The Martingale Model states: if we observed the closing price of the market yesterday, we expect that the market is going to open at the close price yesterday.

Formally:

\begin{equation} E\qty [X_{k}|X_{k-1}, X_{k-2},\ldots] = X_{k-1} \end{equation}

“irrespective of what you know, no matter how long the history, the best expectation of today’s price is yesterday’s price.”

This is not a for sure! modeling statement: this is simply the expected value!! That means, after \(\infty\) times of re-running the universe starting “yesterday”, the new opening price will converge to the last closing price.

hehehe

formal system

a formal system describes a formal language for…

1. writing finite mathematical statements
2. have a definition of what statements are true
3. has a definition of a proof of a statement

examples

Every Turing Machine \(M\) defines some formal system \(\mathcal{F}\) such that \(\Sigma^{*}\) string \(w\) represents the statement “\(M\) accepts \(w\)”

• “true statements \(\mathcal{F}\)” is \(L(M)\)
• a proof that \(M\) accepts \(w\) can be defined to be an accepting computation history on \(M\) for \(w\)

interesting

a formal system \(\mathcal{F}\) is “interesting” if:

matricies are like buckets of numbers. ok, ok, seriously:

matricies are a way of encoding the basis of domain proof: that if Linear Maps are determined uniquely by where they map the basis anyways, why don’t we just make a mathematical object that represents that to encode the linear maps.

definition

Let \(n\), \(m\) be positive integer. An \(m\) by \(n\) matrix \(A\) is a rectangular array of elements of \(\mathbb{F}\) with \(m\) rows and \(n\) columns: