### 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
Random variables (scalars or vectors/matrices): upper case
Examples:
;
;
(realized random variable);
(coefficient).
Exception:  is realized invariant; (not ) is invariant as random variable.
• Dummy counters: (similar for ).
We do not use upper case , to avoid confusion with random variables (such as Poisson counter).
We do not use other letters (except in special cases), to be more succinct and clear.
Examples:
(sum);
for , (pseudo-code).
• Scalars (random or non-random): regular math font
Vectors and matrices (random or non-random): bold math font
Examples:
(random vector);
(non-random matrix).
• Symmetric, positive (semi)definite matrix: square bold math , where are the respective symmetric Riccati roots (47.253).
Examples:
is a generic symmetric and positive (semi)definite matrix;
denotes the “volatility” vector extracted from , i.e. ;
denotes the “correlation” matrix extracted from , i.e. ;
in (multivariate counterpart of univariate );
(matrix-variate counterpart of univariate ).
• Date/point-in-time: Latin letter
Period/time span: Greek letter
Examples:
12/01/2014, 13h:07m:04s:472ms (current time);
12/02/2014, 00h:00m:00s:000ms (investment horizon);
01/07/2010, 00h:00m:00s:000ms (inception date/vintage);
01/07/2018, 00h:00m:00s:000ms (time of expiry/maturity date);
years (time to expiry/maturity);
2.4 years (age of a contract).
• Value at time of one unit of an instrument: (in general value is not the same as price).
Examples:
is value ( price) of one share of stock;
is value ( dirty price, price) of a one-dollar notional coupon bond;
is value ( price) of a one-dollar notional zero-coupon bond;
is value ( price) of one forward contract;
is value ( price) of one futures contract;
is value ( price) of one index (e.g. the S&P 500 index);
is value ( price) of one call option.
Exception: in market microstructure we use prices , because they are more common and equivalent (19.64).
• Short form for a process sampled at discrete times (time series) .
Example:
is a time series of values sampled at unit steps.
• Time interval for flow variables [W]:
Examples:
(return from to );
(P&L of instruments from to ).
• Time interval for processes with multiple monitoring times:
Examples:
(univariate stochastic path from to , with equal-lag sampling);
(multivariate path from to , with equal-lag sampling).
• Time: throughout the present work, time is a continuum, or t∈R. (2)

- On a microscopic scale, such as in market microstructure, events (such as trades) occur at discrete and randomly spaced points in the time continuum, or . Then a value (such as a trading price) occurring at time is the mark associated with the point 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 . On this scale we have continuous-time stochastic processes , which are assumed right continuous with left limits (cadlag [W]). 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 . Regardless, we consider these as instances of a continuum .
- For valuation (or “pricing”) purposes, we focus on two discrete points in time: the current valuation time and the payoff horizon . 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 , although we only focus on the two snapshots and .

#### Operators and special functions

 1P≡{1if P is true0if P is false Iverson brackets [W] 1x∈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(n)ˉn×1,δ(n) canonical basis δ(n)ˉn×1≡⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0⋅1←n-th position⋅0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ F[v](x) Fourier transform (Table 46.5) F−1[v](x) inverse Fourier transform (Table 46.5) Jq[f](x) integration operator applied q times (31.20) c−1η(y) inverse-call transformation with parameter η (E.1.29) (s)+≡max(0,s) positive part function erf(x) error function [W] erf−1(x) inverse error function [W] ΔXt≡Xt−Xt−1 change (difference, increment) LpXt≡Xt−p lag operator of order p (??) ewmaτHLw(t,x.) exponentially weighted moving average at time t with half-life τHL and trailing window of size w diag(x) operator that extracts the main diagonal of an ˉn×ˉn matrix x, i.e. diag(x)≡⎛⎜ ⎜⎝x1,1⋮xˉn,ˉn⎞⎟ ⎟⎠ Diag(x) operator that maps an ˉn×1 vector x into an ˉn×ˉn matrix which has x on the main diagonal and null entries elsewhere, i.e. Diag(x)≡⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝x10⋯00x2⋯0⋅⋅⋱⋅00⋯xˉn⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ |x| absolute value of the scalar x det(x) determinant of the ˉn×ˉn square matrix x |S| cardinality of the set S ⌊x⌋ floor function [W] ⌈x⌉ ceiling function [W] tr(x) trace of the ˉn×ˉn square matrix x, i.e.tr(x)≡x1,1+…+xˉn,ˉn vec(x) operator that maps an arbitrary ˉn×ˉk matrix x into a ˉnˉk×1 vector, stacking the columns of x, i.e. vec(x)=⎛⎜ ⎜ ⎜ ⎜⎝x⋅,1x⋅,2⋅x⋅,ˉk⎞⎟ ⎟ ⎟ ⎟⎠ x† pseudo-inverse (47.140) of the matrix x x∗ conjugate transpose of the complex-valued matrix x ||⋅|| Euclidean (vector) norm ||⋅||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) ∘ Hadamard product [W] ⊗ Kronecker product [W] ⊕ Kronecker sum [W] Kˉk,ˉn ˉkˉn×ˉkˉn commutation matrix (47.147) [f∗g](x) convolution [W] [a∗b]t discrete convolution (??) sortx sorting function which allows to obtain from a generic set {xi}i∈J the set of sorted elements {xsorti}i∈J, where xsorti≡xsortx(i) and i↦sortx(i) is a permutation of the indexes i∈J such that xsort1≤…≤xsorti≤… ∝ directly proportional to [W]

#### Calculus

 g(x)=(g1(x),…,gˉm(x))' multivariate function ∇xg≡(∇xg1|…|∇xgˉm) gradient ∇2x,xg≡(∇2x,xg1|…|∇2x,xgˉm) Hessian [jg]m,n≡∂gm∂xn Jacobian

See Chapter 48.1.3 for details.

#### Probability and general distribution theory

 P real world probability Q risk neutral probability fX probability density function (pdf) of X FX cumulative distribution function (cdf) of X φX characteristic function of X Xd=Y equality in distribution (i.e. fX=fY) fX|z(x) (or f(x|z)) conditional pdf of X|z ϕ univariate standard normal pdf Φ univariate standard normal cdf ϕϱ2 multivariate standard normal pdf (correlation matrix: ϱ2) Φϱ2 multivariate standard normal cdf (correlation matrix: ϱ2) Φν univariate standard Student t cdf (degrees of freedom: ν) fμ,σ2(x) pdf of an elliptical random variable X∼El(μ,σ2,gˉn), including normal (fNμ,σ2(x)), Student t (ftμ,σ2,ν(x)) and Cauchy (fCauchyμ,σ2(x)) φμ,σ2(ω) characteristic function of an elliptical random variable X∼El(μ,σ2,gˉn), including normal (φNμ,σ2(x)), Student t (φtμ,σ2,ν(x)) and Cauchy (φCauchyμ,σ2(x)) It information (generator) fpriΘ prior distribution (pdf) fposΘ posterior distribution (pdf) CopX⇔fU 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 Loc{X} location feature of X Disp{X} dispersion feature of X E{X} expectation vector, each entry is the expectation of the respective entry in X Et{X} or E{X|it} expectation vector conditioned on the information it V{X} variance vector, each entry is the variance of the respective entry in X Sd{X} standard deviation vector, each entry is the standard deviation of the respective entry in X Cv{X,Y} covariance matrix (Cv{X}≡Cv{X,X}) Cv{X,Y}≡⎛⎜ ⎜ ⎜ ⎜ ⎜⎝Cv{X1,Y1}Cv{X1,Y2}⋯⋅Cv{X2,Y2}Cv{X2,Y3}⋯⋅⋱⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ Cr{X,Y} correlation matrix (Cr{X}≡Cr{X,X}=Diag(1.∕Sd{X})Cv{X}Diag(1.∕Sd{X})) Cr{X,Y}≡⎛⎜ ⎜ ⎜ ⎜ ⎜⎝Cr{X1,Y1}Cr{X1,Y2}⋯⋅Cr{X2,Y2}Cr{X2,Y3}⋯⋅⋱⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ Sk{⋅} skewness Ku{⋅} kurtosis Mod{⋅} mode Med{⋅} median Ran{⋅} interquantile range MoDis{⋅} modal dispersion qX(c) quantile at confidence level c of X σ2≽0 symmetric, positive semi-definite (dispersion) σvol “volatility” vector ϱ2≡corr(σ2) “correlation” matrix

#### Distributions

 Bernoulli(p) Bernoulli distribution with parameter p Beta(α,γ) Beta distribution with shape parameters α, γ Cauchy(μ,σ2) Cauchy distribution with location parameter μ, dispersion parameter σ2 Dirichlet(λ) Dirichlet distribution with parameter λ El(μ,σ2,gˉn) Elliptical distribution with location parameter μ, dispersion parameter σ and generator function gˉn Exponential(λ) Exponential distribution with rate parameter λ Gamma(k,θ) Gamma distribution with shape parameter k, scale parameter θ (⋆) LogN(μ,σ2) (Multivariate) lognormal distribution with location parameter μ, dispersion parameter σ2 (⋆⋆) N(μ,σ2) (Multivariate) normal distribution with expectation (vector)μ, (co)variance (matrix)σ2 N(μ,σ2,s2) (Matrix-variate) normal distribution with expectation (vector)μ, dispersion parameterσ2, dispersion parameters2 Poisson(λ) Poisson distribution of intensity λ Poisson(λ,γ) Poisson distribution of intensity λ, grid size γ t(μ,σ2,ν) (Multivariate) Student t-distribution with location parameter μ, dispersion parameter σ2, degrees of freedom ν t(μ,σ2,s2,ν) (Matrix-variate) Student t-distribution with location parameter μ, dispersion parameterσ2, dispersion parameters2, degrees of freedom ν χ2ν Chi-squared distribution with degrees of freedom ν Fν1,ν2 F distribution with degrees of freedom ν1 and ν2 Unif(S) (Multivariate) uniform distribution on the set S Wishart(ν,σ2) Wishart distribution with degrees of freedom ν, dispersion parameter σ2 InvWishart(ν,σ2) Inverse-Wishart distribution with degrees of freedom ν, dispersion parameter σ2 {xt,pt}ˉtt=1 historical with flexible probabilities distribution {x(j),p(j)}ˉjj=1 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 , dispersion parameter and shift parameter is denoted by .

#### Stochastic processes

 Ct Compound Poisson process Nλt Poisson process with intensity parameter λ Bt 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 tstart inception time of a contract (such as a swap), which is a point in time tend time of maturity/expiry of a contract, which is a point in time (as opposed to the time to maturity) tnow current point in time thor investment horizon thor>tnow, which is a point in time s(t) settlement date corresponding to the trade time t &