Chernoff approximation of diffusions and further applications Yana - - PowerPoint PPT Presentation

Chernoff approximation of diffusions and further applications

Yana A. Butko Analysis and Probability 2019

Yana A. Butko Chernoff approximation Analysis and Probability 2019 1 / 61

Survey paper

Yana A. Butko (2019) The method of Chernoff approximation ArXiv: http://arxiv.org/abs/1905.07309.

Yana A. Butko Chernoff approximation Analysis and Probability 2019 2 / 61

Chernoff approximation of Markov evolution

Yana A. Butko Chernoff approximation Analysis and Probability 2019 3 / 61

Chernoff approximation of Markov evolution

(ξt)t⩾0 is a time homogeneous Markov process (⇒ no memory) ⇒ transition kernel P(t,x,dy) ∶= P(ξt ∈ dy ∣ ξ0 = x)

Yana A. Butko Chernoff approximation Analysis and Probability 2019 4 / 61

Chernoff approximation of Markov evolution

(ξt)t⩾0 is a time homogeneous Markov process (⇒ no memory) ⇒ transition kernel P(t,x,dy) ∶= P(ξt ∈ dy ∣ ξ0 = x) Then f (t,x) ∶= ∫ f0(y)P(t,x,dy) ≡ E[f0(ξt) ∣ ξ0 = x] solves the following evolution equation: {

∂f ∂t (t,x)

= Lf (t,x), f (0,x) = f0(x), where Ttf0(x) ∶= ∫ f0(y)P(t,x,dy) ≡ etLf0. (Tt)t⩾0 is an operator semigroup (i.e. T0 = Id, Tt ○ Ts = Tt+s).

Yana A. Butko Chernoff approximation Analysis and Probability 2019 5 / 61

Chernoff approximation of Markov evolution

Stochastics

To determine the transition kernel P(t,x,dy) for a given process (ξt)t⩾0. ⇕

Functional Analysis

To construct the semigroup Tt ≡ etL with a given generator L. ⇕

PDEs

To solve a (Cauchy problem for a) given PDE

∂f ∂t = Lf .

Yana A. Butko Chernoff approximation Analysis and Probability 2019 6 / 61

Chernoff approximation of Markov evolution

Example: Heat equation ∂f ∂t = 1 2∆f , x ∈ Rd. Heat semigroup Ttf0(x) ∶= (2πt)−d/2 ∫

Rd

f0(y)exp{−∣x − y∣2 2t }dy. Transition kernel of Brownian motion P(t,x,dy) = (2πt)−d/2 exp{−∣x − y∣2 2t }dy.

Yana A. Butko Chernoff approximation Analysis and Probability 2019 7 / 61

Chernoff approximation of Markov evolution

Chernoff approximation: To find (F(t))t⩾0 (not a SG!!!) such that Ttf0 = lim

n→∞[F(t/n)]n f0.

Yana A. Butko Chernoff approximation Analysis and Probability 2019 8 / 61

Chernoff approximation of Markov evolution

Chernoff approximation: To find (F(t))t⩾0 (not a SG!!!) such that Ttf0 = lim

n→∞[F(t/n)]n f0.

⇒ discrete time approximation to the solution f (t,x): u0 ∶= f0, uk ∶= F(t/n)uk−1, k = 1,...,n, f (t,⋅) ≈ un.

Yana A. Butko Chernoff approximation Analysis and Probability 2019 9 / 61

Chernoff approximation of Markov evolution

Chernoff approximation: To find (F(t))t⩾0 (not a SG!!!) such that Ttf0 = lim

n→∞[F(t/n)]n f0.

⇒ discrete time approximation to the solution f (t,x): u0 ∶= f0, uk ∶= F(t/n)uk−1, k = 1,...,n, f (t,⋅) ≈ un. ⇒ Markov chain approximation to (ξt)t⩾0 (e.g., Euler scheme), (ξn

k)k=1,...,n, ∶

E[f0(ξn

k)∣ξn k−1] = F(t/n)f0(ξn k−1)

⇒ E[f0(ξt)∣ξ0 = x] = lim

n→∞E[f0(ξn n)∣ξ0 = x]

Yana A. Butko Chernoff approximation Analysis and Probability 2019 10 / 61

Chernoff approximation of Markov evolution

Chernoff approximation: To find (F(t))t⩾0 (not a SG!!!) such that Ttf0 = lim

n→∞[F(t/n)]n f0.

