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

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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;
Image of page 2
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 ) .
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
48 CHAPTER 2. BROWNIAN MOTION Equivalently, we can put it in terms of the Brownian motion u ( x, t ) = E ( f ( x - B t )).
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern