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
 Nonrandom quantities (scalars or vectors/matrices): lower case x,y,α,β,μ,σ2,…
Random variables (scalars or vectors/matrices): upper case X,Y,A,B,M,Σ2,…
Examples:
X∼N(μ,σ2);
Σ2∼Wishart(ν,σ2);
x (realized random variable);
β (coefficient).
Exception: ϵt is realized invariant; εt (not Et) is invariant as random variable.
 Dummy counters: n=1,…,¯n (similar for i,j,k,l,t,…).
We do not use upper case n=1,…,N, to avoid confusion with random variables (such as Nt Poisson counter).
We do not use other letters n=1,…,j (except in special cases), to be more succinct and clear.
Examples:
∑¯nn=1xn (sum);
for i=1,…,¯ı, xi←yi (pseudocode).
 Scalars (random or nonrandom): regular math font x,X,α,β,…
Vectors and matrices (random or nonrandom): bold math font x,X,α,β,…
Examples:
X≡(X1,…,X¯n)' (random vector);
β≡{βn,k}k=1,…,¯kn=1,…,¯n (nonrandom matrix).
 Symmetric, positive (semi)definite matrix: square bold math s2,σ2,Σ2,…, where s,σ,Σ,…are the respective symmetric Riccati roots (15.428).
Examples:
s2 is a generic symmetric and positive (semi)definite ¯n×¯n matrix;
svol≡svol≡vol(s2) denotes the “volatility” vector extracted from s2, i.e. svol≡√diag(s2);
c2≡corr(s2) denotes the “correlation” matrix extracted from s2, i.e. c2≡Diag(1.∕svol)s2Diag(1.∕svol);
σ2 in X∼N(μ,σ2) (multivariate counterpart of univariate X∼N(μ,σ2));
Σ2∼Wishart(ν,σ2) (matrixvariate counterpart of univariate Σ2∼Gamma(k,θ)).
 Date/pointintime: Latin letter t,s,u,…
Period/time span: Greek letter τ,υ,…
Examples:
tnow= 12/01/2014, 13h:07m:04s:472ms (current time);
thor= 12/02/2014, 00h:00m:00s:000ms (investment horizon);
tstart= 01/07/2010, 00h:00m:00s:000ms (inception date/vintage);
tend= 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: Vt (in general value is not the same as price).
Examples:
Vstockt is value (= price) of one share of stock;
Vbondt is value (= dirty price, ≠ price) of a onedollar notional coupon bond;
Vzcbt is value (= price) of a onedollar notional zerocoupon bond;
Vfwdt is value (≠ price) of one forward contract;
Vfuturest is value (≠ price) of one futures contract;
Vindext is value (= price) of one index (e.g. the S&P 500 index);
Vcallt is value (= price) of one call option.
Exception: in market microstructure we use prices Pt, because they are more common and equivalent (0a.64).
 Short form for a process sampled at discrete times (time series) x.={...,xti,xti+1,xti+2,...}.
Example:
x.={…,xt−1,xt,xt+1,…} is a time series of values sampled at unit steps.
 Time interval for flow variables [W]: t→u
Examples:
Rt→u (return from t to u);
Πt→u≡(Π1,t→u,…,Π¯n,t→u)' (P&L of ¯n instruments from t to u).
 Time interval for processes with multiple monitoring times: t⇝u
Examples:
Xt⇝u≡(XtXt+1…,Xu−1Xu) (univariate stochastic path from t to u, with equallag sampling);
xt⇝u≡(xtxt+1…xu−1xu) (multivariate path from t to u, with equallag sampling).
 Time: throughout the present work, time t is a continuum, or
t∈R. (1)  On a microscopic scale, such as in market microstructure, events (such as trades) occur at discrete and randomly spaced points Tk in the time continuum, or Tk∈R. Then a value ˜Xk (such as a trading price) occurring at time Tk is the mark associated with the point Tk 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∈R. On this scale we have continuoustime stochastic processes {Xt}t∈R, which are assumed right continuous with left limits (cadlag [W]). We slice time using the half closehalf 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 {Xt}t∈R over a discrete time grid {Xtk}k∈N. The grid is typically equally spaced, or t0,t1=t0+Δ,t2=t0+2Δ,…, 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,…. Regardless, we consider these {tk}k∈N as instances of a continuum t∈R.
 For valuation (or “pricing”) purposes, we focus on two discrete points in time: the current valuation time tnow and the payoff horizon thor. If we are pricing a rebalancing strategy, such as an optionreplicating strategy, trading and rebalancing is assumed to occur at any point in time t∈[tnow,thor], although we only focus on the two snapshots tnow and thor.
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)¯n×1,δ(n) 
canonical basis 

δ(n)¯n×1≡⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0⋅1←nth position⋅0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ 
F[v](x) 
Fourier transform (Table 17.3) 
F−1[v](x) 
inverse Fourier transform (Table 17.3) 
Iq[f](x) 
integration operator applied q times (24.20) 
c−1η(y) 
inversecall 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 halflife 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(x) 
determinant of the ¯n×¯n square matrix x 
card(S) 
cardinality of the set S 
⌊x⌋ 
floor function [W] 
⌈x⌉ 
ceiling function [W] 
tr(x) 
trace of the ¯n×¯n square matrix x, i.e.tr(x)≡x1,1+…+x¯n,¯n 
vec(x) 
operator that maps an arbitrary ¯n×¯k matrix x into a ¯n¯k×1 vector, stacking the columns of x, i.e. 

vec(x)=⎛⎜ ⎜ ⎜ ⎜⎝x⋅,1x⋅,2⋅x⋅,¯k⎞⎟ ⎟ ⎟ ⎟⎠ 
x† 
pseudoinverse (15.454) of the matrix x 
x∗ 
conjugate transpose of the complexvalued matrix x 
∥⋅∥ 
Euclidean (vector) norm 
∥⋅∥F 
Frobenius (matrix) norm 
.∕ 
elementwise 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] 
K¯k,¯n 
¯k¯n×¯k¯n commutation matrix (15.462) 
[f∗g](x) 
convolution (17.103) 
[a∗b]t 
discrete convolution (17.105) 
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) 
¯ndimensional 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) 
¯ndimensional 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 
¯ndimensional unit hypercube 




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




See Chapter 16.1.3 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) 
fXz(x) (or f(xz)) 
conditional pdf of Xz 
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,g¯n), 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), 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{Xit} 
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{ 