Continuous time Markov Chains: construction and basic tools Conrado - - PowerPoint PPT Presentation

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Continuous time Markov Chains: construction and basic tools Conrado - - PowerPoint PPT Presentation

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Continuous time Markov Chains: construction and basic tools Conrado da Costa Department of Mathematical Sciences (Durham University) email: conrado.da-costa@durham.ac.uk September, 2020 1 / 21 Outline Construction Finite Infinite

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SLIDE 1

Continuous time Markov Chains: construction and basic tools

Conrado da Costa

Department of Mathematical Sciences (Durham University) email: conrado.da-costa@durham.ac.uk September, 2020

1 / 21

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SLIDE 2

Outline

Construction Finite Infinite Interacting Particle Systems Partial overview Tools Kolmogorov equations Martingales Tightness Martingale problems Panorama

2 / 21

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SLIDE 3

Finite state spaces

S = {x1, . . . , xn} Time: t

  • xi

S

  • x1
  • xj
  • xn

3 / 21

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SLIDE 4

Finite state spaces

S = {x1, . . . , xn} Time: t

  • xi

S

  • x1
  • xj
  • xn

pt

i,1

pt

i,j

∗ pt

i,n

3 / 21

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SLIDE 5

Finite state spaces

S = {x1, . . . , xn} Time: t

  • xi

S

  • x1
  • xj
  • xn

pt

i,1

pt

i,j

∗ pt

i,n

3 / 21

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SLIDE 6

Finite state spaces

S = {x1, . . . , xn} Time: t

  • xi

S

  • x1
  • xj
  • xn

pt

i,1

pt

i,j

∗ pt

i,n

Stochastic Matrix Pt         pt

1,1

. . . pt

1,j

. . . pt

1,n

. . . . . . . . . pt

i,1

. . . pt

i,j

. . . pt

i,n

. . . . . . . . . pt

n,1

. . . pt

n,j

. . . pt

n,n

       

3 / 21

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SLIDE 7

Finite state spaces

S = {x1, . . . , xn} Time: t

  • xi

S

  • x1
  • xj
  • xn

pt

i,1

pt

i,j

∗ pt

i,n

Stochastic Matrix Pt         pt

1,1

. . . pt

1,j

. . . pt

1,n

. . . . . . . . . pt

i,1

. . . pt

i,j

. . . pt

i,n

. . . . . . . . . pt

n,1

. . . pt

n,j

. . . pt

n,n

        ∀ i, j pt

i,j ≥ 0,

∀ i,

  • j pt

i,j = 1.

3 / 21

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SLIDE 8

Finite state spaces

S = {x1, . . . , xn} Time: t

  • xi

S

  • x1
  • xj
  • xn

pi,1 pi,j ∗ pi,n Stochastic Matrix P         p1,1 . . . p1,j . . . p1,n . . . . . . . . . pi,1 . . . pi,j . . . pi,n . . . . . . . . . pn,1 . . . pn,j . . . pn,n         ∀ i, j pi,j ≥ 0, ∀ i,

  • j pi,j = 1.

3 / 21

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SLIDE 9

Two constructions

Wait and jump Follow the clock

4 / 21

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SLIDE 10

Two constructions

Wait and jump Follow the clock E = {E k, k ∈ N} iid Exp(1) U = {Uk, k ∈ N} iid U(0,1)

4 / 21

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SLIDE 11

Two constructions

Wait and jump Follow the clock E = {E k, k ∈ N} iid Exp(1) U = {Uk, k ∈ N} iid U(0,1) rate at x: rx ∈ (0, ∞)

4 / 21

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SLIDE 12

Two constructions

Wait and jump Follow the clock E = {E k, k ∈ N} iid Exp(1) U = {Uk, k ∈ N} iid U(0,1) rate at x: rx ∈ (0, ∞) T0 := 0 X x

0 = x

4 / 21

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SLIDE 13

Two constructions

Wait and jump Follow the clock E = {E k, k ∈ N} iid Exp(1) U = {Uk, k ∈ N} iid U(0,1) rate at x: rx ∈ (0, ∞) T0 := 0 X x

0 = x

Tk+1 = Tk +

E k r(X x

k )

X x

k+1 = FP(X x k , Uk)

4 / 21

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SLIDE 14

Two constructions

Wait and jump Follow the clock E = {E k, k ∈ N} iid Exp(1) U = {Uk, k ∈ N} iid U(0,1) rate at x: rx ∈ (0, ∞) T0 := 0 X x

0 = x

Tk+1 = Tk +

E k r(X x

k )

X x

k+1 = FP(X x k , Uk)

Fx(E, U)(t) =

∞

  • k=0

1[Ti,Ti+1)(t)X x

i

4 / 21

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SLIDE 15

Two constructions

Wait and jump Follow the clock E = {E k, k ∈ N} iid Exp(1) independent E k

x,y ∼ Exp(rxy)

ˆ E = {E k

x,y, x, y ∈ E, k ∈ N}

U = {Uk, k ∈ N} iid U(0,1) rate at x: rx ∈ (0, ∞) T0 := 0 X x

0 = x

Tk+1 = Tk +

E k r(X x

k )

X x

k+1 = FP(X x k , Uk)

Fx(E, U)(t) =

∞

  • k=0

1[Ti,Ti+1)(t)X x

i

4 / 21

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SLIDE 16

Two constructions

Wait and jump Follow the clock E = {E k, k ∈ N} iid Exp(1) independent E k

x,y ∼ Exp(rxy)

