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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 (19.36), even when this is not a random variable.

Scalar/vector/matrix

Non-bold for all scalars (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 (13.140)

(10)

where are the respective symmetric matrix roots (13.452).

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 its 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 .

Data/information

We use lower case , rather than the more cumbersome notation (18), for all data, namely observable information variables that are realized at a given time

(22)

where the set of observations points up to time may vary from case to case

(23)

We use upper case , rather than the more cumbersome notation (17), for the random information variables, counterparts of the data variables (22)

(24)

consistently with the convention (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

Iverson brackets [W]

indicator function of a set (19.178), notation consistent with Iverson brackets

Dirac delta function (point mass) concentrated at (16.96)

canonical basis

Fourier transform (16.140)

inverse Fourier transform (16.143)

integration operator applied times (24.48)

inverse-call transformation with parameter (E.1.29)

positive part function

error function (19.100)

inverse error function [W]

change (difference, increment)

lag operator of order

exponentially weighted moving average at time with half-life and trailing window of size

operator that extracts the main diagonal of an matrix , i.e.

operator that maps an vector into an matrix which has on the main diagonal and null entries elsewhere, i.e.

absolute value of the scalar

determinant (13.108) of the square matrix

cardinality (16.17) of the set

floor function [W]

ceiling function [W]

trace of the square matrix , i.e. (13.465)

operator that maps an arbitrary matrix into a vector, stacking the columns of , i.e. (13.459)

pseudo-inverse (13.478) of the matrix

conjugate transpose (13.132) of the complex-valued matrix

complex conjugate of the complex number

Euclidean (vector) norm (13.161)

Frobenius (matrix) norm (13.242)

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)

Hadamard product (13.472)

Kronecker product (13.474)

Kronecker sum [W]

commutation matrix (13.486)

convolution (16.165)

discrete convolution (16.171)

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

directly proportional to [W]

Sets

ellipsoid with center , shape and radius (when the subscript is dropped)

-dimensional unit sphere

filled ellipsoid with center , shape and radius (when the subscript is dropped)

-dimensional filled unit ball

unit simplex in

-dimensional unit hypercube

Calculus

multivariate function (14.58)

gradient (14.60)

Hessian (14.78)

Jacobian (14.61)

See Chapter 14.1.3 for details.

Probability and general distribution theory

real world probability

risk neutral probability

probability density function (pdf) (19.34) of

cumulative distribution function (cdf) (19.36) of

characteristic function (19.39) of

equality in distribution (i.e. ) (19.45)

(or )

conditional pdf (19.57) of

univariate standard normal pdf

univariate standard normal cdf (19.102)

multivariate standard normal pdf (correlation matrix: )

multivariate standard normal cdf (correlation matrix: )

univariate standard Student t cdf (degrees of freedom: )

pdf of an elliptical random variable (19.249), including normal (), Student () and Cauchy ()

characteristic function of an elliptical random variable (19.259), including normal (), Student () and Cauchy (