞ P(ω(Xn, δ) > ) = 0.

Remark 4.13. Notice that (i) follows from (ii) and limC→∞ lim supn→∞ P(|Xn(0)| > C) = 0. To see this, take δ = 1 and = C/2 so that by deﬁnition of the modulus of continuity

T X kXnk[0,T ] ≤ |Xn(0)| + ω(Xn, 1). i=1 Consequently, the triangle inequality gives

P(kXnk[0,T ] > C) ≤ P(|Xn(0)| + dT eω(Xn, 1) > C)

≤ P(|Xn(0)| > C/2) + P(ω(Xn, 1) > C/(2dT e)), where dre is the smallest integer larger than r, from which it is clear that (ii) and limC→∞ lim supn→∞ P(|Xn(0)| > C) = 0 imply (i).

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 12 Clayton Barnes

0 00 Lemma 4.14. Assume that the sequences of processes (Xn), (Xn), (Xn) in D([0,T ], R) satisfy 0 0 00 00 Xn(t) − Xn(s) ≤ Xn(t) − Xn(s) ≤ Xn(t) − Xn(s), 0 ≤ s ≤ t, 0 00 almost surely. If both (Xn) and (Xn) are C-tight, then (Xn) is also C-tight.

Proof. For any C > 0, the triangle inequality gives

0 00 P(kXnk[0,T ] > C) ≤ P(kXnk[0,T ] > C/2) + P(kXnk[0,T ] > C/2), which veriﬁes condition (i) in the statement of Lemma 4.12 by taking limC→∞ lim supn→∞ of both sides. Similarly, for every δ, > 0,

0 00 P(ω(Xn, δ) > ) ≤ P(ω(Xn, δ) > /2) + P(ω(Xn, δ) > /2), and taking limδ→0 lim supn→∞ on both sides veriﬁes condition (ii) in the statement of Lemma 4.12.

The following two lemmas are classical and we omit proofs.

d d d Lemma 4.15. Let X = Exp(λ),Y = Exp(µ) be independent. Then X ∧ Y = Exp(λ + µ) is independent from W = 1{X∧Y =X}. In other words, the minimum of two independent exponential random variables is independent from which exponential r.v. occurred ﬁrst.

n Lemma 4.16. Fix a > 0, and for each n ∈ N let Nn be a Poisson process with intensity α2 . Then −n {2 Nn(t): t ∈ [0,T ]} converges in distribution to the line g(t) = at, in the space D([0,T ], R). In −n particular {2 Nn(t): t ∈ [0,T ], n ∈ N} is C-tight.

Lemma 4.17. There is a ﬁltered probability space (Ω, (Ft)t≥0, P) satisfying the usual conditions, supporting the Ft-adapted process (Sn,Zn,Vn) given in Deﬁnition 4.2, also supporting the Ft- 0 0 0 adapted process (Un,Zn,Ln), such that 0 0 Un = Sn + Zn, 0 U 0 Mn = M n , 0 n 0 0 Ln(t) = 2 Leb(0 < s < t : Un = −Mn).

0 n −n Here Zn is a Poisson point process of intensity |v2 | and jump size sign(v)2 . Furthermore, 0 0 Zn(t) − Zn(s) ≤ Zn(t) − Zn(s), (15) for all 0 ≤ s ≤ t ≤ T , almost surely, and

0 0 0 ≤ Ln(t) − Ln(s) ≤ Ln(t) − Ln(s), (16) 0 for all 0 ≤ s ≤ t ≤ T , almost surely. The construction will yield independence between Zn and Sn.

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 13

Proof. By deﬁnition Vn ≥ −v almost surely. Recall that Zn is a point process with stochastic intensity n −n |2 Vn| and jump size sign(Vn)2 , so by Lemma 3.5 we assume the probability space included a 0 n −n process Zn with downward stochastic jump intensity |v2 | and step size 2 such that 0 0 Zn(t) − Zn(s) ≤ Zn(t) − Zn(s) for all 0 ≤ s ≤ t ≤ T, almost surely. This inequality holds because the negative transitions of Z will n 0 have a rate less than |v|2 , which is the transition rate for negative jumps of Zn. (And by deﬁnition 0 0 Zn makes negative jumps only.) The jump times of Zn are independent of Sn hence the processes are independent. This demonstrates (15). It follows that

0 0 Un := Sn + Zn ≥ Sn + Zn =: Un, 0 0 almost surely. Notice both processes Un + Mn,Un + Mn transition as a continuous time (nonnegative) 0 random walk but with an additional “drift” process of Zn,Zn respectively. For instance, a transition of Un beginning from its running minimum corresponds to a transition from zero for the walk Un + Mn. U 0 U 0 By (15), the process Un + Mn dominates that of Un + Mn . That is, 0 0 Un + Mn ≥ Un + Mn ≥ 0, almost surely. (17) 0 0 0 0 Hence, Un + Mn is zero whenever Un + Mn is zero. Consequently, {s < z < t : Un(z) = −Mn(z)} ⊂ {z : s < z < t, Un(z) = −Mn(z)} for every (s, t) ⊂ [0,T ], almost surely. Therefore,

0 0 0 ≤ Ln(t) − Ln(s) ≤ Ln(t) − Ln(s), (18) for every (s, t) ⊂ [0,T ], almost surely, demonstrating (16).

q √ 0 Lemma 4.18. For every T > 0, E(Mn(T )) ≤ E(Mn(T )) ≤ 2 2T + T |v| 2T + 2|v|T where v is 0 the initial value of Vn and Mn is deﬁned in Lemma 4.17.

