09Stoch5

# 09Stoch5 - Chapter 2 Brownian Motion 2.1 Continuous time...

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Chapter 2 Brownian Motion 2.1 Continuous time stochastic processes We call a family of random variables { X t } t 0 on (Ω , F , P ) a continuous time stochastic process. For each ω Ω, X ( · , ω ) = X ( · ) ( ω ) is called a sample path. Usually we treat X ( · , ω ) = ω ( t ) (this can be justified). There are two most important classes of continuous time stochastic pro- cesses. The first one is the Poisson process { N t } t 0 , the number of arrivals in time [0 , t ] according to an arrival rate λ per unit time. Figure 2.1: 45

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46 CHAPTER 2. BROWNIAN MOTION Recall that a Poisson random variable X with rate λ has distribution P ( X = k ) = e - λ λ k k ! , k = 0 , 1 , 2 · · · Hence N t has distribution P ( N t = k ) = e - λt ( λt ) k k ! , k = 0 , 1 , 2 · · · A Poisson process is characterized by 1. N 0 = 0; 2. Independent increment: for 0 < t 1 < t 2 < · · · < t n , N t 1 , N t 2 - N t 1 , N t 3 - N t 2 , · · · , N t n - N t n - 1 are independent. 3. Poisson increment: for t > s , N t - N s N ( t - s ) , i.e., it has a Poisson distribution with rate λ ( t - s ). The next one is the Brownian motion { B t } t 0 . It is also called a Wiener process due to the pioneer work of Wiener in the 20’s. Recall that a one dimension normal distribution N ( μ, σ 2 ) has density function 1 2 πσ e - ( x - μ ) 2 2 σ 2 , x R and N (0 , 1) is called the standard normal distribution. The Brownian motion is defined by 1. B 0 = 0; 2. Independent increment: for 0 < t 1 < t 2 < · · · < t n , B t 1 , B t 2 - B t 1 , B t 3 - B t 2 , · · · , B t n - B t n - 1 are independent;
2.1. CONTINUOUS TIME STOCHASTIC PROCESSES 47 Figure 2.2: 3. Normal increment: for t > s, B t - B s , has normal distribution N (0 , t - s ). We will see in the next section that almost all sample paths are continuous, but not differentiable anywhere. We can also define in the same way the higher dimensional Brownian motion, i.e., { B t } t 0 has range in R d ; the corresponding density function in (3) is 1 (2 π ( t - s )) d/ 2 e - | x | 2 2( t - s ) , x R d . The Brownian motion was first formulated by Einstein to study diffusion. Heuristically we can realize it as the following: it is direct to check that p ( t, x ) = (2 πt ) - d 2 e -| x | 2 / 2 t satisfies ∂p ( t, x ) ∂t = 1 2 Δ p ( t, x ) where Δ = d i =1 2 ∂x 2 is the Laplacian. Hence it satisfies the heat equation ∂u ∂t = 1 2 Δ u on R d . If we are given an initial condition u ( x, 0) = f ( x ), it is known that the solution is given by u ( x, t ) = Z R d f ( x - y ) p ( t, y ) dy = ( f * p t )( x ) .

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48 CHAPTER 2. BROWNIAN MOTION Equivalently, we can put it in terms of the Brownian motion u ( x, t ) = E ( f ( x - B t )).
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