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Brownian Motion Chuan-Hsiang Han November 24, 2010 Symmetric Random Walk Given Ω∞ , ℱ, ; let = 1 , 2 , 3 ⋯ ∈ Ω∞ and = = , and denotes the outcome of th toss. Define the r.v.'s ∞ that for each =1 +1, = −1, = = A S.R.W. is a process ∞ =0 such that 0 = 0 and = =1 , = 1,2, ⋯ . Independent Increments of S.R.W. Choose 0 = 0 < 1 < ⋯ < , the r.v.s 1 = 1 − 0 , 2 − 1 , ⋯ − −1 are independent, where the increment is defined by +1 +1 − = = +1 . Note: (1) Increments are independent. (2) The increment +1 − has mean 0 and variance +1 − .(Stationarity) Martingale Property of S.R.W. For any nonnegative integers > , ℱ = − + ℱ = ℱ contains all the information of the first coin tosses. If R.W. is not symmetric, it is not a martingale. Markov Property of S.R.W. For any nonnegative integers > and any integrable function ℱ = − + ℱ − = − , =0 1 2 1 2 −− 2 − + Quadratic Variation of S.R.W. The quadratic variation up to time is defined to be , = − −1 2 = =1 Note the difference between = (an average over all paths), and , = . (pathwise property) Scaled S.R.W. Goal: to approximate Brownian Motion 1 = ∈+ 1 1. new time interval is "very small" of instead of 1 2. magnitude is "small" of 1 instead of 1. For any ∈ 0, ∞ , can be defined as a linear interpolation between the nearest such that ≤ < +1 . Properties of Scaled S.R.W. (i) independent increments: for any 0 = 0 < 1 < ⋯ < , 1 − 2 − 1 0 , ,⋯ − (iii) Martingale property ℱ = . (iv) Markov Property: for any function , these exists a function so that ℱ = . (v) Quadratic variation: for any ≥ 0, = =1 , = =1 1 = . − −1 2 = =1 1 2 Limiting (Marginal) Distribution of S.R.W. Theorem 3.2.1. (Central Limit Theorem) For any fixed ≥ 0, ↑∞ ≜ ~ 0, in dist. or ≤ ↑∞ −∞ Proof: shown in class. 1 2 − 2 2 . A Numerical Example : 0 ≤ 100 0.25 ≤ 0.2 = : 0 ≤ 25 ≤ 2 = 0.1555 0.2 2 −2 2 : 0 ≤ 0.25 ≤ 0.2 = 2 0 ≈ 0.1554 Log-Normality as the Limit of the Binomial Model Theorem 3.2.2. (Central Limit Theorem) For any fixed ≥ 0, = 0 = 0 ↑∞ 2 − − 2 in the distribution sense, where = 1 + = 1 − ,and = = 1+− − , What is Brownian Motion? "If is a continuous process with independent increments that are normally distributed, then is a Brownian motion." Standard Brownian Motions check Definition 3.3.1 in the text. Definition of SBM: Let the stochastic process , ≥ 0 under a probability space Ω, ℱ, P be continuous and satisfy: 1. 0 = 0 2. + − ~ 0, 3. + − is independent of − +1 for 0 < ⋯ = . Covariance Matrix Check , = , for any nonnegative and For any vector = 1 , 2 , ⋯ , with 0 ≤ 1 ≤ ⋯ ≤ , 1 1 ⋯ 1 1 2 ⋯ 2 = ≜ ⋮ ⋮ ⋮ 1 2 ⋯ In fact, ~ 0, . Joint Moment-Generating Function of BM 1 , 2 , ⋯ , = ∙ = 1 1 + 2 2 + ⋯ + 1 = 1 + 2 + ⋯ + 2 1 2 1 + 2 + ⋯ + 2 2 − 1 + ⋯ 2 Alternative Characteristics of Brownian Motion (Theorem 3.3.2) For any continuous process , ≥ 0 with 0 = 0, the following three properties are equivalent. (i) increments are independent and normally distributed. (ii) For any 0 ≤ 0 ≤ 1 ≤ ⋯ ≤ , 1 , 2 , ⋯ , are jointly normally distributed. (ii) 1 , 2 , ⋯ , has the joint momentgenerating function as before. If any of the three holds, then , ≥ 0, is a SBM. Filtration for B.M. Definition 3.3.3 Let Ω, ℱ, be a probability space on which the B.M. ≥0 is defined. A filtration for the B.M. is a collection of -algebras ℱ ≥0 , satisfying (i) (Information accumulates) For 0 ≤ ≤ , ℱ ⊆ ℱ . (ii) (Adaptivity) each is ℱ -measurable. (iii) (Independence of future increments) 0 ≤ < , the increment − is independent of ℱ . [Note, this property leads to Efficient Market Hypothesis.] Martingale property Theorem 3.3.4 B.M. is a martingale. Proof: |ℱ = ⋯ = Levy's Characteristics of Brownian Motion The process is SBM iff the conditional characterization function is − |ℱ = 2 − − 2 Variations: First-Order (Total) Variation Given a function defined on 0, , the total variation is defined by −1 = lim Π →0 +1 − =0 where the partition Π = 0 = 0, 1 , ⋯ , = and Π = =0,⋯,−1 +1 − If is differentiable, +1 − = ′ ⋆ +1 − for some ⋆ +1 , . Then = lim lim Π →0 Π →0 −1 ⋆ ′ =0 −1 =0 +1 − +1 − = 0 = ′ . Quadratic Variation Def. 3.4.1 The quadratic variation of up to time is defined by −1 , = lim Π →0 +1 − =0 2 If is continuous differentiable, +1 − = ′ ⋆ +1 − for some ⋆ +1 , . Then , = lim −1 +1 =0 Π →0 −1 ⋆ 2 lim =0 ′ +1 − Π →0 2 ′ 0 − = lim 2 Π →0 ≤ Π × Quadratic Variation of B.M. Thm. 3.4.3 Let ≥0 be a Brownian Motion. Then , = for all ≥ 0 a.s.. B.M. accumulates quadratic variation at rate one per unit time. Informal notion: ∙ = , ∙ = 0, ∙ = 0 Geometric Brownian Motion The geometric Brownian motion is a process of the following form = 0 + − 2 2 . where 0 is the current value, ≥0 is a B.M., is the drift and > 0 is the volatility. For each partition 0 = 0, 1 , ⋯ , = , define the log returns +1 = +1 − + − 2 2 +1 − Volatility Estimation of GBM The realized variance is defined by 2 −1 +1 =0 which converges to 2 as Π → 0 BM is a Markov process Thm. 3.5.1 Let ≥0 be a B.M. and ℱ≥0 be a filtration for this B.M.. Then (1)Wt0 is a Markov process. Thm. 3.6.1. (2) = − 1 2 2 is martingale. (We call exponential martingale.) Note: = 1