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.

 - if and only if. - hence. - for all . - there exists . - such that. - set of such that takes place. - converges to . - acts from to . - jump of . . - integer part of real number . - the number of elements in the set . -scalar product of and or with being mathematical expectation or duality action of on . -scalar product of and defined via a summation or integration over the variable (if there is a possibility of confusion with another variable). -scalar product in Hilbert space . -norm of in space . -energy norm, for some . -matrix with elements . -closure of set . - complex conjugate of number . -dual space of space or adjoint operator of operator or polar cone to cone . - Fourier or Laplace transform of . - seminorm consisting of only the highest derivatives, see the section ( Sobolev spaces ). . -function restricted to the interval . . . - Indicator of event or indicator function of the set . It is equal to 1 if the is true ( occurs or ) and zero otherwise. In particular, Prob . - is asymptotically equal to : or for a constant (in Bayesian analysis context) or random variable is distributed like or is asymptotic expansion of . (greater of the two as in "union"). - the minimal -algebra containing the union of the set collections and . (as in "intersection"). . . is the limit - the set of points where the is achieved. -indexes of active constraints at the feasible point . -affine hull of the set . a.s. - almost surely. a.e. - almost everywhere. - drift of the process at time : . - price of a risky bond with zero recovery as observed at The is the maturity of the bond. - ball of radius centered at . -volume of the ball. - (matrix of) volatility of the process at time : . - Borel field on real line. - Borel field over the topological space . - complement of the set . ch.f. - characteristic function. -closure of , see ( Convex Hull, Cone, Relative Interior ). -convex hull of , see ( Convex hull ). see the section ( Function spaces section ). binomial coefficients, , . or - Kronecker's delta. - boundary of the set . -subdifferential of the function at the point . -differential of the function applied to the argument . - in finite element, PDE and Sobolev space sections " " refers to diameter of (spacial set under consideration). d.f. - distribution function. - expectation of taken with respect to the probability measure . - expectation of applied to the random quantity . - risk neutral expectation of , see ( Risk neutral pricing ). - expectation of with respect to the -forward probability measure, see ( T-forward probability measure ). -vector . - forward price as observed at with maturity . - forward LIBOR observed at and effective during , see ( Forward LIBOR ). = , given the settlement dates structure . -feasible direction cone of the set at the point . - characteristic function of a random variable . - the -algebra containing information available at time . - the -algebra generated by the r.v. . GMRA - generalized multiresolution analysis. - -algebra containing both (cross product of) and . - Schwartz space (see the formula ( Schwartz space )). - chunkiness parameter in finite elements sections, hazard rate in financial sections. - -algebra generated by credit events or Poisson jumps. - see the section ( Sobolev spaces section ). iff - if and only if. iid - independent identically distributed. - interior of set . i.o. - infinitely often. -condition number of matrix . - finite difference approximation of the second derivative at the -th knot of the lattice. LHS - left hand side (of equation). see the section ( Function spaces section ). - linear operator, usually differential operator of elliptic type or a generator of Markov process. - space of linear bounded operators acting from to . -strong operator norm from to . MRA - multiresolution analysis. - geometric drift of the process at time : . -geometric drift of the process under numeraire , - some martingale. - normal variable with mean and standard deviation . -normal cone of the set at the point . -null space of matrix or operator . -positive integers, . -non-negative integers, . -integer interval, , for . OST - orthonormal system of translates. - orthogonal projection on the finite element space (see the definition ( Orthogonal L2 projection )). - price of the riskless bond with zero recovery as observed at The is the maturity of the bond. p.m. - probability measure. in pr. - in probability. For example, in pr. means " converges to in probability". -distribution of the random variable . -distribution of the normal variable . - probability of the event . Prob - probability of the event . - projection of A on B. -the set of all absolutely continuous measures with respect to the measure of in real variable context or the class of "shape functions" in finite element context. in finite element context. -set of permutations of integers taken from the range . -set of permutations of integers taken from the range . RHS - right hand side (of equation). - Ritz projection on finite element space (see the definition ( Elliptic Ritz projection )). - riskless rate. r.v. - random variable. - real line. - -dimensional space. -non-negative quadrant of the . . -range of matrix or operator . - correlation of quantities and . - swap rate for a vanilla fixed-for-floating LIBOR swap with payments occurring at ,..., . - finite element space. span - linear span of the set . s.p.m. - subprobability measure. - support set of the function . supp - support set of the function . s.t. - such that. - stopping time or default time. -tangent cone of the set at the point . - (column of) geometric volatility of the process at time : . - sigma algebra generated by path of the process up to the time . -minimal sigma algebra containing the components and . - spectrum of operator . - point spectrum of operator . - a value of a derivative at time . - (column of) standard Brownian motion at . - (column of) standard Brownian motion with respect to the -forward probability measure. - (column of) standard Brownian motion with respect to the risk neutral probability measure. - (column of) standard Brownian motion with respect to the numeraire . - see the section ( Sobolev spaces section ). - event space (complete description of what may happen). - spot dollar price of a pound. -dual space if is a space with linearity and topology, polar cone if is a cone, conjugate operator or matrix if is an operator or a matrix, convex conjugate function if is a function. or - mesh in the chapters on wavelets and finite elements. or - scale and transport operations applied to functions or (wavelet chapter). - standard normal variable. , spot pound price of a dollar. - set of all integers, .
 Notation. Index. Contents.