ˆ E = {E k

x,y, x, y ∈ E, k ∈ N}

U = {Uk, k ∈ N} iid U(0,1) rate at x: rx ∈ (0, ∞) T0 := 0 X x

0 = x

T0 := 0 X x

0 = x

Tk+1 = Tk +

E k r(X x

k )

X x

k+1 = FP(X x k , Uk)

Fx(E, U)(t) =

∞

  • k=0

1[Ti,Ti+1)(t)X x

i

4 / 21

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SLIDE 17

Two constructions

Wait and jump Follow the clock E = {E k, k ∈ N} iid Exp(1) independent E k

x,y ∼ Exp(rxy)

ˆ E = {E k

x,y, x, y ∈ E, k ∈ N}

U = {Uk, k ∈ N} iid U(0,1) rate at x: rx ∈ (0, ∞) T0 := 0 X x

0 = x

T0 := 0 X x

0 = x

Tk+1 = Tk +

E k r(X x

k )

X x

k+1 ∈ {y, E k X x

k ,y ≤ E k

X x

k z, ∀z}

X x

k+1 = FP(X x k , Uk)

Tk+1 = Tk + E k

X x

k ,X x k+1

Fx(E, U)(t) =

∞

  • k=0

1[Ti,Ti+1)(t)X x

i

4 / 21

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SLIDE 18

Two constructions

Wait and jump Follow the clock E = {E k, k ∈ N} iid Exp(1) independent E k

x,y ∼ Exp(rxy)

ˆ E = {E k

x,y, x, y ∈ E, k ∈ N}

U = {Uk, k ∈ N} iid U(0,1) rate at x: rx ∈ (0, ∞) T0 := 0 X x

0 = x

T0 := 0 X x

0 = x

Tk+1 = Tk +

E k r(X x

k )

X x

k+1 ∈ {y, E k X x

k ,y ≤ E k

X x

k z, ∀z}

X x

k+1 = FP(X x k , Uk)

Tk+1 = Tk + E k

X x

k ,X x k+1

Fx(E, U)(t) =

∞

  • k=0

1[Ti,Ti+1)(t)X x

i

Gx(E)(t) =

∞

  • k=0

1[Ti,Ti+1)(t)X x

i

4 / 21

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SLIDE 19

Two constructions

Wait and jump Follow the clock E = {E k, k ∈ N} iid Exp(1) independent E k

x,y ∼ Exp(rxy)

ˆ E = {E k

x,y, x, y ∈ E, k ∈ N}

U = {Uk, k ∈ N} iid U(0,1) rate at x: rx ∈ (0, ∞) T0 := 0 X x

0 = x

T0 := 0 X x

0 = x

Tk+1 = Tk +

E k r(X x

k )

X x

k+1 ∈ {y, E k X x

k ,y ≤ E k

X x

k z, ∀z}

X x

k+1 = FP(X x k , Uk)

Tk+1 = Tk + E k

X x

k ,X x k+1

Fx(E, U)(t) =

∞

  • k=0

1[Ti,Ti+1)(t)X x

i

Gx(E)(t) =

∞

  • k=0

1[Ti,Ti+1)(t)X x

i

Let Px (ˆ Px) be the law on D([0, ∞), S) induced by Fx (Gx).

4 / 21

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SLIDE 20

Two constructions

Wait and jump Follow the clock E = {E k, k ∈ N} iid Exp(1) independent E k

x,y ∼ Exp(rxy)

ˆ E = {E k

x,y, x, y ∈ E, k ∈ N}

U = {Uk, k ∈ N} iid U(0,1) rate at x: rx ∈ (0, ∞) T0 := 0 X x

0 = x

T0 := 0 X x

0 = x

Tk+1 = Tk +

E k r(X x

k )

X x

k+1 ∈ {y, E k X x

k ,y ≤ E k

X x

k z, ∀z}

X x

k+1 = FP(X x k , Uk)

Tk+1 = Tk + E k

X x

k ,X x k+1

Fx(E, U)(t) =

∞

  • k=0

1[Ti,Ti+1)(t)X x

i

Gx(E)(t) =

∞

  • k=0

1[Ti,Ti+1)(t)X x

i

Let Px (ˆ Px) be the law on D([0, ∞), S) induced by Fx (Gx). If rxy = pxyrx for all x, y ∈ S then Px = ˆ Px

4 / 21

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SLIDE 21

graphical construction

Poisson point processes + colouring = Graphical construction

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SLIDE 22

graphical construction

Poisson point processes + colouring = Graphical construction Superpose colored Poisson processes with rate rxy

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SLIDE 23

graphical construction

Poisson point processes + colouring = Graphical construction Superpose colored Poisson processes with rate rxy start from a PPP r(x) color marks in x as y: with probability pxy

5 / 21

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SLIDE 24

graphical construction

Poisson point processes + colouring = Graphical construction Superpose colored Poisson processes with rate rxy start from a PPP r(x) color marks in x as y: with probability pxy These are all equivalent.

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SLIDE 25

graphical construction

Poisson point processes + colouring = Graphical construction

5 / 21

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SLIDE 26

An intrinsic construction

Operator on Cb(S): Lf (x) =

y∈S rxy[f (y) − f (x)].

6 / 21

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SLIDE 27

An intrinsic construction

Operator on Cb(S): Lf (x) =

y∈S rxy[f (y) − f (x)].

When S is finite, L defines a semi-group on Cb(S)

6 / 21

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SLIDE 28

An intrinsic construction

Operator on Cb(S): Lf (x) =

y∈S rxy[f (y) − f (x)].

When S is finite, L defines a semi-group on Cb(S) Stf = exp(tL)f = (tL)n n! f for f ∈ Cb(S)

6 / 21

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SLIDE 29

An intrinsic construction

Operator on Cb(S): Lf (x) =

y∈S rxy[f (y) − f (x)].

When S is finite, L defines a semi-group on Cb(S) Stf = exp(tL)f = (tL)n n! f for f ∈ Cb(S) By duality S∗

t acts on P(S):

6 / 21

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SLIDE 30

An intrinsic construction

Operator on Cb(S): Lf (x) =

y∈S rxy[f (y) − f (x)].

When S is finite, L defines a semi-group on Cb(S) Stf = exp(tL)f = (tL)n n! f for f ∈ Cb(S) By duality S∗

t acts on P(S):

S∗

t µ0 = µt is defined by

µt, f := µ0, Stf =

  • x

µ0(x)(Stf )(x).

6 / 21

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SLIDE 31

An intrinsic construction

Operator on Cb(S): Lf (x) =

y∈S rxy[f (y) − f (x)].

When S is finite, L defines a semi-group on Cb(S) Stf = exp(tL)f = (tL)n n! f for f ∈ Cb(S) By duality S∗

t acts on P(S):

S∗

t µ0 = µt is defined by

µt, f := µ0, Stf =

  • x

µ0(x)(Stf )(x). To complete the construction:

6 / 21

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SLIDE 32

An intrinsic construction

Operator on Cb(S): Lf (x) =

y∈S rxy[f (y) − f (x)].

When S is finite, L defines a semi-group on Cb(S) Stf = exp(tL)f = (tL)n n! f for f ∈ Cb(S) By duality S∗

t acts on P(S):

S∗

t µ0 = µt is defined by

µt, f := µ0, Stf =

  • x

µ0(x)(Stf )(x). To complete the construction: extension and regularization.

6 / 21

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SLIDE 33

Infinite state spaces

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SLIDE 34

Infinite state spaces

Challenges of infinite state spaces: supx r(x) = ∞.

7 / 21

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SLIDE 35

Infinite state spaces

Challenges of infinite state spaces: supx r(x) = ∞. Explosion, Implosions

7 / 21

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SLIDE 36

Infinite state spaces

Challenges of infinite state spaces: supx r(x) = ∞. Explosion, Implosions Solutions: Construction up to explosion, Construction via limits.

7 / 21

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SLIDE 37

Infinite state spaces

Challenges of infinite state spaces: supx r(x) = ∞. Explosion, Implosions Solutions: Construction up to explosion, Construction via limits. This is the case of Interacting Particle Systems (IPS)

7 / 21

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SLIDE 38

Interacting Particle Systems

S = X V , η ∈ S, x, y ∈ V

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SLIDE 39

Interacting Particle Systems

S = X V , η ∈ S, x, y ∈ V Simple cases {0, 1}N - not infinite, but very big and sparse.

8 / 21

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SLIDE 40

Interacting Particle Systems

S = X V , η ∈ S, x, y ∈ V Simple cases {0, 1}N - not infinite, but very big and sparse. Local interaction. Local transformation maps: Γm : S → S, m ∈ M.

8 / 21

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SLIDE 41

Interacting Particle Systems

S = X V , η ∈ S, x, y ∈ V Simple cases {0, 1}N - not infinite, but very big and sparse. Local interaction. Local transformation maps: Γm : S → S, m ∈ M. rates {r(η, m), m ∈ S}.

8 / 21

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SLIDE 42

Interacting Particle Systems

S = X V , η ∈ S, x, y ∈ V Simple cases {0, 1}N - not infinite, but very big and sparse. Local interaction. Local transformation maps: Γm : S → S, m ∈ M. rates {r(η, m), m ∈ S}. Formal generators Lf (η) =

  • m

r(η, m)[f (Γmη) − f (η)]

8 / 21

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SLIDE 43

Interacting Particle Systems

S = X V , η ∈ S, x, y ∈ V Simple cases {0, 1}N - not infinite, but very big and sparse. Local interaction. Local transformation maps: Γm : S → S, m ∈ M. rates {r(η, m), m ∈ S}. Formal generators Lf (η) =

  • m

r(η, m)[f (Γmη) − f (η)] V = Z:

8 / 21

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SLIDE 44

Interacting Particle Systems

S = X V , η ∈ S, x, y ∈ V Simple cases {0, 1}N - not infinite, but very big and sparse. Local interaction. Local transformation maps: Γm : S → S, m ∈ M. rates {r(η, m), m ∈ S}. Formal generators Lf (η) =

  • m

r(η, m)[f (Γmη) − f (η)] V = Z: uncountable state space

8 / 21

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SLIDE 45

Overview

What is the object of interest here?

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SLIDE 46

Overview

What is the object of interest here?

  • d

dtµt

= L∗µt µ0 = δx

9 / 21

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SLIDE 47

Overview

What is the object of interest here?

  • d

dtµt

= L∗µt µ0 = δx

Goal: To understand the behavior of µt in the relevant scales.

9 / 21

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SLIDE 48

Questions

The type of questions we look:

10 / 21

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SLIDE 49

Questions

The type of questions we look: ◮ evolution of St(f ) for test functions f .

