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A. Notation.
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Notation.


$\Leftrightarrow$ - if and only if.

$\Rightarrow$ - hence.

$\forall x$ - for all $x$ .

$\exists x$ - there exists $x$ .

$:~$ - such that.

MATH - set of $x$ such that $A$ takes place.

$a\rightarrow b$ - $a$ converges to $b$ .

$f:A\rightarrow B$ - $\ f$ acts from $A$ to $B$ .

$\left[ U\right] $ - jump of $U$ . MATH .

$\left[ x\right] $ - integer part of real number $x$ .

$\#S$ - the number of elements in the set $S$ .

MATH -scalar product of $a$ and $b$ or MATH with $E$ being mathematical expectation or duality action of $a\in X^{\ast}$ on $b\in X$ .

MATH -scalar product of $a$ and $b$ defined via a summation or integration over the variable $x$ (if there is a possibility of confusion with another variable).

MATH -scalar product in Hilbert space $H$ .

MATH -norm of $x$ in space $X$ .

MATH -energy norm, MATH for some $X$ .

MATH -matrix with elements $x_{kp}$ .

$\bar{X}$ -closure of set $X$ .

$\bar{x}$ - complex conjugate of number $x$ .

$X^{\ast}$ -dual space of space $X$ or adjoint operator of operator $X$ or polar cone to cone $X$ .

$\hat{x}$ - Fourier or Laplace transform of $x$ .

MATH - seminorm consisting of only the highest derivatives, see the section ( Sobolev spaces ).

MATH .

MATH -function $u$ restricted to the interval $\left[ a,b\right] $ .

MATH .

MATH .

$1_{A}$ - Indicator of event $A$ or indicator function of the set $A$ . It is equal to 1 if the $A$ is true ( $A$ occurs or MATH ) and zero otherwise. In particular, Prob MATH .

$a\symbol{126}b$ - $a$ is asymptotically equal to $b$ : $\lim\frac{a}{b}=1$ or $a=Cb$ for a constant $C$ (in Bayesian analysis context) or random variable $a$ is distributed like $b$ or $b$ is asymptotic expansion of $a$ .

MATH (greater of the two as in "union").

MATH - the minimal $\sigma$ -algebra containing the union of the set collections $\QTR{cal}{F}_{1}$ and $\QTR{cal}{F}_{2}$ .

MATH (as in "intersection").

$x^{+}=x\vee0$ .

MATH .

$y_{t\pm}$ is the limit MATH

MATH - the set of points $x\in C$ where the MATH is achieved.

MATH -indexes of active constraints at the feasible point $x^{\ast}$ .

MATH -affine hull of the set $X$ .

a.s. - almost surely.

a.e. - almost everywhere.

$a_{X,t}$ - drift of the process $X_{t}$ at time $t$ : MATH .

MATH - price of a risky bond with zero recovery as observed at $t.$ The $T$ is the maturity of the bond.

MATH - ball of radius $r$ centered at $x_{0}$ . MATH -volume of the ball.

$b_{X,t}$ - (matrix of) volatility of the process $X_{t}$ at time $t$ : MATH .

$\QTR{cal}{B}$ - Borel field on real line.

MATH - Borel field over the topological space $X$ .

$A^{c}$ - complement of the set $A$ .

ch.f. - characteristic function.

$cl\left( A\right) $ -closure of $A$ , see ( Convex Hull, Cone, Relative Interior ).

MATH -convex hull of $A$ , see ( Convex hull ).

MATH see the section ( Function spaces section ).

$C_{n}^{k}-$ binomial coefficients, MATH , MATH .

$\delta_{pq}$ or $\delta_{p,q}$ - Kronecker's delta.

$\partial X$ - boundary of the set $X$ .

MATH -subdifferential of the function $f$ at the point $x$ .

MATH -differential of the function MATH applied to the argument $t$ .

$d$ - in finite element, PDE and Sobolev space sections " $d$ " refers to diameter of $\Omega$ (spacial set under consideration).

d.f. - distribution function.

MATH - expectation of $X$ taken with respect to the probability measure $P$ .

MATH - expectation of $f\left( y\right) $ applied to the random quantity $y$ .

MATH - risk neutral expectation of $X$ , see ( Risk neutral pricing ).

MATH - expectation of $X$ with respect to the $T$ -forward probability measure, see ( T-forward probability measure ).

$e_{m}$ -vector MATH .

MATH - forward price as observed at $t$ with maturity $T$ .

MATH - forward LIBOR observed at $t$ and effective during MATH , see ( Forward LIBOR ).

MATH = MATH , given the settlement dates structure MATH .

MATH -feasible direction cone of the set $X$ at the point $x$ .

$f_{X}$ - characteristic function of a random variable $X$ .

$\QTR{cal}{F}_{t}$ - the $\sigma$ -algebra containing information available at time $t$ .

MATH - the $\sigma$ -algebra generated by the r.v. $X$ .

GMRA - generalized multiresolution analysis.

$\QTR{cal}{G}_{t}$ - $\sigma$ -algebra containing both (cross product of) $\QTR{cal}{F}_{t}$ and $\QTR{cal}{H}_{t}$ .

MATH - Schwartz space (see the formula ( Schwartz space )).

$\gamma$ - chunkiness parameter in finite elements sections, hazard rate in financial sections.

