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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

Non-random quantities (scalars or vectors/matrices): lower case $x, y, α, β, μ, σ^{2}, \dots$
Random variables (scalars or vectors/matrices): upper case $X, Y, A, B, M, Σ^{2}, \dots$
Examples:
$X \sim N (μ, σ^{2})$ ;
$Σ^{2} \sim Wishart (ν, σ^{2})$ ;
$x$ (realized random variable);
$β$ (coefficient).
Exception: $ϵ_{t}$ is realized invariant; $ε_{t}$ (not $E_{t}$ ) is invariant as random variable.

Dummy counters: $n = 1, \dots, ˉ n$ (similar for $i, j, k, l, t, \dots$ ).
We do not use upper case $n = 1, \dots, N$ , to avoid confusion with random variables (such as $N_{t}$ Poisson counter).
We do not use other letters $n = 1, \dots, j$ (except in special cases), to be more succinct and clear.
Examples:
$\sum_{n = 1}^{ˉ n} x_{n}$ (sum);
for $i = 1, \dots, ˉ ı$ , $x_{i} \leftarrow y_{i}$ (pseudo-code).

Scalars (random or non-random): regular math font $x, X, α, β, \dots$
Vectors and matrices (random or non-random): bold math font $x, X, α, β, \dots$
Examples:
$X \equiv {(X_{1}, \dots, X_{ˉ n})}^{'}$ (random vector);
$β \equiv {β_{n, k}}_{n = 1, \dots, ˉ n}^{k = 1, \dots, ˉ k}$ (non-random matrix).

Symmetric, positive (semi)definite matrix: square bold math $s^{2}, σ^{2}, Σ^{2}, \dots$ , where $s, σ, Σ, \dots$ are the respective symmetric Riccati roots (47.253).
Examples:
$s^{2}$ is a generic symmetric and positive (semi)definite $ˉ n \times ˉ n$ matrix;
$s_{vol} \equiv s^{vol} \equiv vol (s^{2})$ denotes the “volatility” vector extracted from $s^{2}$ , i.e. $s_{vol} \equiv \sqrt{diag (s^{2})}$ ;
$c^{2} \equiv corr (s^{2})$ denotes the “correlation” matrix extracted from $s^{2}$ , i.e. $c^{2} \equiv Diag (1 . ∕ s_{vol}) s^{2} Diag (1 . ∕ s_{vol})$ ;
$σ^{2}$ in $X \sim N (μ, σ^{2})$ (multivariate counterpart of univariate $X \sim N (μ, σ^{2})$ );
$Σ^{2} \sim Wishart (ν, σ^{2})$ (matrix-variate counterpart of univariate $Σ^{2} \sim Gamma (k, θ)$ ).

Date/point-in-time: Latin letter $t, s, u, \dots$
Period/time span: Greek letter $τ, υ, \dots$
Examples:
$t_{now} =$ 12/01/2014, 13h:07m:04s:472ms (current time);
$t_{hor} =$ 12/02/2014, 00h:00m:00s:000ms (investment horizon);
$t_{start} =$ 01/07/2010, 00h:00m:00s:000ms (inception date/vintage);
$t_{end} =$ 01/07/2018, 00h:00m:00s:000ms (time of expiry/maturity date);
$τ = 3$ years (time to expiry/maturity);
$α =$ 2.4 years (age of a contract).

Value at time $t$ of one unit of an instrument: $V_{t}$ (in general value is not the same as price).
Examples:
$V_{t}^{stock}$ is value ( $=$ price) of one share of stock;
$V_{t}^{bond}$ is value ( $=$ dirty price, $\neq$ price) of a one-dollar notional coupon bond;
$V_{t}^{zcb}$ is value ( $=$ price) of a one-dollar notional zero-coupon bond;
$V_{t}^{fwd}$ is value ( $\neq$ price) of one forward contract;
$V_{t}^{futures}$ is value ( $\neq$ price) of one futures contract;
$V_{t}^{index}$ is value ( $=$ price) of one index (e.g. the S&P 500 index);
$V_{t}^{call}$ is value ( $=$ price) of one call option.
Exception: in market microstructure we use prices $P_{t}$ , because they are more common and equivalent (19.64).

Short form for a process sampled at discrete times (time series) $x . = {. . ., x_{t_{i}}, x_{t_{i + 1}}, x_{t_{i + 2}}, . . .}$ .
Example:
$x . = {\dots, x_{t - 1}, x_{t}, x_{t + 1,} \dots}$ is a time series of values sampled at unit steps.

Time interval for flow variables [W]: $t \to u$
Examples:
$R_{t \to u}$ (return from $t$ to $u$ );
$Π_{t \to u} \equiv {(Π_{1, t \to u}, \dots, Π_{ˉ n, t \to u})}^{'}$ (P&L of $ˉ n$ instruments from $t$ to $u$ ).

Time interval for processes with multiple monitoring times: $t ⇝ u$
Examples:
$X_{t ⇝ u} \equiv (X_{t} | X_{t + 1} | \dots, X_{u - 1} | X_{u})$ (univariate stochastic path from $t$ to $u$ , with equal-lag sampling);
$x_{t ⇝ u} \equiv (x_{t} | x_{t + 1} | \dots | x_{u - 1} | x_{u})$ (multivariate path from $t$ to $u$ , with equal-lag sampling).

Time: throughout the present work, time $t$ is a continuum, or
$t \in R .$ (2)
- On a microscopic scale, such as in market microstructure, events (such as trades) occur at discrete and randomly spaced points $T_{k}$ in the time continuum, or $T_{k} \in R$ . Then a value ${˜ X}_{k}$ (such as a trading price) occurring at time $T_{k}$ is the mark associated with the point $T_{k}$ in a marked point process.
- On a macroscopic scale, such as in dynamic strategies, trading within or across instruments can occur at any point in time $t \in R$ . On this scale we have continuous-time stochastic processes ${X_{t}}_{t \in R}$ , which are assumed right continuous with left limits (cadlag [W]). 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 ${X_{t}}_{t \in R}$ over a discrete time grid ${X_{t_{k}}}_{k \in N}$ . The grid is typically equally spaced, or $t_{0}, t_{1} = t_{0} + Δ, t_{2} = t_{0} + 2 Δ, \dots$ , 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 $t_{0} = 0, t_{1} = 1, t_{2} = 2, \dots$ . Regardless, we consider these ${t_{k}}_{k \in N}$ as instances of a continuum $t \in R$ .
- For valuation (or “pricing”) purposes, we focus on two discrete points in time: the current valuation time $t_{now}$ and the payoff horizon $t_{hor}$ . If we are pricing a rebalancing strategy, such as an option-replicating strategy, trading and rebalancing is assumed to occur at any point in time $t \in [t_{now}, t_{hor}]$ , although we only focus on the two snapshots $t_{now}$ and $t_{hor}$ .

