Motivation: SPT Reflected BM Sticky BM Our results
Multidimensional sticky Brownian motions as limits of exclusion processes Joint work with Mykhaylo Shkolnikov
Miklos´ Z. Racz´
UC Berkeley
UW-Madison Probability Seminar November 14, 2013. Motivation: SPT Reflected BM Sticky BM Our results Symmetric simple exclusion process
Particles try to jump
I with rate 1 to the left, and
I with rate 1 to the right.
Exclusion: the jump is suppressed if the destination site is occupied. Topic of today’s talk: Describing the diffusive scaling limit of such particle systems.
Motivation: SPT Reflected BM Sticky BM Our results Symmetric simple exclusion with stickiness When apart, particles jump as usual:
When there is a collision, the whole system slows down: Motivation: SPT Reflected BM Sticky BM Our results Symmetric simple exclusion with stickiness When apart, particles jump as usual:
When there is a collision, the whole system slows down:
Topic of today’s talk: Describing the diffusive scaling limit of such particle systems. Motivation: SPT Reflected BM Sticky BM Our results Outline
Motivation: stochastic portfolio theory
Queueing theory and reflected Brownian motion
Sticky Brownian motion in 1D
Our results Motivation: SPT Reflected BM Sticky BM Our results Outline
Motivation: stochastic portfolio theory
Queueing theory and reflected Brownian motion
Sticky Brownian motion in 1D
Our results Motivation: SPT Reflected BM Sticky BM Our results Capital distribution curves
log(market rank) vs. log(market capitalization) for US stock markets in 1929,..., 2009 Question: why is this curve stable? Motivation: SPT Reflected BM Sticky BM Our results Stochastic portfolio theory: a descriptive theory
I Developed by Fernholz (2002), Banner, Fernholz, and Karatzas (2005), and coauthors.
I Rank-based models:
d (log Xi (t)) = γrt (i) (t) dt + σrt (i) (t) dWi (t) ,
where rt (i) is the rank of particle i at time t.
I Can describe the observed capital distribution curves.
However, rank-based models have shortcomings. To address this:
I A second-order stock market model: rank- and name-based (Fernholz, Ichiba, Karatzas, 2013)
I Brownian particles with asymmetric collisions (Karatzas, Pal, Shkolnikov, 2012) Motivation: models that describe events that slow down the market (e.g., a court trial)? Motivation: SPT Reflected BM Sticky BM Our results Outline
Motivation: stochastic portfolio theory
Queueing theory and reflected Brownian motion
Sticky Brownian motion in 1D
Our results Motivation: SPT Reflected BM Sticky BM Our results A simple flow model
I X0: initial inventory level,
I A, B: increasing, continuous stochastic processes,
I At : cumulative input until time t,
I Bt : cumulative potential output until time t,
I Lt : amount of potential output lost until time t,
I Xt := X0 + At − Bt : netput process
I Zt := X0 + At − (Bt − Lt ) = Xt + Lt : inventory process
Model: X is a Brownian motion; reasonable for balanced, high volume flows. Then: Z is a reflected Brownian motion, and L is its local time process at zero. Motivation: SPT Reflected BM Sticky BM Our results Reflected Brownian motion
Definition d Let η ∈ R , let Γ be a d × d non-degenerate covariance matrix, let R be a d d × d matrix. For x ∈ (R+) , a semimartingale reflecting Brownian motion d (SRBM) in the orthant (R+) associated with the data (η, Γ, R) that starts fromx is a continuous, (Ft )-adapted, d-dimensional process Z s.t.
d Z (t) = X (t) + RY (t) ∈ (R+) for all t ≥ 0,
(i) X is a d-dimensional Brownian motion with drift vector η and covariance matrix Γ such that {X (t) − ηt, Ft , t ≥ 0} is a martingale and X (0) = x Px -a.s.,
(ii) Y is an (Ft )-adapted, d-dimensional process such that Px -a.s. for each th i ∈ [d], the i component Yi of Y satisfies
(a) Yi (0) = 0, (b) Yi is continuous and nondecreasing, (c) Yi can increase only when Z is on the face Fi . Y is referred to as the “pushing” process of Z. Motivation: SPT Reflected BM Sticky BM Our results Reflected Brownian motion
I Studied heavily in the ’80s: Harrison, Reiman, Williams et al.
