 ### 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,α,β,μ,σ2,… (1)

Upper case for all random variables (scalars or vectors/matrices) X,Y,A,B,M,Σ2,… (2)

Examples:

- Random variable with normal distribution X∼N(μ,σ2); (3)

- Realization of a random vector xrealization⇐=X. (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 ϵrealization⇐ε; (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) x,X,α,β,…. (6)

Bold for all vectors and matrices (random or non-random) x,X,α,β,…. (7)

Examples:

- Matrix of coefficients β≡{βn,k}k=1,…,¯kn=1,…,¯n; (8)

- Normal random vector X≡⎛⎜ ⎜⎝X1⋮X¯n⎞⎟ ⎟⎠∼N(μ,σ2), (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) s2,σ2,Σ2,…, (10)

where are the respective symmetric matrix roots (13.452).

Examples:

- Covariance matrix σ2≡{[σ2]n,m}¯nn,m=1 (11)

of a normally distributed multivariate random variable X∼N(μ,σ2); (12)

- Random matrix Σ2≡{[Σ2]n,m}¯nn,m=1 (13)

with Wishart distribution Σ2∼Wishart(ν,σ2). (14)

Dummy counters

Same base letter for a dummy counter and its end point n=1,…,¯n (15)

and similar for all other letters/counters

Examples:

- Summation ∑¯nn=1cn; (16)

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

Collection of variables

Calligraphic subscript for collections of (multivariate) variables XJ≡{Xj}j∈J, (17)

where is a continuum or discrete set of indices. The counterpart of the above (17) for realized variables is xJ≡{xj}j∈J. (18)

Examples:

- Entries of a vector xN≡{xn}n∈N, (19)

where ;

- (Multivariate) stochastic processes monitored at selected times XT≡{Xt}t∈T, (20)

where the monitoring times can be discrete , or a continuum ;

- (Multivariate) random fields monitored at selected locations XU≡{Xt}t∈U, (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 it≡{xs}s∈Tt, (22)

where the set of observations points up to time may vary from case to case Tt≡⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩{t1,t2,…,t¯¯¯m}≤tfinite observations, (un)evenly spaced{−∞,…t−1,t}infinite discrete observations, evenly spaced, includes current[t0,t]continuous observations, finite interval(−∞,t]continuous observations, infinite interval...... (23)

We use upper case , rather than the more cumbersome notation (17), for the random information variables, counterparts of the data variables (22) It≡{Xs}s∈Tt, (24)

consistently with the convention (17).

Examples:

- Equally (unit) spaced data samples of a stochastic process from inception to the latest available realization it≡{xtstart,xtstart+1,…,xtend},t−1

Time

We think of time as a continuum, and we use latin letters to denote a point-in-time t,s,u,…∈R. (26)

We use greek letters for a period or time span τ=t−s,υ=v−u,…. (27)

Examples:

- Current time tnow=27 December 2014, 18:07:04:472; (28)

- Time to expiry/maturity τ=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 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

 t0=0,t1=1,t2=2,…. (30)

#### Operators and special functions 1P≡{1if P is true0if P is false Iverson brackets [W] 1x∈A indicator function of a set A (19.178), notation consistent with Iverson brackets δ(u),δ(u)(t) Dirac delta function (point mass) concentrated at u (16.96) δ(n)¯n×1,δ(n) canonical basis δ(n)¯n×1≡⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0⋅1⋅0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠←n-th position F[g](ω) Fourier transform (16.140) F−1[˜g](t) inverse Fourier transform (16.143) Iq[f](x) integration operator applied q times (24.48) c−1η(y) inverse-call transformation with parameter η (E.1.29) (s)+≡max(0,s) positive part function erf(x) error function (19.100) erf−1(x) inverse error function [W] ΔXt≡Xt−Xt−1 change (difference, increment) Sp[X]t≡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 ¯n×¯n matrix x, i.e. diag(x)≡⎛⎜ ⎜⎝x1,1⋮x¯n,¯n⎞⎟ ⎟⎠ Diag(x) operator that maps an ¯n×1 vector x into an ¯n×¯n matrix which has x on the main diagonal and null entries elsewhere, i.e. Diag(x)≡⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝x10⋯00x2⋯0⋅⋅⋱⋅00⋯x¯n⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ |x| absolute value of the scalar x det(q) determinant (13.108) of the ¯n×¯n square matrix q card(A) cardinality (16.17) of the set A ⌊x⌋ floor function [W] ⌈x⌉ ceiling function [W] tr(q) trace of the ¯n×¯n square matrix q, i.e.tr(q)≡x1,1+…+x¯n,¯n (13.465) vec(a) operator that maps an arbitrary ¯n×¯k matrix a into a ¯n¯k×1 vector, stacking the columns of a, i.e. (13.459) vec(a)=⎛⎜ ⎜ ⎜ ⎜⎝a⋅,1a⋅,2⋅a⋅,¯k⎞⎟ ⎟ ⎟ ⎟⎠ x† pseudo-inverse (13.478) of the matrix x xH conjugate transpose (13.132) of the complex-valued matrix x ¯¯¯z=x−iy complex conjugate of the complex number z=x+iy ∥⋅∥2 Euclidean (vector) norm (13.161) ∥⋅∥F 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] K¯k,¯n ¯k¯n×¯k¯n commutation matrix (13.486) [f∗g](x) convolution (16.165) [a∗b]t discrete convolution (16.171) sortx sorting function which allows to obtain from a generic set {xi}i∈I the set of sorted elements {xsorti}i∈I, where xsorti≡xsortx(i) and i↦sortx(i) is a permutation of the indexes i∈I such that xsort1≤…≤xsorti≤… ∝ directly proportional to [W]

#### Sets ∂Er(m,σ2)≡{x∈R¯n:(x−m)'(σ2)−1(x−m)=r2} ellipsoid with center m, shape σ2 and radius r (when r=1 the subscript is dropped) ∂B¯n≡∂E(0,I¯n) ¯n-dimensional unit sphere Er(m,σ2)≡{x∈R¯n:(x−m)'(σ2)−1(x−m)≤r} filled ellipsoid with center m, shape σ2 and radius r (when r=1 the subscript is dropped) B¯n≡E(0,I¯n) ¯n-dimensional filled unit ball S¯n−1≡{x∈R¯n:∑¯nn=1xn=1 and xn≥0 for all n} unit simplex in R¯n [0,1]¯n≡[0,1]×⋯×[0,1]¯n times ¯n-dimensional unit hypercube

#### Calculus g(x)=(g1(x),…,g¯¯¯m(x))' multivariate function (14.58) ∇f(x)≡(∇f1(x)|…|∇f¯¯¯m(x)) gradient (14.60) ∇2f(x)≡(∇2f1(x)|⋯|∇2f¯¯¯m(x)) Hessian (14.78) jf≡(∇f(x))' Jacobian (14.61)

See Chapter 14.1.3 for details.

#### Probability and general distribution theory P real world probability Q risk neutral probability fX probability density function (pdf) (19.34) of X FX cumulative distribution function (cdf) (19.36) of X φX characteristic function (19.39) of X Xd=Y equality in distribution (i.e. fX=fY) (19.45) fX|z(x) (or f(x|z)) conditional pdf (19.57) of X|z ϕ univariate standard normal pdf Φ univariate standard normal cdf (19.102) 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,g¯n) (19.249), 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,g¯n) (19.259), including normal (φNμ,σ2(x)), Student t (φtμ,σ2,ν(x)) and Cauchy (φCauchyμ,σ2