Math 288 - Probability Theory and Stochastic Process Taught by Horng-Tzer Yau Notes by Dongryul Kim Spring 2017 This course was taught by Horng-Tzer Yau. The lectures were given at MWF 12-1 in Science Center 310. The textbook was Brownian Motion, Martingales, and Stochastic Calculus by Jean-Fran¸coisLe Gall. There were 11 undergrad- uates and 12 graduate students enrolled. There were 5 problem sets and the course assistant was Robert Martinez. Contents 1 January 23, 2017 5 1.1 Gaussians . .5 1.2 Gaussian vector . .6 2 January 25, 2017 8 2.1 Gaussian process . .8 2.2 Conditional expectation for Gaussian vectors . .8 3 January 27, 2017 10 3.1 Gaussian white noise . 10 3.2 Pre-Brownian motion . 11 4 January 30, 2017 13 4.1 Kolmogorov's theorem . 13 5 February 1, 2017 15 5.1 Construction of a Brownian motion . 15 5.2 Wiener measure . 16 6 February 3, 2017 17 6.1 Blumenthal's zero-one law . 17 6.2 Strong Markov property . 18 1 Last Update: August 27, 2018 7 February 6, 2017 19 7.1 Reflection principle . 19 7.2 Filtrations . 20 8 February 8, 2017 21 8.1 Stopping time of filtrations . 21 8.2 Martingales . 22 9 February 10, 2017 23 9.1 Discrete martingales - Doob's inequality . 23 10 February 13, 2017 25 10.1 More discrete martingales - upcrossing inequality . 25 11 February 15, 2017 27 11.1 Levi's theorem . 27 11.2 Optional stopping for continuous martingales . 28 12 February 17, 2017 29 13 February 22, 2017 30 13.1 Submartingale decomposition . 30 14 February 24, 2017 31 14.1 Backward martingales . 31 14.2 Finite variance processes . 32 15 February 27, 2017 33 15.1 Local martingales . 33 16 March 1, 2017 35 16.1 Quadratic variation . 35 17 March 3, 2017 37 17.1 Consequences of the quadratic variation . 37 18 March 6, 2017 39 18.1 Bracket of local martingales . 39 18.2 Stochastic integration . 40 19 March 8, 2017 41 19.1 Elementary process . 41 20 March 20, 2017 43 20.1 Review . 43 2 21 March 22, 2017 45 21.1 Stochastic integration with respect to a Brownian motion . 45 21.2 Stochastic integration with respect to local martingales . 45 22 March 24, 2017 47 22.1 Dominated convergence for semi-martingales . 47 22.2 It^o'sformula . 48 23 March 27, 2017 50 23.1 Application of It^ocalculus . 50 24 March 29, 2017 52 24.1 Burkholder{Davis{Gundy inequality . 52 25 March 31, 2017 54 25.1 Representation of martingales . 54 25.2 Girsanov's theorem . 55 26 April 3, 2017 56 26.1 Cameron{Martin formula . 56 27 April 5, 2017 58 27.1 Application to the Dirichlet problem . 58 27.2 Stochastic differential equations . 59 28 April 7, 2017 61 28.1 Existence and uniqueness in the Lipschitz case . 61 28.2 Kolomogorov forward and backward equations . 62 29 April 10, 2017 64 29.1 Girsanov's formula as an SDE . 65 29.2 Ornstein{Uhlenbeck process . 65 29.3 Geometric Brownian motion . 66 30 April 12, 2017 67 30.1 Bessel process . 67 31 April 14, 2017 69 31.1 Maximum principle . 69 31.2 2-dimensional Brownian motion . 70 32 April 17, 2017 71 32.1 More on 2-dimensional Brownian motion . 71 33 April 19, 2017 73 33.1 Feynman{Kac formula . 73 33.2 Girsanov formula again . 74 3 34 April 21, 2017 76 34.1 Kipnis{Varadhan cutoff lemma . 76 35 April 24, 2017 78 35.1 Final review . 78 4 Math 288 Notes 5 1 January 23, 2017 The textbook is J. F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus. n A Brownian motion is a random function X(t): R≥0 ! R . This is a model for the movement of one particle among many in \billiard" dymanics. For instance, if there are many particles in box and they bump into each other. Robert Brown gave a description in the 19th century. X(t) is supposed to describe the trajectory of one particle, and it is random. To simplify this, we assume X(t) is a random walk. For instance, let N ( X 1 with probability 1=2 SN = Xi;Xi = −1 with probability 1=2: i=1 For non-integer values, you interpolate it linearly and let bNtc X SN (t) = Xi + (Nt − bNtc)XbNtc+1: i=1 2 Then E(SN ) = N. As we let N ! 