⇒ discrete time approximation to the solution f (t,x): u0 ∶= f0, uk ∶= F(t/n)uk−1, k = 1,...,n, f (t,⋅) ≈ un. ⇒ Markov chain approximation to (ξt)t⩾0 (e.g., Euler scheme), (ξn

k)k=1,...,n, ∶

E[f0(ξn

k)∣ξn k−1] = F(t/n)f0(ξn k−1)

⇒ E[f0(ξt)∣ξ0 = x] = lim

n→∞E[f0(ξn n)∣ξ0 = x]

⇒ approximation of path integrals in Feynman-Kac formulae.

Yana A. Butko Chernoff approximation Analysis and Probability 2019 11 / 61

Chernoff approximation of Markov evolution

Chernoff Theorem (1968): Let (Tt)t⩾0 be a strongly continuous semigroup on X with generator (L,Dom(L)). Let (F(t))t⩾0 be a family of bounded linear operators on X. Assume that

• F(0) = Id,
• ∥F(t)∥ ⩽ ewt for some w ∈ R, and all t ⩾ 0,
• lim

t→0

F(t)ϕ − ϕ t = Lϕ for all ϕ ∈ D, where D is a core for (L,Dom(L)). Then it holds Ttϕ = lim

n→∞[F(t/n)]n ϕ,

∀ ϕ ∈ X, and the convergence is locally uniform with respect to t ⩾ 0.

Yana A. Butko Chernoff approximation Analysis and Probability 2019 12 / 61

Chernoff approximation of Markov evolution

Chernoff Theorem (1968): Let (Tt)t⩾0 be a strongly continuous semigroup on X with generator (L,Dom(L)). Let (F(t))t⩾0 be a family of bounded linear operators on X. Assume that

• F(0) = Id,

(consistency)

• ∥F(t)∥ ⩽ ewt for some w ∈ R, and all t ⩾ 0,

(stability)

• lim

t→0

F(t)ϕ − ϕ t = Lϕ (consistency) for all ϕ ∈ D, where D is a core for (L,Dom(L)). Then it holds Ttϕ = lim

n→∞[F(t/n)]n ϕ,

∀ ϕ ∈ X, and the convergence is locally uniform with respect to t ⩾ 0. Meta-theorem of Numerics: Consistency + stability ⇒ convergence.

Yana A. Butko Chernoff approximation Analysis and Probability 2019 13 / 61

Chernoff approximation of Markov evolution

Chernoff Theorem (1968): Let (Tt)t⩾0 be a strongly continuous semigroup on X with generator (L,Dom(L)). Let (F(t))t⩾0 be a family of bounded linear operators on X. Assume that

• F(0) = Id,
• ∥F(t)∥ ⩽ ewt for some w ∈ R, and all t ⩾ 0,
• lim

t→0

F(t)ϕ − ϕ t = Lϕ for all ϕ ∈ D, where D is a core for (L,Dom(L)). Then it holds Ttϕ = lim

n→∞[F(t/n)]n ϕ,

∀ ϕ ∈ X, and the convergence is locally uniform with respect to t ⩾ 0. L is bdd ⇒ F(t) ∶= Id+tL ∼ etL ⇒ etL = lim

n→∞(Id+ t

nL)

n

Yana A. Butko Chernoff approximation Analysis and Probability 2019 14 / 61

Chernoff approximation of Markov evolution

Chernoff Theorem (1968): Let (Tt)t⩾0 be a strongly continuous semigroup on X with generator (L,Dom(L)). Let (F(t))t⩾0 be a family of bounded linear operators on X. Assume that