$\QTR{cal}{H}_{t}$ - $\sigma$ -algebra generated by credit events or Poisson jumps.

$H^{k},H_{0}^{k}$ - see the section ( Sobolev spaces section ).

iff - if and only if.

iid - independent identically distributed.

MATH - interior of set $A$ .

i.o. - infinitely often.

MATH -condition number of matrix $A$ .

$\Lambda_{k}$ - finite difference approximation of the second derivative at the $k$ -th knot of the lattice.

LHS - left hand side (of equation).

$L^{p},L_{loc}^{p}$ see the section ( Function spaces section ).

$L$ - linear operator, usually differential operator of elliptic type or a generator of Markov process.

MATH - space of linear bounded operators acting from $X$ to $Y$ .

MATH -strong operator norm from $X$ to $Y$ .

MRA - multiresolution analysis.

$\mu_{X,t}$ - geometric drift of the process $X_{t}$ at time $t$ : MATH .

$\mu_{X,t}^{Z}$ -geometric drift of the process $X_{t}$ under numeraire $Z$ ,

$M_{t}$ - some martingale.

MATH - normal variable with mean $\mu$ and standard deviation $\sigma$ .

MATH -normal cone of the set $X$ at the point $x$ .

MATH -null space of matrix or operator $A$ .

$\QTR{cal}{N}_{+}$ -positive integers, MATH .

$\QTR{cal}{N}_{0}$ -non-negative integers, MATH .

MATH -integer interval, MATH , MATH for $n\leq m$ .

OST - orthonormal system of translates.

$P_{h}$ - orthogonal projection on the finite element space $S_{h}$ (see the definition ( Orthogonal L2 projection )).

MATH - price of the riskless bond with zero recovery as observed at $t.$ The $T$ is the maturity of the bond.

p.m. - probability measure.

in pr. - in probability. For example, $X_{n}\rightarrow X$ in pr. means " $X_{n}$ converges to $X$ in probability".

$p_{X}$ -distribution of the random variable $X$ .

MATH -distribution of the normal variable MATH .

$P\left( A\right) $ - probability of the event $A$ .

Prob $\left( A\right) $ - probability of the event $A$ .

$\Pr_{B}A$ - projection of A on B.

$\QTR{cal}{P}$ -the set of all absolutely continuous measures with respect to the measure $P$ of MATH in real variable context or the class of "shape functions" in finite element context.

MATH in finite element context.

MATH -set of permutations of $k$ integers taken from the range $1,...,N$ .

MATH -set of permutations of $N$ integers taken from the range $1,...,N$ .

RHS - right hand side (of equation).

$R_{h}$ - Ritz projection on finite element space $S_{h}$ (see the definition ( Elliptic Ritz projection )).

$r_{t}$ - riskless rate.

r.v. - random variable.

$\QTR{cal}{R}$ - real line.

$\QTR{cal}{R}^{m}$ - $m$ -dimensional space.

MATH -non-negative quadrant of the $\QTR{cal}{R}^{m}$ . MATH .

$R\left( A\right) $ -range of matrix or operator $A$ .

$\rho_{XY}$ - correlation of quantities $X$ and $Y$ .

MATH - swap rate for a vanilla fixed-for-floating LIBOR swap with payments occurring at $T_{\alpha}$ ,..., $T_{\beta-1}$ .

$S_{h}$ - finite element space.

span $X$ - linear span of the set $X$ .

s.p.m. - subprobability measure.

$spt~u$ - support set of the function $u$ .

supp $~u$ - support set of the function $u$ .

s.t. - such that.

$\tau$ - stopping time or default time.

MATH -tangent cone of the set $X$ at the point $x$ .

$\sigma_{X,t}$ - (column of) geometric volatility of the process $X_{t}$ at time $t$ : MATH .

MATH - sigma algebra generated by path of the process $B_{u\text{ }}$ up to the time $s$ .

MATH -minimal sigma algebra containing the components $A$ and $B$ .

MATH - spectrum of operator $A$ .

MATH - point spectrum of operator $A$ .

$V_{t}$ - a value of a derivative at time $t$ .

$W_{t}$ - (column of) standard Brownian motion at $t$ .

$W_{t}^{T}$ - (column of) standard Brownian motion with respect to the $T$ -forward probability measure.

$W_{t}^{\ast}$ - (column of) standard Brownian motion with respect to the risk neutral probability measure.

$W^{X}$ - (column of) standard Brownian motion with respect to the numeraire $X$ .

MATH - see the section ( Sobolev spaces section ).

$\Omega$ - event space (complete description of what may happen).

$X_{t}$ MATH - spot dollar price of a pound.

$X^{\ast}$ -dual space if $X$ is a space with linearity and topology, polar cone if $X$ is a cone, conjugate operator or matrix if $X$ is an operator or a matrix, convex conjugate function if $X$ is a function.

$x_{d,k}$ or $x_{k}^{d}$ - mesh in the chapters on wavelets and finite elements.

$\psi_{d,k}$ or $\phi_{d,k}$ - scale and transport operations applied to functions $\psi$ or $\phi$ (wavelet chapter).

$\xi$ - standard normal variable.

$Y_{t}=$ MATH , spot pound price of a dollar.

$\QTR{cal}{Z}$ - set of all integers, MATH .





Notation. Index. Contents.


















Copyright 2007