10 / 21

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SLIDE 50

Questions

The type of questions we look: ◮ evolution of St(f ) for test functions f . ◮ invariant measures

10 / 21

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SLIDE 51

Questions

The type of questions we look: ◮ evolution of St(f ) for test functions f . ◮ invariant measures ◮ convergence

10 / 21

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SLIDE 52

Questions

The type of questions we look: ◮ evolution of St(f ) for test functions f . ◮ invariant measures ◮ convergence ◮ effects of space-time rescaling for Sn

·

10 / 21

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SLIDE 53

Questions

The type of questions we look: ◮ evolution of St(f ) for test functions f . ◮ invariant measures ◮ convergence ◮ effects of space-time rescaling for Sn

·

◮ An attempt of explaining physical phenomena:

10 / 21

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SLIDE 54

Questions

The type of questions we look: ◮ evolution of St(f ) for test functions f . ◮ invariant measures ◮ convergence ◮ effects of space-time rescaling for Sn

·

◮ An attempt of explaining physical phenomena:

◮ fluid equation

10 / 21

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SLIDE 55

Questions

The type of questions we look: ◮ evolution of St(f ) for test functions f . ◮ invariant measures ◮ convergence ◮ effects of space-time rescaling for Sn

·

◮ An attempt of explaining physical phenomena:

◮ fluid equation ◮ state of matter

10 / 21

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SLIDE 56

Questions

The type of questions we look: ◮ evolution of St(f ) for test functions f . ◮ invariant measures ◮ convergence ◮ effects of space-time rescaling for Sn

·

◮ An attempt of explaining physical phenomena:

◮ fluid equation ◮ state of matter ◮ phase transition

10 / 21

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SLIDE 57

Tools

11 / 21

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SLIDE 58

Tools

◮ Kolmogorov equations

11 / 21

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SLIDE 59

Tools

◮ Kolmogorov equations ◮ Dynkin Martingales

11 / 21

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SLIDE 60

Tools

◮ Kolmogorov equations ◮ Dynkin Martingales ◮ Tightness criteria

11 / 21

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SLIDE 61

Tools

◮ Kolmogorov equations ◮ Dynkin Martingales ◮ Tightness criteria ◮ Martingale problems

11 / 21

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SLIDE 62

Tools

◮ Kolmogorov equations ◮ Dynkin Martingales ◮ Tightness criteria ◮ Martingale problems ◮ Limits of IPS

11 / 21

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SLIDE 63

Forward Kolmogorov equations

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SLIDE 64

Forward Kolmogorov equations

  • ∂tP(s, x; t, •) = L∗P(s, x; t, •)

limt↓s P(s, x; t, •) = δx(•)

12 / 21

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SLIDE 65

Forward Kolmogorov equations

  • ∂tP(s, x; t, •) = L∗P(s, x; t, •)

limt↓s P(s, x; t, •) = δx(•) Let v(t, x) := E[ϕ(Xt)|Xs = x] =

  • P(s, x; t, dy)ϕ(y)

12 / 21

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SLIDE 66

Forward Kolmogorov equations

  • ∂tP(s, x; t, •) = L∗P(s, x; t, •)

limt↓s P(s, x; t, •) = δx(•) Let v(t, x) := E[ϕ(Xt)|Xs = x] =

  • P(s, x; t, dy)ϕ(y)

∂tv(t, x) =

  • ∂tP(s, x; t, dy)ϕ(y)

12 / 21

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SLIDE 67

Forward Kolmogorov equations

  • ∂tP(s, x; t, •) = L∗P(s, x; t, •)

limt↓s P(s, x; t, •) = δx(•) Let v(t, x) := E[ϕ(Xt)|Xs = x] =

  • P(s, x; t, dy)ϕ(y)

∂tv(t, x) =

  • ∂tP(s, x; t, dy)ϕ(y)

∂tv(t, x) = lim

h

  • P(s, x; t, dy)h−1
  • P(t, y; t + h, dz)(ϕ(z) − ϕ(y))

12 / 21

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SLIDE 68

Forward Kolmogorov equations

  • ∂tP(s, x; t, •) = L∗P(s, x; t, •)

limt↓s P(s, x; t, •) = δx(•) Let v(t, x) := E[ϕ(Xt)|Xs = x] =

  • P(s, x; t, dy)ϕ(y)

∂tv(t, x) =

  • ∂tP(s, x; t, dy)ϕ(y)

∂tv(t, x) = lim

h

  • P(s, x; t, dy)h−1
  • P(t, y; t + h, dz)(ϕ(z) − ϕ(y))

=

  • P(s, x; t, dy)Lϕ(y)

12 / 21

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SLIDE 69

Forward Kolmogorov equations

  • ∂tP(s, x; t, •) = L∗P(s, x; t, •)

limt↓s P(s, x; t, •) = δx(•) Let v(t, x) := E[ϕ(Xt)|Xs = x] =

  • P(s, x; t, dy)ϕ(y)

∂tv(t, x) =

  • ∂tP(s, x; t, dy)ϕ(y)

∂tv(t, x) = lim

h

  • P(s, x; t, dy)h−1
  • P(t, y; t + h, dz)(ϕ(z) − ϕ(y))

=

  • P(s, x; t, dy)Lϕ(y) =
  • L∗P(s, x; t, dy)ϕ(y)

12 / 21

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SLIDE 70

Backward Kolmogorov equations

  • −∂sP(s, x; t, •) = LP(s, x; t, •)

lims↑t P(s, x; t, •) = δx(•)

13 / 21

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SLIDE 71

Backward Kolmogorov equations

  • −∂sP(s, x; t, •) = LP(s, x; t, •)

lims↑t P(s, x; t, •) = δx(•) Let w(s, x) := E[ϕ(Xt)|Xs = x] =

  • P(s, x; t, dy)ϕ(y)

13 / 21

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SLIDE 72

Backward Kolmogorov equations

  • −∂sP(s, x; t, •) = LP(s, x; t, •)

lims↑t P(s, x; t, •) = δx(•) Let w(s, x) := E[ϕ(Xt)|Xs = x] =

  • P(s, x; t, dy)ϕ(y)

∂sw(s, x) =

  • −∂sP(s, x; t, dy)ϕ(y)

13 / 21

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SLIDE 73

Backward Kolmogorov equations

  • −∂sP(s, x; t, •) = LP(s, x; t, •)

lims↑t P(s, x; t, •) = δx(•) Let w(s, x) := E[ϕ(Xt)|Xs = x] =

  • P(s, x; t, dy)ϕ(y)

∂sw(s, x) =

  • −∂sP(s, x; t, dy)ϕ(y)

− ∂sw(s, x) = lim

h h−1

  • P(s − h, x; s, dy)[w(s, y) − w(s, x)]

13 / 21

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SLIDE 74

Backward Kolmogorov equations

  • −∂sP(s, x; t, •) = LP(s, x; t, •)

lims↑t P(s, x; t, •) = δx(•) Let w(s, x) := E[ϕ(Xt)|Xs = x] =

  • P(s, x; t, dy)ϕ(y)

∂sw(s, x) =

  • −∂sP(s, x; t, dy)ϕ(y)

− ∂sw(s, x) = lim

h h−1

  • P(s − h, x; s, dy)[w(s, y) − w(s, x)]

= lim

h h−1[Lw(s − h, x)h + o(h)]

13 / 21

slide-75
SLIDE 75

Backward Kolmogorov equations

  • −∂sP(s, x; t, •) = LP(s, x; t, •)

lims↑t P(s, x; t, •) = δx(•) Let w(s, x) := E[ϕ(Xt)|Xs = x] =

  • P(s, x; t, dy)ϕ(y)

∂sw(s, x) =

  • −∂sP(s, x; t, dy)ϕ(y)

− ∂sw(s, x) = lim

h h−1

  • P(s − h, x; s, dy)[w(s, y) − w(s, x)]

= lim

h h−1[Lw(s − h, x)h + o(h)] = Lw(s, x)

13 / 21

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SLIDE 76

Dynkin Martingales

If F : R+ × S → R sup

(s,x)

  • ∂j

sF(s, x)

  • < C

14 / 21

slide-77
SLIDE 77

Dynkin Martingales

If F : R+ × S → R sup

(s,x)

  • ∂j

sF(s, x)

  • < C

then

  • MF(t) := F(t, Xt) − F(0, X0) −

t

0 (∂s + L)F(s, Xs) ds

NF(t) := [MF(t)]2 − t

0 (QF)(s, Xs) ds

are martingales,

14 / 21

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SLIDE 78

Dynkin Martingales

If F : R+ × S → R sup

(s,x)

  • ∂j

sF(s, x)

  • < C

then

  • MF(t) := F(t, Xt) − F(0, X0) −

t

0 (∂s + L)F(s, Xs) ds

NF(t) := [MF(t)]2 − t

0 (QF)(s, Xs) ds

are martingales, where QF(s, x) := LF 2(s, x)−2F(s, x)LF(s, x)

14 / 21

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SLIDE 79

Dynkin Martingales

If F : R+ × S → R sup

(s,x)

  • ∂j

sF(s, x)

  • < C

then

  • MF(t) := F(t, Xt) − F(0, X0) −

t

0 (∂s + L)F(s, Xs) ds

NF(t) := [MF(t)]2 − t

0 (QF)(s, Xs) ds

are martingales, where QF(s, x) := LF 2(s, x)−2F(s, x)LF(s, x) =

  • y

rxy[F(s, y)−F(s, x)]2

14 / 21

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SLIDE 80

Proof (Dynkin)

MF(t) − MF(s) = F(t, Xt) − F(s, Xs) − t

s

(∂r + L)F(r, Xr) dr

15 / 21

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SLIDE 81

Proof (Dynkin)

MF(t) − MF(s) = F(t, Xt) − F(s, Xs) − t

s

(∂r + L)F(r, Xr) dr E[MF(t) − MF(s)|Fs] = 0

15 / 21

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SLIDE 82

Proof (Dynkin)

MF(t) − MF(s) = F(t, Xt) − F(s, Xs) − t

s

(∂r + L)F(r, Xr) dr E[MF(t) − MF(s)|Fs] = 0 ϕ(r) := E[MF(r) − MF(s)|Fs] ϕ(s) = 0 ϕ′(r) = 0

15 / 21

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SLIDE 83

Proof (Dynkin)

MF(t) − MF(s) = F(t, Xt) − F(s, Xs) − t

s

(∂r + L)F(r, Xr) dr E[MF(t) − MF(s)|Fs] = 0 ϕ(r) := E[MF(r) − MF(s)|Fs] ϕ(s) = 0 ϕ′(r) = 0

h−1E[ r+h

r

(∂u + L)F(u, Xu) du|Fs] → E[(∂r + L)F(r, Xr)|Fs]

15 / 21

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SLIDE 84

Proof (Dynkin)

MF(t) − MF(s) = F(t, Xt) − F(s, Xs) − t

s

(∂r + L)F(r, Xr) dr E[MF(t) − MF(s)|Fs] = 0 ϕ(r) := E[MF(r) − MF(s)|Fs] ϕ(s) = 0 ϕ′(r) = 0

h−1E[ r+h

r

(∂u + L)F(u, Xu) du|Fs] → E[(∂r + L)F(r, Xr)|Fs] h−1E[F(r + h, Xr+h) − F(r, Xr)|Fs] =

15 / 21

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SLIDE 85

Proof (Dynkin)

MF(t) − MF(s) = F(t, Xt) − F(s, Xs) − t

s

(∂r + L)F(r, Xr) dr E[MF(t) − MF(s)|Fs] = 0 ϕ(r) := E[MF(r) − MF(s)|Fs] ϕ(s) = 0 ϕ′(r) = 0

h−1E[ r+h

r

(∂u + L)F(u, Xu) du|Fs] → E[(∂r + L)F(r, Xr)|Fs] h−1E[F(r + h, Xr+h) − F(r, Xr)|Fs] = h−1E[F(r + h, Xr+h) − F(r, Xr)|Fs] + h−1E[F(r, Xr+h) − F(r, Xr)|Fs]

15 / 21

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SLIDE 86

Proof (Dynkin)

MF(t) − MF(s) = F(t, Xt) − F(s, Xs) − t

s

(∂r + L)F(r, Xr) dr E[MF(t) − MF(s)|Fs] = 0 ϕ(r) := E[MF(r) − MF(s)|Fs] ϕ(s) = 0 ϕ′(r) = 0

h−1E[ r+h

r

(∂u + L)F(u, Xu) du|Fs] → E[(∂r + L)F(r, Xr)|Fs] h−1E[F(r + h, Xr+h) − F(r, Xr)|Fs] = h−1E[F(r + h, Xr+h) − F(r, Xr)|Fs] + h−1E[F(r, Xr+h) − F(r, Xr)|Fs] → E[∂rF(r, Xr)|Fs] + E[LF(r, Xr)|Fs]

15 / 21

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SLIDE 87

Proof (Dynkin)

NF (t) := [MF (t)]2 − t (QF)(s, Xs) ds MF (t) = F(t, Xt) − F(0, X0) − t (∂s + L)F(s, Xs) ds 16 / 21

slide-88
SLIDE 88

Proof (Dynkin)

NF (t) := [MF (t)]2 − t (QF)(s, Xs) ds MF (t) = F(t, Xt) − F(0, X0) − t (∂s + L)F(s, Xs) ds F 2(t, Xt) ∼ t (∂s + L)F 2(s, Xs) ds MF

0 (t) := MF (t) + F(0, X0) = F(t, Xt) −

t (∂s + L)F(s, Xs) ds 16 / 21

slide-89
SLIDE 89

Proof (Dynkin)

NF (t) := [MF (t)]2 − t (QF)(s, Xs) ds MF (t) = F(t, Xt) − F(0, X0) − t (∂s + L)F(s, Xs) ds F 2(t, Xt) ∼ t (∂s + L)F 2(s, Xs) ds MF

0 (t) := MF (t) + F(0, X0) = F(t, Xt) −

t (∂s + L)F(s, Xs) ds Integration by parts: GtMt = t

0 GsdMs +

t

0 MsdGs

16 / 21

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SLIDE 90

Proof (Dynkin)

NF (t) := [MF (t)]2 − t (QF)(s, Xs) ds MF (t) = F(t, Xt) − F(0, X0) − t (∂s + L)F(s, Xs) ds F 2(t, Xt) ∼ t (∂s + L)F 2(s, Xs) ds MF

0 (t) := MF (t) + F(0, X0) = F(t, Xt) −

t (∂s + L)F(s, Xs) ds Integration by parts: GtMt = t

0 GsdMs +

t

0 MsdGs

MF

0 (t)

t (∂s + L)F(s, Xs) ds ∼ t

  • F(s, Xs) −

s (∂r + L)F(r, Xr ) dr

  • (∂s + L)F(s, Xs) ds

16 / 21

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SLIDE 91

Proof (Dynkin)

NF (t) := [MF (t)]2 − t (QF)(s, Xs) ds MF (t) = F(t, Xt) − F(0, X0) − t (∂s + L)F(s, Xs) ds F 2(t, Xt) ∼ t (∂s + L)F 2(s, Xs) ds MF

0 (t) := MF (t) + F(0, X0) = F(t, Xt) −

t (∂s + L)F(s, Xs) ds Integration by parts: GtMt = t

0 GsdMs +

t

0 MsdGs

MF

0 (t)

t (∂s + L)F(s, Xs) ds ∼ t

  • F(s, Xs) −

s (∂r + L)F(r, Xr ) dr

  • (∂s + L)F(s, Xs) ds

[MF (t)]2 = F 2(t, Xt) − 2F(t, Xt) t (∂s + L)F(s, Xs) ds + t (∂s + L)F(s, Xs) ds 2

16 / 21

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SLIDE 92

Proof (Dynkin)

NF (t) := [MF (t)]2 − t (QF)(s, Xs) ds MF (t) = F(t, Xt) − F(0, X0) − t (∂s + L)F(s, Xs) ds F 2(t, Xt) ∼ t (∂s + L)F 2(s, Xs) ds MF

0 (t) := MF (t) + F(0, X0) = F(t, Xt) −

t (∂s + L)F(s, Xs) ds Integration by parts: GtMt = t

0 GsdMs +

t

0 MsdGs

MF

0 (t)

t (∂s + L)F(s, Xs) ds ∼ t

  • F(s, Xs) −

s (∂r + L)F(r, Xr ) dr

  • (∂s + L)F(s, Xs) ds

[MF (t)]2 = F 2(t, Xt) − 2F(t, Xt) t (∂s + L)F(s, Xs) ds + t (∂s + L)F(s, Xs) ds 2 ∼ t (∂s + L)F 2(s, Xs) ds − 2MF

0 (t)

t (∂s + L)F(s, Xs) ds − t (∂s + L)F(s, Xs) ds 2

16 / 21

slide-93
SLIDE 93

Proof (Dynkin)

NF (t) := [MF (t)]2 − t (QF)(s, Xs) ds MF (t) = F(t, Xt) − F(0, X0) − t (∂s + L)F(s, Xs) ds F 2(t, Xt) ∼ t (∂s + L)F 2(s, Xs) ds MF

0 (t) := MF (t) + F(0, X0) = F(t, Xt) −

t (∂s + L)F(s, Xs) ds Integration by parts: GtMt = t

0 GsdMs +

t

0 MsdGs

MF

0 (t)

t (∂s + L)F(s, Xs) ds ∼ t

  • F(s, Xs) −

s (∂r + L)F(r, Xr ) dr

  • (∂s + L)F(s, Xs) ds

[MF (t)]2 = F 2(t, Xt) − 2F(t, Xt) t (∂s + L)F(s, Xs) ds + t (∂s + L)F(s, Xs) ds 2 ∼ t (∂s + L)F 2(s, Xs) ds − 2MF

0 (t)

t (∂s + L)F(s, Xs) ds − t (∂s + L)F(s, Xs) ds 2 ∼ t (∂s + L)F 2(s, Xs) ds − 2 t F(s, Xs)(∂s + L)F(s, Xs) ds + 2 t s (∂r + L)F(r, Xr) dr(∂s + L)F(s, Xs) ds − t (∂s + L)F(s, Xs) ds 2

16 / 21

slide-94
SLIDE 94

Proof (Dynkin)

NF (t) := [MF (t)]2 − t (QF)(s, Xs) ds MF (t) = F(t, Xt) − F(0, X0) − t (∂s + L)F(s, Xs) ds F 2(t, Xt) ∼ t (∂s + L)F 2(s, Xs) ds MF

0 (t) := MF (t) + F(0, X0) = F(t, Xt) −

t (∂s + L)F(s, Xs) ds Integration by parts: GtMt = t

0 GsdMs +

t

0 MsdGs

MF

0 (t)

t (∂s + L)F(s, Xs) ds ∼ t

  • F(s, Xs) −

s (∂r + L)F(r, Xr ) dr

  • (∂s + L)F(s, Xs) ds

[MF (t)]2 = F 2(t, Xt) − 2F(t, Xt) t (∂s + L)F(s, Xs) ds + t (∂s + L)F(s, Xs) ds 2 ∼ t (∂s + L)F 2(s, Xs) ds − 2MF

0 (t)

t (∂s + L)F(s, Xs) ds − t (∂s + L)F(s, Xs) ds 2 ∼ t (∂s + L)F 2(s, Xs) ds − 2 t F(s, Xs)(∂s + L)F(s, Xs) ds

16 / 21

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SLIDE 95

Proof (Dynkin)

NF (t) := [MF (t)]2 − t (QF)(s, Xs) ds MF (t) = F(t, Xt) − F(0, X0) − t (∂s + L)F(s, Xs) ds F 2(t, Xt) ∼ t (∂s + L)F 2(s, Xs) ds MF

0 (t) := MF (t) + F(0, X0) = F(t, Xt) −

t (∂s + L)F(s, Xs) ds Integration by parts: GtMt = t

0 GsdMs +

t

0 MsdGs

MF

0 (t)

t (∂s + L)F(s, Xs) ds ∼ t

  • F(s, Xs) −

s (∂r + L)F(r, Xr ) dr

  • (∂s + L)F(s, Xs) ds

[MF (t)]2 = F 2(t, Xt) − 2F(t, Xt) t (∂s + L)F(s, Xs) ds + t (∂s + L)F(s, Xs) ds 2 ∼ t (∂s + L)F 2(s, Xs) ds − 2MF

0 (t)

t (∂s + L)F(s, Xs) ds − t (∂s + L)F(s, Xs) ds 2 ∼ t (∂s + L)F 2(s, Xs) ds − 2 t F(s, Xs)(∂s + L)F(s, Xs) ds ∼ t (QF)(s, Xs) ds

16 / 21

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SLIDE 96

Tightness criterion

Family of processes (X N

· , N ∈ N)

17 / 21

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SLIDE 97

Tightness criterion

Family of processes (X N

· , N ∈ N) is tight if

∀ε > 0, ∃K(ε); inf

N P(X N ∈ K(ε)) > 1 − ε.

17 / 21

slide-98
SLIDE 98

Tightness criterion

Family of processes (X N

· , N ∈ N) is tight if

∀ε > 0, ∃K(ε); inf

N P(X N ∈ K(ε)) > 1 − ε.

Aldous’s criterion.

17 / 21

slide-99
SLIDE 99

Tightness criterion

Family of processes (X N

· , N ∈ N) is tight if

∀ε > 0, ∃K(ε); inf

N P(X N ∈ K(ε)) > 1 − ε.

Aldous’s criterion. Probabilistic version of Arzel´ a-Ascoli:

17 / 21

slide-100
SLIDE 100

Tightness criterion

Family of processes (X N

· , N ∈ N) is tight if

∀ε > 0, ∃K(ε); inf

N P(X N ∈ K(ε)) > 1 − ε.