1, we have three properties: 2 SN (t) • lim E p = t N!1 N E(SN (t)SN (s)) • lim p 2 = t ^ s = min(t; s) N!1 N p • SN (t)= N ! N(0; t) This is the model we want. 1.1 Gaussians Definition 1.1. X is a standard Gaussian random variable if 1 Z 2 P (X 2 A) = p e−x =2dx 2π A for a Borel set A ⊆ R. For a complex variable z 2 C, Z zX 1 zx −x2=2 z2=2 E(e ) = p e e dx = e : 2π This is the moment generating function. When z = iξ, we have 1 2 k 2 X (−ξ ) 2 (iξ) = e−ξ =2 = 1 + iξ X + X2 + ··· : k! E 2 E k=0 Math 288 Notes 6 From this we can read off the moments of the Gaussian EXn. We have (2n)! X2n = ; X2n+1 = 0: E 2nn! E Given X ∼ N (0; 1), we have Y = σX + m ∼ N (m; σ2). Its moment gener- ating function is given by zY imξ 2 2 Ee = e − σ ξ =2: 2 2 Also, if Y1 ∼ N (m1; σ1) and Y2 ∼ N (m2; σ2) and they are independent, then z(Y +Y ) zY zY i(m +m )ξ−(σ2+σ2)ξ2=2 Ee 1 2 = Ee 1 Ee 2 = e 1 2 1 2 : 2 2 So Y1 + Y2 ∼ N (m1 + m2; σ1 + σ2). 2 2 Proposition 1.2. If Xn ∼ N (mn; σn) and Xn ! X in L , then 2 2 (i) X is Gaussian with mean m = limn!1 mn and variance σ = limn!1 σn. p (ii) Xn ! X in probability and Xn ! X in L for p ≥ 1. 2 Proof. (i) By definition, E(Xn − X) ! 0 and so jEXn − EXj ≤ (EjXn − Xj2)1=2 ! 0. So EX = m. Similarly because L2 is complete, we have Var X = σ2. Now the characteristic function of X is iξX iξX im ξ−σ2 ξ2=2 imξ−σ2ξ2=2 e = lim e n = lim e n n = e : E n!1 n!1 So X ∼ N (m; σ2). (ii) Because EX2p can be expressed in terms of the mean and variance, we 2p 2p p have supn EjXnj < 1 and so supn EjXn − Xj < 1. Define Yn = jXn − Xj . 2 Then Yn is bounded in L and hence is uniformly integrable. Also it converges 1 to 0 in probability. It follows that Yn ! 0 in L , which means that Xn ! X in p L . It then follows that Xn ! X in probability. Definition 1.3. We say that Xn is uniformly integrable if for every > 0 there exists a K such that supn E[jXnj · 1jXnj>K ] < . 1 Proposition 1.4. A sequence of random variables Xn converge to X in L if and only if Xn are uniformly integrable and Xn ! X in probability. 1.2 Gaussian vector Definition 1.5. A vector X 2 Rn is a Gaussian vector if for every n 2 Rn, hu; Xi is a Gaussian random variable. Example 1.6. Define X1 ∼ N (0; 1) and = ±1 with probability 1=2 each. Let X2 = X1. Then X = (X1;X2) is not a Gaussian vector. Math 288 Notes 7 Note that u 7! E[u · X] is a linear map and so E[u · X] = u · mX for some n Pn mX 2 R . We call mX the mean of X. If X = i=1 Xiei for an orthonormal n normal basis ei of R , then the mean of X is N X mX = (EXi)ei: i=1 Likewise, the map u 7! Var u · X is a quadratic form and so Var uX = hu; qX ui for some matrix qX . We all qX the covariant matrix of X. We write qX (n) = hu; qX ui. Then the characteristic function of X is zi(u·X) 2 Ee = exp(izmX − z qX (u)=2): Proposition 1.7. If Cov(Xj;Xk)1≤j;k≤n is diagonal, and X = (X1;:::;Xn) is a Gaussian vector, then Xis are independent. Proof. First check that n X qX (u) = uiuj Cov(Xi;Xj): i;j=1 Then n 1 1 X exp(iu · X) = exp − q (u) = exp − u2 Var(X ) E 2 X 2 i i i=1 n Y 1 = exp − u2 Var(X ) : 2 i i i=1 This gives a full description of X. Theorem 1.8. Let γ : Rn ! Rn be a positive definite symmetric n × n matrix. Then there exists a Gaussian vector X with Cov X = γ. Proof. Diagonalize γ and let λj ≥ 0 be the eigenvalues. Choose v1; : : : ; vn an n orthonormal basis of eigenvectors of R and let wj = λjvj = γvj. Choose Yj to be Gaussians N (0; 1). Then the vector n X X = λjvjYj j=1 is a Gaussian vector with the right covariance. Math 288 Notes 8 2 January 25, 2017 We start with a probability space (Ω; F;P ) and joint Gaussians (or a Gaussian vector) X1;:::;Xd. We have Cov(Xj;Xk) = E[XjXk] − E[Xj]E[Xk]; γX = (Cov(Xj;Xk))jk = C: Then the probability distribution is given by 1 −hx;C−1xi=2 PX (x) = p e (2π)d=2 det C where C > 0 and the characteristic function is itx −ht;Cti=2 E[e ] = e : 2.1 Gaussian process A (centered) Gaussian space is a closed subspace of L2(Ω; F;P ) containing only Gaussian random variables.