• F(0) = Id,
• ∥F(t)∥ ⩽ ewt for some w ∈ R, and all t ⩾ 0,
• lim

t→0

F(t)ϕ − ϕ t = Lϕ for all ϕ ∈ D, where D is a core for (L,Dom(L)). Then it holds Ttϕ = lim

n→∞[F(t/n)]n ϕ,

∀ ϕ ∈ X, and the convergence is locally uniform with respect to t ⩾ 0. L is unbdd ⇒ F(t) ∶= (Id−tL)

−1 ≡ 1 t RL(1/t)

∼ etL ⇒ etL = lim

n→∞(Id− t

nL)

−n

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LEGO principle for Chernoff approximation:

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LEGO principle for Chernoff approximation:

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LEGO principle for Chernoff approximation:

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LEGO principle for Chernoff approximation:

Nice processes to start with:

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LEGO principle for Chernoff approximation:

Nice processes to start with:

1 ξt with known P(t,x,dy) Yana A. Butko Chernoff approximation Analysis and Probability 2019 20 / 61

LEGO principle for Chernoff approximation:

Nice processes to start with:

1 ξt with known P(t,x,dy):

Brownian motion on a star graph with Wentzell conditions at the vertex (Kostrykin, Potthoff, Schrader, 2012)

Yana A. Butko Chernoff approximation Analysis and Probability 2019 21 / 61

LEGO principle for Chernoff approximation:

Nice processes to start with:

1 ξt with known P(t,x,dy):

Brownian motion on a star graph with Wentzell conditions at the vertex (Kostrykin, Potthoff, Schrader, 2012) New families of subordinators with explicit transition probability semigroup (Burridge, Kuznetsov, Kwa´ snicki, Kyprianou, 2014)

Yana A. Butko Chernoff approximation Analysis and Probability 2019 22 / 61

LEGO principle for Chernoff approximation:

Nice processes to start with:

1 ξt with known P(t,x,dy):

Brownian motion on a star graph with Wentzell conditions at the vertex (Kostrykin, Potthoff, Schrader, 2012) New families of subordinators with explicit transition probability semigroup (Burridge, Kuznetsov, Kwa´ snicki, Kyprianou, 2014)

2 ξt whose P(t,x,dy) is already Chernoff approximated Yana A. Butko Chernoff approximation Analysis and Probability 2019 23 / 61

LEGO principle for Chernoff approximation:

Nice processes to start with:

1 ξt with known P(t,x,dy):

Brownian motion on a star graph with Wentzell conditions at the vertex (Kostrykin, Potthoff, Schrader, 2012) New families of subordinators with explicit transition probability semigroup (Burridge, Kuznetsov, Kwa´ snicki, Kyprianou, 2014)

2 ξt whose P(t,x,dy) is already Chernoff approximated

Feller processes in Rd (B¨

• ttcher, Butko, Schilling, Schnurr, Smolyanov 2009-2012)

Yana A. Butko Chernoff approximation Analysis and Probability 2019 24 / 61

LEGO principle for Chernoff approximation:

Nice processes to start with:

1 ξt with known P(t,x,dy):

Brownian motion on a star graph with Wentzell conditions at the vertex (Kostrykin, Potthoff, Schrader, 2012) New families of subordinators with explicit transition probability semigroup (Burridge, Kuznetsov, Kwa´ snicki, Kyprianou, 2014)

2 ξt whose P(t,x,dy) is already Chernoff approximated

Feller processes in Rd (B¨

• ttcher, Butko, Schilling, Schnurr, Smolyanov 2009-2012)

Brownian motion in a compact Riemannian manifold (Smolyanov, Weizs¨ acker, Wittich 1999-2007)

Yana A. Butko Chernoff approximation Analysis and Probability 2019 25 / 61

LEGO principle for Chernoff approximation:

Chernoff approximations are known for the following operations:

Yana A. Butko Chernoff approximation Analysis and Probability 2019 26 / 61

LEGO principle for Chernoff approximation:

Chernoff approximations are known for the following operations:

• perator splitting:

L∗ ∶= L1 + ... + Lm averaging of generators: L∗ ∶= ∫ Lεµ(dε) multiplicative perturbations of L ⇔ random time change of ξt via an additive functional: L∗ ∶= aL subordination: L∗ ∶= −f (−L), ξ∗

t ∶= ξηt

“rotation”: L∗ ∶= iL killing of ξt upon leaving a domain G ⊂ Rd: L∗ ∶= L+ Dirichlet boundary / external conditions (oblique) reflection of ξt from ∂G: L∗ ∶= L+ Robin boundary conditions

Yana A. Butko Chernoff approximation Analysis and Probability 2019 27 / 61

LEGO principle for Chernoff approximation:

Chernoff approximations are known for the following operations:

• perator splitting:

L∗ ∶= L1 + ... + Lm Butko, Schilling, Smolyanov 2012 averaging of generators: L∗ ∶= ∫ Lεµ(dε) Borisov, Orlov, Sakbaev, Smolyanov 2014-2018 multiplicative perturbations of L ⇔ random time change of ξt via an additive functional: L∗ ∶= aL Butko, Schilling, Smolyanov 2012, Butko 2019 subordination: L∗ ∶= −f (−L), ξ∗

t ∶= ξηt

Butko 2018 “rotation”: L∗ ∶= iL Remizov 2016 killing of ξt upon leaving a domain G ⊂ Rd: L∗ ∶= L+ Dirichlet boundary / external conditions Butko, Grothaus, Smolyanov 2010, Butko 2018 (oblique) reflection of ξt from ∂G: L∗ ∶= L+ Robin boundary conditions Nittka 2009, Butko 2019

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Operator splitting

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Operator splitting

L∗ ∶= L1 + ... + Lm

• n D:= a core for L∗

Yana A. Butko Chernoff approximation Analysis and Probability 2019 30 / 61

Operator splitting

L∗ ∶= L1 + ... + Lm

• n D:= a core for L∗

If Fk(t) ∶ Fk(0) = Id, ∥Fk(t)∥ ⩽ etwk, F ′

k(0) = Lk

• n D

then F ∗(t) ∶= F1(t) ○ ... ○ Fm(t) ∼ etL∗

Yana A. Butko Chernoff approximation Analysis and Probability 2019 31 / 61

Operator splitting

L∗ ∶= L1 + ... + Lm

• n D:= a core for L∗

If Fk(t) ∶ Fk(0) = Id, ∥Fk(t)∥ ⩽ etwk, F ′

k(0) = Lk

• n D

then F ∗(t) ∶= F1(t) ○ ... ○ Fm(t) ∼ etL∗ Corollary (Daletskii–Lie–Trotter formula): etL1 ○ etL2 ∼ et(L1+L2) i.e. et(L1+L2) = lim

n→∞[etL1/n ○ etL2/n] n

Yana A. Butko Chernoff approximation Analysis and Probability 2019 32 / 61

Operator splitting

L∗ ∶= L1 + ... + Lm

• n D:= a core for L∗

If Fk(t) ∶ Fk(0) = Id, ∥Fk(t)∥ ⩽ etwk, F ′

k(0) = Lk

• n D

then F ∗(t) ∶= F1(t) ○ ... ○ Fm(t) ∼ etL∗ Let m = 2. Then F ∗(t) ∼ etL∗ for F ∗(t) ∶= τF1(t) ○ F2(t) + (1 − τ)F2(t) ○ F1(t), τ ∈ [0,1] and F ∗(t) ∶= F1(θt) ○ F2(t) ○ F1((1 − θ)t), θ ∈ [0,1]

Yana A. Butko Chernoff approximation Analysis and Probability 2019 33 / 61

Operator splitting

L∗ ∶= L1 + ... + Lm

• n D:= a core for L∗

If Fk(t) ∶ Fk(0) = Id, ∥Fk(t)∥ ⩽ etwk, F ′

k(0) = Lk

• n D

then F ∗(t) ∶= F1(t) ○ ... ○ Fm(t) ∼ etL∗ Let m = 2. Then F ∗(t) ∼ etL∗ for F ∗(t) ∶= τF1(t) ○ F2(t) + (1 − τ)F2(t) ○ F1(t), τ ∈ [0,1] and F ∗(t) ∶= F1(θt) ○ F2(t) ○ F1((1 − θ)t), θ ∈ [0,1] θ = 1

