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Reading 6: Stochastic Processes

Reading 6: Stochastic Processes#

For the class on Wednesday, January 31st

Reading assignments#

  1. Read the following sections of [CT14]

    • Sec. 1.4.1 “The binomial distribution”

    • Sec. 1.4.2 “The Gaussian distribution” (Skip this subsection if you are already familiar with the Central Limit Theorem. This subsection just proves the Central Limit Theorem in the special case of a simple random walk.)

    • Sec. 1.5.2 “General results for random walks in one dimension”

    • Sec. 1.5.3 “Random walks in three dimensions” (You can skip the beginning of this subsection and start from right after Eq. (1.90) where it discusses example physical systems. You can also skip “The central limit theorem” portion if you are familiar with it.)

  2. Consider a simple one-dimensional random walk. The increment at each step, \(s_i\), is either \(+1\) or \(-1\), each of which occurs with a probability of 0.5. The increments are all independent. Let \(x_N = \sum_{i=1}^{N} s_i\) denote the location of this random walk after \(N\) steps.

    We define the diffusively rescaled random walk as follows. For any positive real number \(t\),

    \[ W_N(t) = \frac{x_{\lfloor Nt \rfloor}}{\sqrt{N}}. \]

    Here \(\lfloor Nt \rfloor\) is the largest integer not greater than \(Nt\).

    The Brownian motion is defined as \(W(t) = \lim_{N \to\infty} W_N(t)\).

  3. A Markov chain is a stochastic process where the transition from Step \(i\) to Step \(i+1\) only depends on the state of Step \(i\), and not the steps before Step \(i\).

Questions#

Hint

Submit your answer on Canvas. Due at noon, Wednesday, January 31st.

  1. List anything from your reading that confuses you. Explain why they confuse you. If nothing confuses you, briefly summarize what you have learned from this reading assignment.

  2. Consider the diffusively rescaled random walk defined in (2) at \(t=1\), \(W_N(t=1)\). What are its expected value and variance? (Use the definition of \(W_N\) and what you read from Sec. 1.4.1 or 1.5.2!)

Discussion Preview#

Note

We will discuss the following questions in class. They are included here so that you have a chance to think about them before class. You need not submit your answers as part of this assignment.

  1. We can simulate the Brownian motion numerically using the diffusively rescaled random walk defined above. Typically we choose a very large value for \(N\) to approximate the limit. Let’s choose \(N=10,000\).

    • How many possible paths exist for \(W_N(t)\) from \(t=0\) to \(1\)?

    • If we simulate 1,000 different realizations (paths) of \(W_N(t)\) from \(t=0\) to \(1\), roughly what fraction of all possible paths of \(W_N(t)\) from \(t=0\) to \(1\) we would have sampled with 1,000 realizations?

  2. Discuss how you would check the diffusively rescaled random walk converges.

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