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39
18.175: Lecture 26 Last lecture Scott Sheffield MIT 18.175 Lecture 26

Transcript of 18.175: Lecture 26 .1in Last lecture - Mathematicssheffield/2016175/Lecture26.pdf · 18.175:...

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18.175: Lecture 26

Last lecture

Scott Sheffield

MIT

18.175 Lecture 26

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Outline

Recollections

Strong Markov property

18.175 Lecture 26

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Outline

Recollections

Strong Markov property

18.175 Lecture 26

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More σ-algebra thoughts

I Write Fos = σ(Br : r ≤ s).

I Write F+s = ∩t>sFo

t

I Note right continuity: ∩t>sF+t = F+

s .

I F+s allows an “infinitesimal peek at future”

18.175 Lecture 26

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More σ-algebra thoughts

I Write Fos = σ(Br : r ≤ s).

I Write F+s = ∩t>sFo

t

I Note right continuity: ∩t>sF+t = F+

s .

I F+s allows an “infinitesimal peek at future”

18.175 Lecture 26

Page 6: 18.175: Lecture 26 .1in Last lecture - Mathematicssheffield/2016175/Lecture26.pdf · 18.175: Lecture 26 Last lecture Scott She eld MIT 18.175 Lecture 26. Outline Recollections Strong

More σ-algebra thoughts

I Write Fos = σ(Br : r ≤ s).

I Write F+s = ∩t>sFo

t

I Note right continuity: ∩t>sF+t = F+

s .

I F+s allows an “infinitesimal peek at future”

18.175 Lecture 26

Page 7: 18.175: Lecture 26 .1in Last lecture - Mathematicssheffield/2016175/Lecture26.pdf · 18.175: Lecture 26 Last lecture Scott She eld MIT 18.175 Lecture 26. Outline Recollections Strong

More σ-algebra thoughts

I Write Fos = σ(Br : r ≤ s).

I Write F+s = ∩t>sFo

t

I Note right continuity: ∩t>sF+t = F+

s .

I F+s allows an “infinitesimal peek at future”

18.175 Lecture 26

Page 8: 18.175: Lecture 26 .1in Last lecture - Mathematicssheffield/2016175/Lecture26.pdf · 18.175: Lecture 26 Last lecture Scott She eld MIT 18.175 Lecture 26. Outline Recollections Strong

Looking ahead

I Expectation equivalence theorem If Z is bounded andmeasurable then for all s ≥ 0 and x ∈ Rd have

Ex(Z |F+s ) = Ex(Z |Fo

s ).

I Proof idea: Consider case that Z =∑m

i=1 fm(B(tm)) and thefm are bounded and measurable. Kind of obvious in this case.Then use same measure theory as in Markov property proof toextend general Z .

I Observe: If Z ∈ F+s then Z = Ex(Z |Fo

s ). Conclude that F+s

and Fos agree up to null sets.

18.175 Lecture 26

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Looking ahead

I Expectation equivalence theorem If Z is bounded andmeasurable then for all s ≥ 0 and x ∈ Rd have

Ex(Z |F+s ) = Ex(Z |Fo

s ).

I Proof idea: Consider case that Z =∑m

i=1 fm(B(tm)) and thefm are bounded and measurable. Kind of obvious in this case.Then use same measure theory as in Markov property proof toextend general Z .

I Observe: If Z ∈ F+s then Z = Ex(Z |Fo

s ). Conclude that F+s

and Fos agree up to null sets.

18.175 Lecture 26

Page 10: 18.175: Lecture 26 .1in Last lecture - Mathematicssheffield/2016175/Lecture26.pdf · 18.175: Lecture 26 Last lecture Scott She eld MIT 18.175 Lecture 26. Outline Recollections Strong

Looking ahead

I Expectation equivalence theorem If Z is bounded andmeasurable then for all s ≥ 0 and x ∈ Rd have

Ex(Z |F+s ) = Ex(Z |Fo

s ).

I Proof idea: Consider case that Z =∑m

i=1 fm(B(tm)) and thefm are bounded and measurable. Kind of obvious in this case.Then use same measure theory as in Markov property proof toextend general Z .

I Observe: If Z ∈ F+s then Z = Ex(Z |Fo

s ). Conclude that F+s

and Fos agree up to null sets.

18.175 Lecture 26

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Blumenthal’s 0-1 law

I If A ∈ F+0 , then P(A) ∈ {0, 1} (if P is probability law for

Brownian motion started at fixed value x at time 0).

I There’s nothing you can learn from infinitesimal neighborhoodof future.

I Proof: If we have A ∈ F+0 , then previous theorem implies

1A = Ex(1A|F+0 ) = Ex(1A|Fo

0 ) = Px(A) Pxa.s.

18.175 Lecture 26

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Blumenthal’s 0-1 law

I If A ∈ F+0 , then P(A) ∈ {0, 1} (if P is probability law for

Brownian motion started at fixed value x at time 0).

I There’s nothing you can learn from infinitesimal neighborhoodof future.

I Proof: If we have A ∈ F+0 , then previous theorem implies

1A = Ex(1A|F+0 ) = Ex(1A|Fo

0 ) = Px(A) Pxa.s.

18.175 Lecture 26

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Blumenthal’s 0-1 law

I If A ∈ F+0 , then P(A) ∈ {0, 1} (if P is probability law for

Brownian motion started at fixed value x at time 0).

I There’s nothing you can learn from infinitesimal neighborhoodof future.

I Proof: If we have A ∈ F+0 , then previous theorem implies

1A = Ex(1A|F+0 ) = Ex(1A|Fo

0 ) = Px(A) Pxa.s.

18.175 Lecture 26

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Markov property

I If s ≥ 0 and Y is bounded and C-measurable, then for allx ∈ Rd , we have

Ex(Y ◦ θs |F+s ) = EBsY ,

where the RHS is function φ(x) = ExY evaluated at x = Bs .

I Proof idea: First establish this for some simple functions Y(depending on finitely many time values) and then usemeasure theory (monotone class theorem) to extend togeneral case.

18.175 Lecture 26

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Markov property

I If s ≥ 0 and Y is bounded and C-measurable, then for allx ∈ Rd , we have

Ex(Y ◦ θs |F+s ) = EBsY ,

where the RHS is function φ(x) = ExY evaluated at x = Bs .

I Proof idea: First establish this for some simple functions Y(depending on finitely many time values) and then usemeasure theory (monotone class theorem) to extend togeneral case.

18.175 Lecture 26

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More observations

I If τ = inf{t ≥ 0 : Bt > 0} then P0(τ = 0) = 1.

I If T0 = inf{t > 0 : Bt = 0} then P0(T0 = 0) = 1.

I If Bt is Brownian motion started at 0, then so is processdefined by X0 = 0 and Xt = tB(1/t). (Proved by checkingE (XsXt) = stE (B(1/s)B(1/t)) = s when s < t. Then checkcontinuity at zero.)

18.175 Lecture 26

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More observations

I If τ = inf{t ≥ 0 : Bt > 0} then P0(τ = 0) = 1.

I If T0 = inf{t > 0 : Bt = 0} then P0(T0 = 0) = 1.

I If Bt is Brownian motion started at 0, then so is processdefined by X0 = 0 and Xt = tB(1/t). (Proved by checkingE (XsXt) = stE (B(1/s)B(1/t)) = s when s < t. Then checkcontinuity at zero.)

18.175 Lecture 26

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More observations

I If τ = inf{t ≥ 0 : Bt > 0} then P0(τ = 0) = 1.

I If T0 = inf{t > 0 : Bt = 0} then P0(T0 = 0) = 1.

I If Bt is Brownian motion started at 0, then so is processdefined by X0 = 0 and Xt = tB(1/t). (Proved by checkingE (XsXt) = stE (B(1/s)B(1/t)) = s when s < t. Then checkcontinuity at zero.)

18.175 Lecture 26

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Outline

Recollections

Strong Markov property

18.175 Lecture 26

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Outline

Recollections

Strong Markov property

18.175 Lecture 26

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Stopping time

I A random variable S taking values in [0,∞] is a stoppingtime if for all t ≥ 0, we have {S > t} ∈ Ft .

I Distinction between {S < t} and {S ≤ t} doesn’t make adifference for a right continuous filtration.

I Example: let S = inf{t : Bt ∈ A} for some open (or closed)set A.

18.175 Lecture 26

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Stopping time

I A random variable S taking values in [0,∞] is a stoppingtime if for all t ≥ 0, we have {S > t} ∈ Ft .

