Stochastic Calculus Notes 3: Brownian Motion

This?note of stochastic calculus covers Brownian motion and its basic properties.

1  Random Walks

1.1  Symmetric random walk

1.1.1  Definition

Let \omega=\omega_1\omega_2\omega_3\cdots be a successive outcomes of the fair coin toss experiment (\mathbb{P}(H)=\mathbb{P}(T)=\frac{1}{2}). Let’s define a random variable

    \[X_j=\left\{\begin{tabular}{ll}1 & \text{if }\omega_j=H\\ -1?& \text{if }\omega_j=T\\ \end{tabular} \right.\]

and define M_0=0, then we can construct following process M_k,k=0,1,2,\cdots, which is called a symmetric random walk.

    \[M_k=\sum^k_{j=1}X_j,\,k=1,2,\cdots\]

With each toss, this process either steps up one unit or down one unit, and each of the two possibilities is equally likely.

Read more Stochastic Calculus Notes 3: Brownian Motion

Stochastic Calculus Notes 2: Information and Conditioning

This note of stochastic calculus covers mathematical descriptions of information and conditioning.

1  Information

1.1  Background

In dynamic hedging, the position of underlying security at each future time is contigent on how the uncertainty between the present time and that future time is resolved. In order to make contingency plans, we need a way to mathematically model the information on which future decisions can be based.

A discrete example:

  • We imagine some random experiment is performed. We denote \Omega as the set of all possible outcomes, and denote \omega as one particular outcome.
  • Suppose we are given some information. Such information is not enough to tell the exact value of \omega, but is enough for us to narrow down the possibilities. Specifically, based on such given information, we can construct a list of sets of outcomes, called the sets resolved by the information, within which we know what sets are sure to contain \omega and what other sets are sure not to contain it.
  • At each time, the list of sets resolved from the given information form a σ-algebra, so we obtain a series of σ-sigma, \mathcal{F}_0,\mathcal{F}_1,\mathcal{F}_2,\cdots, indexed by time. As time moves forward, we obtain finer resolution, meaning that \mathcal{F}_{t_2} contains more information than \mathcal{F}_{t_1} for any t_1<t_2.
  • Above collection of σ-algebra, \mathcal{F}_0,\mathcal{F}_1,\mathcal{F}_2,\cdots, is an example of a filtration.

Below is the definition of infiltration in the continuous-time sense. Read more Stochastic Calculus Notes 2: Information and Conditioning

Asset Pricing Lecture 1: Stochastic Calculus Review

This lecture is a review covering some basic concepts in stochastic calculus and time-series processes.

1  Readings

Read more Asset Pricing Lecture 1: Stochastic Calculus Review

Stochastic Calculus Notes 1: General Probability Theory

This note of stochastic calculus covers general probability theory.

1  Infinite probability spaces

1.1  Definition: σ-algebra

Let \Omega\, be a non-empty set, and \mathcal{F}\, be a collection of subsets of \Omega. We say \mathcal{F}\,  is a \sigma-algebra or \sigma-field if:

  1. The empty set \emptyset\in\mathcal{F}
  2. For any set A\in\mathcal{F}\,\Longrightarrow\,A^c\in\mathcal{F}
  3. Whenever a sequence of sets A_1,A_2,\cdots\in\mathcal{F}\,\Longrightarrow\,\bigcup_{n=1}^\infty A_n\in\mathcal{F}.

Note:

  • If we have a \sigma-algebra of sets, then all operations of the sets will give other sets that are still in the \sigma-algebra.
  • The whole space \Omega\in\mathcal{F}\, since \Omega=\emptyset^c.

Read more Stochastic Calculus Notes 1: General Probability Theory