# Mathematics and Economics

## Bubbles and Crashes: A Survey of the Efficient Market Hypothesis, Part II

In Bubbles and Crashes, Economics, Finance on February 12, 2011 at 3:48 pm

1. Empirical Tests of the Efficient Market Hypothesis and their Results

The models developed so far provide the theoretic basis for our study. We have clearly defined efficient markets and discussed a paradox in their construction, the joint hypothesis problem. We have used the language of probability theory to imbed rigor into our definition. But, we also find that much of the inspiration for these definitions arose from experience. Germinated by a broad intuition for the structure of asset returns and supported by practical experience with traders and trading, the theory is something of a hybrid: equal parts practice and principle. It was sculpted by finely-tuned economic insight, and followed statistical study or grew alongside it. Mathematical rigor was one fiber in the thread of its development.

Historically, the empirical study of the efficient market hypothesis focused on whether prices fully reflect particular subsets of information, while the unpredictability of stock market returns played a lesser role. Weaker forms of the EMH were tested first. The first subset of information considered was that of past prices, and financial economists sought to determine the validity of the weak form efficient market hypothesis. The next subset of information considered concerned the speed at which asset prices adjusted to news and other information as well as past prices; these studies tested the semi-strong form of the hypothesis. Later work considered past prices, fundamental data, news, and insider information, in an attempt to study the strong form efficient market hypothesis.
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## 246A: Complex Analysis, Notes 2 – Meromorphic Functions and Properties of Analytic Functions

In Mathematics on February 10, 2011 at 2:24 am

We continue with a discussion about meromorphic functions and the properties of analytic functions. Later notes will consider the Riemann mapping theorem, harmonic functions and the Dirichlet problem among other topics.

Definition 2.1 A function ${f}$ on an open set ${\Omega}$ is meromorphic if there exists a discrete set of points ${S = \left\{z: z \in \Omega\right\}}$ such that ${f}$ is holomorphic on ${\Omega\setminus S}$ and has poles at each ${z \in S}$. Furthermore, ${f}$ is meromorphic in the extended complex plane if ${F(z) = f(1/z)}$ is either meromorphic or holomorphic at ${0}$. In this case we say that ${f}$ has a pole or is holomorphic at infinity.

By collecting results from the previous section, we are immediately led to the following proposition regarding the Laurent expansions of complex valued functions.

Proposition 2.2 Let ${S}$ be the discrete set of singularities of a complex function ${f:\Omega \rightarrow \mathbb{C}}$ where ${\Omega}$ is an open set in ${\mathbb{C}}$. For a fixed ${z_{0} \in S}$, suppose the Laurent expansion for ${f}$ in an annulus about ${z_{0}}$ is given by ${\sum_{-\infty}^{\infty}a_{n}(z-z_{0})^{n}}$. Then,

1. The function ${f}$ has a removable singularity at ${z_{0}}$ if and only if ${a_{n} = 0}$ for all ${n < 0}$.
2. The function ${f}$ has a pole at ${z_{0}}$ if and only if there exists ${N \in \mathbb{Z}}$ with ${N < 0}$ such that ${a_{n} = 0}$ for all ${n < N}$; that is, the Laurent expansion of ${f}$ about ${z_{0}}$ has only finitely many negative terms.
3. The function ${f}$ has an essential singularity at ${z_{0}}$ if and only if the Laurent expansion of ${f}$ about ${z_{0}}$ has infinitely many negative terms.
4. Furthermore, ${f}$ is meromorphic on the extended complex plane if and only if there exists ${N \in \mathbb{Z}^{+}}$ such that ${a_{n} = 0}$ for ${n > N}$.