I Distinction between {S < t} and {S ≤ t} doesn’t make adifference for a right continuous filtration.

I Example: let S = inf{t : Bt ∈ A} for some open (or closed)set A.

18.175 Lecture 26

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Stopping time

I A random variable S taking values in [0,∞] is a stoppingtime if for all t ≥ 0, we have {S > t} ∈ Ft .

I Distinction between {S < t} and {S ≤ t} doesn’t make adifference for a right continuous filtration.

I Example: let S = inf{t : Bt ∈ A} for some open (or closed)set A.

18.175 Lecture 26

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Strong Markov property

I Let (s, ω)→ Ys(ω) be bounded and R× C-measurable. If Sis a stopping time, then for all x ∈ Rd

Ex(YS ◦ θS |FS) = EB(S)YS on {S <∞},

where RHS means function φ(x , t) = ExYt evaluated atx = B(S), and t = S .

I In fact, similar result holds for more general Markov processes(Feller processes).

I Proof idea: First consider the case that S a.s. belongs to anincreasing countable sequence (e.g., S is a.s. a multiple of2−n). Then this essentially reduces to discrete Markovproperty proof. Then approximate a general stopping time bya discrete time by rounding down to multiple of 2−n. Usesome continuity estimates, bounded convergence, monotoneclass theorem to conclude.

I Extend optional stopping to continuous martingales similarly.

18.175 Lecture 26

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Strong Markov property

I Let (s, ω)→ Ys(ω) be bounded and R× C-measurable. If Sis a stopping time, then for all x ∈ Rd

Ex(YS ◦ θS |FS) = EB(S)YS on {S <∞},

where RHS means function φ(x , t) = ExYt evaluated atx = B(S), and t = S .

I In fact, similar result holds for more general Markov processes(Feller processes).

I Proof idea: First consider the case that S a.s. belongs to anincreasing countable sequence (e.g., S is a.s. a multiple of2−n). Then this essentially reduces to discrete Markovproperty proof. Then approximate a general stopping time bya discrete time by rounding down to multiple of 2−n. Usesome continuity estimates, bounded convergence, monotoneclass theorem to conclude.

I Extend optional stopping to continuous martingales similarly.

18.175 Lecture 26

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Strong Markov property

I Let (s, ω)→ Ys(ω) be bounded and R× C-measurable. If Sis a stopping time, then for all x ∈ Rd

Ex(YS ◦ θS |FS) = EB(S)YS on {S <∞},

where RHS means function φ(x , t) = ExYt evaluated atx = B(S), and t = S .

I In fact, similar result holds for more general Markov processes(Feller processes).

I Proof idea: First consider the case that S a.s. belongs to anincreasing countable sequence (e.g., S is a.s. a multiple of2−n). Then this essentially reduces to discrete Markovproperty proof. Then approximate a general stopping time bya discrete time by rounding down to multiple of 2−n. Usesome continuity estimates, bounded convergence, monotoneclass theorem to conclude.

I Extend optional stopping to continuous martingales similarly.

18.175 Lecture 26

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Strong Markov property

I Let (s, ω)→ Ys(ω) be bounded and R× C-measurable. If Sis a stopping time, then for all x ∈ Rd

Ex(YS ◦ θS |FS) = EB(S)YS on {S <∞},

where RHS means function φ(x , t) = ExYt evaluated atx = B(S), and t = S .

I In fact, similar result holds for more general Markov processes(Feller processes).

I Proof idea: First consider the case that S a.s. belongs to anincreasing countable sequence (e.g., S is a.s. a multiple of2−n). Then this essentially reduces to discrete Markovproperty proof. Then approximate a general stopping time bya discrete time by rounding down to multiple of 2−n. Usesome continuity estimates, bounded convergence, monotoneclass theorem to conclude.

I Extend optional stopping to continuous martingales similarly.

18.175 Lecture 26

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Continuous martingales

I Question: If Bt is a Brownian motion, then is B2t − t a

martingale?

I Question: If Bt and Bt are independent Brownian motions,then is BtBt a martingale?

I Question: If Bt is a martingale, then is eBt−t/2 a martingale?I Question: If Bt is a Brownian motion in C (i.e., real and

imaginary parts are independent Brownian motions) and f isan analytic function on C, is f (Bt) a complex martingale?