I Plays a key role in understanding queues with high traffic. Existence and uniqueness: Theorem (Taylor and Williams (1993)) d There exists a SRBM in the orthant (R+) with data (η, Γ, R) that d starts fromx ∈ (R+) if and only ifR is completely- S. Moreover, when it exists, the joint law of any SRBM, together with its associated pushing process, is unique.
Intuitively: completely-S condition necessary to stay in the orthant. Definition (Completely-S matrix) A matrix d × d matrix A is completely-S if there exists λ ∈ [0, ∞)d such that Aλ ∈ (0, ∞)d , and the same property is shared by every principal submatrix of A. Motivation: SPT Reflected BM Sticky BM Our results Reflected Brownian motion
No time on the boundary:
n d o P Leb t ≥ 0 : Z (t) ∈ ∂ (R+) = 0 = 1.
Another boundary property: for every J ⊆ {1,..., d} s.t. |J| ≥ 2, and for any k ∈ J, we have Z ∞
1{ZJ (s)=0} dYk (s) = 0. 0
Stationary distribution: If the skew-symmetry condition is satisfied, and Z has a stationary distribution, then it is given by a product of exponentials: p (z) = C (µ) exp {γ (µ) · z} . Motivation: SPT Reflected BM Sticky BM Our results Outline
Motivation: stochastic portfolio theory
Queueing theory and reflected Brownian motion
Sticky Brownian motion in 1D
Our results Motivation: SPT Reflected BM Sticky BM Our results Sticky Brownian motion on the half-line Governed by the SDE:
dX (t) = λ1{X(t)=0} dt + 1{X(t)>0} dB (t)
I Weak existence and uniqueness hold, but strong existence and pathwise uniqueness fail.
I Arises through a stochastic time change of reflected Brownian motion
I If B has negative drift, X has an exponential stationary distribution with an added mass at zero.
Well studied:
I Feller (1950’s), Itoˆ and McKean (1963)
I Harrison and Lemoine (1981)
I Chitasvili (1989)
I Amir (1991)
I Warren (1997, 1999) Motivation: SPT Reflected BM Sticky BM Our results Outline
Motivation: stochastic portfolio theory
Queueing theory and reflected Brownian motion
Sticky Brownian motion in 1D
Our results Motivation: SPT Reflected BM Sticky BM Our results Sticky particles
When apart, particles jump as usual:
When there is a collision, the whole system slows down: Motivation: SPT Reflected BM Sticky BM Our results Sticky particles Let M > 0 denote the scaling parameter. Consider independent Poisson processes P , Q , L , and R : √ i i i,j i,j I all have jump size1 / M; I Pi , Qi have jump rates√ Ma, √ L R I Li,j has jump rate Mθi,j , Ri,j has jump rate Mθi,j . √ M M M The particle system X (·) = X1 (·) ,..., Xn (·) evolves on Z/ M: M dX (t) = 1n o d (P (t) − Q (t)) i X M (t)+ √1 vi,j = velocity of particle i when particles j and j + 1 are adjacent (and no others) I The reflection matrix Q = qj,j0 : qj,j0 = vj+1,j0 − vj,j0 , 0 0 qj,j0 = velocity of gap j when particles j and j + 1 are adjacent (and no others) Motivation: SPT Reflected BM Sticky BM Our results Main convergence theorem Theorem (R., Shkolnikov (2013)) Suppose that I Q is completely-S, [...], and M M I the initial conditions X1 (0) ,..., Xn (0) , M > 0 are deterministic and converge to a limit (x1,..., xn) ∈ W asM → ∞. Then the laws of the paths of the particle systems M M M n X1 (·) , X2 (·) ,..., Xn (·) , M > 0 onD ([0, ∞), R ) converge to the law of the unique weak solution of the system of SDEs n−1 √ X dXi (t) = 1{ ( )< ( )<...