Proof. According to Lemma 4.17 we assume our probability space supports (Sn,Zn,Vn) as well as 0 the Zn given in that lemma’s statement. Consequently, 0 0 Un := Sn(t) + Zn(t) ≥ Sn(t) + Zn(t) =: Un, for all t ∈ [0,T ], almost surely, implying

0 Un U 0 Mn(T ) := M (T ) ≤ M n (T ) =: Mn(T ), (19)

−n almost surely. We can express the continuous time random walk Sn as 2 (N1(t) − N2(t)) where 2n 2 Ni are independent Poisson processes of rate 2 , and consequently E(Sn(T ) ) = Var(S√n(T )) = −2n 2 (Var(N1(T )) + Var(N2(T ))) = 2T. By Cauchy-Schwarz this yields E|Sn(T )| ≤ 2T. By 0 −n n Doob’s Martingale inequality, the fact that Zn(T ) is distributed as −2 Poisson(|v2 T |), and in- 0 dependence of Zn and Sn, we compute

(19) 0 E(Mn(T )) ≤ E(Mn(T )) q 0 2 ≤ E(Mn(T ) ), Cauchy-Schwarz

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 14 Clayton Barnes q q 0 2 0 2 ≤ 4E(Un(T ) ) = 2 E(|Sn(T ) + Zn(T )| ), by Doob’s Maximal Inequality q q √ 2 0 0 2 ≤ 2 E(Sn(T )) + E|Sn(T )Zn(T )| + E(Zn(T ) ) ≤ 2 2T + T |v| 2T + 2|v|T.

0 B−v Lemma 4.19. In the notation of Lemma 4.17, Ln converges in distribution to M in the space −v C([0,T ], R) with the uniform norm. Here, B (t) := B(t) − vt where B is a Brownian motion. In 0 0 particular, {Ln : n ∈ N} is C-tight. Furthermore, supn ELn(T ) ≤ supn ELn(T ) < ∞.

Proof. We begin by showing the weak convergence for which we use a similar technique in the proof 0 of Lemma 4.8 (iii). We record the amount of time Un spends at each level of the running minimum, and 0 0 −n express Ln(t) as the sum of these times. By Lemma 4.16, Zn(t) = −2 Nn(t) converges in distribution to g(t) = −vt in the space D([0,T ], R). By Donsker’s theorem, Sn converges in distribution to a 0 0 0 Brownian motion B and consequently Un := Sn + Zn converges in distribution to B + g =: U . This 0 U 0 U 0 f implies Mn := M n converges to M in distribution because f 7→ M is continuous in the uniform n 0 norm and the limiting processes are continuous. See Remark 4.9. Note that 2 Mn(t) + 1 is the number 0 (j) 0 −n of levels the running minimum of Un has reached by time t. Let τ = inf{t > 0 : Un(t) = −j2 }, (j) 0 −n so that τ is the ﬁrst time Mn reaches j2 . Deﬁne (j) 0 0 −n Tj := Leb(s ≥ τ | −Un(s) = Mn(s) = j2 ). Then,

n 0 n 0 2 Mn(s) 2 Mn(s)+1 n X 0 n X 2 Tj ≤ Ln(s) ≤ 2 Tj, (20) j=0 j=0

0 −n 0 −n for all s ∈ [0,T ], almost surely. When Mn = j2 , after the process Un arrives at −j2 for the kth 2n (j) time, it makes a positive jump upon the arrival of an Exp(2 ) random variable, call it µk , while it 0 2n n (j) makes a negative jump upon the arrival of an Exp(2 + |v2 |) random variable µk . Consider the (j) (j)0 pair (µk ∧ µk ,Ij) where Ij = 1 (j) (j)0 (j) . By Lemma 4.15, Ij is independent from the i.i.d. {µk ∧µk =µk } 0 (j) (j) 0 −n sequence (µk ∧ µk : k ≥ 1). Then, Wj := inf{k : Ik = 0} is the number of times Un visits −j2 0 −n d while the signed running minimum Mn is j2 . Because Ij = Bernoulli(p) with 22n p = , 22n+1 + v2n and Wj is Geometrice(p) (since it is the ﬁrst time this sequence of Bernoulli random variables is zero), (j) (j)0 and Lemma 4.15 implies that Wj is independent of µi , µi . Thus,

Wj X (j) (j)0 Tj = µi ∧ µi i=1 is a Geometric sum of i.i.d. exponential random variables of rate λ = 22n+1 + v2n that are in- d dependent of the number of the sums Wj. Such a sum is exponential of rate pλ. That is, Tj =

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 15

2n 0 0 (j) (j+1) Exp(pλ) = Exp(2 ). Each Tj is measurable with respect to σ{(Un(s),Mn(s)) : s ∈ (τ , τ ]}, 0 (j) 0 (j) and (Tj : j ≥ 1) is independent of (Un(τ ),Mn(τ )). (In other words, Tj depends only on the excursions between the stopping times and not the initial position.) Thus (Tj : j ≥ 1) is a sequence of i.i.d. Exp(22n) random variables. We show the left hand side of (20) converges in probability to M 0(s) in the uniform norm for s ∈ [0,T ]. The proof for the right hand side is essentially identical. Without 0 0 loss of generality, we may assume Mn converges almost surely to M by the Skorohod representation 0 0 theorem and the fact shown above that Mn converges to M in distribution. Therefore

n 0 2 Mn(s) X −n 0 sup 2 − M (s) −→ 0, almost surely.

s∈[0,T ] j=0

n 0 2 Mn(s) n X 0 To show the sequence 2 Tj, n ≥ 1, converges in probability to M (s) uniformly for s ∈ [0,T ], j=0 it sufﬁces to show n 0 n 0 2 Mn(s) 2 Mn(s) n X X −n P sup 2 Tj − 2 −→ 0.

s∈[0,T ] j=0 j=0

d 2n n d n Because Tj = Exp(2 ), we know zj := 2 Tj = Exp(2 ). Then,

n 0 n 0 n 0 2 Mn(s) 2 Mn(s) 2 Mn(s) n X X −n X −n 2 Tj − 2 = (zj − 2 ), j=0 j=0 j=0