Aldous’s criterion. Probabilistic version of Arzel´ a-Ascoli: 1) sup

N

P[X N

t

/ ∈ K(t, ǫ)] ≤ ǫ 2) lim

γ→0 lim sup N

P[X N : ω′(X N, γ) > ǫ] = 0

17 / 21

slide-101
SLIDE 101

Tightness criterion

Family of processes (X N

· , N ∈ N) is tight if

∀ε > 0, ∃K(ε); inf

N P(X N ∈ K(ε)) > 1 − ε.

Aldous’s criterion. Probabilistic version of Arzel´ a-Ascoli: 1) sup

N

P[X N

t

/ ∈ K(t, ǫ)] ≤ ǫ 2) lim

γ→0 lim sup N

P[X N : ω′(X N, γ) > ǫ] = 0 Then there is a subsequence {Nk, k ∈ N} and X ∗

· ∈ D s.t.

X Nk

·

→ X ∗

·

17 / 21

slide-102
SLIDE 102

Martingale problems

18 / 21

slide-103
SLIDE 103

Martingale problems

A continuous adapted process (Mt, t ≥ 0) is a d-dimensional Brownian Motion if and only if f (Mt) − f (M0) − 1 2 t ∆f (Ms) ds is a local martingale for all f ∈ C 2(R).

18 / 21

slide-104
SLIDE 104

Martingale problems

A continuous adapted process (Mt, t ≥ 0) is a d-dimensional Brownian Motion if and only if f (Mt) − f (M0) − 1 2 t ∆f (Ms) ds is a local martingale for all f ∈ C 2(R). Theorem:(L´ evy) If (Mt, t ≥ 0) is a continuous real valued local martingale with Mk, Mjt = tδk,t then Mt is a Brownian motion.

18 / 21

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SLIDE 105

Proof (L´ evy)

Goal: to show that Mt − Ms ∼ N(0, t − s) and Mt − Ms ⊥ Fs

19 / 21

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SLIDE 106

Proof (L´ evy)

Goal: to show that Mt − Ms ∼ N(0, t − s) and Mt − Ms ⊥ Fs E[eiλ(Mt−Ms)1A] = P(A)e− λ2

2 (t−s)

∀A ∈ Fs.

19 / 21

slide-107
SLIDE 107

Proof (L´ evy)

Goal: to show that Mt − Ms ∼ N(0, t − s) and Mt − Ms ⊥ Fs E[eiλ(Mt−Ms)1A] = P(A)e− λ2

2 (t−s)

∀A ∈ Fs. By Itˆ

  • ’s formula

eiλMt = eiλMs + iλ t

s

eiλMu dMu − λ2 2 t

s

eiλMu du.

19 / 21

slide-108
SLIDE 108

Proof (L´ evy)

Goal: to show that Mt − Ms ∼ N(0, t − s) and Mt − Ms ⊥ Fs E[eiλ(Mt−Ms)1A] = P(A)e− λ2

2 (t−s)

∀A ∈ Fs. By Itˆ

  • ’s formula

eiλMt = eiλMs + iλ t

s

eiλMu dMu − λ2 2 t

s

eiλMu du. Multiply by e−iλMs1A, and take expectation E[eiλ(Mt−Ms)1A] = P(A) − λ2 2 t

s

E[eiλ(Mu−Ms)1A] du.

19 / 21

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SLIDE 109

Proof (L´ evy)

Goal: to show that Mt − Ms ∼ N(0, t − s) and Mt − Ms ⊥ Fs E[eiλ(Mt−Ms)1A] = P(A)e− λ2

2 (t−s)

∀A ∈ Fs. By Itˆ

  • ’s formula

eiλMt = eiλMs + iλ t

s

eiλMu dMu − λ2 2 t

s

eiλMu du. Multiply by e−iλMs1A, and take expectation E[eiλ(Mt−Ms)1A] = P(A) − λ2 2 t

s

E[eiλ(Mu−Ms)1A] du.

19 / 21

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SLIDE 110

Panorama

Particle systems SDE’s

20 / 21

slide-111
SLIDE 111

Panorama

Particle systems Martingale Problems SDE’s

20 / 21

slide-112
SLIDE 112

Panorama

Particle systems X n

t − X n 0 −

t

0 Ln(X n s ) ds = Mn t

Martingale Problems SDE’s

20 / 21

slide-113
SLIDE 113

Panorama

Particle systems Xt − X0 − t

0 b(Xs) ds = Mt

Martingale Problems SDE’s

20 / 21

slide-114
SLIDE 114

Panorama

Particle systems Xt − X0 − t

0 b(Xs) ds = Mt

Mt = t

0 σ(Xs) dBs

Martingale Problems SDE’s

20 / 21

slide-115
SLIDE 115

Panorama

Particle systems Xt − X0 − t

0 b(Xs) ds = Mt

Mt = t

0 σ(Xs) dBs

Xt = X0 + t

0 b(Xs) ds +

t

0 σ(Xs) dBs

Martingale Problems SDE’s

20 / 21

slide-116
SLIDE 116

Thank you!

E[eiλ(Mt−Ms)1A] = P(A)e−λ2/2(t−s)

Particle systems Xt − X0 − t

0 b(Xs) ds = Mt

Mt = t

0 σ(Xs) dBs

Xt = X0 + t

0 b(Xs) ds +

t

0 σ(Xs) dBs

Martingale Problems SDE’s

21 / 21