< ( )} 2a dWi (t) + 1 vi,j dt, X1 t X2 t Xn t {Xj (t)=Xj+1(t)} j=1 i ∈ [n], in W starting from (x1,..., xn). n HereW = (W1,..., Wn) is a standard Brownian motion in R . Motivation: SPT Reflected BM Sticky BM Our results Assumptions Intuitively: completely-S condition necessary to stay in the wedge W. 2 (2) (2) Define the (n − 1) × (n − 1) matrix Q = qi,(k,`) : −θL if k = i − 1, ` 6= i − 1, i,` θL + θR if k = i, ` 6= i, (2) := i+1,` i,` qi,(k,`) R −θi+1,` if k = i + 1, ` 6= i + 1, 0 otherwise. Our additional assumption Assume that the matrices Q and Q(2) (restricted to nonzero columns) are “jointly completely-S”, in the following sense. |J| For every J ⊆ [n − 1], J 6= ∅, there exists γ ∈ (R+) s.t. J γ · q·,j ≥ 1 for every j ∈ J, and (2),J (2) γ · q·,(k,`) ≥ 1 for every k, ` ∈ J such that q·,(k,`) is nonzero. L R Satisfied in many natural cases, e.g., when θi,j = θi,j = θ > 0. Motivation: SPT Reflected BM Sticky BM Our results Proof steps In order to prove our convergence result: I weak existence and uniqueness of the SDE, M I tightness of X (·) , M > 0 , I identification of the limit. Motivation: SPT Reflected BM Sticky BM Our results Existence and uniqueness of the SDE Consider the system of SDEs n−1 X dXi (t) = 1{ ( )< ( )<...< ( )} (bi dt + dWi (t))+ 1 vi,j dt. X1 t X2 t Xn t {Xj (t)=Xj+1(t)} j=1 Here: I bi , i ∈ [n] are real constants; I W = (W1,..., Wn) is an n-dim. Brownian motion w/ zero drift, and a strictly positive definite diffusion matrix C = (ci,i0 ); I V = vi,j is a matrix with real entries; I the initial conditions Xi (0) = xi , i ∈ [n], satisfy (x1,..., xn) ∈ W. Theorem IfQ is completely- S, then there exists a unique weak solution to the system of SDEs above. Moreover, ifQ is not completely- S, then there is no weak solution. Note: the diffusion matrix of X is both discontinuous, and degenerate, so classical results do not apply. Since Q is completely-S, there exists a weak solution to n−1 X dZbj (t) = bj+1 − bj dt + dBj (t) + qj,j0 dΛj0 (t) , j ∈ [n − 1] , j0=1 I with initial conditions Zbj (0) = xj+1 − xj , j ∈ [n − 1], I B = (B1,..., Bn−1) is a Brownian motion with zero drift and diffusion matrix A = aj,j0 given by aj,j0 = cj,j0 + cj+1,j0+1 − cj,j0+1 − cj+1,j0 , I Λj (·) is the semimartingale local time at zero of Zbj (·), I and recall that qj,j0 = vj+1,j0 − vj,j0 . Motivation: SPT Reflected BM Sticky BM Our results Weak existence of the SDE n−1 X dXi (t) = 1{ ( )< ( )<...< ( )} (bi dt + dWi (t))+ 1 vi,j dt. X1 t X2 t Xn t {Xj (t)=Xj+1(t)} j=1 Main ideas for weak existence: n−1 I Start from an appropriate SRBM in the orthant (R+) , and I apply an appropriate stochastic time change. Motivation: SPT Reflected BM Sticky BM Our results Weak existence of the SDE n−1 X dXi (t) = 1{ ( )< ( )<...< ( )} (bi dt + dWi (t))+ 1 vi,j dt. X1 t X2 t Xn t {Xj (t)=Xj+1(t)} j=1 Main ideas for weak existence: n−1 I Start from an appropriate SRBM in the orthant (R+) , and I apply an appropriate stochastic time change. Since Q is completely-S, there exists a weak solution to n−1 X dZbj (t) = bj+1 − bj dt + dBj (t) + qj,j0 dΛj0 (t) , j ∈ [n − 1] , j0=1 I with initial conditions Zbj (0) = xj+1 − xj , j ∈ [n − 1], I B = (B1,..., Bn−1) is a Brownian motion with zero drift and diffusion matrix A = aj,j0 given by aj,j0 = cj,j0 + cj+1,j0+1 − cj,j0+1 − cj+1,j0 , I Λj (·) is the semimartingale local time at zero of Zbj (·), I and recall that qj,j0 = vj+1,j0 − vj,j0 . Motivation: SPT Reflected BM Sticky BM Our results Weak existence of the SDE I Let βb = βb1, βb2,..., βbn be a Brownian motion with zero drift and diffusion matrix C such that Bi (·) = βbi+1(·) − βbi (·). I Define Xb = (Xb1, Xb2,..., Xbn) as the unique process satisfying n n n n−1 X X X X Xbi (t) = xi + bi t + βbi (t) + vi,j Λj (t) , i=1 i=1 i=1 j=1 Xb2 (t) − Xb1 (t) ,..., Xbn (t) − Xbn−1 (t) = Zb1 (t) ,..., Zbn−1 (t) . I Finally, define n−1 X T (t) := t + Λ(t) := t + Λj (t), t ≥ 0, j=1 τ(t) := inf{s ≥ 0 : T (s) = t}, t ≥ 0, and apply the stochastic time change X (·) = Xb (τ (·)) . Motivation: SPT Reflected BM Sticky BM Our results Properties of sticky Brownian motion Behavior at the boundary: does not spend a non-empty time interval on ∂W, however, P (Leb ({t ≥ 0 : X(t) ∈ ∂W}) > 0) = 1. Stationary distributions: under certain conditions, the gap process Z (·) has a stationary distribution as a weighted sum of exponentials. −1 T Suppose Q (b2 − b1, b3 − b2,..., bn − bn−1) < 0 componentwise and2 A = QD + DQT , where D = diag(A). Let −1 −1 T γ = 2 D Q (b2 − b1, b3 − b2,..., bn − bn−1) . n−1 Then the stationary distribution of Z in the orthant (R+) is n−1 √ 1 hγ,zi X aj,j j e dz + 1 dz . C 2 {z∈Fj } j=1 Motivation: SPT Reflected BM Sticky BM Our results Decomposition To obtain our convergence result, we study the decomposition n−1 n−1 n−1 n−1 M M M X R,M X L,M X R,M X L,M Xi (t) = Xi (0)+Ai (t)+ Ci,j (t)− Ci,j (t)+ ∆i,j (t)− ∆i,j (t) , j=1 j=1 j=1 j=1 where Z t M A (t) := 1n o d (P (s) − Q (s)) , i X M (s)+ √1 Main technical issue: dealing with the indicator functions. Motivation: SPT Reflected BM Sticky BM Our results Convergence statement Under the assumptions stated before, the family n o X M , AM , IL,M , IR,M , ∆L,M , ∆R,M , M > 0 2 is tight in D [0, ∞), R4n −2n . Moreover, every limit point satisfies Z · √ n−1 Z · ∞ X X (·) = 1 ∞ ∞ 2a dW (s) + v 1n o ds, i X (s)<... with a suitable n-dimensional standard Brownian motion W = (W1,... Wn). Motivation: SPT Reflected BM Sticky BM Our results Convergence proof sketch, I Step 1. Tightness X Step 2. A few observations: √ I The jumps of all components are bounded by1 / M, so all components of the limit point must have continuous paths. M I The family A (t), M > 0 is uniformly integrable: Z t h M 2i 1 A (t) = 1n o √ (dP + dQ )(s) ≤ 2at, E i E X M (s)+ √1 ∞ ∞ R · Second: hX i (·) = hA i (·) = 2a 1 ∞ ∞ ds i i 0 {X1 (s)<... Z t2 (ω1, ω2, . . . , ωn) 7→ 1{ω1(s)<···<ωn(s)} ds; t1 I Use the occupation time formula to show that the measure ∞ ∞ ∞ d hXi i assigns zero mass to the sets {t ≥ 0 : Xj (t) = Xj+1(t)}. Motivation: SPT Reflected BM Sticky BM Our results Convergence proof sketch, II Step 3. Determining the quadratic covariation processes ∞ ∞ ∞ ∞ hXi , Xi0 i = hAi , Ai0 i . ∞ ∞ R · Second: hX i (·) = hA i (·) = 2a 1 ∞ ∞ ds i i 0 {X1 (s)<... Z t2 (ω1, ω2, . . . , ωn) 7→ 1{ω1(s)<···<ωn(s)} ds; t1 I Use the occupation time formula to show that the measure ∞ ∞ ∞ d hXi i assigns zero mass to the sets {t ≥ 0 : Xj (t) = Xj+1(t)}. Motivation: SPT Reflected BM Sticky BM Our results Convergence proof sketch, II Step 3. Determining the quadratic covariation processes ∞ ∞ ∞ ∞ hXi , Xi0 i = hAi , Ai0 i . ∞ ∞ 0 First: hAi , Ai0 i = 0 when i 6= i . M M I For any t ≥ 0, the family Ai (t)Ai0 (t), M > 0 is uniformly M 2 M 2 2 2 integrable due to E Ai (t) Ai0 (t) ≤ 4a t ; ∞ ∞ I Ai (·)Ai0 (·) is the limit in D ([0, ∞), R) of the family of M M martingales Ai (·)Ai0 (·), M > 0 ; ∞ ∞ I So Ai (·)Ai0 (·) is a martingale w.r.t. its