−n −2n where zj − 2 are i.i.d. mean zero random variables with variance 2 . By Kolmogorov’s maximal inequality, for each C, > 0

 n 0  2 Mn(s) X −n P  sup (zj − 2 ) > 

s∈[0,T ] j=0   k X −n 0 ≤ P  sup (zj − 2 ) >  + P(Mn(T ) > C) n 1≤k≤2 C j=0

2nC  −2 X −n 0 ≤ Var  (zj − 2 ) + P(Mn(T ) > C) j=0 −2 n −2n 0 ≤ 2 C2 + P(Mn(T ) > C). 0 0 Because Mn converges to M almost surely, this implies

 n 0  2 Mn(s) X −n lim sup P  sup (zj − 2 ) >  n→∞ s∈[0,T ] j=0 0 ≤ lim sup P(Mn(T ) > C) n→∞

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 16 Clayton Barnes

0 = P(M (T ) > C), 0 0 where C > 0 is arbitrary. Since M (T ) is ﬁnite a.s., P(M (T ) > C) can be made arbitrarily small with a large choice for C. Hence

n 0 2 Mn(s) X −n P sup (zj − 2 ) −→ 0,

s∈[0,T ] j=0 and the left hand side of (20) converges in probability to M 0. That is,

n 0 2 Mn(s) n X 0 P sup 2 Tj − M (s) −→ 0. (21)

s∈[0,T ] j=0 The convergence in probability for the right hand side is similar,

n 0 2 Mn(s)+1 n X 0 P sup 2 Tj − M (s) −→ 0. (22)

s∈[0,T ] j=0 Because (20) is an almost sure bound,

0 0 P sup |Ln(s) − M (s)| > s∈[0,T ]

 n 0   n 0  2 Mn(s) 2 Mn(s)+1 n X 0 n X 0 ≤ P  sup 2 Tj − M (s) > /2 + P  sup 2 Tj − M (s) > /2 ,

s∈[0,T ] j=0 s∈[0,T ] j=0 and taking lim supn→∞ on both sides, (21) and (22) imply

0 0 lim sup P sup |Ln(s) − M (s)| > = 0, n→∞ s∈[0,T ]

0 0 for every > 0. This complete the proof that Ln converges in probability to M in the uniform norm. 0 0 Since M is a continuous process, the sequence {Ln : n ∈ N} is C-tight; see Deﬁnition 4.10. 0 To demonstrate the uniform moment bound, note that Ln(T ) ≤ Ln(T ), equation (20), and Wald’s lemma imply 0 0 −n E(Ln(T )) ≤ E(Ln(T )) = E(Mn(T )) + 2 . 0 Applying the uniform moment bound on Mn(T ) given in Lemma 4.18, we see 0 0 sup E(Ln(T )) ≤ sup E(Ln(T )) ≤ sup E(Mn(T )) + 1 < ∞. n n n

Corollary 4.20. The collection of processes {Ln : n ∈ N} is C-tight.

0 Proof. This follows directly from (18), the fact that {Ln : n ∈ N} is C-tight by Lemma 4.19 (and that the zero process is trivially C-tight), and Lemma 4.12.

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 17

Lemma 4.21. The collection of processes {Zn : n ∈ N} is C-tight.

Proof. We will use a localization argument by stopping the stochastic intensity of Zn when it becomes n large. Recall that |2 Vn| is the stochastic intensity of Zn. For C > v ≥ 0 set v + C T n = inf{t > 0 : V (t) > C} = inf t > 0 : L (t) > . C n n K

n Deﬁne a process Zn such that Zn| n = Zn| n , almost surely, while after time T let Zn have e e [0,TC ] [0,TC ] C e positive jump intensity C2n and jump size 2−n (and make only positive jumps). By Lemma 3.5 we can stochastically dominate the number of transitions made by Zen in a given time interval by the 1 n −n number of transitions made by a point process Zn(t) of intensity C2 and jump size 2 (in the same 1 −n n time interval). More precisely, we may assume there exists a process Zn(t) = 2 N(C2 t) on our probability space, where N is Poisson process of unit intensity, and

1 1 0 ≤ |Zn(t) − Zn(s)| ≤ |Zn(t) − Zn(s)| (23)

(n) 1 for every interval (s, t) ⊂ [0,TC ], almost surely. By monotonicity of Zn and Doob’s maximal inequality, for ﬁxed , δ, C > 0 we have

P sup sup |Zn(u) − Zn(v)| > t∈[0,T −δ] t

(n) (n) ≤ P sup sup |Zen(u) − Zen(v)| > , TC > T + P (TC < T ) t∈[0,T −δ] t

1 1 (n) ≤ P sup sup |Zn(u) − Zn(v)| > + P (TC < T ) t∈[0,T −δ] t

1 1 = P sup |Zn(t + δ) − Zn(t)| > + P (Vn(T ) > C) t∈[0,T −δ]

1 1 |v| + K supn E(Ln(T )) ≤ P sup |Zn(t + δ) − Zn(t)| > + . t∈[0,T −δ] C

By Lemma 4.16 and Lemma 4.12 we know

1 1 lim lim sup P sup |Zn(t + δ) − Zn(t)| > = 0. δ→0 n→∞ t∈[0,T −δ]

By this and the uniform moment bound of supn E(Ln(T )) in Lemma 4.19, there exists a constant A independent of C, , n such that A lim lim sup P sup sup |Zn(u) − Zn(v)| > ≤ . δ→0 n→∞ t∈[0,T −δ] t

By choosing C arbitrarily large, we see lim lim sup P sup sup |Zn(u) − Zn(v)| > = 0. δ→0 n→∞ t∈[0,T −δ] t

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 18 Clayton Barnes

Hence Zn satisfies (ii) of Lemma 4.12. Condition (i) follows from Remark 4.13 and the fact that Zn(0) = 0 almost surely. This completes our verification of conditions (i)-(ii) of Lemma 4.12 sufficient for C-tightness of {Zn : n ∈ N}.