Operators and special functions

$1_{P} \equiv {\begin{matrix} 1 & if P is true 0 & if P is false \end{matrix}$	Iverson brackets [W]
$1_{x \in A}$	indicator function of a set $A$ [W], (notation consistent with Iverson brackets
$δ^{(y)}, δ^{(y)} (x)$	Dirac delta function (point mass) concentrated at $y$ [W]
$δ_{ˉ n \times 1}^{(n)}, δ^{(n)}$	canonical basis
	$δ_{ˉ n \times 1}^{(n)} \equiv ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} 0 \cdot 1 & \leftarrow n -th position \cdot 0 \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠$
$F [v] (x)$	Fourier transform (Table 46.5)
$F^{- 1} [v] (x)$	inverse Fourier transform (Table 46.5)
$J^{q} [f] (x)$	integration operator applied $q$ times (31.20)
$c_{η}^{- 1} (y)$	inverse-call transformation with parameter $η$ (E.1.29)
${(s)}^{+} \equiv max (0, s)$	positive part function
$erf (x)$	error function [W]
${erf}^{- 1} (x)$	inverse error function [W]
$Δ X_{t} \equiv X_{t} - X_{t - 1}$	change (difference, increment)
$L^{p} X_{t} \equiv X_{t - p}$	lag operator of order $p$ (??)
${ewma}_{w}^{τ_{HL}} (t, x .)$	exponentially weighted moving average at time $t$ with half-life $τ_{HL}$ and trailing window of size $w$
$diag (x)$	operator that extracts the main diagonal of an $ˉ n \times ˉ n$ matrix $x$ , i.e.
	$diag (x) \equiv ⎛ ⎜ ⎜ ⎝ \begin{matrix} x_{1, 1} ⋮ x_{ˉ n, ˉ n} \end{matrix} ⎞ ⎟ ⎟ ⎠$
$Diag (x)$	operator that maps an $ˉ n \times 1$ vector $x$ into an $ˉ n \times ˉ n$ matrix which has $x$ on the main diagonal and null entries elsewhere, i.e.
	$Diag (x) \equiv ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} x_{1} & 0 & \dots & 0 0 & x_{2} & \dots & 0 \cdot & \cdot & ⋱ & \cdot 0 & 0 & \dots & x_{ˉ n} \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠$
$\| x \|$	absolute value of the scalar $x$

$det (x)$	determinant of the $ˉ n \times ˉ n$ square matrix $x$
$\| S \|$	cardinality of the set $S$
$⌊ x ⌋$	floor function [W]
$⌈ x ⌉$	ceiling function [W]
$tr (x)$	trace of the $ˉ n \times ˉ n$ square matrix $x$ , i.e. $tr (x) \equiv x_{1, 1} + \dots + x_{ˉ n, ˉ n}$
$vec (x)$	operator that maps an arbitrary $ˉ n \times ˉ k$ matrix $x$ into a $ˉ n ˉ k \times 1$ vector, stacking the columns of $x$ , i.e.
	$vec (x) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} x_{\cdot, 1} x_{\cdot, 2} \cdot x_{\cdot, ˉ k} \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠$
$x^{†}$	pseudo-inverse (47.140) of the matrix $x$
$x^{*}$	conjugate transpose of the complex-valued matrix $x$
$\| \| \cdot \| \|$	Euclidean (vector) norm
$\| \| \cdot \| \|_{F}$	Frobenius (matrix) norm
$. ∕$	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)
$\circ$	Hadamard product [W]
$\otimes$	Kronecker product [W]
$\oplus$	Kronecker sum [W]
$K_{ˉ k, ˉ n}$	$ˉ k ˉ n \times ˉ k ˉ n$ commutation matrix (47.147)
$[f * g] (x)$	convolution [W]
${[a * b]}_{t}$	discrete convolution (??)
${sort}_{x}$	sorting function which allows to obtain from a generic set ${x_{i}}_{i \in J}$ the set of sorted elements ${x_{i}^{sort}}_{i \in J}$ , where $x_{i}^{sort} \equiv x_{{sort}_{x} (i)}$ and $i \mapsto {sort}_{x} (i)$ is a permutation of the indexes $i \in J$ such that $x_{1}^{sort} \leq \dots \leq x_{i}^{sort} \leq \dots$
$\propto$	directly proportional to [W]

Calculus

$g (x) = {(g_{1} (x), \dots, g_{ˉ m} (x))}^{'}$	multivariate function
$\nabla_{x} g \equiv (\nabla_{x} g_{1} \| \dots \| \nabla_{x} g_{ˉ m})$	gradient
$\nabla_{x, x}^{2} g \equiv (\nabla_{x, x}^{2} g_{1} \| \dots \| \nabla_{x, x}^{2} g_{ˉ m})$	Hessian
${[j_{g}]}_{m, n} \equiv \frac{\partial g_{m}}{\partial x_{n}}$	Jacobian

See Chapter 48.1.3 for details.

Probability and general distribution theory

$P$	real world probability
$Q$	risk neutral probability
$f_{X}$	probability density function (pdf) of $X$
$F_{X}$	cumulative distribution function (cdf) of $X$
$φ_{X}$	characteristic function of $X$
$X d = Y$	equality in distribution (i.e. $f_{X} = f_{Y}$ )
$f_{X \| z} (x)$ (or $f (x \| z)$ )	conditional pdf of $X \| z$
$ϕ$	univariate standard normal pdf
$Φ$	univariate standard normal cdf
$ϕ_{ϱ^{2}}^{2}$	multivariate standard normal pdf (correlation matrix: $ϱ^{2}$ )
$Φ_{ϱ^{2}}^{2}$	multivariate standard normal cdf (correlation matrix: $ϱ^{2}$ )
$Φ_{ν}$	univariate standard Student t cdf (degrees of freedom: $ν$ )
$f_{μ, σ^{2}}^{2} (x)$	pdf of an elliptical random variable $X \sim El (μ, σ^{2}, g_{ˉ n})$ , including normal ( $f_{μ, σ^{2}}^{N} (x)$ ), Student $t$ ( $f_{μ, σ^{2}, ν}^{t} (x)$ ) and Cauchy ( $f_{μ, σ^{2}}^{Cauchy} (x)$ )
$φ_{μ, σ^{2}}^{2} (ω)$	characteristic function of an elliptical random variable $X \sim El (μ, σ^{2}, g_{ˉ n})$ , including normal ( $φ_{μ, σ^{2}}^{N} (x)$ ), Student $t$ ( $φ_{μ, σ^{2}, ν}^{t} (x)$ ) and Cauchy ( $φ_{μ, σ^{2}}^{Cauchy} (x)$ )
$I_{t}$	information (generator)
$f_{Θ}^{pri}$	prior distribution (pdf)
$f_{Θ}^{pos}$	posterior distribution (pdf)
${Cop}_{X} \Leftrightarrow f_{U}$	copula of $X$ represented by the pdf of the grades $U$
$θ$	vector of parameters

Summary statistical features

$S {X}$	generic statistical feature of $X$
$L oc {X}$	location feature of $X$
$D isp {X}$	dispersion feature of $X$
$E {X}$	expectation vector, each entry is the expectation of the respective entry in $X$
$E_{t} {X}$ or $E {X \| i_{t}}$	expectation vector conditioned on the information $i_{t}$
$V {X}$	variance vector, each entry is the variance of the respective entry in $X$
$S d {X}$	standard deviation vector, each entry is the standard deviation of the respective entry in $X$
$C v {X, Y}$	covariance matrix ( $C v {X} \equiv C v {X, X}$ )
	$C v {X, Y} \equiv ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} C v {X_{1}, Y_{1}} & C v {X_{1}, Y_{2}} & \dots \cdot & C v {X_{2}, Y_{2}} & C v {X_{2}, Y_{3}} & \dots \cdot & ⋱ \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠$
$C r {X, Y}$	correlation matrix ( $C r {X} \equiv C r {X, X} = Diag (1 . ∕ S d {X}) C v {X} Diag (1 . ∕ S d {X})$ )
	$C r {X, Y} \equiv ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} C r {X_{1}, Y_{1}} & C r {X_{1}, Y_{2}} & \dots \cdot & C r {X_{2}, Y_{2}} & C r {X_{2}, Y_{3}} & \dots \cdot & ⋱ \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠$
$S k {\cdot}$	skewness
$K u {\cdot}$	kurtosis
$M od {\cdot}$	mode
$M ed {\cdot}$	median
$R an {\cdot}$	interquantile range
$M o D is {\cdot}$	modal dispersion
$q_{X} (c)$	quantile at confidence level $c$ of $X$
$σ^{2} ≽ 0$	symmetric, positive semi-definite (dispersion)
$σ_{vol}$	“volatility” vector

