Lecture Notes for Stochastic Differential Equations --- A parapharse from Professor Xue's Lectures (Lecture 2)

3
5
2011

Lecture Notes for Stochastic Differential Equations --- A parapharse from Professor Xue's Lectures (Lecture 2)

Introduction

In this lecture, we reviewed the definition of the conditional probability with respect to a sub- $\sigma$ -field. And then gave several special examples to compute the conditional probability. After mentioning the conditional probability with respect to a random variable, and some function measurability, an easier way to check a conditional probability is given. Finally, we introduce the so-called $\pi$ - $\lambda$ theorem and given an application of it.

General Case

Consider the probability measure space $(\Omega, \mathcal F, \mathbb P)$ . Event $A \in \mathcal F$ and $\mathcal G$ is a sub- $\sigma$ -field. The conditional probability $\mathbb P\{A|\mathcal G\}$ satiesfies

$\mathbb P\{A|\mathcal G\}$ is $\mathcal G$ -measurable
$\int_G\mathbb P\{A|\mathcal G\}dP = \mathbb P\{A\cap G\}, \forall G \in \mathcal G$ .

In the sense of a.e. equal, the random variable is uniquely determined.

These two requirements may seemingly being surprising at the first sight, yet they really have their probabilistic interpretations.

The first entry tells us that by the knowledge of $\mathcal G$ , we could define $\mathbb P\{A|\mathcal G\}$ .

The second entry is understood while determing the price of a gamble. If the expected gain is zero, then the price of entering a gamble must be $\mathbb P\{A|\mathcal G\}$ . To see this, if $A$ happens, the gain is $1 - \mathbb P\{A|\mathcal G\}$ , and if $A$ does not happen, the gain will be $- \mathbb P\{A|\mathcal G\}$ . Together, the gain will be

$(1 - \mathbb P\{A|\mathcal G\})\mathbf 1_A - \mathbb P\{A|\mathcal G\} \mathbf 1_{A^c} = \mathbf 1_A - \mathbb P\{A|\mathcal G\}$

which will lead to the result, if we integral over $G$ .

Ex. What is $\mathbb P\{A|\mathcal G\}$ if $A \in \mathcal G$ ?

Solution. This is Example 33.3. The key is $\mathbf 1_A$ .

Ex. What is $\mathbb P\{A|\mathcal G\}$ if $\mathcal G = \{\emptyset, \Omega\}$ ?

Solution. This is Example 33.4. The key is $\mathbb P\{A\}$ .

Ex. Suppose $A$ is independent of $\mathcal G$ , i.e., $\mathbb P\{A\cap G\} = \mathbb P\{A\} \mathbb P\{G\}, \forall G\in\mathcal G$ , then what is $\mathbb P\{A|\mathcal G\}$ ?

Solution. Intuitively it is $\mathbb P\{A\}$ .

Ex. Suppose $\Omega = \mathbb R^2$ , $\omega = (x, y)$ , $\mathcal F = \mathcal B(\mathbb R^2)$ , and

$\mathbb P\{A\} = \int_A f(x, y)\,dx dy, \quad \forall A \in \mathcal F.$

Here, $f(x, y)$ is Borel-measurable functon, $f: \mathbb R^2 \to \mathbb R$ , and $f(x, y) \ge 0$ almost everywhere, and $\iint_{\mathbb R^2} f(x, y) \,dxdy =1$ . Countable additivity is trivial to verify. It follows that $\mathbb P$ is a probability measure. Suppose $\mathcal G$ is a sub- $\sigma$ -field generated by the form $E\times \mathbb R^1 = \{(x, y): x\in E\}, \forall E\in \mathcal B(\mathbb R^1)$ . Let $A$ be $\mathbb R^1\times F = \{(x,y): y\in F\}, \forall F \in \mathcal B(\mathbb R^1)$ . What is $\mathbb P\{A|\mathcal G\}$ ?

Solution. One claims that

$\varphi(x, y) = \frac{\int_Ff(x,t)d\,t}{\int_{\mathbb R^1}f(x,t)\,dt}$

is a version of $\mathbb P\{A|\mathcal G\}$ . To verify this, one should verify second property of the definition. Since a general element in $\mathcal G$ takes the form $E \times \mathbb R^1$ , it is essential to prove that

$\int_{E \times \mathbb R^1} \varphi(x, y)\,dP(x, y) = \mathbb P\{A\cap(E\times\mathbb R^1)\}.$

Notice that the right hand side of the equation is exactly $\mathbb P\{E \times F\}$ . To evaluate the value on the left hand side, one must resort to the Fubini's Theorem and a previous Theorem.

It is also possible to define a conditionial probability with respect to random variable $X$ . This could be done by define $\sigma(X) = \{X^{-1}(B), B\in \mathcal B(\mathbb R^1)\}$ . And define $\mathbb P\{A|X\} = \mathbb P\{A|\sigma(X)\}$ .

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Lecture Notes for Stochastic Differential Equations --- A parapharse from Professor Xue's Lectures (Lecture 2)

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