Diffusion scaling of a limit-order book model Steven E. Shreve - - PowerPoint PPT Presentation

Diffusion scaling of a limit-order book model

Steven E. Shreve Department of Mathematical Sciences Carnegie Mellon University shreve@andrew.cmu.edu nearly complete work with Christopher Almost John Lehoczky Xiaofeng Yu Thera Stochastics In Honor of Ioannis Karatzas May 31 – June 2, 20-17

Goals of this work

Build a “zero-intelligence” Poisson model of the limit-order book and determine its diffusion limit.

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Goals of this work

Build a “zero-intelligence” Poisson model of the limit-order book and determine its diffusion limit.

◮ “Zero-intelligence” — No strategic play by the agents

submitting orders.

1 / 50

Goals of this work

Build a “zero-intelligence” Poisson model of the limit-order book and determine its diffusion limit.

◮ “Zero-intelligence” — No strategic play by the agents

submitting orders.

◮ Poisson — Arrivals of buy and sell limit and market orders are

Poisson processes. Exponentially distributed waiting times before cancellations.

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Goals of this work

Build a “zero-intelligence” Poisson model of the limit-order book and determine its diffusion limit.

◮ “Zero-intelligence” — No strategic play by the agents

submitting orders.

◮ Poisson — Arrivals of buy and sell limit and market orders are

Poisson processes. Exponentially distributed waiting times before cancellations.

◮ Maybe a little intelligence — Locations of arrivals and

cancellations depend on the state of the limit-order book.

1 / 50

Goals of this work

Build a “zero-intelligence” Poisson model of the limit-order book and determine its diffusion limit.

◮ “Zero-intelligence” — No strategic play by the agents

submitting orders.

◮ Poisson — Arrivals of buy and sell limit and market orders are

Poisson processes. Exponentially distributed waiting times before cancellations.

◮ Maybe a little intelligence — Locations of arrivals and

cancellations depend on the state of the limit-order book.

◮ Diffusion scaling — Accelerate time by a factor of n, divide

volume by √n, and pass to the limit as n → ∞.

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Goals of this work

Build a “zero-intelligence” Poisson model of the limit-order book and determine its diffusion limit.

◮ “Zero-intelligence” — No strategic play by the agents

submitting orders.

◮ Poisson — Arrivals of buy and sell limit and market orders are

Poisson processes. Exponentially distributed waiting times before cancellations.

◮ Maybe a little intelligence — Locations of arrivals and

cancellations depend on the state of the limit-order book.

◮ Diffusion scaling — Accelerate time by a factor of n, divide

volume by √n, and pass to the limit as n → ∞.

◮ Diffusion limit — Evolution of the limiting limit-order book is

described in terms of Brownian motions.

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Goals of this work

Build a “zero-intelligence” Poisson model of the limit-order book and determine its diffusion limit.

◮ “Zero-intelligence” — No strategic play by the agents

submitting orders.

◮ Poisson — Arrivals of buy and sell limit and market orders are

Poisson processes. Exponentially distributed waiting times before cancellations.

◮ Maybe a little intelligence — Locations of arrivals and

cancellations depend on the state of the limit-order book.

◮ Diffusion scaling — Accelerate time by a factor of n, divide

volume by √n, and pass to the limit as n → ∞.

◮ Diffusion limit — Evolution of the limiting limit-order book is

described in terms of Brownian motions.

◮ Computation of statistics – Use the limiting limit-order model

to compute statistics of model dynamics.

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Partial history

◮ Cont, R., Stoikov, S. & Talreja, R. (2010) “A

stochastic model for order book dynamics,” Operations Research 58, 549–563. Poisson arrivals of buy and sell orders and exponential waiting times before cancellations. Use Laplace transforms to compute statistics of the order book dynamics.

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Partial history

◮ Cont, R., Stoikov, S. & Talreja, R. (2010) “A

stochastic model for order book dynamics,” Operations Research 58, 549–563. Poisson arrivals of buy and sell orders and exponential waiting times before cancellations. Use Laplace transforms to compute statistics of the order book dynamics.

◮ Cont, R. & de Larrard, A. (2013) “Price dynamics in a

Markovian limit order market,” SIAM J. Financial Mathematics 4, 1–25. Always a one tick spread and orders queue only at the best bid and best ask prices. If one of these is depleted, both move

• ne tick and the book reinitializes. Derive the diffusion-scaled

limit.