3 Corollary 4.22. The collection of processes {(Sn,Zn,Vn): n ∈ N} is C-tight in D([0,T ], R ) with the Skorohod topology.

d Remark 4.23. As topological spaces, D([0,T ], R ) with the Skorohod topology is not equivalent to d D([0,T ], R) with the product topology. However, because the marginals are C-tight this a non-issue essentially because uniform convergence to a continuous function becomes the same in both spaces. See the comment in Jacod and Shiryaev [9, VI.1.21].

Proof. Note that Sn is C-tight by Donsker’s theorem, while C-tightness of Vn = v − KLn, and Zn, follow from Corollary 4.20, and Lemma 4.21, respectively. By Skorohod’s representation theorem, every subsequence of (Sn,Zn,Vn) converging to a limiting process (S, Z, V ) can be assumed to con- 3 verge almost surely in the product metric d×d×d, the product metric on D([0,T ], R) . By C-tightness of the marginals, S, Z, V are all continuous, so that (S, Z, V ) is a continuous process. As in Remark 4.6, this implies the subsequence of (Sn,Zn,Vn) converges to (S, Z, V ) almost surely in the uni- 3 form norm, which implies almost sure convergence in D([0,T ], R ) under the Skorohod metric. Thus, 3 (Sn,Zn,Vn) is C-tight as a collection of processes with paths in D([0,T ], R ).

4.5. Lemma 4.8: Characterization of subsequential limits

We prove items (i)-(iv) in Lemma 4.8 separately. The proof of (iii) was inspired by the proof of Lévy’s theorem given in [12, Chapter 6], where the authors essentially note the equivalence of the processes X (Un + Mn,Vn) and (Xn,Vn ), which we described in subsection 4.1, for the case K = 0. Lévy’s B B 2 theorem is the statement that (L, |B|) and (M ,B +M ) yield the same distribution on C([0,T ], R ). Here L is the local time of |B| at zero. See [10, Chapter 3.6] for a detailed statement.

Proof of (i). This follows trivially from the deﬁnition of Un = Sn + Zn.

Proof of (ii). Recall Sn is a continuous time scaled simple random walk. Since Snk converges to S, S is a Brownian motion by Donsker’s theorem.

n We give a brief heuristic for the proof of (iii). Recall Ln = 2 · Leb(0 < s < t : Un(s) = −Mn(s)) and Vn = −v +KLn. Each time Mn increases, Un will make approximately a Geometric(1/2) number of visits to this new minimum value before Mn increases again. Also, Un will spend approximately an 2n Exp(2 ) amount of time at each one of these visits. Therefore, Ln, which is the total amount of time n Un spends on −Mn scaled by 2 , is approximately

2nM 2nM Xn Xn 2nExp(22n) = Exp(2n), i=1 i=1 where Exp(2n) indicates independent exponential random variables of rate 2n. If you suppose this sum is concentrated around its expectation conditional on Mn, then

n 2 Mn(t) X −n Ln(t) ≈ 2 = Mn(t). i=1

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 19

Furthermore, if Mn converges almost surely to a process M in the uniform norm, one expects Ln to converge to M as well.

Proof of (iii). By tightness, and without loss of generality, assume Lnk −→ L and Unk −→ U almost surely in the uniform norm of continuous functions. See Remark 4.6. We will use a localization argument by stopping Ln after it reaches a large value. For a positive constant C > |v|, deﬁne

(nk) TC = inf{t > 0 : Lnk (t) > C}.

n , (S ,Z ,U ) (SC ,ZC ,U C ), For each k consider a modiﬁcation of nk nk nk , denoted nk nk nk solving the system (i)-(iii) in subsection 4.1 but replacing (iii) with

C n C C Vn (t) = −v + K[2 Leb(0 < s < t : Un (s) = −Mn (s)) ∧ C], C C C Un = Sn + Zn , C U C Mn (t) := M n (t).

In other words we are stopping Ln when it reaches C while keeping the other dynamics of the system (SC ,ZC ,U C ) (S ,Z ,U ) [0,T (nk)] the same. Therefore nk nk nk is equal to nk nk nk on the interval C , a.s., while [T (nk), ∞) LC C U on C the process nk is constantly , and nk is the sum of a (scaled) continuous time random walk and an independent jump process of rate −v + KC2nk and jump size 2−n. Fix m ∈ 2−nk U C −m −m N. We bound the number of positive excursions of nk above conditional on being the running minimum. Let

τ = inf{t > τ : U C (t) = −m = −M C } ∧ T, m,i+1 m,i nk nk

U C −m m M C T be the consecutive times nk visits when is the current value of nk , up until time . (Set τ = 0 U C −m. m,0 ). That is, the consecutive times nk visits its running minimum at Let

z+ = inf{s > 0 : U C (τ (nk) + s) > U C (τ (nk))} m,j nk m,j nk m,j and z− = inf{s > 0 : U C (τ (nk) + s) < U C (τ (nk))} m,j nk m,j nk m,j U C denote the amount of time until the next positive and negative, respectively, jump of nk after the jth −m z+ ∧ z− U C −m jth excursion starting at . Then m,j m,j is the time nk spends on during its visit −m |LC | ≤ C ZC 2nk |V C | to its running minimum . Since n , the stochastic intensity of nk is nk , which is bounded below by 22nk −v and above by 22nk +C2nk . We set v = 0 in the remaining computations for U C = SC + ZC 22nk + 2nk |V C | convenience. Therefore the positive jump times of nk nk nk have intensity nk d 2nk − 2nk and the negative jump times arrive with intensity 2 . In other words, zm,j = Exp(2 ). By Lemma 3.1 we assume the probability space contains two independent sequences of i.i.d. expo- − nential random variables {νi,j}i,j∈N, {ei,j}i,j∈N that are also independent of zm,j and of the position m , with rates 22nk and C2nk , respectively, and where