own filtration. Motivation: SPT Reflected BM Sticky BM Our results Convergence proof sketch, II Step 3. Determining the quadratic covariation processes ∞ ∞ ∞ ∞ hXi , Xi0 i = hAi , Ai0 i . ∞ ∞ 0 First: hAi , Ai0 i = 0 when i 6= i . M M I For any t ≥ 0, the family Ai (t)Ai0 (t), M > 0 is uniformly M 2 M 2 2 2 integrable due to E Ai (t) Ai0 (t) ≤ 4a t ; ∞ ∞ I Ai (·)Ai0 (·) is the limit in D ([0, ∞), R) of the family of M M martingales Ai (·)Ai0 (·), M > 0 ; ∞ ∞ I So Ai (·)Ai0 (·) is a martingale w.r.t. its own filtration. ∞ ∞ R · Second: hX i (·) = hA i (·) = 2a 1 ∞ ∞ ds i i 0 {X1 (s)<... Z t2 (ω1, ω2, . . . , ωn) 7→ 1{ω1(s)<···<ωn(s)} ds; t1 I Use the occupation time formula to show that the measure ∞ ∞ ∞ d hXi i assigns zero mass to the sets {t ≥ 0 : Xj (t) = Xj+1(t)}. Motivation: SPT Reflected BM Sticky BM Our results Convergence proof sketch, II Step 4. Dealing with the finite variation terms Want to show that Z · ∞,2 Ii,j (·) := 1{X ∞(s)=X ∞ (s),X ∞(s)=X ∞ (s)} ds 0 i i+1 j j+1 is identically zero when i 6= j. The limiting processes inherit many properties of the prelimit processes, and after the stochastic time change the process of spacings can be written as Zb (·) = Zb (0) + Bb (·) + QbI∞,1 (·) + Q(2)bI∞,2 (·) . Can finish the argument as Reiman and Williams (1988) did for the boundary property of SRBM, but need to assume that Q and Q(2) are “jointly completely-S”. Motivation: SPT Reflected BM Sticky BM Our results More general particle systems M 1 R L dX (t) = √ 1n o d S (M t) − S (M t) i X M (t)+ √1 L R I Si and Si , i ∈ [n], are renewal processes, not necessarily independent, L R I Ti,j and Ti,j , i ∈ [n], j ∈ [n − 1], are independent renewal processes. Motivation: SPT Reflected BM Sticky BM Our results More general particle systems Under the previous assumptions and an additional moment assumption on the interarrival times between jumps, we have: The laws of the paths of the particle systems X M (·) , M > 0 converge in D ([0, ∞), Rn) to the law of the unique weak solution of the system of SDEs R L 3/2 dXi (t) = 1{X1(t)<... L,L L,R L,R R,R C = (ci,i0 ) = (ci,i0 + ci,i0 + ci0,i + ci,i0 ), where these terms come from the covariances of the random variables describing the interarrival times related to the renewal L R processes Si and Si , i ∈ [n]. Thank you! Open questions: I Other types of sticky interaction? I In particular, local slowdown? Motivation: SPT Reflected BM Sticky BM Our results Summary and open questions Takeaways: I Exclusion processes with stickiness (global slowdown) ⇓ diffusive limit multidimensional sticky Brownian motion I Potential applications I stochastic portfolio theory I queueing theory Thank you! Motivation: SPT Reflected BM Sticky BM Our results Summary and open questions Takeaways: I Exclusion processes with stickiness (global slowdown) ⇓ diffusive limit multidimensional sticky Brownian motion I Potential applications I stochastic portfolio theory I queueing theory Open questions: I Other types of sticky interaction? I In particular, local slowdown? Motivation: SPT Reflected BM Sticky BM Our results Summary and open questions Takeaways: I Exclusion processes with stickiness (global slowdown) ⇓ diffusive limit multidimensional sticky Brownian motion I Potential applications I stochastic portfolio theory I queueing theory Open questions: I Other types of sticky interaction? I In particular, local slowdown? Thank you!