2

⇒ symmetric Strang splitting (scheme of 2nd order)

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Averaging

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Averaging

E is a parameter set, µ is a probability measure on E L∗ ∶= ∫

E

Lεµ(dε) Then F(t) ∶= ∫

E

etLεµ(dε) ∼ et ∫E Lεµ(dε) ≡ etL∗

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Chernoff approximation for Feller processes

Yana A. Butko Chernoff approximation Analysis and Probability 2019 37 / 61

Chernoff approximation for Feller processes

Feller process ξt ↭ Tt ≡ etL

• n C∞(Rd)

with L ≡ − ̂ H: ̂ Hϕ(x) ∶ = (2π)−d ∫

Rd

∫

Rd

eip⋅(x−q)H(x,p)ϕ(q)dqdp, ≡ (F−1 ○ H(x,⋅) ○ Fϕ)(x) where H(x,⋅) is given by the L´ evy–Khintchine formula H(x,p) = C(x) + iB(x) ⋅ p + p ⋅ A(x)p + ∫

y≠0

(1 − eiy⋅p + iy ⋅ p 1 + ∣y∣2 )N(x,dy).

Yana A. Butko Chernoff approximation Analysis and Probability 2019 38 / 61

Chernoff approximation for Feller processes

Feller process ξt ↭ Tt ≡ etL

• n C∞(Rd)

with L ≡ − ̂ H: ̂ Hϕ(x) ∶= (2π)−d ∫

Rd

∫

Rd

eip⋅(x−q)H(x,p)ϕ(q)dqdp, where H(x,⋅) is given by the L´ evy–Khintchine formula H(x,p) = C(x) + iB(x) ⋅ p + p ⋅ A(x)p + ∫

y≠0

(1 − eiy⋅p + iy ⋅ p 1 + ∣y∣2 )N(x,dy). Remark: If H = H(x,p) then ̂ e−tH ≠ e−t ̂

H

Yana A. Butko Chernoff approximation Analysis and Probability 2019 39 / 61

Chernoff approximation for Feller processes

Feller process ξt ↭ Tt ≡ etL

• n C∞(Rd)

with L ≡ − ̂ H: ̂ Hϕ(x) ∶= (2π)−d ∫

Rd

∫

Rd

eip⋅(x−q)H(x,p)ϕ(q)dqdp, where H(x,⋅) is given by the L´ evy–Khintchine formula H(x,p) = C(x) + iB(x) ⋅ p + p ⋅ A(x)p + ∫

y≠0

(1 − eiy⋅p + iy ⋅ p 1 + ∣y∣2 )N(x,dy), Remark: If H = H(x,p) then ̂ e−tH ≠ e−t ̂

H

Theorem: F(t) ∶= ̂ e−tH ∼ e−t ̂

H,

i.e. e−t ̂

H = lim n→∞[̂

e−tH/n]

n

Yana A. Butko Chernoff approximation Analysis and Probability 2019 40 / 61

Chernoff approximation for Feller processes

Feller process ξt ↭ Tt ≡ etL

• n C∞(Rd)

with L ≡ − ̂ H: ̂ Hϕ(x) ∶= (2π)−d ∫

Rd

∫

Rd

eip⋅(x−q)H(x,p)ϕ(q)dqdp, where H(x,⋅) is given by the L´ evy–Khintchine formula H(x,p) = C(x) + iB(x) ⋅ p + p ⋅ A(x)p + ∫

y≠0

(1 − eiy⋅p + iy ⋅ p 1 + ∣y∣2 )N(x,dy). Theorem: F(t) ∶= ̂ e−tH ∼ e−t ̂

H,

i.e. e−t ̂

H = lim n→∞[̂

e−tH/n]

n

Remark: Let µx

t ∶ F [µx t ](p) = e−tH(x,−p)−ip⋅x. Then

F(t)ϕ(x) = (2π)−d ∫

Rd

∫

Rd

eip⋅(x−q)e−tH(x,p)ϕ(q)dqdp = ∫

Rd

ϕ(q)µx

t (dq).