+ νm,j ∧ em,j ≤ zm,j ≤ νm,j,

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 20 Clayton Barnes almost surely. Consequently,

− − + − zm,j ∧ νm,j ∧ em,j ≤ zm,j ∧ zm,j ≤ zm,j ∧ νm,j, almost surely. (24) Deﬁne

m m m Aj := 1 − ,Bj := 1 − + + ,Cj := 1 − . {zm,j ∧νm,j =νm,j } {zm,j ∧zm,j =zm,j } {zm,j ∧νm,j ∧em,j =νm,j ∧em,j }

m C th Note that Bj is the indicator for whether Un jumped in the positive direction during its j visit to its m k m m running minimum −m. By construction Aj ,Bj ,Cj are Bernoulli random variables and are coupled so that

m m m Aj ≤ Bj ≤ Cj , almost surely. (25) m m m The deﬁnition of Aj ,Bj ,Cj depend on nk, which is hidden from notation. While the sequence m (Bj : j ≥ 1) is not an i.i.d. sequence, and is not a sequence of independent random variables m m since the jump rate changes with time, both (Aj : j ≥ 1) and (Cj : j ≥ 1) are i.i.d. sequences of −n −n Bernoulli(1/2), Bernoulli([1 + C2 k ]/[2 + C2 k ]), respectively. For each i ∈ N, denote Qi as the −i2−nk U M C = i2−nk . U C number of visits to by nk while nk This is the number of visits nk makes to −n −n −i2 k when −i2 k is the running minimum. We will use (25) to sandwich Qi above and below by −n geometric random variables. Denote mi = i2 k . Then

mi Qi = inf{j ≥ 1 : Bj = 0}.

Q U C −m That is, i is the number of visits nk makes to its running minimum i because once a negative mi −nk jump occurs, i.e. Bj reaches 0, the running minimum decreases to −(i + 1)2 . Consider

mi Wi = inf{j ≥ 1 : Aj = 0},

mi Vi = inf{j ≥ 1 : Cj = 0}.

mi mi Because (Aj ), (Cj ) are each i.i.d. sequences of Bernoulli random variables, Wi,Vi are geometric random variables, and

Wi ≤ Qi ≤ Vi, almost surely, 1k (W = k) = , P i 2 1 1 + C/2nk k−1 P(Vi = k) = . 2 + C/2nk 2 + C/2nk

−n −n That is, Wi,Vi are geometrically distributed with parameters 1/2, (1 + C2 k )/(2 + C2 k ) respectively. C Now that we have sandwiched the number of steps Un makes at a certain level of its running minimum, we will analyze the Lebesgue time the process spends at its running minimum. Since the 2−nk M C 2nk m − 1 2nk m + 1 τ size of each step is , nk has visited between and sites up until time m,1.

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 21

T M C m 0 ≤ i2−nk =: m < M C (s) Let i be the time that nk spends at the site i, for i nk and a given s ∈ [0,T ]. ν , e , z 0 ≤ i2−nk < M C (s) By deﬁnition of m,j m,j m,j and the inequality (24), for nk we have

W W V Xi Xi Xi Φi := z− ∧ ν ∧ e ≤ T ≤ z− ∧ ν + z− ∧ ν enk mi,j mi,j mi,j i mi,j mi,j mi,j mi,j j=1 j=1 j=Wi (26) V Xi =: Φi + z− ∧ ν , nk mi,j mi,j j=Wi

C almost surely. This is because Un spends at least Wi steps at the running minimum −mi, each step spending at least z− ∧ ν ∧ e Lebesgue amount of time for each 1 ≤ j ≤ W , giving the mi,j mi,j mi,j i lower bound. The upper bound is the same reasoning. By Lemma 4.15, z− ∧ν is an Exp(22n+1) mi,j mi,j random variable independent from Ami . Because z− ∧ ν ∧ e is a measurable function of j mi,j mi,j mi,j z− ∧ν and e , both of which are independent from Ami , z− ∧ν ∧e is independent mi,j mi,j mi,j j mi,j mi,j mi,j from Ami as well. Consequently z− ∧ ν ∧ e , for 1 ≤ j ≤ W , are independent from W . j mi,j mi,j mi,j i i Similarly V is independent of z− ∧ ν for 1 ≤ j ≤ V . Since by deﬁnition, i mi,j mi,j i

n C 2 k Mn (s) Xk LC (s) = 2nk T , nk i i=1

LC (s) (26) implies we can sandwich nk by summing these upper bounds and lower bounds of times spent at each intermediate level. That is,

n C 2 k Mn (s)−1 Xk 2nk Φi ≤ LC (s) enk nk i=1

n C n C 2 k Mn (s)+1 2 k Mn (s)+1 V (27) Xk Xk Xi ≤ 2nk Φi + z− ∧ ν nk mi,j mi,j i=1 i=1 j=Wi

