Stochastic Dynamical Systems and SDE s. An Informal Introduction

Transcription

1 Stochastic Dynamical Systems and SDE s An Informal Introduction Olav Kallenberg Graduate Student Seminar, April 18, / 33

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3 Simple recursion: Deterministic system, discrete time x n+1 = f (x n ), n = 0, 1,... More generally: x n+1 = f n (x 0, x 1,..., x n ) Reduction to the previous case: y n = (n, x 0, x 1,..., x n ) 3 / 33

4 Recursion with randomness (Markov chain) Recursion: X n+1 = f (X n, ξ n+1 ), n = 0, 1,... with ξ 1, ξ 2,... independent randomization variables More generally: X n+1 = f n (X 0,..., X n ; ξ 1,..., ξ n+1 ) Reduction to previous case: Y n = (n, X 0,..., X n ; ξ 1,..., ξ n ) 4 / 33

5 Recursion in terms of increments Recursion: X n = X n+1 X n = f (X n, ξ n+1 ) X n = g(x n, ξ n+1 ) Summation: n 1 X n = X 0 + g(x k, ξ k+1 ) k=0 5 / 33

6 Space-homogeneous case (random walk) Assume g(x, u) is independent of x : X n = g(ξ n+1 ) = η n+1 Here η 1, η 2,... are i.i.d. random variables. Distribution: ν(b) = P{η B} = P{g(ξ) B} Summation: X n = X 0 + η η n (Return to general case...) 6 / 33

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8 Transition kernel, Markov property Conditional distribution: µ(x, B) = P[ X n+1 B X n = x ] = P{f (x, ξ) B} Conditioning on (X 0,..., X n ) gives same result! Conditional independence: (X 0,..., X n 1 ) (X n+1, X n+2,... ) X n In words: (past) (future) (present) 8 / 33

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10 Dynamical system, continuous time Simple recursion: x t+h = f h (x t ), t, h 0 Iteration: x t+h+k = f k (x t+h ) = f k f h (x t ), t, h, k 0 Semigroup property: f s+t = f s f t, s, t 0 10 / 33

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12 Differential equation Increments: Now let h 0: x t+h x t h In differential form: = f h(x t ) f 0 (x t ) h dx t dt = b(x t), t 0 dx t = b(x t ) dt = f h f 0 h (x t ) 12 / 33

13 Dynamical system with randomness (Markov process) Recursion: Transition kernel: X t+h = f h (X t, ξ t+h t ), t, h 0 µ h (x, B) = P[ X t+h B X t = x ] = P{f h (x, ξ) B} Markov property: {X s, s < t} {X u, u > t} Semigroup property: X t µ s+t = µ s µ t, s, t 0 Generator: µ h µ 0 h A 13 / 33

14 Space-homogeneous case (Lévy process) Stationary, independent increments: Increment distributions: Semigroup property: Paths may have jumps... X t+h X t = g h (ξ t+h t ), t, h 0 ν h (B) = P{g h (ξ) B}, h 0 ν s+t = ν s ν t, s, t 0 14 / 33

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18 Continuous paths (Brownian motion) By central limit theorem and semigroup property: X t N(b t, σ 2 t), t 0 for some constants: b drift σ diffusion rate Now take b = 0 and σ = 1 (standardize): B t N(0, t), t 0 Then in general: X t = b t + σb t, t 0 In differential form: dx t = b dt + σdb t 18 / 33

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20 Stochastic differential equation, diffusion processes Now let b and σ depend on location: dx t = b(x t ) dt + σ(x t ) db t In integrated form: t t X t = X 0 + b(x s ) ds + σ(x s ) db s 0 0 The first integral is elementary. The second is not, since B is: nowhere differentiable has unbounded variation has extremely irregular paths How to make sense of this? 20 / 33

21 Itô stochastic integrals and SDE s We can define the process: (Y B) t = t 0 Y s db s, t 0 in a weak probabilistic sense, provided that: Y is non-anticipating (depends only on the past) t 0 Y 2 s ds <, t 0 Then we can prove existence and uniqueness of solutions to: or more generally: dx t = b(x t ) dt + σ(x t ) db t dx t = b(t, X ) dt + σ(t, X ) db t for suitable functions b and σ. 21 / 33

22 Itô s formula, stochastic calculus Semimartingale decomposition: X t = X 0 + t 0 σ s (X ) db s + t M t martingale part (centered process) A t compensator (drift component) 0 b s (X ) ds = X 0 + M t + A t Itô s formula (transformation of semimartingales): f (X t ) = f (X 0 ) + t t 0 f (X s ) dx s where [X ] t denotes the quadratic variation of X t. 0 f (X s ) d[x ] s In ordinary calculus, [X ] t 0, and the last term vanishes. 22 / 33

23 Reductions to Brownian motion 1. Diffusion continuous martingale: X t diffusion M t = f (X t ) a continuous martingale for a suitable function f 2. Continuous martingale Brownian motion: M t continuous martingale B t = M τ t a Brownan motion for a suitable process τ t Similarly, any point process can be time-changed into a Poisson process. Thus, Brownian motion and Poisson processes emerge as the basic building blocks of stochastic processes. 23 / 33

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26 Connection: Brownian motion has generator / 33 Probability and potential theory Diffusion (or heat) equation u = 1 u, or: 2 Fundamental solution: 2 u t = 2 u x u x u x3 2 u(x, t) = (2πt) 3/2 exp( x 2 /2t) This is also the probability density of B t N(0, t). The PDE describes the average motion of a huge number of particles. Brownian motion describes the individual motion of each particle.

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28 Some early history Bachelier ( ) random walk, Brownian motion Markov (1906) Markov property, Markov chains Wiener (1923) existence of Brownian motion Bernstein ( ) martingale property Kolmogorov ( ) conditioning, Markov processes Lévy ( ) Brownian paths, Lévy processes Feller ( ) Markov semigroups and generators Doob ( ) modern martingale theory Itô ( ) stochastic integration and SDE s Dynkin ( ) modern Markov process theory 28 / 33

29 Glossary of probability terms stochastic model involving randomness stochastic process randomly evolving function probability theory study of stochastic processes Markov chain recursion involving randomness random walk space-homogeneous random recursion Markov process stochastic dynamical system diffusion continuous Markov process Brownian motion space-homogeneous diffusion 29 / 33

30 Glossary of probability terms (continued) Lévy process space-homogeneous Markov process Poisson process independent-increment point process Itô integral stochastic integral w.r.t. Brownian motion SDE stochastic differential equation martingale process centered to have drift zero potential theory PDE-theory involving the Laplacian semigroup functions f t satisfying f s+t = f s f t 30 / 33

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32 Basic graduate course in probability Discrete time (fall): elements of measure theory random variables and processes conditioning, independence, 0 1 laws strong limits, law of large numbers weak limits, central limit theorem martingales Markov property and chains Poisson and related processes stationary processes and ergodic theory 32 / 33

33 Basic graduate course in probability (continued) Continuous time (spring): random walk, Brownian motion Skorohod embedding, weak convergence Markov processes and semigroups Itô integrals and calculus stochastic differential equations continuous-time martingales change of time, space, measure 33 / 33