Yana A. Butko Chernoff approximation Analysis and Probability 2019 41 / 61

Chernoff approximation for Feller processes

Feller process ξt ↭ Tt ≡ etL

• n C∞(Rd)

with L ≡ − ̂ H, where H(x,⋅) is given by the L´ evy–Khintchine formula H(x,p) = C(x) + iB(x) ⋅ p + p ⋅ A(x)p + ∫

y≠0

(1 − eiy⋅p + iy ⋅ p 1 + ∣y∣2 )N(x,dy) Theorem: F(t) ∶= ̂ e−tH ∼ e−t ̂

H,

i.e. e−t ̂

H = lim n→∞[̂

e−tH/n]

n

Remark: Let µx

t ∶ F [µx t ](p) = (2π)−d/2e−tH(x,−p)−ip⋅x. Then

F(t)ϕ(x) = (2π)−d ∫

Rd

∫

Rd

eip⋅(x−q)e−tH(x,p)ϕ(q)dqdp = ∫

Rd

ϕ(q)µx

t (dq).

Example: non-degenerate diffusion (N ≡ 0, C ≡ 0): F(t)ϕ(x) = 1 √ (4πt)d detA(x) ∫

Rd

ϕ(q)e− (x−q−tB(x))⋅A−1(x)(x−q−tB(x))

4t

dq

Yana A. Butko Chernoff approximation Analysis and Probability 2019 42 / 61

Chernoff approximation for Feller processes

Example: non-degenerate diffusion: F(t)ϕ(x) = 1 √ (4πt)d detA(x) ∫

Rd

ϕ(q)e− (x−q−tB(x))⋅A−1(x)(x−q−tB(x))

4t

dq

Yana A. Butko Chernoff approximation Analysis and Probability 2019 43 / 61

Chernoff approximation for Feller processes

Example: non-degenerate diffusion: F(t)ϕ(x) = 1 √ (4πt)d detA(x) ∫

Rd

ϕ(q)e− (x−q−tB(x))⋅A−1(x)(x−q−tB(x))

4t

dq F(t) ∼ etL, L ∶= tr(A∇2) − B ⋅ ∇

Yana A. Butko Chernoff approximation Analysis and Probability 2019 44 / 61

Chernoff approximation for Feller processes

Example: non-degenerate diffusion: F(t)ϕ(x) = 1 √ (4πt)d detA(x) ∫

Rd

ϕ(q)e− (x−q−tB(x))⋅A−1(x)(x−q−tB(x))

4t

dq F(t) ∼ etL, L ∶= tr(A∇2) − B ⋅ ∇ Consider SDE dξt = −B(ξt)dt + √ 2A(ξt)dWt,

Yana A. Butko Chernoff approximation Analysis and Probability 2019 45 / 61

Chernoff approximation for Feller processes

Example: non-degenerate diffusion: F(t)ϕ(x) = 1 √ (4πt)d detA(x) ∫

Rd

ϕ(q)e− (x−q−tB(x))⋅A−1(x)(x−q−tB(x))

4t

dq F(t) ∼ etL, L ∶= tr(A∇2) − B ⋅ ∇ Consider SDE dξt = −B(ξt)dt + √ 2A(ξt)dWt, Euler–Maruyama scheme on [0,t] with time step t/n: X0 ∶= ξ0, Xk+1 ∶= Xk − B(Xk) t n + √ 2t n A(Xk)Zk, k = 0,...,n − 1, (Zk)k=0,...,n−1 are i.i.d., ∼ N(0,id), Xk ⊥ Zk.

Yana A. Butko Chernoff approximation Analysis and Probability 2019 46 / 61

Chernoff approximation for Feller processes

Example: non-degenerate diffusion: F(t)ϕ(x) = 1 √ (4πt)d detA(x) ∫

Rd

ϕ(q)e− (x−q−tB(x))⋅A−1(x)(x−q−tB(x))

4t

dq F(t) ∼ etL, L ∶= tr(A∇2) − B ⋅ ∇ Consider SDE dξt = −B(ξt)dt + √ 2A(ξt)dWt, Euler–Maruyama scheme on [0,t] with time step t/n: X0 ∶= ξ0, Xk+1 ∶= Xk − B(Xk) t n + √ 2t n A(Xk)Zk, k = 0,...,n − 1. Then, for all k = 0,...,n − 1 holds: E[f0(Xk+1)∣Xk] = E ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ f0 ⎛ ⎝x − B(x) t n + √ 2t n A(x)Zk ⎞ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

• x∶=Xk

= F(t/n)f0(Xk)

Yana A. Butko Chernoff approximation Analysis and Probability 2019 47 / 61

Chernoff approximation for Feller processes

Non-degenerate diffusion: F(t) ∼ etL, L ∶= tr(A∇2) − B ⋅ ∇ F(t)ϕ(x) = 1 √ (4πt)d detA(x) ∫

Rd

ϕ(q)e− (x−q−tB(x))⋅A−1(x)(x−q−tB(x))

4t

dq Consider SDE dξt = −B(ξt)dt + √ 2A(ξt)dWt, Euler–Maruyama scheme on [0,t] with time step t/n: X0 ∶= ξ0, Xk+1 ∶= Xk − B(Xk) t n + √ 2t n A(Xk)Zk, k = 0,...,n − 1. Then, for all k = 0,...,n − 1 holds: E[f0(Xk+1)∣Xk] = E ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ f0 ⎛ ⎝x − B(x) t n + √ 2t n A(x)Zk ⎞ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

• x∶=Xk

= F(t/n)f0(Xk) Hence (tower property) E[f0(Xn)∣X0 = x] = E[E[f0(Xn)∣Xn−1]∣X0 = x] = ... = F n(t/n)f0(x).

Yana A. Butko Chernoff approximation Analysis and Probability 2019 48 / 61

Chernoff approximation for Feller processes

Non-degenerate diffusion: F(t) ∼ etL, L ∶= tr(A∇2) − B ⋅ ∇ F(t)ϕ(x) = 1 √ (4πt)d detA(x) ∫

Rd

ϕ(q)e− (x−q−tB(x))⋅A−1(x)(x−q−tB(x))

4t

dq Consider SDE dξt = −B(ξt)dt + √ 2A(ξt)dWt, Euler–Maruyama scheme on [0,t] with time step t/n: X0 ∶= ξ0, Xk+1 ∶= Xk − B(Xk) t n + √ 2t n A(Xk)Zk, k = 0,...,n − 1. Hence (tower property) E[f0(Xn)∣X0 = x] = E[E[f0(Xn)∣Xn−1]∣X0 = x] = ... = F n(t/n)f0(x). And E[f0(ξt)∣ξ0 = x] = etLf0(x) = lim

n→∞F n(t/n)f0(x) = lim n→∞E[f0(Xn)∣X0 = x].

Yana A. Butko Chernoff approximation Analysis and Probability 2019 49 / 61

Chernoff approximation for diffusions with BC

Yana A. Butko Chernoff approximation Analysis and Probability 2019 50 / 61

Chernoff approximation for diffusions with BC

ξt process in Rd ⇒ etL

• n

X ∶= C∞(Rd)

Yana A. Butko Chernoff approximation Analysis and Probability 2019 51 / 61

Chernoff approximation for diffusions with BC

ξt process in Rd ⇒ etL

• n

X ∶= C∞(Rd) G is a nice domain in Rd

Yana A. Butko Chernoff approximation Analysis and Probability 2019 52 / 61

Chernoff approximation for diffusions with BC

ξt process in Rd ⇒ etL

• n

X ∶= C∞(Rd) G is a nice domain in Rd ξ∗

t ∶= process ξt in G + BC

⇒ etL∗

• n Y ∶= C(G) or C0(G)