I Question: If Bt is a Brownian motion on Rd and f is aharmonic function on Rd , is f (Bt) a martingale?

I Question: Suppose Bt is a one dimensional Brownian motion,and gt : C→ C is determined by solving the ODE

∂tgt(z) =

2

gt(z)− 2Bt, g0(z) = z .

Is arg(gt(z)−Wt) a martingale?

18.175 Lecture 26

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Continuous martingales

I Question: If Bt is a Brownian motion, then is B2t − t a

martingale?I Question: If Bt and Bt are independent Brownian motions,

then is BtBt a martingale?

I Question: If Bt is a martingale, then is eBt−t/2 a martingale?I Question: If Bt is a Brownian motion in C (i.e., real and

imaginary parts are independent Brownian motions) and f isan analytic function on C, is f (Bt) a complex martingale?

I Question: If Bt is a Brownian motion on Rd and f is aharmonic function on Rd , is f (Bt) a martingale?

I Question: Suppose Bt is a one dimensional Brownian motion,and gt : C→ C is determined by solving the ODE

∂tgt(z) =

2

gt(z)− 2Bt, g0(z) = z .

Is arg(gt(z)−Wt) a martingale?

18.175 Lecture 26

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Continuous martingales

I Question: If Bt is a Brownian motion, then is B2t − t a

martingale?I Question: If Bt and Bt are independent Brownian motions,

then is BtBt a martingale?I Question: If Bt is a martingale, then is eBt−t/2 a martingale?

I Question: If Bt is a Brownian motion in C (i.e., real andimaginary parts are independent Brownian motions) and f isan analytic function on C, is f (Bt) a complex martingale?

I Question: If Bt is a Brownian motion on Rd and f is aharmonic function on Rd , is f (Bt) a martingale?

I Question: Suppose Bt is a one dimensional Brownian motion,and gt : C→ C is determined by solving the ODE

∂tgt(z) =

2

gt(z)− 2Bt, g0(z) = z .

Is arg(gt(z)−Wt) a martingale?

18.175 Lecture 26

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Continuous martingales

I Question: If Bt is a Brownian motion, then is B2t − t a

martingale?I Question: If Bt and Bt are independent Brownian motions,

then is BtBt a martingale?I Question: If Bt is a martingale, then is eBt−t/2 a martingale?I Question: If Bt is a Brownian motion in C (i.e., real and

imaginary parts are independent Brownian motions) and f isan analytic function on C, is f (Bt) a complex martingale?

I Question: If Bt is a Brownian motion on Rd and f is aharmonic function on Rd , is f (Bt) a martingale?

I Question: Suppose Bt is a one dimensional Brownian motion,and gt : C→ C is determined by solving the ODE

∂tgt(z) =

2

gt(z)− 2Bt, g0(z) = z .

Is arg(gt(z)−Wt) a martingale?

18.175 Lecture 26

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Continuous martingales

I Question: If Bt is a Brownian motion, then is B2t − t a

martingale?I Question: If Bt and Bt are independent Brownian motions,

then is BtBt a martingale?I Question: If Bt is a martingale, then is eBt−t/2 a martingale?I Question: If Bt is a Brownian motion in C (i.e., real and

imaginary parts are independent Brownian motions) and f isan analytic function on C, is f (Bt) a complex martingale?

I Question: If Bt is a Brownian motion on Rd and f is aharmonic function on Rd , is f (Bt) a martingale?

I Question: Suppose Bt is a one dimensional Brownian motion,and gt : C→ C is determined by solving the ODE

∂tgt(z) =

2

gt(z)− 2Bt, g0(z) = z .

Is arg(gt(z)−Wt) a martingale?

18.175 Lecture 26

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Continuous martingales

I Question: If Bt is a Brownian motion, then is B2t − t a

martingale?I Question: If Bt and Bt are independent Brownian motions,

then is BtBt a martingale?I Question: If Bt is a martingale, then is eBt−t/2 a martingale?I Question: If Bt is a Brownian motion in C (i.e., real and

imaginary parts are independent Brownian motions) and f isan analytic function on C, is f (Bt) a complex martingale?

I Question: If Bt is a Brownian motion on Rd and f is aharmonic function on Rd , is f (Bt) a martingale?

