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)

$$x,y,\alpha ,\beta ,\mu ,{\sigma }^2,\dots$$

(1)

Upper case for all random variables (scalars or vectors/matrices)

$$X,Y,A,B,M,{\Sigma }^2,\dots$$

(2)

Examples:

- Random variable with normal distribution

$$X\sim N(\mu ,{\sigma }^2)$$

(3)

- Realization of a random vector

$$x\stackrel{\text{realization}}{\Leftarrow }X$$

(4)

Exceptions:

- Although we consistently use lower case $ϵ$ for a non-random realized residual, we use the font $\epsilon$, as opposed to the upper case $E$, for the residual as a random variable

$$ϵ\stackrel{\text{realization}}{\Leftarrow }\epsilon$$

(5)

- Consistently with most literature, we prefer to use $F$for the cdf (2a.9), even when this is not a random variable.

Scalar/vector/matrix

Non-bold for all scalar (random or non-random)

$$x,X,\alpha ,\beta ,\dots$$

(6)

Bold for all vectors and matrices (random or non-random)

$$x,X,\alpha ,\beta ,\dots$$

(7)

Examples:

- Matrix of coefficients

$$\beta \equiv {\left\{{\beta }_{n,k}\right\}}_{n=1,\dots ,\overline{n}}^{k=1,\dots ,\overline{k}}$$

(8)

- Normal random vector

$$X\equiv \left(\begin{array}{c}{X}_1\\ ⋮\\ X_{\overline{n}}\end{array}\right)\sim N(\mu ,{\sigma }^2)$$

(9)

where $\mu$ is a vector and ${\sigma }^2$ is a matrix, see also (12) below.

Symmetric, positive (semi)definite matrix

Squared bold for all symmetric, positive (semi)definite matrices (31.140)

$$s^2,{\sigma }^2,{\Sigma }^2,\dots$$

(10)

where $s,\sigma ,\Sigma ,\dots$are the respective symmetric matrix roots (31.453).

Examples:

- Covariance matrix

$${\sigma }^2\equiv {\left\{{\left[{\sigma }^2\right]}_{n,m}\right\}}_{n,m=1}^{\overline{n}}$$

(11)

of a normally distributed multivariate random variable

$$X\sim N(\mu ,{\sigma }^2)$$

(12)

- Random matrix

$${\Sigma }^2\equiv {\left\{{\left[{\Sigma }^2\right]}_{n,m}\right\}}_{n,m=1}^{\overline{n}}$$

(13)

with Wishart distribution

$${\Sigma }^2\sim Wishart(\nu ,{\sigma }^2)$$

(14)

Dummy counters

Same base letter for a dummy counter $n$ and end point $\overline{n}$

$$n=1,\dots ,\overline{n}$$

(15)

and similar for all other letters/counters $i,j,k,l,t,\dots$

Examples:

- Summation

$${\sum }_{n=1}^{\overline{n}}{c}_n$$

(16)

- Indices of a rectangular matrix (8) or square matrix (11).

Collection of variables

Calligraphic subscript for collections of (multivariate) variables

$$X_J\equiv {\left\{X_j\right\}}_{j\in J}$$

(17)

where $J\equiv \left\{j_1,j_2,\dots \right\}$ is a continuum or discrete set of indices. The counterpart of the above (17) for realized variables is

$$x_J\equiv {\left\{x_j\right\}}_{j\in J}$$

(18)

Examples:

- Entries of a vector $x\equiv {\left(x_1,\dots ,x_{\overline{n}}\right)}^{\text{'}}$

$$x_N\equiv {\left\{x_n\right\}}_{n\in N}$$

(19)

where $T\equiv N\subseteq \left\{1,\dots ,\overline{n}\right\}$;

- (Multivariate) stochastic processes monitored at selected times

$$X_T\equiv {\left\{X_t\right\}}_{t\in T}$$

(20)

where the monitoring times can be discrete $J\equiv T\subseteq Z$, or a continuum $J\equiv T\subseteq R$;

- (Multivariate) random fields monitored at selected locations

$$X_U\equiv {\left\{X_t\right\}}_{t\in U}$$

(21)

where $J\equiv U\subseteq R^k$.

Data/information

We use the letter $i_t$ (lower case) for all data, namely observable information variables that are realized at a given time $t$

$$i_t\equiv {\left\{x_s\right\}}_{s\in T_t}\text{,}$$

(22)

rather than the more cumbersome notation $x_{T_t}$ (18), where the set of observations points up to time $t$ may vary from case to case

$$T_t\equiv \{\begin{array}{cc}\left\{t_1,t_2,\dots ,t_{\overline{m}}\right\}\le t& \text{finite observations, (un)evenly spaced}\\ \left\{-\infty ,\dots t-1,t\right\}& \text{infinite discrete observations, evenly spaced, includes current}\\ [t_0,t]& \text{continuous observations, finite interval}\\ (-\infty ,t]& \text{continuous observations, infinite interval}\\ \text{...}& \text{...}\end{array}$$

(23)

We use the letter $I_t$ (upper case) for the random information variables counterparts of the data variables (22), namely observable information variables that are not yet realized

$$I_t\equiv {\left\{X_s\right\}}_{s\in T_t}\text{,}$$

(24)

rather than the more cumbersome notation $X_{T_t}$ (17).

Examples:

- Equally (unit) spaced data samples of a stochastic process from inception to the latest available realization

$$i_t\equiv \{x_{t_{start}},x_{t_{start}+1},\dots ,x_{t_{end}}\},t-1<t_{end}\le t$$

(25)

Time

We think of time as a continuum, and we use latin letters to denote a point-in-time

$$t,s,u,\dots \in R$$

(26)

We use greek letters for a period or time span

$$\tau =t-s,\upsilon =v-u,\dots$$

(27)

Examples:

- Current time

$$t_{now}=\text{27 December 2014, 18:07:04:472}$$

(28)

- Time to expiry/maturity

$$\tau =\text{2 days + 1 hour + 36 minutes + 32.247 seconds}$$

(29)

Notes:

- On a microscopic scale, such as in market microstructure, events (such as trades) occur at discrete time points $t_k$ 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 $t\in R$. On this scale we have continuous-time stochastic processes ${\left\{X_t\right\}}_{t\in R}$. We slice time using the half close-half open interval convention $[t,u)$. Under this setup the last value of a process in $[t,u)$ (such as the closing price over a day) is defined as the left limit.

- For estimation purposes, we sample continuous time processes ${\left\{X_t\right\}}_{t\in R}$ over a discrete time grid ${\left\{X_{t_k}\right\}}_{k\in N}$. The grid is typically equally spaced, or $t_0,t_1=t_0+\Delta ,t_2=t_0+2\Delta ,\dots$, 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

Operators and special functions