Stochastic Computation

Transcription

1 Stochastic Computation A Brief Introduction Klaus Ritter Computational Stochastics TU Kaiserslautern 1/1

2 Objectives Objectives of Computational Stochastics: Construction and analysis of (A) algorithms for computational problems arising in stochastics. 2/6

3 Objectives Objectives of Computational Stochastics: Construction and analysis of (A) algorithms for computational problems arising in stochastics. Example: Computation of probabilities, expectations, etc. in complex stochastic models. Applications in fluid dynamics, statistical physics, finance, etc. 2/5

4 Objectives Objectives of Computational Stochastics: Construction and analysis of (A) algorithms for computational problems arising in stochastics. (B) stochastic algorithms for computational problems. 2/4

5 Objectives Objectives of Computational Stochastics: Construction and analysis of (A) algorithms for computational problems arising in stochastics. (B) stochastic algorithms for computational problems. Question: When and how to use a random number generator for problems from analysis, optimization, stochastics, etc. 2/3

6 Objectives Objectives of Computational Stochastics: Construction and analysis of (A) algorithms for computational problems arising in stochastics. (B) stochastic algorithms for computational problems. Note: Computational tools vs. computational problems. 2/2

7 Objectives Objectives of Computational Stochastics: Construction and analysis of (A) algorithms for computational problems arising in stochastics. (B) stochastic algorithms for computational problems. Note: Computational tools vs. computational problems. In the sequel, for illustration: infinite-dimensional integration for stochastic differential equations. 2/1

8 Stochastic Differential Equations Consider an autonomous, scalar SDE dx t = µ(x t )dt+σ(x t )dw t, t [0,T], X 0 = x 0 with a scalar Brownian motionw = (W t ) t [0,T], drift and diffusion coefficientsµ,σ : R R, and initial valuex 0 R. 3/3

9 Stochastic Differential Equations Consider an autonomous, scalar SDE dx t = µ(x t )dt+σ(x t )dw t, t [0,T], X 0 = x 0 with a scalar Brownian motionw = (W t ) t [0,T], drift and diffusion coefficientsµ,σ : R R, and initial valuex 0 R. Assumption µ,σ Lip 1 (R,R). 3/2

10 Stochastic Differential Equations Consider an autonomous, scalar SDE dx t = µ(x t )dt+σ(x t )dw t, t [0,T], X 0 = x 0 with a scalar Brownian motionw = (W t ) t [0,T], drift and diffusion coefficientsµ,σ : R R, and initial valuex 0 R. The solutionx = (X t ) t [0,T] is a Markov process with continuous paths, and lim h 0 h 1 E(X t+h X t X t = x) = µ(x), lim h 0 h 1/2 (E((X t+h X t ) 2 X t = x) ) 1/2 = σ (x). 3/1

11 The Euler Scheme The Euler scheme with time-discretization fork = 0,...,K is defined by ˆX t0 = x 0, t k = t K k = k/k T ˆX tk+1 = ˆX tk +µ( ˆX tk ) (t k+1 t k ) and piecewise linear interpolation. +σ( ˆX tk ) (W tk+1 W tk ), }{{} iid,n(0,t/k) This yields a stochastic process ˆX = ˆX K = ( ˆX t ) t [0,T]. 4/1

12 Givenx 0 = 1,µ,σ, and The Computational Problem f : C([0,T]) R, compute S(µ,σ,f) = E(f(X)) = C([0,T]) f dp X. 5/5

13 The Computational Problem Givenx 0 = 1,µ,σ, and f : C([0,T]) R, compute S(µ,σ,f) = E(f(X)) = C([0,T]) f dp X. Infinite-dimensional integration w.r.t. a measure that is given only implicitly. 5/4

14 Givenx 0 = 1,µ,σ, and The Computational Problem f : C([0,T]) R, compute S(µ,σ,f) = E(f(X)) = C([0,T]) f dp X. Assumption:(µ,σ,f) F for F = Lip 1 (R,R) Lip 1 (R,R) Lip 1 (C([0,T]),R). 5/3

15 Givenx 0 = 1,µ,σ, and The Computational Problem f : C([0,T]) R, compute S(µ,σ,f) = E(f(X)) = C([0,T]) f dp X. Monte Carlo Euler, with independent copies ˆX K 1,..., ˆX K M of ˆXK, S K,M (µ,σ,f) = 1/M M m=1 f( ˆX K m) 5/2

16 Givenx 0 = 1,µ,σ, and The Computational Problem f : C([0,T]) R, compute S(µ,σ,f) = E(f(X)) = C([0,T]) f dp X. Monte Carlo Euler, with independent copies ˆX K 1,..., ˆX K M of ˆXK, S K,M (µ,σ,f) = 1/M M m=1 f( ˆX K m) Questions: Quality ofs K,M? Are there better algorithms? 5/1

17 Resources real number machine; cost one per operation, ideal random number generator; cost one per call, oracles forµ(x) andσ(x) for anyx R; cost one per call, oracle forf(x) for any piecewise linear functionx C([0,T]) with equidistant breakpoints; cost = #breakpoints /3

18 Resources real number machine; cost one per operation, ideal random number generator; cost one per call, oracles forµ(x) andσ(x) for anyx R; cost one per call, oracle forf(x) for any piecewise linear functionx C([0,T]) with equidistant breakpoints; cost = #breakpoints + 2. Hereby, we get the definition of a randomized algorithmaand its cost, cost(a,(µ,σ,f),ω). 6/2

19 Resources real number machine; cost one per operation, ideal random number generator; cost one per call, oracles forµ(x) andσ(x) for anyx R; cost one per call, oracle forf(x) for any piecewise linear functionx C([0,T]) with equidistant breakpoints; cost = #breakpoints + 2. Hereby, we get the definition of a randomized algorithmaand its cost, cost(a,(µ,σ,f),ω). Maximal error and cost ofa error(a) = sup (µ,σ,f) F cost(a) = sup (µ,σ,f) F ( E(S(µ,σ,f) A(µ,σ,f)) 2 ) 1/2, E(cost(A,(µ,σ,f), )). 6/1

20 Results Thm. With suitable parameters, Monte Carlo Euler A achieves error(a) log cost(a) 1/4. 7/3

21 Results Thm. With suitable parameters, Monte Carlo Euler A achieves error(a) log cost(a) 1/4. Thm. With suitable parameters, multi-level Monte Carlo Euler A achieves error(a) log cost(a) 1/2. See Heinrich (1998), Giles (2008),... 7/2

22 Results Thm. With suitable parameters, Monte Carlo Euler A achieves error(a) log cost(a) 1/4. Thm. With suitable parameters, multi-level Monte Carlo Euler A achieves error(a) log cost(a) 1/2. See Heinrich (1998), Giles (2008),... Thm. For the n-th minimal error e(n) = inf{error(a) : A randomized alg. withcost(a) n} we have e(n) log n 1/2. See Creutzig, Dereich, Müller-Gronbach, R (2008). 7/1