=: Rnk (s), and this inequality holds for all s ∈ [0,T ], almost surely. Now we will apply the squeeze theorem to (27) and show the left hand and right hand of that in- U quality, and hence Lnk , converge to M on [0,T ] in probability, hence for some subsequence of nk the convergence holds almost surely. The sum of a Geometric(p) number of independent exponen- tials of rate λ is exponential with rate pλ provided the number of exponential random variables being summed is independent of the exponential random variables themselves. Therefore, since Wi (resp. V ) is geometric and independent of z− ∧ ν ∧ e for 1 ≤ j ≤ W (resp. z− ∧ ν for i mi,j mi,j mi,j i mi,j mi,j 1 ≤ j ≤ V Φi (22nk+1 + C2nk )/2 Φi i), enk is distributed as an exponential of rate . Similarly nk has ex- 22nk . 2nk Φi ponential rate We think of enk as an exponential random variable with rate approximately 2nk , 2nk Φi 22nk . Φi while in fact nk is an exponential with rate exactly Because enk is measurable with i respect to F and independent from Fτ , (Φe : 1 ≤ i ≤ Wi) is a collection of inde- (τmi,1,τmi+1,1] mi,1 nk

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 22 Clayton Barnes

Φi pendent exponential random variables by the strong Markov property. In other words, enk depends only on the excursion between these two hitting times and does not depend on the initial position of these excursions. Deﬁne

TC := inf{t > 0 : L(t) > C}

(nk) where L = limnk Lnk . For any 0 < < C, it is clear that lim infnk TC ≥ TC−, and as a result

LC = L ,U C = U nk nk nk nk

[0,T ∧ T ] n U C (·) [0,T ] on C− for large enough k, almost surely. Because nk converges uniformly on to U M C (·) [0,T ∧ T ] the continuous process , almost surely, we know nk converges uniformly on C− to U U M . We will show that the left and right hand sides of (27) converge almost surely to M (s ∧ TC−) for each ﬁxed s ∈ [0,T ]. We go through the details for the left hand side, and the right hand is similar. 0 For ease of notation we denote s = s ∧ TC−. Note

n C 0 C 2 k Mn (s ) 2nk M U (s0) Xk X 2nk Φi − 2nk Φi enk enk

i=1 i=1 (28) n C 0 U 0 n C 0 U 0 2 k [Mn (s )∨M (s )] 2 k |Mn (s )−M (s )| k 1 k X nk i X ≤ 2 Φen ≤ ei, k 2nk i=2nk [M C (s0)∧M U (s0)] i=1 nk

e (1) M C almost surely, where i are i.i.d. Exp , and that are independent from nk . The last inequality comes d 2nk Φi = exp(2nk +C/2) |M C (s0)−M U (s0)| → 0 from Lemma 3.1 and the fact that enk are i.i.d. Since nk , almost surely, the strong law of large numbers implies that

n C 0 U 0 2 k |Mn (s )−M (s )| 1 k X ei −→ 0, almost surely. 2nk i=1

d 2nk Φi 2−nk uk u = exp(1 + C2−(nk+1)) We can express enk as i where i are i.i.d., so

2nk M U (s0) 2nk M U (s0) 1 X nk i X k 2 Φen = ui . k 2nk i=1 i=1

We condition on M U (s0) to compute

 2nk M U (s0)  1 1 1 X k U 0 nk U 0 Var  ui M (s ) = 2 M (s ) 2nk 22nk (1 + C2−(nk+1))2 i=1 M U (s0) = , 2nk + C + C22−(nk+2)

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 23 which approaches zero. The expectation conditional on M U (s0) is

 2nk M U (s0)  1 X k U 0 U 0 1 E  ui M (s ) = M (s ) . 2nk 1 + C2−(nk+1) i=1

Consequently,

2nk M U (s0) 2nk M U (s0) 1 X nk i X k U 0 2 Φen = ui −→ M (s ), k 2nk i=1 i=1 in probability, and by (28),

n C 0 2 k Mn (s ) Xk 2nk Φi −→ M U (s0), enk (29) i=1 in probability. Similarly for the right side of (27), one can show

n C 0 2 k Mn (s ) Xk 2nk Φi −→ M U (s0), nk (30) i=1

0 U 0 in probability. Now the term Rnk (s ) converges to M (s ) once we demonstrate

n C 0 2 k Mn (s )+1 V Xk Xi z− ∧ ν mi,j mi,j i=1 j=Wi converges to zero in probability. But this follows from two applications of Wald’s lemma, the second of which uses the ﬁltration generated (for ﬁxed i) by z− , ν and e to compute mi,j mi,j mi,j

 V  Xi z− ∧ ν = ( (V − W ) + 1) (z− ∧ ν ) E  mi,j mi,j E i i E mi,j mi,j j=Wi 2 + C2−nk 1 = 2 − · =: an , 1 + C2−nk 22nk+1 + C2nk k which clearly approaches zero. Using Lemma 4.18, the moment bound hypothesis of Wald’s equation C 0 C 1/2 1/2 is satisﬁed since we have E(Mn (s )) ≤ E(Mn (T )) ≤ 2(2T + T |v|(2T ) + 2|v|T ) , a uniform bound with respect to n. Thus, the second application of Wald’s lemma gives

2nk M C (s0)+1  nk Vi X X − n C 0  z ∧ ν  = (2 k M (s ) + 1) · an , E  mi,j mi,j E nk k i=1 j=Wi

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 24 Clayton Barnes which approaches zero as nk → ∞. Consequently,

n C 0 2 k Mn (s )+1 V Xk Xi z− ∧ ν mi,j mi,j i=1 j=Wi

0 U C 0 does indeed converge to zero in probability, and therefore Rnk (s ) converges to M (s ) in probability. Because convergence in probability implies almost sure convergence for some subsequence, we 0 can find a common subsequence nk where both (29) and (30) occur almost surely, for fixed s, where 0 0 s = s ∧ TC−. We relabel nk as nk. Similarly we can use a cantor diagonalization to find a further subsequence where (29) and (30) occur for all rationals in [0,T ∧ TC−], almost surely. Applying the squeeze theorem to the inequality (27) then yields

0 = lim |LC (s0) − M U (s0)| = lim |L (s0) − M U (s0)| = |L(s0) − M U (s0)| nk nk (31) nk→∞ nk→∞

s0 = s ∧ T s ∈ ∩ [0,T ] L = lim L = lim LC = where C−, for any Q , almost surely. Therefore nk nk nk nk C M U = M U [0,T ∧ T ] ⊂ [0, lim inf T C ∧ T ], for all rational numbers in C− nk nk almost surely. So U L = M on [0,TC− ∧ T ], almost surely, since both processes are continuous. U Letting C approach inﬁnity, P(TC ≥ T ) −→ 1, since L is a ﬁnite process, which yields M = L on [0,T ], almost surely, completing the proof of (iii).