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This talk

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This talk

◮ We derive the diffusion-scaled limit.

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This talk

◮ We derive the diffusion-scaled limit. ◮ The order book consists of more than queues at the best bid

and best ask prices.

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This talk

◮ We derive the diffusion-scaled limit. ◮ The order book consists of more than queues at the best bid

and best ask prices.

◮ We continue through the price change without reinitializing

the model.

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This talk

◮ We derive the diffusion-scaled limit. ◮ The order book consists of more than queues at the best bid

and best ask prices.

◮ We continue through the price change without reinitializing

the model.

◮ Our limiting model has a two-tick spread at almost every

time, contrary to empirical observations.

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This talk

◮ We derive the diffusion-scaled limit. ◮ The order book consists of more than queues at the best bid

and best ask prices.

◮ We continue through the price change without reinitializing

the model.

◮ Our limiting model has a two-tick spread at almost every

time, contrary to empirical observations.

◮ The pre-limit model, which the limiting model approximates,

has more realistic behavior. For the parameters considered here, there is a one-tick spread 76% of the time.

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This talk

◮ We derive the diffusion-scaled limit. ◮ The order book consists of more than queues at the best bid

and best ask prices.

◮ We continue through the price change without reinitializing

the model.

◮ Our limiting model has a two-tick spread at almost every

time, contrary to empirical observations.

◮ The pre-limit model, which the limiting model approximates,

has more realistic behavior. For the parameters considered here, there is a one-tick spread 76% of the time.

◮ We present some computations in the limiting model.

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Arrivals and cancellations of buy orders

c c λ2 λ1 λ0 ask bid

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Arrivals and cancellations of buy orders

c c λ2 λ1 λ0 ask bid

◮ All arriving and departing orders are of size 1.

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Arrivals and cancellations of buy orders

c c λ2 λ1 λ0 ask bid

◮ All arriving and departing orders are of size 1. ◮ Poisson arrivals of market buys at rate λ0. These execute at

the best ask price.

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Arrivals and cancellations of buy orders

c c λ2 λ1 λ0 ask bid

◮ All arriving and departing orders are of size 1. ◮ Poisson arrivals of market buys at rate λ0. These execute at

the best ask price.

◮ Poisson arrivals of limit buys at one and two ticks below the

best ask price at rates λ1 and λ2, respectively.

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Arrivals and cancellations of buy orders

c c λ2 λ1 λ0 ask bid

◮ All arriving and departing orders are of size 1. ◮ Poisson arrivals of market buys at rate λ0. These execute at

the best ask price.

◮ Poisson arrivals of limit buys at one and two ticks below the

best ask price at rates λ1 and λ2, respectively.

◮ Cancellations of limit buys two or more ticks below the best

bid price, at rate θ/√n per order.

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Arrivals and cancellations of buy orders

c c λ2 λ1 λ0 ask bid

◮ All arriving and departing orders are of size 1. ◮ Poisson arrivals of market buys at rate λ0. These execute at

the best ask price.

◮ Poisson arrivals of limit buys at one and two ticks below the

best ask price at rates λ1 and λ2, respectively.

◮ Cancellations of limit buys two or more ticks below the best

bid price, at rate θ/√n per order.

◮ All processes are independent of one another.

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Arrivals and cancellations of sell orders

µ2 µ1 µ0 c c λ2 λ1 λ0 ask bid

◮ All orders are of size 1. ◮ Poisson arrivals of market sells at rate µ0. These execute at

the best bid price.

◮ Poisson arrivals of limit sells at one and two ticks above the

best bid price at rates µ1 and µ2, respectively.

◮ Cancellations of limit sells two or more ticks above the best

ask price, at rate θ/√n per order.

◮ All processes are independent of one another.

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Constraints on parameters

In order to have a diffusion limit, among the six parameters λ0, λ1, λ2, µ0, µ1, and µ2, there are three degrees of freedom. Let a and b be positive constants satisfying a + b > ab. Then λ1 = (a − 1)λ0, λ2 = (a + b − ab)λ0, µ1 = (b − 1)µ0, µ2 = (a + b − ab)µ0, aλ0 = bµ0. In addition to a and b, there is a scale parameter, which can be set by choosing µ0.

