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 ( ξ t ) t ⩾ 0 is a time homogeneous Markov process ( ⇒ no memory) ⇒ transition kernel P ( t , x , dy ) ∶= P ( ξ t ∈ dy ∣ ξ 0 = x ) Yana A. Butko Chernoff approximation Analysis and Probability 2019 4 / 61

Chernoff approximation of Markov evolution ( ξ t ) t ⩾ 0 is a time homogeneous Markov process ( ⇒ no memory) ⇒ transition kernel P ( t , x , dy ) ∶= P ( ξ t ∈ dy ∣ ξ 0 = x ) Then f ( t , x ) ∶= ∫ f 0 ( y ) P ( t , x , dy ) ≡ E [ f 0 ( ξ t ) ∣ ξ 0 = x ] solves the following evolution equation: ∂ t ( t , x ) = Lf ( t , x ) , { ∂ f f ( 0 , x ) = f 0 ( x ) , where T t f 0 ( x ) ∶= ∫ f 0 ( y ) P ( t , x , dy ) ≡ e tL f 0 . ( T t ) t ⩾ 0 is an operator semigroup (i.e. T 0 = Id , T t ○ T s = T t + s ). Yana A. Butko Chernoff approximation Analysis and Probability 2019 5 / 61

Chernoff approximation of Markov evolution Stochastics P ( t , x , dy ) for a given process ( ξ t ) t ⩾ 0 . To determine the transition kernel ⇕ Functional Analysis T t ≡ e tL To construct the semigroup with a given generator L . ⇕ PDEs ∂ t = Lf . ∂ f To solve a (Cauchy problem for a) given PDE Yana A. Butko Chernoff approximation Analysis and Probability 2019 6 / 61

Chernoff approximation of Markov evolution Example: Heat equation ∂ t = 1 x ∈ R d . ∂ f 2∆ f , Heat semigroup f 0 ( y ) exp {−∣ x − y ∣ 2 T t f 0 ( x ) ∶= ( 2 π t ) − d / 2 ∫ } dy . 2 t R d Transition kernel of Brownian motion P ( t , x , dy ) = ( 2 π t ) − d / 2 exp {−∣ x − y ∣ 2 } dy . 2 t 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 T t f 0 = lim n →∞ [ F ( t / n )] n f 0 . 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 T t f 0 = lim n →∞ [ F ( t / n )] n f 0 . ⇒ discrete time approximation to the solution f ( t , x ) : u 0 ∶= f 0 , u k ∶= F ( t / n ) u k − 1 , k = 1 ,..., n , f ( t , ⋅ ) ≈ u n . 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 T t f 0 = lim n →∞ [ F ( t / n )] n f 0 . ⇒ discrete time approximation to the solution f ( t , x ) : u 0 ∶= f 0 , u k ∶= F ( t / n ) u k − 1 , k = 1 ,..., n , f ( t , ⋅ ) ≈ u n . ⇒ Markov chain approximation to ( ξ t ) t ⩾ 0 (e.g., Euler scheme), ( ξ n k ) k = 1 ,..., n , ∶ E [ f 0 ( ξ n k )∣ ξ n k − 1 ] = F ( t / n ) f 0 ( ξ n k − 1 ) ⇒ E [ f 0 ( ξ t )∣ ξ 0 = x ] = lim n →∞ E [ f 0 ( ξ n n )∣ ξ 0 = 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 T t f 0 = lim n → ∞ [ F ( t / n )] n f 0 . ⇒ discrete time approximation to the solution f ( t , x ) : u 0 ∶= f 0 , u k ∶= F ( t / n ) u k − 1 , k = 1 ,..., n , f ( t , ⋅ ) ≈ u n . ⇒ Markov chain approximation to ( ξ t ) t ⩾ 0 (e.g., Euler scheme), ( ξ n k ) k = 1 ,..., n , ∶ E [ f 0 ( ξ n k )∣ ξ n k − 1 ] = F ( t / n ) f 0 ( ξ n k − 1 ) ⇒ E [ f 0 ( ξ t )∣ ξ 0 = x ] = lim n → ∞ E [ f 0 ( ξ n n )∣ ξ 0 = 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 ( T t ) 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 )∥ ⩽ e wt for some w ∈ R , and all t ⩾ 0 , F ( t ) ϕ − ϕ ● = L ϕ lim t t → 0 for all ϕ ∈ D , where D is a core for ( L , Dom ( L )) . Then it holds T t ϕ = 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 ( T t ) 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 )∥ ⩽ e wt for some w ∈ R , and all t ⩾ 0 , ( stability ) F ( t ) ϕ − ϕ ● = L ϕ ( consistency ) lim t t → 0 for all ϕ ∈ D , where D is a core for ( L , Dom ( L )) . Then it holds T t ϕ = 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 ( T t ) 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 )∥ ⩽ e wt for some w ∈ R , and all t ⩾ 0 , F ( t ) ϕ − ϕ ● = L ϕ lim t t → 0 for all ϕ ∈ D , where D is a core for ( L , Dom ( L )) . Then it holds T t ϕ = lim n → ∞ [ F ( t / n )] n ϕ, ∀ ϕ ∈ X , and the convergence is locally uniform with respect to t ⩾ 0. ⇒ F ( t ) ∶= Id + tL ∼ ⇒ e tL L is bdd e tL = 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 ( T t ) 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 )∥ ⩽ e wt for some w ∈ R , and all t ⩾ 0 , F ( t ) ϕ − ϕ ● = L ϕ lim t t → 0 for all ϕ ∈ D , where D is a core for ( L , Dom ( L )) . Then it holds T t ϕ = lim n → ∞ [ F ( t / n )] n ϕ, ∀ ϕ ∈ X , and the convergence is locally uniform with respect to t ⩾ 0. − 1 ≡ 1 ⇒ F ( t ) ∶= ( Id − tL ) t R L ( 1 / t ) ∼ ⇒ e tL L is unbdd e tL = lim − n n → ∞ ( Id − t nL ) Yana A. Butko Chernoff approximation Analysis and Probability 2019 15 / 61

LEGO principle for Chernoff approximation: Yana A. Butko Chernoff approximation Analysis and Probability 2019 16 / 61

LEGO principle for Chernoff approximation: Yana A. Butko Chernoff approximation Analysis and Probability 2019 17 / 61

LEGO principle for Chernoff approximation: Yana A. Butko Chernoff approximation Analysis and Probability 2019 18 / 61

LEGO principle for Chernoff approximation: Nice processes to start with: Yana A. Butko Chernoff approximation Analysis and Probability 2019 19 / 61

LEGO principle for Chernoff approximation: Nice processes to start with: 1 ξ t 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 ξ t 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 ξ t 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 ξ t 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 ξ t whose P ( t , x , dy ) is already Chernoff approximated Yana A. Butko Chernoff approximation Analysis and Probability 2019 23 / 61