Notation
We spend a tremendous amount of time on notation and conventions, to make them both intuitive for readers from different disciplines and consistent across a wide set of topics. In the short Section 1 here below we point out a few key tenets that appear throughout all the present work. For the full list of notations, keep reading further below.
Key notation tenets
Random/non-random
Lower case for all non-random quantities (scalars or vectors/matrices)
![]() | (1) |
Upper case for all random variables (scalars or vectors/matrices)
![]() | (2) |
Examples:
- Random variable with normal distribution
![]() | (3) |
- Realization of a random vector
![]() | (4) |
Exceptions:
- Although we consistently use lower case for a non-random realized residual, we use the font , as opposed to the upper case , for the residual as a random variable
![]() | (5) |
- Consistently with most literature, we prefer to use for the cdf (2b.7), even when this is not a random variable.
Scalar/vector/matrix
Non-bold for all scalar (random or non-random)
![]() | (6) |
Bold for all vectors and matrices (random or non-random)
![]() | (7) |
Examples:
- Matrix of coefficients
![]() | (8) |
- Normal random vector
![]() | (9) |
where is a vector and is a matrix, see also (12) below.
Symmetric, positive (semi)definite matrix
Squared bold for all symmetric, positive (semi)definite matrices (33.140)
![]() | (10) |
where are the respective symmetric matrix roots (33.450).
Examples:
- Covariance matrix
![]() | (11) |
of a normally distributed multivariate random variable
![]() | (12) |
- Random matrix
![]() | (13) |
with Wishart distribution
![]() | (14) |
Dummy counters
Same base letter for a dummy counter and end point
![]() | (15) |
and similar for all other letters/counters
Examples:
- Summation
![]() | (16) |
- Indices of a rectangular matrix (8) or square matrix (11).
Collection of variables
Calligraphic subscript for collections of (multivariate) variables
![]() | (17) |
where is a continuum or discrete set of indices. The counterpart of the above (17) for realized variables is
![]() | (18) |
Examples:
- Entries of a vector
![]() | (19) |
where ;
- (Multivariate) stochastic processes monitored at selected times
![]() | (20) |
where the monitoring times can be discrete , or a continuum ;
- (Multivariate) random fields monitored at selected locations
![]() | (21) |
where .
Mean-covariance/probabilistic environment
Mean-covariance | Probabilistic
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s mean-covariance class (2a.27) is | ’s distribution is | ||
have the same mean-covariance class | have the same distribution | ||
are uncorrelated (3a.6) | are independent (3b.6) | ||
projected on evaluated at (3a.39) | conditioned on evaluated at (3b.11) | ||
are partially uncorrelated with respect to (3a.79) | are conditionally independent given (3b.41) | ||

Data/information
We use the letter (lower case) for all data, namely observable information variables that are realized at a given time
![]() | (22) |
rather than the more cumbersome notation (18), where the set of observations points up to time may vary from case to case
![]() | (23) |
We use the letter (upper case) for the random information variables counterparts of the data variables (22), namely observable information variables that are not yet realized
![]() | (24) |
rather than the more cumbersome notation (17).
Examples:
- Equally (unit) spaced data samples of a stochastic process from inception to the latest available realization
![]() | (25) |
Time
We think of time as a continuum, and we use latin letters to denote a point-in-time
![]() | (26) |
We use greek letters for a period or time span
![]() | (27) |
Examples:
- Current time
![]() | (28) |
- Time to expiry/maturity
![]() | (29) |
Notes:
- On a microscopic scale, such as in market microstructure, events (such as trades) occur at discrete time points in the time continuum.
- On a macroscopic scale, such as in dynamic strategies of for valuation/pricing, trading within or across instruments can occur at any point in time . On this scale we have continuous-time stochastic processes . We slice time using the half close-half open interval convention . Under this setup the last value of a process in (such as the closing price over a day) is defined as the left limit.
- For estimation purposes, we sample continuous time processes over a discrete time grid . The grid is typically equally spaced, or , although it need not be. If the grid is equally spaced, we may shift and rescale, for ease of notation and without loss of generality, the grid to
(30) |
Operators and special functions
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Iverson brackets [W] |
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indicator function of a set (39.173), notation consistent with Iverson brackets |
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Dirac delta function (point mass) concentrated at (36.96) |
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canonical basis |
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Fourier transform (36.145) |
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inverse Fourier transform (36.148) |
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integration operator applied times (6b.48) |
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inverse-call transformation with parameter (E.41.29) |
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positive part function |
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error function (39.10) |
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inverse error function [W] |
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change (difference, increment) |
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lag operator of order |
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exponentially weighted moving average at time with half-life and trailing window of size |
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operator that extracts the main diagonal of an matrix , i.e. |
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operator that maps an vector into an matrix which has on the main diagonal and null entries elsewhere, i.e. |
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absolute value of the scalar |
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determinant (33.108) of the square matrix |
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cardinality (36.17) of the set |
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floor function [W] |
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ceiling function [W] |
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trace of the square matrix , i.e. (33.463) |
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operator that maps an arbitrary matrix into a vector, stacking the columns of , i.e. (33.457) |
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pseudo-inverse (33.476) of the matrix |
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conjugate transpose (33.132) of the complex-valued matrix |
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complex conjugate of the complex number |
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Euclidean (vector) norm (33.162) |
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Frobenius (matrix) norm (33.243) |
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element-wise division (if one operand is a scalar and the other is not, the scalar expands into an array of the same size as the other operand) |
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Hadamard product (33.470) |
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Kronecker product (33.472) |
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Kronecker sum [W] |
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commutation matrix (33.484) |
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convolution (36.170) |
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discrete convolution (36.176) |
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sorting function which allows to obtain from a generic set the set of sorted elements , where and is a permutation of the indexes such that |
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directly proportional to [W] |
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Sets
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ellipsoid with center , shape and radius (when the subscript is dropped) |
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-dimensional unit sphere |
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filled ellipsoid with center , shape and radius (when the subscript is dropped) |
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-dimensional filled unit ball |
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