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Constraints on parameters

In order to have a diffusion limit, among the six parameters λ0, λ1, λ2, µ0, µ1, and µ2, there are three degrees of freedom. Let a and b be positive constants satisfying a + b > ab. Then λ1 = (a − 1)λ0, λ2 = (a + b − ab)λ0, µ1 = (b − 1)µ0, µ2 = (a + b − ab)µ0, aλ0 = bµ0. In addition to a and b, there is a scale parameter, which can be set by choosing µ0. To simplify the presentation, we set λ1 = λ2 = µ1 = µ2 = 1, λ0 = µ0 = λ := (1 + √ 5)/2.

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Limit-order book arrivals and departures

1 λ 1 c 1 1 λ c c 1 1 λ λ 1 1 1 λ 1 λ 1 1 c c λ 1 1 1 1 λ c 1 λ 1 1 λ 1 c λ 1 1 c c λ 1 1 1 1 λ λ 1 1 c 1 1 λ 1 c c 1 λ U V W X Y Z

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Transitions of (W , X)

X W λ 1 1 λ 1 1 1 1 1 λ 1 1 λ λ 1 1 λ 1 1 λ 1 1 λ 1 1 1 1 1

Transitions of (W , X)

X W λ 1 1 λ 1 1 1 1 1 λ 1 1 λ λ 1 1 λ 1 1 λ 1 1 λ 1 1 1 1 1 (W , X) is null recurrent ⇔ λ = 1+

√ 5 2

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Split Brownian motion

The diffusion scaling of a generic process Q is defined to be

• Qn(t) :=

1 √nQ(nt).

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Split Brownian motion

The diffusion scaling of a generic process Q is defined to be

• Qn(t) :=

1 √nQ(nt).

Theorem

Conditional on the bracketing processes V and Y remaining nonzero, ( W n, X n) converges in distribution to a split Brownian motion (W ∗, X ∗) = (max{G ∗, 0}, min{G ∗, 0}), where G ∗ is a one-dimensional Brownian motion with variance 4λ per unit time.

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Split Brownian motion

X W λ 1 1 λ 1 1 1 1 1 λ 1 1 λ λ 1 1 λ 1 1 λ 1 1 λ 1 1 1 1 1

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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Split Brownian motion

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

• Br. Motion

d dt W ∗, W ∗t = 4λ,

The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

• Br. Motion

d dt W ∗, W ∗t = 4λ,

• Br. Motion

d dt Y ∗, Y ∗t = 4λ

The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

• Br. Motion

d dt W ∗, W ∗t = 4λ,

• Br. Motion

d dt Y ∗, Y ∗t = 4λ d dt W ∗, Y ∗t = 4

The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

• Br. Motion

d dt W ∗, W ∗t = 4λ,

• Br. Motion

d dt Y ∗, Y ∗t = 4λ d dt W ∗, Y ∗t = 4

Frozen at 1

θ

Frozen at − 1

θ

The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

• Br. Motion

d dt W ∗, W ∗t = 4λ,

• Br. Motion

d dt Y ∗, Y ∗t = 4λ d dt W ∗, Y ∗t = 4

Frozen at 1

θ

Frozen at − 1

θ

Frozen at 0

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The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

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The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗ Frozen at 1

θ

Frozen at − 1

θ

Frozen at 0

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The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗ Frozen at 1

θ

Frozen at − 1

θ

Frozen at 0 U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗ Jumps to 1

θ

Jumps to − 1

θ

Jumps to 0 Starts to diffuse

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The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗ V ∗ and X ∗ are in a race to zero.

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The other queues

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗ V ∗ and X ∗ are in a race to zero.

• A. Metzler, Stat. & Probab. Letters, 2010: “On the first

passage problem for correlated Brownian motion.”

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The other queues

Suppose V ∗ wins. T ∗ U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

◮ Reset the “bracketing processes” to be U∗ and X ∗. ◮ (V ∗, W ∗) begins executing a split Brownian motion.

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Snapped Brownian motion

Let’s consider the V ∗ process in more detail.

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Snapped Brownian motion

Let’s consider the V ∗ process in more detail. As long as the “bracketing processes” V ∗ and Y ∗ remain nonzero, (W ∗, X ∗) executes a split Brownian motion: (W ∗, X ∗) = (max{G ∗, 0}, min{G ∗, 0}), where G ∗ is a one-dimensional Brownian motion with variance 4λ per unit time. U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗ Frozen at 1

θ

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Snapped Brownian motion

Still have the split Brownian motion, (W ∗, X ∗) = (max{G ∗, 0}, min{G ∗, 0}), but now V ∗ is diffusing. U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

• Br. Motion

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Snapped Brownian motion

t G ∗(t)

Snapped Brownian motion

t G ∗(t)

1 θ

t V ∗(t)

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Summary of properties of the limiting model

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Summary of properties of the limiting model

◮ At almost every time, there is a two-tick spread (i.e., one

empty tick), but this happens only 24% of the time in the pre-limit model.

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Summary of properties of the limiting model

◮ At almost every time, there is a two-tick spread (i.e., one

empty tick), but this happens only 24% of the time in the pre-limit model.

◮ The queues at the best bid and best ask in the limiting model

form a two-dimensional correlated Brownian motion.

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Summary of properties of the limiting model

◮ At almost every time, there is a two-tick spread (i.e., one

empty tick), but this happens only 24% of the time in the pre-limit model.

◮ The queues at the best bid and best ask in the limiting model

form a two-dimensional correlated Brownian motion.

◮ The queues behind the best bid and best ask in the limiting

model are frozen at 1

θ and − 1 θ.

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Summary of properties of the limiting model

◮ At almost every time, there is a two-tick spread (i.e., one

empty tick), but this happens only 24% of the time in the pre-limit model.

◮ The queues at the best bid and best ask in the limiting model

form a two-dimensional correlated Brownian motion.

◮ The queues behind the best bid and best ask in the limiting

model are frozen at 1

θ and − 1 θ. ◮ When the queue at the best bid or the best ask is depleted,

we have a three-tick spread.

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Summary of properties of the limiting model

◮ At almost every time, there is a two-tick spread (i.e., one

empty tick), but this happens only 24% of the time in the pre-limit model.

◮ The queues at the best bid and best ask in the limiting model

form a two-dimensional correlated Brownian motion.

◮ The queues behind the best bid and best ask in the limiting

model are frozen at 1

θ and − 1 θ. ◮ When the queue at the best bid or the best ask is depleted,

we have a three-tick spread.

◮ We transition through the three-tick spread using the concept

• f a snapped Brownian motion.
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Renewal states

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗

Renewal states

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗ T ∗ U∗ V ∗ W ∗ X ∗ Y ∗

Renewal states

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗ T ∗ U∗ V ∗ W ∗ X ∗ Y ∗ V ∗ W ∗ X ∗ Y ∗ Z ∗ A∗

Renewal states

U∗ V ∗ W ∗ X ∗ Y ∗ Z ∗ T ∗ U∗ V ∗ W ∗ X ∗ Y ∗ V ∗ W ∗ X ∗ Y ∗ Z ∗ A∗ Which way? How long?

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How long to transition between renewal states?

Recall W ∗ = max{G ∗, 0}, X ∗ = min{G ∗, 0}.

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How long to transition between renewal states?

Recall W ∗ = max{G ∗, 0}, X ∗ = min{G ∗, 0}. Negative excursions of G ∗: V ∗ diffuses; Y ∗ frozen at -1/θ.

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How long to transition between renewal states?

Recall W ∗ = max{G ∗, 0}, X ∗ = min{G ∗, 0}. Negative excursions of G ∗: V ∗ diffuses; Y ∗ frozen at -1/θ. Positive excursions of G ∗: Y ∗ diffuses; V ∗ frozen at 1/θ.

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How long to transition between renewal states?

Recall W ∗ = max{G ∗, 0}, X ∗ = min{G ∗, 0}. Negative excursions of G ∗: V ∗ diffuses; Y ∗ frozen at -1/θ. Positive excursions of G ∗: Y ∗ diffuses; V ∗ frozen at 1/θ. Lengths of positive excursions of G ∗ Lengths of negative excursions of G ∗ Local time

• f G ∗ at 0

How long to transition between renewal states?

Recall W ∗ = max{G ∗, 0}, X ∗ = min{G ∗, 0}. Negative excursions of G ∗: V ∗ diffuses; Y ∗ frozen at -1/θ. Positive excursions of G ∗: Y ∗ diffuses; V ∗ frozen at 1/θ. Lengths of positive excursions of G ∗ Lengths of negative excursions of G ∗ Local time

• f G ∗ at 0

τV

How long to transition between renewal states?

Recall W ∗ = max{G ∗, 0}, X ∗ = min{G ∗, 0}. Negative excursions of G ∗: V ∗ diffuses; Y ∗ frozen at -1/θ. Positive excursions of G ∗: Y ∗ diffuses; V ∗ frozen at 1/θ. Lengths of positive excursions of G ∗ Lengths of negative excursions of G ∗ Local time

• f G ∗ at 0

τV τY

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Calculation of renewal time distribution

◮ Let p(ℓ) be the probability V ∗ reaches zero during a negative

excursion of G ∗ of length ℓ. Can be computed by adapting Metzler.

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Calculation of renewal time distribution

◮ Let p(ℓ) be the probability V ∗ reaches zero during a negative

excursion of G ∗ of length ℓ. Can be computed by adapting Metzler.

◮ p(ℓ) is also the probability Y ∗ reaches zero during a positive

excursion of G ∗.

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Calculation of renewal time distribution

◮ Let p(ℓ) be the probability V ∗ reaches zero during a negative

excursion of G ∗ of length ℓ. Can be computed by adapting Metzler.

◮ p(ℓ) is also the probability Y ∗ reaches zero during a positive

excursion of G ∗.

◮ Four independent Poisson random measures:

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Calculation of renewal time distribution

◮ Let p(ℓ) be the probability V ∗ reaches zero during a negative

excursion of G ∗ of length ℓ. Can be computed by adapting Metzler.

◮ p(ℓ) is also the probability Y ∗ reaches zero during a positive

excursion of G ∗.

◮ Four independent Poisson random measures:

◮ ν±

0 (dt dℓ) – Lengths of positive (negative) excursion of G ∗

during which Y ∗ (V ∗) reaches zero. L´ evy measure is µ0(dℓ) = p(ℓ) dℓ 2 √ 2πℓ3 .

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Calculation of renewal time distribution

◮ Let p(ℓ) be the probability V ∗ reaches zero during a negative

excursion of G ∗ of length ℓ. Can be computed by adapting Metzler.

◮ p(ℓ) is also the probability Y ∗ reaches zero during a positive

excursion of G ∗.

◮ Four independent Poisson random measures:

◮ ν±

0 (dt dℓ) – Lengths of positive (negative) excursion of G ∗

during which Y ∗ (V ∗) reaches zero. L´ evy measure is µ0(dℓ) = p(ℓ) dℓ 2 √ 2πℓ3 .

◮ ν±

×(dt dℓ) – Lengths of positive (negative) excursions of G ∗

during which Y ∗ (V ∗) does not reach zero. L´ evy measure is µ×(dℓ) = (1 − p(ℓ)) dℓ 2 √ 2πℓ3 .

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Calculation of renewal time distribution

◮ τY = min{t ≥ 0 : ν+

• (0, t] × (0, ∞)
• > 0.

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Calculation of renewal time distribution

◮ τY = min{t ≥ 0 : ν+

• (0, t] × (0, ∞)
• > 0.

◮ τV = min{t ≥ 0 : ν−

• (0, t] × (0, ∞)
• > 0.

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Calculation of renewal time distribution

◮ τY = min{t ≥ 0 : ν+

• (0, t] × (0, ∞)
• > 0.

◮ τV = min{t ≥ 0 : ν−

• (0, t] × (0, ∞)
• > 0.

◮ τY and τV are independent.

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Calculation of renewal time distribution

◮ τY = min{t ≥ 0 : ν+

• (0, t] × (0, ∞)
• > 0.

◮ τV = min{t ≥ 0 : ν−

• (0, t] × (0, ∞)
• > 0.

◮ τY and τV are independent. ◮ We want to know the distribution of

(i) the chronological time T1 corresponding to local time τY ∧ τV ,

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Calculation of renewal time distribution

◮ τY = min{t ≥ 0 : ν+

• (0, t] × (0, ∞)
• > 0.

◮ τV = min{t ≥ 0 : ν−

• (0, t] × (0, ∞)
• > 0.

◮ τY and τV are independent. ◮ We want to know the distribution of

(i) the chronological time T1 corresponding to local time τY ∧ τV , (ii) plus the chronological elapsed time T2 in the “last excursion” beginning at local time τY ∧ τV before Y ∗ or V ∗ reaches zero.

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Calculation of renewal time distribution

◮ τY = min{t ≥ 0 : ν+

• (0, t] × (0, ∞)
• > 0.

◮ τV = min{t ≥ 0 : ν−

• (0, t] × (0, ∞)
• > 0.

◮ τY and τV are independent. ◮ We want to know the distribution of

(i) the chronological time T1 corresponding to local time τY ∧ τV , (ii) plus the chronological elapsed time T2 in the “last excursion” beginning at local time τY ∧ τV before Y ∗ or V ∗ reaches zero.

◮ (i) is

T1 := ∞

ℓ=0

τY ∧τV

t=0

ℓν+

×(dt dℓ) +

∞

ℓ=0

τY ∧τV

t=0

ℓν−

×(dt dℓ).

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Calculation of renewal time distribution

◮ τY = min{t ≥ 0 : ν+

• (0, t] × (0, ∞)
• > 0.

◮ τV = min{t ≥ 0 : ν−

• (0, t] × (0, ∞)
• > 0.

◮ τY and τV are independent. ◮ We want to know the distribution of

(i) the chronological time T1 corresponding to local time τY ∧ τV , (ii) plus the chronological elapsed time T2 in the “last excursion” beginning at local time τY ∧ τV before Y ∗ or V ∗ reaches zero.

◮ (i) is

T1 := ∞

ℓ=0

τY ∧τV

t=0

ℓν+

×(dt dℓ) +

∞

ℓ=0

τY ∧τV

t=0

ℓν−

×(dt dℓ). ◮ For (ii), we observe that the distribution of the length of the

“last excursion” is µ0(dℓ)/µ0((0, ∞)).

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Calculation of renewal time distribution

◮ τY = min{t ≥ 0 : ν+

• (0, t] × (0, ∞)
• > 0.

◮ τV = min{t ≥ 0 : ν−

• (0, t] × (0, ∞)
• > 0.

◮ τY and τV are independent. ◮ We want to know the distribution of

(i) the chronological time T1 corresponding to local time τY ∧ τV , (ii) plus the chronological elapsed time T2 in the “last excursion” beginning at local time τY ∧ τV before Y ∗ or V ∗ reaches zero.

◮ (i) is

T1 := ∞

ℓ=0

τY ∧τV

t=0

ℓν+

×(dt dℓ) +

∞

ℓ=0

τY ∧τV

t=0

ℓν−

×(dt dℓ). ◮ For (ii), we observe that the distribution of the length of the

“last excursion” is µ0(dℓ)/µ0((0, ∞)).

◮ Adapt Metzler again to compute the distribution of the

elapsed time T2 in the “last excursion,” conditioned on its length.

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Calculation of renewal time distribution

The moment-generating function of T1 + T2 is E

• e−α(T1+T2)

= ∞ ℓ e−αs p(s, ℓ) √ 2πℓ3 ds dℓ α 2 + ∞ e−αℓ p(ℓ) dℓ 2 √ 2πℓ3

• ,

where p(s, ℓ) is the conditional density in s of the elapsed time T2 given that the “last excursion” has length ℓ.

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0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0

Probability Density Function

x

Concluding remarks

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Φρ´ εδoς Παπαγγ´ ελou

Fredos Papangelou Emeritus Professor School of Mathematics University of Manchester

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Vorlesungen ¨ uber Maβtheorie von F. Papangelou

“Sei I ∗

n das nach beiden Seiten um seine zweifache L¨

ange erweiterte Intervall In.”

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Vorlesungen ¨ uber Maβtheorie von F. Papangelou

“Sei I ∗

n das nach beiden Seiten um seine zweifache L¨

ange erweiterte Intervall In.” English translation: “Let I ∗

n be the

toward-both-sides-for-its-doubled-length-extended interval In.”

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∆ηµ´ ητρης Mπερτσεκ´ ας

Dimitri Bertsekas Jerry Mcafee Professor Electrical and Computer Engineering Massachusetts Institute of Technology

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´ Iω´ αννης Kαρατζ´ ας

Ioannis Karatzas Eugene Higgins Professor of Applied Probability Department of Mathematics Columbia University

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“Every valley shall be exalted, and every mountain and hill shall be made low.” — Isaiah 40:4.

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“Every valley shall be exalted, and every mountain and hill shall be made low.” — Isaiah 40:4. “Martingales sprang fully armed from the forehead of Joseph Doob.” — Karatzas

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If you enounter Greeks bearing gifts....

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If you enounter Greeks bearing gifts.... welcome them with open arms.

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