Proof of (iv). As in the other proofs, Remark 4.6 and Lemma 4.7 allow us to assume without loss of generality that for the subsequence (Snm ,Znm ,Vnm ,Lnm ),

(Snm ,Znm ,Vnm ,Lnm ) −→ (S, Z, V, L) (32) almost surely, in the uniform norm on C([0,T ], R); we have Unm → U on C([0,T ], R) as well. In the previous proof of (iii) we showed L(t) = M U (t) for each t ∈ [0,T ], almost surely. In this proof we wish to show t Z Z(t) = V (x) dx, for t ∈ [0,T ], (33) 0 almost surely, where V = KM U − v. We take v ≤ 0 for the time being and reduce to this case at the 0 end. It sufﬁces to demonstrate that for each s ∈ [0,T ] there is a subsequence nm such that s Z Z 0 (s)−→ V (x) dx, almost surely. (34) nm 0 Z · By a Cantor diagonalization Z(·) and M U (s) ds will agree for all rationals in [0,T ], almost surely. 0 The two processes will then agree on [0,T ], almost surely, because both processes are continuous. For a given n, n Zen(s) := 2 Zn(s)

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 25 counts the number of jumps of Zn by time s. Equivalently, this counts the number of arrival times {uk : k ≥ 1} of jumps by the process Zn. (We hide the dependence of uk on n for convenience). For C > 0, let

(n) τC = inf{s > 0 : Vn(s) > C},

αk = sup Vn(s), s∈[uk,uk+1]

αk = inf Vn(s). s∈[uk,uk+1] Assume for the time being that for every ﬁxed δ > 0,

sup{(ui+1 − ui): Zenm (δ ∧ s) ≤ i ≤ Zenm (s)} −→ 0, in probability. (35)

0 Then there exists a subsequence nm, which we relabel as nm, such that sup{(ui+1 − ui): Zenm (δ ∧ s) ≤ i ≤ Zenm (s)} −→ 0, almost surely. We use the time between jumps, uk+1 −uk, as the time step in a Riemann sum approximation of the integral in (34). By the deﬁnition of αk, αk and the exponential n representation of the gap times given in (8), there is a sequence µk of i.i.d. Exp(2 ) random variables such that

αk(uk+1 − uk) ≤ µk ≤ αk(uk+1 − uk), almost surely. Therefore,

Zen (s) Zen (s) Zen (s) Xm Xm Xm αk(uk+1 − uk) ≤ µk ≤ αk(uk+1 − uk) (36)

k=Zenm (δ∧s) k=Zenm (δ∧s) k=Zenm (δ∧s) where we deﬁne the left and right sums to be zero should the set of such indices Zenm (δ ∧ s) ≤ k ≤

Zenm (s) be empty. From (35) together with (32) and Riemann integrability of the limiting function V ,

Zen (s) s Zen (s) Xm Z Xm lim α (u − u ) = V (x) dx = lim α (u − u ), (37) 0 k k+1 k 0 k k+1 k nm→∞ nm→∞ k=Zenm (δ∧s) δ∧s k=Zenm (δ∧s) where convergence holds uniformly on [0,T ], almost surely. By the squeeze theorem,

Zen (s) s Xm Z µk −→ V (x) dx, (38)

k=Zenm (δ∧s) δ∧s

nm nm almost surely, as well. Since the µk are i.i.d. exponential r.v.’s of rate 2 and Zenm (s) = 2 Znm (s) with Znm (·) → Z(·) almost surely, the law of large numbers implies

Zenm (s) X a.s. µk −→ Z(s) − Z(δ ∧ s),

k=Zenm (δ∧s)

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 26 Clayton Barnes for each s ∈ [0,T ]. Therefore

s Z Z(s) − Z(δ ∧ s) = V (x) dx (39) δ∧s

R δ for each s in [0,T ], almost surely. Since Z(δ) −→ 0 almost surely and 0 V (x) dx −→ 0, as δ → 0, this gives Z s Z(s) = V (x) dx 0 as desired.

To demonstrate (35), recall the jump process Znk determining the gap between jump times ui+1 −ui nk nk (nk) has an intensity process 2 |Vnk | that is bounded below by 2 on the interval [τ , ∞). Heuris- tically, on this interval the intensity cannot be too small so the inter-arrival times are not too large.

(This is where we use the fact that v ≤ 0, so that |Vnk | = Vnk . In the case that v > 0 the intensity nk 2 |Vnk | will cross zero, which we handle at the end.) By Lemma 3.1 there exists an i.i.d. sequence vi n of exponential random variables with rate 2 m that stochastically dominate ui+1 − ui. We have

(nm) {τ > δ} = {Vnm (δ) ≤ }. (40)

For 0 < η 1, C > 0,

P(sup{(ui+1 − ui): Zenm (δ) ≤ i ≤ Zenm (t)} > η)

(nm) (nm) (nm) ≤ P(sup{(ui+1 − ui): Zenm (τ ) ≤ i ≤ Zenm (t)} > η, τ ≤ δ) + P(τ > δ)

(nm) ≤ P(sup{vi : 1 ≤ i ≤ Zenm (t)} > η) + P(τ > δ)

nm nm nm (nm) ≤ P(sup{vi, 1 ≤ i ≤ C2 } > η, Zenm (t) ≤ C2 ) + P(Zenm (t) > C2 ) + P(τ > δ)

nm (nm) ≤ P(vi > η : some 1 ≤ i ≤ C2 ) + P(Znm (t) > C) + P(τ > δ)

nm (nm) ≤ C2 P(vi > η) + P(Znm (t) > C) + P(τ > δ)

nm nm (nm) ≤ C2 exp(−η2 ) + P(Znm (t) > C) + P(τ > δ), nm nm = C2 exp(−η2 ) + P(Znm (t) > C) + P(Vnm (δ) ≤ ), by (40).

Taking lim sup with respect to nm on both sides and applying the assumption that Zn → Z and Vn → V almost surely, we have

lim sup P(sup{(ui+1 − ui): Zenm (δ) ≤ i ≤ Zenm (t)} > η) ≤ P(Z(t) > C) + P(V (δ) ≤ ). nm→∞

Since C, > 0 are arbitrary and V (0) ≥ 0 on our assumption v ≤ 0,

lim sup P(sup{(ui+1 − ui): Zenm (δ) ≤ i ≤ Zenm (t)} > η) = 0, nm→∞ for every ﬁxed δ > 0, proving (35).

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 Approximating Brownian motion with inert drift by jump processes 27

To show the case v > 0 reduces to v = 0, notice that

(nm) (nm) T−3/2 ≤ lim inf τ− ≤ lim sup τ ≤ T3/2, (41) nm→∞ nm→∞ where

Ta := inf{t > 0 : V (t) > a}.

For almost each ω in our probability space there is an N(ω) such that |Vnm | is monotone and bounded (nm) (nm) away from zero on the intervals [0, lim infnm→∞ τ− ] ⊃ [0,T−3/2 ∧T ] and [lim supnm→∞ τ ,T ] ⊃ [T3/2 ∧ T,T ], for all nm ≥ N(ω). With this fact and (41) we can apply the proof thus far to show Z t Z(t) − Z(s) = V (x) dx for s, t ∈ [0,T−3/2 ∧ T ], or s, t ∈ [T3/2 ∧ T,T ]. s

In addition to this, an L∞ bound gives

Z T3/2∧T |V (x)| dx ≤ (3/2)T, almost surely, T−3/2∧T Z s which goes to zero as → 0. It follows that Z(s) = V (x) dx for s ∈ [0,T ] in the case v < 0 as 0 well.

5. Mutlidimensional Analog of BMID

In 2007, White constructed a multidimensional analog, see [13], whose stationary distribution was found by Bass, Burdzy, Chen and Hairer [3]. This multidimensional analog is a pair of processes n (Z,V ) where Z is a diffusion reflecting inside a sufficiently smooth domain D ⊂ R , and V is its drift. This drift is the inward normal integrated against the local time Z spends on ∂D. That is, Z t Z t Z(t) = B(t) + η(Z(s)) dL(s) + V (s) ds, 0 0 (42) Z t V (t) = V0 + η(Z(s)) dL(s), 0 where η(x) is the inward unit normal for x ∈ ∂D and t → L(t) is a nondecreasing continuous function flat off of ∂D. By this we mean L increases only on Z−1(∂D). The authors show (Z,V ) has a stationary distribution of µ × γ, where µ is the uniform distribution on D and γ is the Gaussian distribution n on R . This is interesting in part because the stationary distribution of the drift is always Gaussian and does not depend on D, and also because the stationary distribution is always a product form. When Z is one dimensional, and D = [0, ∞), the process Z is one dimensional reflected BMID which is the process introduced by Knight.

Acknowledgements

CB is a Zuckerman Postdoctoral scholar at Technion-Israel’s Institute of Technology, Industrial En- gineering and Management, Haifa, Israel, 32000. The preparation of this manuscript was partially

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021 28 Clayton Barnes supported by FNS 200021_175728/1. During this research the author was graduate student at the Uni- versity of Washington and visited Universidad de Chile.

References

[1] C. Barnes. (2020). Hydrodynamic limit and propagation of chaos for Brownian particles reﬂect- ing from a Newtonian barrier. Ann. Appl. Probab., 30(4), 1582–1613. [2] C. Barnes, K. Burdzy, and C.-E., Gauthier. (2019). Billiards with Markovian reﬂection laws. Electron. J. Probab., 24. [3] R. F. Bass, K. Burdzy, Z.-Q. Chen, and M. Hairer. (2010). Stationary distributions for diffusions with inert drift. Probab. Theory Related Fields, 146(1-2):1. [4] P. Billingsley. (1968). Convergence of Probability Measures. Wiley. [5] P. Brémaud. (1981). Point Processes and Queues: Martingale Dynamics. Springer New York. [6] K. Burdzy and D. White. (2008). Markov processes with product-form stationary distribution. Electron. Commun. Probab., 13:614–627. [7] S. Ethier and T. Kurtz. (1986). Markov Processes: Characterization and Convergence. Wiley. [8] C.-E. Gauthier. (2018). Central Limit Theorem for one and two dimensional Self-Repelling Dif- fusions. ALEA, 15:691–702. [9] J. Jacod and A. Shiryaev. (2002) Limit Theorems for Stochastic Processes. Springer Berlin Heidelberg, 2nd edition. [10] I. Karatzas, S. Shreve. (1991) Brownian Motion and Stochastic Calculus. Springer-Verlag, 2nd edition. [11] F. B. Knight. (2001). On the path of an inert object impinged on one side by a Brownian particle. Probab. Theory Related Fields, 121(4):577–598. [12] P. Mörters and Y. Peres. (2010). Brownian Motion. Cambridge University Press. [13] D. White. (2007). Processes with inert drift. Electron. J. Probab., 12:1509–1546.

imsart-bj ver. 2020/08/06 file: DiscreteApprox2.tex date: February 18, 2021