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Okay, so the logarithmic return on a stock is given by:

$$r_τ (t) = \ln P(t+τ) - \ln P(t),$$ where τ is the interval of time.

I have no problem calculating that. My question comes to the following formula:

$$ρ(T) \sim 〈r_τ (t+T) \cdot r_τ (t)〉$$

This is supposedly the autocorrelation function of log-returns. What's the deal with the brackets?

MJD
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Darragh
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    Not too sure about stocks, but the angular brackets commonly indicate expected value in statistics. Would that make sense in this context? – EuYu Apr 18 '13 at 03:56
  • In physics, it's suppose to be an inner product. http://math.stackexchange.com/questions/134156/what-does-langle-y-lm-y-lambda-mu-rangle-delta-l-lambda-delta http://math.stackexchange.com/questions/173407/a-notation-question-langle-x-y-rangle – Roman Luštrik Aug 21 '13 at 13:38
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    The angle brackets probably represent a time-average. – Lukasz Wiklendt Jan 10 '18 at 00:35

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In Bra-ket notation, we denote an inner product of two vectors $|u\rangle,\,|v\rangle$ as $\langle u|v\rangle$, where the map from vectors $|v\rangle$ in the inner product space to scalars $\langle u|v\rangle$ is a linear map labelled $\langle u|$. (The set of such linear maps is a vector space called the dual space; its elements are bras $\langle u|$, whereas the original space has ket elements $|v\rangle$.)

If the matrix $M$ keeps $|v\rangle$ in the same vector space by mapping it to $M|v\rangle$, $\langle u|M|v\rangle$ is not only an inner product; if $|u\rangle,\,|v\rangle$ are elements of some basis, this inner product is a matrix element in that basis.

In quantum mechanics, to each observable $O$ there corresponds an $\widehat{O}$ such that in the pure state $|\psi\rangle$ the expectation of $O$ is $\langle\psi|\widehat{O}|\psi\rangle$. If the state can be taken for granted, we abbreviate this as $\langle\widehat{O}\rangle$.

Even with classical probability, we can express the covariance of two random variables as an inner product on a vector space: namely, a quotient space of equivalence classes of finite-variance random variables. I go through the details here. Then $A,\,B$ has covariance $\langle AB\rangle-\langle A\rangle\langle B\rangle$, but the second term can be deleted if at least one of $A,\,B$ has mean $0$.

If variables have variance $1$, their covariance is also their correlation. Autocorrelation is one special case.

J.G.
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A clear and concise definition I found in Wolfram MathWorld which states a below, Angle Bracket

Its history is discussed by Professor Brailsford in Computerphile youtube channel here.

Another question was asked here which is more in the context of statistics and can be helpful to understand.

TheRimalaya
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