Drunken Birds, Brownian Motion, and Other Random Fun Michael Perlmutter Department of Mathematics Purdue University 1 M. Perlmutter(Purdue) Brownian Motion and Martingales
Outline Review of Basic Probability Random Walks Brownian Motion Martingales Theorems from MA 530 2 M. Perlmutter(Purdue) Brownian Motion and Martingales
Random Variables Informal A Probability space is a the set of possible outcomes of a random event. A Random Variable is some number that depends on the outcome. Formal A Probability Space is a measure space (Ω, F, P) such that P(Ω) = 1. A Random Variable is a measurable function defined on Ω. Measurable sets are called events. 3 M. Perlmutter(Purdue) Brownian Motion and Martingales
Discrete Random Variables Take countably many values, {x n } n=0 µ = E(X ) = n=0 np(x = n) Var(X ) = E((X µ) 2 ) = E(X 2 ) E(X ) 2 Ex: Binomial(n, p): P(X = k) = ( ) n k p k (1 p) n k Continuous Random Variables Has density f (x) so P(X A) = A f (x)dx. Cumulative distribution function F (x) = x f (t)dt = P(X x) µ = E(X ) = xf (x)dx = xdf. Ex: N(µ, σ 2 ) : f (x) = 1 2πσ e (x µ)2 /2σ 2 Facts: E(aX + by ) = ae(x ) + be(y ). Linear Combinations of normal R.V. s are normal. 4 M. Perlmutter(Purdue) Brownian Motion and Martingales
Conditional Probabiliy and Filtrations Conditional Probability P(A B) = P(A B) P(B) A and B are independent if P(A B) = P(A)P(B). E(X A) = E(X 1 A) P(A) Let X n a sequence of R.V. s. Filtrations F n = σ(x 0,..., X n ) = { event s known from observing X 0,..., X n. } E(X n+k F n ) = E(X n+k X 0,..., X n ) = Best guess of X n+k after watching first n steps. X is independent of F n if it is independent of all random variables whose values can be know at time n. 5 M. Perlmutter(Purdue) Brownian Motion and Martingales
Central Limit Theorem Central Limit Theorem Let X 1, X 2,..., be i.i.d. R.V. s with finite mean, µ, and variance, σ 2 ; let S n = X 0+...+X n n+1. Theorem: For large n, S n N(µ, σ 2 /n). Formally, S n nµ σ n N(0, 1). (1) This is why real world data is often normally distributed, e.g., heights, blood pressures, lengths of manufactured objects. 6 M. Perlmutter(Purdue) Brownian Motion and Martingales
Simple Symmetric Random Walk on the Integers Definition P( k = ±1) = 1/2 (Coin flip) X n = n k=0 k is called a Random Walk. Properties 1 E( X n ) < for all n 2 E(X n+1 F n ) = X n 3 P(X n+k = x F n ) = P(X n+k = x X n ) 4 X 0 = 0 almost surely 5 X n+k X n is independent of F n 6 X n+k X n d X k for all n Any Process with Properties 1 and 2 is called a Martingale. Any Process with property 3 is called a Markov Chain. 7 M. Perlmutter(Purdue) Brownian Motion and Martingales
History and Applications of BM History Robert Brown 1827 Bachelier 1900 Einstein 1905 Wiener 1923 Applications Black Scholes Equation Gas diffusion Other Stochastic Processes 8 M. Perlmutter(Purdue) Brownian Motion and Martingales
Brownian Motion Definition A (Standard) Brownian Motion is any continuous time stochastic process, (B t ) t 0, which satisfies 1 B 0 = 0 2 t B t is a.s. continuous 3 B t B s is independent of F s 4 B t B s d N(0, t s). Fact: Brownian Motions exist. Note: B 1 d N(0, 1) but B 1 = (B 1 B 1/2 ) + B 1/2 d N(0, 1/2) + N(0, 1/2). Note: Brownian Motion is the continuous time analog of a random walk. It is both a Markov Process and a martingale. 9 M. Perlmutter(Purdue) Brownian Motion and Martingales
Properties of Brownian Motion Let 0 < α < 1/2, then, a.s. C(ω) so B t (ω) B s (ω) < C(ω) t s α. B t lim sup t 2tloglogt = 1 a.s. Brownian path s are a.s. nowhere differentiable. Intuitively, differentiability would imply B t+h B t B t B t h for small h. B 2 t t is a martingale. So is exp(b t t/2). If f (x, t) satisfies ( 1 2 t)f (x, t) = 0, f (B t, t) is a martingale. If u = 0, and B t is complex BM then u(b t ) is a martingale. Thus, if f is complex analytic, f (B t ) is a complex martingale. 10 M. Perlmutter(Purdue) Brownian Motion and Martingales