I Question: Suppose Bt is a one dimensional Brownian motion,and gt : C→ C is determined by solving the ODE

∂tgt(z) =

2

gt(z)− 2Bt, g0(z) = z .

Is arg(gt(z)−Wt) a martingale?

18.175 Lecture 26

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Farewell... and for future reference..

I Course has reached finite stopping time. Process goes on.

I Future probability graduate courses include 18.176 and18.177. (See also statistics courses 18.655 and 18.657.)

I Probability seminar: Mondays at 4:15.

I I am happy to help with quals and reading.

I Talk to other friendly postdocs and faculty in probability:Stephane Benoist, Vadim Gorin, Alexei Borodin, PhilippeRigollet, Elchanan Borodin, Boris Hanin, Alexey Bufetov, etc.

I Thanks for taking the class!

18.175 Lecture 26

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Farewell... and for future reference..

I Course has reached finite stopping time. Process goes on.

I Future probability graduate courses include 18.176 and18.177. (See also statistics courses 18.655 and 18.657.)

I Probability seminar: Mondays at 4:15.

I I am happy to help with quals and reading.

I Talk to other friendly postdocs and faculty in probability:Stephane Benoist, Vadim Gorin, Alexei Borodin, PhilippeRigollet, Elchanan Borodin, Boris Hanin, Alexey Bufetov, etc.

I Thanks for taking the class!

18.175 Lecture 26

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Farewell... and for future reference..

I Course has reached finite stopping time. Process goes on.

I Future probability graduate courses include 18.176 and18.177. (See also statistics courses 18.655 and 18.657.)

I Probability seminar: Mondays at 4:15.

I I am happy to help with quals and reading.

I Talk to other friendly postdocs and faculty in probability:Stephane Benoist, Vadim Gorin, Alexei Borodin, PhilippeRigollet, Elchanan Borodin, Boris Hanin, Alexey Bufetov, etc.

I Thanks for taking the class!

18.175 Lecture 26

Page 37: 18.175: Lecture 26 .1in Last lecture - Mathematicssheffield/2016175/Lecture26.pdf · 18.175: Lecture 26 Last lecture Scott She eld MIT 18.175 Lecture 26. Outline Recollections Strong

Farewell... and for future reference..

I Course has reached finite stopping time. Process goes on.

I Future probability graduate courses include 18.176 and18.177. (See also statistics courses 18.655 and 18.657.)

I Probability seminar: Mondays at 4:15.

I I am happy to help with quals and reading.

I Talk to other friendly postdocs and faculty in probability:Stephane Benoist, Vadim Gorin, Alexei Borodin, PhilippeRigollet, Elchanan Borodin, Boris Hanin, Alexey Bufetov, etc.

I Thanks for taking the class!

18.175 Lecture 26

Page 38: 18.175: Lecture 26 .1in Last lecture - Mathematicssheffield/2016175/Lecture26.pdf · 18.175: Lecture 26 Last lecture Scott She eld MIT 18.175 Lecture 26. Outline Recollections Strong

Farewell... and for future reference..

I Course has reached finite stopping time. Process goes on.

I Future probability graduate courses include 18.176 and18.177. (See also statistics courses 18.655 and 18.657.)

I Probability seminar: Mondays at 4:15.

I I am happy to help with quals and reading.

I Talk to other friendly postdocs and faculty in probability:Stephane Benoist, Vadim Gorin, Alexei Borodin, PhilippeRigollet, Elchanan Borodin, Boris Hanin, Alexey Bufetov, etc.

I Thanks for taking the class!

18.175 Lecture 26

Page 39: 18.175: Lecture 26 .1in Last lecture - Mathematicssheffield/2016175/Lecture26.pdf · 18.175: Lecture 26 Last lecture Scott She eld MIT 18.175 Lecture 26. Outline Recollections Strong

Farewell... and for future reference..

I Course has reached finite stopping time. Process goes on.

I Future probability graduate courses include 18.176 and18.177. (See also statistics courses 18.655 and 18.657.)

I Probability seminar: Mondays at 4:15.

I I am happy to help with quals and reading.

I Talk to other friendly postdocs and faculty in probability:Stephane Benoist, Vadim Gorin, Alexei Borodin, PhilippeRigollet, Elchanan Borodin, Boris Hanin, Alexey Bufetov, etc.

I Thanks for taking the class!

18.175 Lecture 26