$ϱ^{2} \equiv corr (σ^{2})$	“correlation” matrix

Distributions

$Bernoulli (p)$	Bernoulli distribution with parameter $p$
$Beta (α, γ)$	Beta distribution with shape parameters $α$ , $γ$
$Cauchy (μ, σ^{2})$	Cauchy distribution with location parameter $μ$ , dispersion parameter $σ^{2}$
$Dirichlet (λ)$	Dirichlet distribution with parameter $λ$
$El (μ, σ^{2}, g_{ˉ n})$	Elliptical distribution with location parameter $μ$ , dispersion parameter $σ$ and generator function $g_{ˉ n}$
$Exponential (λ)$	Exponential distribution with rate parameter $λ$
$Gamma (k, θ)$	Gamma distribution with shape parameter $k$ , scale parameter $θ$ ( $⋆$ )
$LogN (μ, σ^{2})$	(Multivariate) lognormal distribution with location parameter $μ$ , dispersion parameter $σ^{2}$ ( $⋆ ⋆$ )
$N (μ, σ^{2})$	(Multivariate) normal distribution with expectation (vector) $μ$ , (co)variance (matrix) $σ^{2}$
$N (μ, σ^{2}, s^{2})$	(Matrix-variate) normal distribution with expectation (vector) $μ$ , dispersion parameter $σ^{2}$ , dispersion parameter $s^{2}$
$Poisson (λ)$	Poisson distribution of intensity $λ$
$Poisson (λ, γ)$	Poisson distribution of intensity $λ$ , grid size $γ$
$t (μ, σ^{2}, ν)$	(Multivariate) Student $t$ -distribution with location parameter $μ$ , dispersion parameter $σ^{2}$ , degrees of freedom $ν$
$t (μ, σ^{2}, s^{2}, ν)$	(Matrix-variate) Student $t$ -distribution with location parameter $μ$ , dispersion parameter $σ^{2}$ , dispersion parameter $s^{2}$ , degrees of freedom $ν$
$χ_{ν}^{2}$	Chi-squared distribution with degrees of freedom $ν$
$F_{ν_{1}, ν_{2}}$	F distribution with degrees of freedom $ν_{1}$ and $ν_{2}$
$Unif (S)$	(Multivariate) uniform distribution on the set $S$
$Wishart (ν, σ^{2})$	Wishart distribution with degrees of freedom $ν$ , dispersion parameter $σ^{2}$
$InvWishart (ν, σ^{2})$	Inverse-Wishart distribution with degrees of freedom $ν$ , dispersion parameter $σ^{2}$
${x_{t}, p_{t}}_{t = 1}^{ˉ t}$	historical with flexible probabilities distribution

${x^{(j)}, p^{(j)}}_{j = 1}^{ˉ j}$	scenario-probability distribution

( $⋆$ ) The gamma distribution has different equivalent parametrizations [W]. Furthermore, the gamma is a special case of Wishart (27.196).

( $⋆ ⋆$ ) The (multivariate) shifted lognormal distribution with location parameter $μ$ , dispersion parameter $σ^{2}$ and shift parameter $a$ is denoted by $X + a \sim LogN (μ, σ^{2})$ .

Stochastic processes

$C_{t}$	Compound Poisson process
$N_{t}^{λ}$	Poisson process with intensity parameter $λ$
$B_{t}$	Wiener process (standard Brownian motion)

Time conventions and counting indexes

$t$	generic point in time (e.g. "12/01/2014, 13h:07m:04s:472ms")
$u$	generic point in time $u > t$
$t_{start}$	inception time of a contract (such as a swap), which is a point in time
$t_{end}$	time of maturity/expiry of a contract, which is a point in time (as opposed to the time to maturity)
$t_{now}$	current point in time
$t_{hor}$	investment horizon $t_{hor} > t_{now}$ , which is a point in time
$s (t)$	settlement date corresponding to the trade time $t$
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