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Brownian Motion Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory 1 / 86 Outline 1 Stochastic processes 2 Definition and construction of the Wiener process 3 First properties 4 Martingale property 5 Markov property 6 Pathwise properties Samy T. Brownian motion Probability Theory 2 / 86 Outline 1 Stochastic processes 2 Definition and construction of the Wiener process 3 First properties 4 Martingale property 5 Markov property 6 Pathwise properties Samy T. Brownian motion Probability Theory 3 / 86 Stochastic processes Definition 1. Let: (Ω, F, P) probability space. I ⊂ R+ interval. n {Xt ; t ∈ I} family of random variables, R -valued Then: 1 If ω 7→ Xt (ω) measurable, X is a stochastic process 2 t 7→ Xt (ω) is called a path 3 X is continuous if its paths are continuous a.s. Samy T. Brownian motion Probability Theory 4 / 86 Modifications of processes Definition 2. Let X and Y be two processes on (Ω, F, P). 1 X is a modification of Y if P (Xt = Yt ) = 1, for all t ∈ I 2 X and Y are non-distinguishable if P (Xt = Yt for all t ∈ I) = 1 Remarks: (i) Relation (2) implicitly means that (Xt = Yt for all t ∈ I) ∈ F (ii) (2) is much stronger than (1) (iii) If X and Y are continuous, (2) ⇐⇒ (1) Samy T. Brownian motion Probability Theory 5 / 86 Filtrations Filtration: Increasing sequence of σ-algebras, i.e ,→ If s < t, then Fs ⊂ Ft ⊂ F. Interpretation: Ft summarizes an information obtained at time t Negligible sets: N = {F ∈ F; P(F ) = 0} Complete filtration: Whenever N ⊂ Ft for all t ∈ I Stochastic basis: (Ω, F, (Ft )t∈I , P) with a complete (Ft )t∈I Remark: Filtration (Ft )t∈I can always be thought of as complete ,→ One replaces Ft by Fˆt = σ(Ft , N ) Samy T. Brownian motion Probability Theory 6 / 86 Adaptation Definition 3. Let (Ω, F, (Ft )t∈I , P) stochastic basis {Xt ; t ∈ I} stochastic process We say that X is Ft -adapted if for all t ∈ I: m m Xt : (Ω, Ft ) −→ (R , B(R )) is measurable Remarks: X (i) Let Ft = σ{Xs ; s ≤ t} the natural filtration of X. X ,→ Process X is always Ft -adapted. X (ii) A process X is Ft -adapted iff Ft ⊂ Ft Samy T. Brownian motion Probability Theory 7 / 86 Outline 1 Stochastic processes 2 Definition and construction of the Wiener process 3 First properties 4 Martingale property 5 Markov property 6 Pathwise properties Samy T. Brownian motion Probability Theory 8 / 86 Definition of the Wiener process Notation: For a function f , δfst ≡ ft − fs Definition 4. Let (Ω, F, P) probability space {Wt ; t ≥ 0} stochastic process, R-valued We say that W is a Wiener process if: 1 W0 = 0 almost surely 2 Let n ≥ 1 and 0 = t0 < t1 < ··· < tn. The increments δWt0t1 , δWt1t2 , . , δWtn−1tn are independent 3 For 0 ≤ s < t we have δWst ∼ N (0, t − s) 4 W has continuous paths almost surely Samy T. Brownian motion Probability Theory 9 / 86 Illustration: chaotic path 1.5 1.0 0.5 0.0 Brownian motion -0.5 -1.0 -1.5 0.0 0.2 0.4 0.6 0.8 1.0 Time Samy T. Brownian motion Probability Theory 10 / 86 Illustration: random path 1.5 1.0 0.5 0.0 Brownian motion -0.5 -1.0 -1.5 0.0 0.2 0.4 0.6 0.8 1.0 Time Samy T. Brownian motion Probability Theory 11 / 86 Existence of the Wiener process Theorem 5. There exists a probability space (Ω, F, P) on which one can construct a Wiener process. Classical constructions: Kolmogorov’s extension theorem Limit of a renormalized random walk Lévy-Ciesilski’s construction Samy T. Brownian motion Probability Theory 12 / 86 Haar functions Definition 6. We define a family of functions {hk :[0, 1] → R; k ≥ 0}: h0(t) = 1 h1(t) = 1[0,1/2](t) − 1(1/2,1](t), and for n ≥ 1 and 2n ≤ k < 2n+1: n/2 n/2 hk (t) = 2 1 k−2n k−2n+1/2 (t) − 2 1 k−2n+1/2 k−2n+1 (t) [ 2n , 2n ] ( 2n , 2n ] Lemma 7. The functions {hk :[0, 1] → R; k ≥ 0} form an orthonormal basis of L2([0, 1]). Samy T. Brownian motion Probability Theory 13 / 86 h4 h5 h6 h7 2 √ h2 h3 2 1 h1 1 1 3 1 5 3 7 1 8 4 8 2 8 4 8 −1 √ − 2 −2 Haar functions: illustration 2 √ 2 1 h1 1 1 3 1 5 3 7 1 8 4 8 2 8 4 8 −1 √ − 2 −2 Samy T. Brownian motion Probability Theory 14 / 86 h4 h5 h6 h7 2 √ h2 h3 2 1 h1 1 1 3 1 5 3 7 1 8 4 8 2 8 4 8 −1 √ − 2 −2 Haar functions: illustration 2 √ h2 h3 2 1 h1 1 1 3 1 5 3 7 1 8 4 8 2 8 4 8 −1 √ − 2 −2 Samy T. Brownian motion Probability Theory 14 / 86 2 √ h2 h3 2 1 h1 1 1 3 1 5 3 7 1 8 4 8 2 8 4 8 −1 √ − 2 −2 Haar functions: illustration h4 h5 h6 h7 2 √ h2 h3 2 1 h1 1 1 3 1 5 3 7 1 8 4 8 2 8 4 8 −1 √ − 2 −2 Samy T. Brownian motion Probability Theory 14 / 86 Proof Norm: For 2n ≤ k < 2n+1, we have k−2n+1 Z 1 Z n h2(t) dt = 2n 2 dt = 1. k k−2n 0 2n Orthogonality: If k < l, we have two situations: (i) Supp(hk ) ∩ Supp(hl ) = ?. Then trivially hhk , hl iL2([0,1]) = 0 (ii) Supp(hl ) ⊂ Supp(hk ). Then if 2n ≤ k < 2n+1 we have: Z 1 n/2 hhk , hl iL2([0,1]) = ±2 hl (t) dt = 0. 0 Samy T. Brownian motion Probability Theory 15 / 86 Proof (2) 2 Complete system: Let f ∈ L ([0, 1]) s.t hf , hk i = 0 for all k. ,→ We will show that f = 0 almost everywhere. Step 1: Analyzing the relations hf , hk i = 0 R t ,→ We show that s f (u) du = 0 for dyadic r, s. R t Step 2: Since s f (u) du = 0 for dyadic r, s, we have Z t f (t) = ∂t f (u) du = 0, almost everywhere, 0 according to Lebesgue’s derivation theorem. Samy T. Brownian motion Probability Theory 16 / 86 Schauder functions Definition 8. We define a family of functions {sk :[0, 1] → R; k ≥ 0}: Z t sk (t) = hk (u) du 0 Lemma 9. n n+1 Functions {sk :[0, 1] → R; k ≥ 0} satisfy for 2 ≤ k < 2 : n n 1 k−2 k−2 +1 Supp(sk ) = Supp(hk ) = [ 2n , 2n ] 2 1 ksk k∞ = 2n/2+1 Samy T. Brownian motion Probability Theory 17 / 86 Schauder functions: illustration 1 2 s1 2−3/2 s2 s3 1 1 3 1 5 3 7 1 8 4 8 2 8 4 8 Samy T. Brownian motion Probability Theory 18 / 86 Gaussian supremum Lemma 10. Let {Xk ; k ≥ 1} i.i.d sequence of N (0, 1) r.v. We set: Mn ≡ sup {|Xk |; 1 ≤ k ≤ n} . Then q Mn = O ln(n) almost surely Samy T. Brownian motion Probability Theory 19 / 86 Proof Gaussian tail: Let x > 0. We have: Z ∞ 2 2 2 − z − z P (|Xk | > x) = e 4 e 4 dz (2π)1/2 x 2 Z ∞ 2 2 − x − z − x ≤ c1e 4 e 4 dz ≤ c2e 4 . x 1/2 Application of Borel-Cantelli: Let Ak = (|Xk | ≥ 4(ln(k)) ). According to previous step we have: ∞ c X P (Ak ) ≤ 4 =⇒ P (Ak ) < ∞ =⇒ P (lim sup Ak ) = 0 k k=1 Conclusion: ω-a.s there exists k0 = k0(ω) such that 1/2 ,→ |Xk (ω)| ≤ 4[ln(k)] for k ≥ k0. Samy T. Brownian motion Probability Theory 20 / 86 Concrete construction on [0, 1] Proposition 11. Let {sk ; k ≥ 0} Schauder functions family {Xk ; k ≥ 0} i.i.d sequence of N (0, 1) random variables. We set: X Wt = Xk sk (t). k≥0 Then W is a Wiener process on [0, 1] ,→ In the sense of Definition 4. Samy T. Brownian motion Probability Theory 21 / 86 Proof: uniform convergence P Step 1: Show that k≥0 Xk sk (t) converges ,→ Uniformly in [0, 1], almost surely. ,→ This also implies that W is continuous a.s Problem reduction: See that for all ε > 0 ,→ there exists n0 = n0(ω) such that for all n0 ≤ m < n we have: n 2 −1 X Xk sk ≤ ε. k=2m ∞ Samy T. Brownian motion Probability Theory 22 / 86 Proof: uniform convergence (2) Useful bounds: 1 Let η > 0. We have (Lemma 10): η |Xk | ≤ c k , with c = c(ω) p p+1 2 For 2 ≤ k < 2 , functions sk have disjoint support. Thus 2p+1−1 X 1 s ≤ p . k +1 p 2 2 k=2 ∞ Samy T. Brownian motion Probability Theory 23 / 86 Proof: uniform convergence (3) Uniform convergence: for all t ∈ [0, 1] we have: n p+1 2 2 −1 X X X Xk sk (t) ≤ |Xk | sk (t) k=2m p≥m k=2p ! 2p+1−1 X X ≤ sup |X | s (t) ` k p≥m 2p≤`≤2p+1−1 k=2p X 1 ≤ c1 p( 1 −η) p≥m 2 2 c ≤ 2 , m 1 −η 2 ( 2 ) which shows uniform convergence.
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