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 (2b.7), even when this is not a random variable.

Scalar/vector/matrix

Non-bold for all scalar (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 (33.140)

 s2,σ2,Σ2,… (10)

where are the respective symmetric matrix roots (33.450).

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 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 .

Mean-covariance/probabilistic environment

Data/information

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

 it≡{xs}s∈Tt, (22)

rather than the more cumbersome notation (18), 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 the letter (upper case) for the random information variables counterparts of the data variables (22), namely observable information variables that are not yet realized

 It≡{Xs}s∈Tt, (24)

rather than the more cumbersome notation (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 (39.173), notation consistent with Iverson brackets δ(u),δ(u)(t) Dirac delta function (point mass) concentrated at u (36.96) δ(n)¯n×1,δ(n) canonical basis δ(n)¯n×1≡⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0⋅1⋅0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠←n-th position F[g](ω) Fourier transform (36.145) F−1[˜g](t) inverse Fourier transform (36.148) Iq[f](x) integration operator applied q times (6b.48) c−1η(y) inverse-call transformation with parameter η (E.41.29) (s)+≡max(0,s) positive part function erf(x) error function (39.10) 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 (33.108) of the ¯n×¯n square matrix q card(A) cardinality (36.17) 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 (33.463) vec(a) operator that maps an arbitrary ¯n×¯k matrix a into a ¯n¯k×1 vector, stacking the columns of a, i.e. (33.457) vec(a)=⎛⎜ ⎜ ⎜ ⎜⎝a⋅,1a⋅,2⋅a⋅,¯k⎞⎟ ⎟ ⎟ ⎟⎠ x† pseudo-inverse (33.476) of the matrix x xH conjugate transpose (33.132) of the complex-valued matrix x ¯¯¯z=x−iy complex conjugate of the complex number z=x+iy ∥⋅∥2 Euclidean (vector) norm (33.162) ∥⋅∥F Frobenius (matrix) norm (33.243) .∕ 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 (33.470) ⊗ Kronecker product (33.472) ⊕ Kronecker sum [W] K¯k,¯n ¯k¯n×¯k¯n commutation matrix (33.484) [f∗g](x) convolution (36.170) [a∗b]t discrete convolution (36.176) 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