Yana A. Butko Chernoff approximation Analysis and Probability 2019 53 / 61

Chernoff approximation for diffusions with BC

ξt process in Rd ⇒ etL

• n

X ∶= C∞(Rd) G is a nice domain in Rd ξ∗

t ∶= process ξt in G + BC

⇒ etL∗

• n Y ∶= C∞(G) or C0(G)

We have: F(t) ∼ etL

• n X

(⇒ F ′(0) = L on D)

Yana A. Butko Chernoff approximation Analysis and Probability 2019 54 / 61

Chernoff approximation for diffusions with BC

ξt process in Rd ⇒ etL

• n

X ∶= C∞(Rd) G is a nice domain in Rd ξ∗

t ∶= process ξt in G + BC

⇒ etL∗

• n Y ∶= C(G) or C0(G)

We have: F(t) ∼ etL

• n X

(⇒ F ′(0) = L on D) Task: to find F ∗(t) ∼ etL∗

• n Y

Yana A. Butko Chernoff approximation Analysis and Probability 2019 55 / 61

Chernoff approximation for diffusions with BC

ξt process in Rd ⇒ etL

• n

X ∶= C∞(Rd) G is a nice domain in Rd ξ∗

t ∶= process ξt in G + BC

⇒ etL∗

• n Y ∶= C(G) or C0(G)

We have: F(t) ∼ etL

• n X

(⇒ F ′(0) = L on D) Task: to find F ∗(t) ∼ etL∗

• n Y

Solution:

1 Find an extension E ∗ ∶ Y → X such that E ∗ is a linear contraction

and there is a core D∗ of L∗ with E ∗(D∗) ⊂ D

Yana A. Butko Chernoff approximation Analysis and Probability 2019 56 / 61

Chernoff approximation for diffusions with BC

ξt process in Rd ⇒ etL

• n

X ∶= C∞(Rd) G is a nice domain in Rd ξ∗

t ∶= process ξt in G + BC

⇒ etL∗

• n Y ∶= C(G) or C0(G)

We have: F(t) ∼ etL

• n X

(⇒ F ′(0) = L on D) Task: to find F ∗(t) ∼ etL∗

• n Y

Solution:

1 Find an extension E ∗ ∶ Y → X such that E ∗ is a linear contraction

and there is a core D∗ of L∗ with E ∗(D∗) ⊂ D

2 We have

F ∗(t) ∶= Rt ○ F(t) ○ E ∗ ∼ etL∗ Rt ∶ X → Y is a restriction to G / multiplication with ψt → 1G

Yana A. Butko Chernoff approximation Analysis and Probability 2019 57 / 61

Chernoff approximation for diffusions with BC

ξt process in Rd ⇒ etL

• n

X ∶= C∞(Rd) G is a nice domain in Rd ξ∗

t ∶= process ξt in G + BC

⇒ etL∗

• n Y ∶= C(G) or C0(G)

We have: F(t) ∼ etL

• n X

(⇒ F ′(0) = L on D) Task: to find F ∗(t) ∼ etL∗

• n Y

Solution:

1 Find an extension E ∗ ∶ Y → X such that E ∗ is a linear contraction

and there is a core D∗ of L∗ with E ∗(D∗) ⊂ D E ∗ are known for:

Dirichlet BC ϕ = 0

• n ∂G

Robin BC

∂ϕ ∂n + βϕ = 0

• n ∂G,

β ⩾ 0 smooth

2 We have

F ∗(t) ∶= Rt ○ F(t) ○ E ∗ ∼ etL∗ Rt ∶ X → Y is a restriction to G / multiplication with ψt → 1G

Yana A. Butko Chernoff approximation Analysis and Probability 2019 58 / 61

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Yana A. Butko Chernoff approximation Analysis and Probability 2019 59 / 61

Yana A. Butko (2019) The method of Chernoff approximation ArXiv: http://arxiv.org/abs/1905.07309. Yana A. Butko (2018) Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker-Planck-Kolmogorov equations

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