Examples of Brownian Paths 11 M. Perlmutter(Purdue) Brownian Motion and Martingales
Recurrence and Transience Random Walks If X n is a random walk one or two dimensions, then a.s. X n is recurrent, i.e. it takes every value infinitely often. In three or more dimensions, X n is transient. It takes every value only finitely many times. Moreover, X n. Brownian Motion In one dimension, B t is recurrent. In two dimensions, it is open set recurrent, i.e., B t visits each open set i.o.. In three or more dimensions, B t is transient and B t. 12 M. Perlmutter(Purdue) Brownian Motion and Martingales
Martingale Transforms Discrete Case A sequence of R.V.s, v k, is predictable if the value of v k can be known at time k 1, i.e., v k F k 1. Theorem: Let X n a martingale, d k = X k X k 1 so X n = n k=0 d k. Then, (v X ) n = n v k d k k=0 is a martingale for all v k bounded and predictable. Continuous Case Theorem: (H X ) t = t 0 H s dx s is a martingale, for all martingales X with continuous paths and H s bounded predictable processes. 13 M. Perlmutter(Purdue) Brownian Motion and Martingales
Representation Theorems Time Change If X t is a continuous martingale, there is a unique predictable increasing process X t so that X 0 = 0 and X 2 t X t is a martingale. Theorem: If X t is a continuous-path martingale with X =., then X t is a time change of a Brownian Motion, in particular, there exists a Brownian Motion so that X t = B X t. Ito s Representation Theorem Theorem: Let B t a BM with filtration F t. Then, if X t is a martingale adapted to F t, then there exists a predictable sequence H s so X t = X 0 + t 0 H sdb s. Note: In this case, X t = t 0 H2 s ds. 14 M. Perlmutter(Purdue) Brownian Motion and Martingales
Ito s formula Fake Proof of the Fundamental theorem of calculus Let {t k } n k=0 a partition of (0, t). Taylor: f (x(t k+1 )) f (x(t k )) = f (x(t k ))dt + O(dx(t) 2 ) Summing: f (x(t)) f (x(0)) n k=0 f (x(t k ))dx b a f (x(t))dx because dx 2 is small. Ito s formula f (B(t k+1 )) f (B(t k )) = f (B(t k ))db + 1 2 f (B(t k ))db 2 + O(dB 3 ). t f (B t ) = f (B 0 ) + f (B s )db s + 1 t f (B s )ds (2) 0 2 0 f (B t ) = f (B 0 ) + n i=1 t 0 f xi (B s )db s + 1 2 Note: db 2 t = dt because B 2 t t is a martingale. 15 M. Perlmutter(Purdue) Brownian Motion and Martingales t 0 f (B s )ds (3)
Stopping Times Definition and Examples Let X t be a stochastic process with filtration F t = σ{(x x ) s t }. T is called a Stopping Time if for all t, the event {T t} is in F t. Examples: If S is a (measurable) set and B t is a BM, T = inf{t > 0 : B t T }. If S and T are stopping times, so is S T. Results Theorem: If X t is a martingale and T a stopping time, the stopped process, X t T is a martingale. Theorem: If X t is a martingale and T is a bounded stopping time, E(X T ) = E(X 0 ). 16 M. Perlmutter(Purdue) Brownian Motion and Martingales
The Averaging Property of Harmonic Functions Theorem: Let 0 < r < R. Let u be harmonic on B(z, R). Then, u(z) = 1 2π u(z + re iθ )dθ. 2π 0 Proof: Let B t be complex BM starting at z. Let T = inf{t > 0 : B t z r}. Then, u(z) = u(b 0 ) = Eu(B 0 ) = Eu(B T ) = 1 2π u(z + re iθ )dθ. 2π 0 17 M. Perlmutter(Purdue) Brownian Motion and Martingales
Louiville s Theorem Theorem: Let u be a bounded harmonic function defined on C. Then u is constant. Proof: Let u be bounded, harmonic, and non-constant. Let B t complex BM, X t = u(b t ). X t is a real-valued martingale, so it suffices to show X =. Then X t is a time change of a BM and thus open set recurrent. By Ito s formula, u(b t ) = t 0 u(b s) db s. So, X t = t 0 u(b s) 2 ds. X = 0 u(b s ) 2 ds =. Since if the integral converged, that would imply that lim u(b) s = 0, but u(b) s is open set recurrent and there is an open set on which u is non-vanishing. 18 M. Perlmutter(Purdue) Brownian Motion and Martingales
THANK YOU! 19 M. Perlmutter(Purdue) Brownian Motion and Martingales