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Subject Index

American option, 67, 229 backward SDE, 269 binomial tree, 286, 295 hedging convergence, 237 jump diffusion, 322 modified binomial tree, 302

Asian option, 67 - hedging in complete case, 247

Barrier option correction terms, 287 discrete, 288 lattice, 292, 293, 301, 304 nodes, 293 two-dimensional, 288

Black-Karasinski Model, 383 Brownian motion

geometric, 196 - martingale problem, 81

Cox-Ingersoll-Ross model, 55, 284, 382

Donsker's theorem, 94 Doob's theorems, 8

Heath-Jarrow-Morton model, 375, 383, 384, 386, 399

Ho-Lee model, 383 Hull-White

short-rate, 377, 378 stochastic volatility, III, 314 trinomial tree, 306, 379, 380 two-dimensional tree, 307, 387

Levy Processes, 57, 321 financial modelling, 321 approximations, 328 characteristic functions, 57 Levy-Khintchine formula, 75 subordinators, 335

Lindeberg condition, 93 Lindeberg-Feller theorem, 93

Lookback option valuation, 267

- discrete case, 288 examples, 247 hedging in complete case, 246 lattice models, 294

Marked Point Process, 29 convergence, 351 counting measure, 30 definition, 29 examples, 36 intensity measure, 33 portfolio rebalancing, 349

Markov Process - strong, 42 - Clark-Hauss mann formula, 251

continuous strong, 40 convergence, 46 diffusion with jumps, 80 Dynkin formula, 53 interpolation, 153 limit theorems, 88 Markov jump process, 44 martingale approach, 48, 81 risk-minimization, 260 semigroup, 42 stochastic differential equation, 40 term structure, 377, 381, 384, 385 transition function, 41

Martingale, 7, 8 no-arbitrage, 161 orthogonality, 13 uniformly integrable, 9, 10 and stochastic integral, 28 central limit theorem, 87 characterization of Markov processes, 48,88 continuous, 13 decomposition, 14 difference triangular array, 95

420 Subject Index

discontinuous, 13 discrete time, 58 Gaussian, 79, 87 generators, 35 limits of , 87 local, 9, 10 measure, 32, 161, 162 method,75 multiplicity, 35 of class (D), 15 problem, 48, 81

- representation, 34 special, 59 square-integrable, 9

- uniformly integrable, 9, 12 Minimal martingale measure, 202, 203,

259, 260, 262, 265, 317, 360, 396 discretization, 361 ARCH, 314, 321 bond pricing, 396 definition, 203 density, 204 discrete time, 204

- existence, 204 multinomial, 209 point process, 350 Radon-Nikodym derivative, 205 random discretization, 366 risk-minimization, 255, 257, 262 robustness, 209, 211, 242 weak convergence of options prices, 205

Poisson Process as limit, 329 compound, 260

- Levy process, 322 simulation, 330

Quadratic variation, 20, 21, 60, 107, 148, 153-155, 158, 193, 203, 250, 257

Radon-Nikodym, 23 bond pricing, 396 convergence, 169, 170 discrete time, 163 Levy processes, 324 minimal martingale measure, 205, 350 optimal strategy, 169

- trinomial case, 189 Risk-Minimizing Strategy, 255

- continuous time, 257 -- computation, 259 -- definition, 258 - convergence, 262, 263, 265, 319, 351,

361 discrete time, 255 locally, 258

- Markov case, 260 pseudo-locally, 260

- random portfolio rebalancing, 350 Ritchken and Sankarasubramanian

model,384

Semimartingale, 14 characteristics, 75 decomposition, 14 definition, 14

- Ito formula, 22 quadratic variation, 20 special, 14 stochastic integration, 19 weak convergence, 75

Stochastic Integral, 15 covariation, 22 dominated convergence, 28 first kind, 15 second kind, 24 weak convergence, 100

Transaction Costs, 217 Leland's model, 218 Boyle and Vorst Model, 221

- convergence, 219, 220, 228 hedging, 218 option pricing, 217 rebalancing, 227, 370

Triangular Arrays, 92 general semimartingale, 95

- independent, 92 - infinitesimal, 92

martingale-difference, 95 strongly mixing, 97

Two-factor Gaussian model G2++, 387

Uniform Tightness Condition, 101 implied by, 101 convergence of SDE, 105 definition, 101 equivalent conditions, 102 stability by stochastic integration, 104 sufficient conditions, 102

Symbol Index

(B, C, v): triplet of characteristics, 76 (M, N)1-e: scalar product in 1-t2, 13 11 A: indicator of the set A, 5 Af: infinitesimal generator applied to

function f, 43 B(JRd): set of all Borel-measurable and

bounded functions from JRd to JR, 42 Cl(JRd): subset of C2(JRd) which

contains only nonnegative functions, 83

C2(JRd): set of all continuous bounded functions from JRd to JR which are equal to 0 around 0 and have a limit at infinity, 83

D;f: first partial derivative of the function f, 22

Dij f: second partial derivative of the function f, 22

H (JP, Q): Hellinger integral of the probabilities JP and Q, 108

H.X: integral of the process H with respect to the process X, 11

H * J.!: integral of process H with respect to the random measure J.!, 31

Ix: linear operator associated to the stochastic integral with respect to process X, 16

In '---> I,o: convergence of the sequence of intervals In to the limit interval I, 215

J x: linear mapping associated to stochastic integral with respect to process X, 16

Ps,t: Markov process transition function, 41

51\ T, 5 V T: minimum and maximum of two stopping times 5 and T, 3

T(t): Markov process semigroup, 42 TA: first entry time in the set A, 6 Var(A): variation process of A, 11

X = Y a.s.: X and Y equal almost surely, 5

X'" Y: X and Y equivalent, 4 X*: X; = sUPo<.<t IX.I, 16 XT: X stopped-at time T, 4

L(E) Xn ==> X.: weak convergence of the

sequence X n , with values in the Polish space E, to the process X, 70

L(iJl(JRd )) Xn ==> X: weak convergence of the

sequence of rcll processes Xn to the process X, 71

Xn S X: finite-dimensional weak convergence, 71

X_: left limit of X, 4 [X, Y]: quadratic co-variation of two

semimartingales X and Y, 20 [[5, T]]: stochastic interval, 5 C(j[J)(JRd»: bounded and continuous

mappings from j[J)(JRd) to JR, 1 C(JRd): space of all continuous functions

from JR + to JRd, 65 j[J)(JRd): right-continuous functions

having left limits, 1, 64 j[J)"cp, 16 .1X: jump of X, 4 IEiP[XI9]: conditional expectation of X

with respect to the sub-field 9 under the probability JP, 8

IEiP[X]: expectation of X under the probability JP, 8

IF = (:Ft, t E T): increasing family of sub-sigma-fields, 2

IL: set of all adapted processes, left continuous and right limited (led), 16

IL 2 (X): set of processes H such that

(it 2 , XTn-) is integrable for each stopping time Tn, 26

422 Symbol Index

ILl: set of random variables X such that IXIP is integrable, 9

lLuep , 16 M drn : set of k x m matrices with

real-valued coefficients, 69 Q 1- lP': Q and lP' are singular, 109 Qn <J lP' n: the sequence of probabilities

(Qn)n is contiguous to the sequence (lP'n)n, 115

Qn6lP'n: the sequence of probabilities (Qn)n is entirely separated from the sequence (lP'n)n, 115

§uep, 16 Ate' 12 Aloe, 12 B= (n,F,IF,lP'): stochastic basis, 2 Dt(lRd): D(JRd) = Vt:o:oD~(JRd), 64 Dt(JRd): Dt(JRd) = ns:O:tD~(JRd), 64 D~(JRd): a-field generated by all maps

a ---> a(s) for s :::; t, 64 [(X): Doleans-Dade exponential of X,

23 F X : filtration generated by the process

X,5 FT: filtration and stopping time T, 2 F'f: filtration generated by the MPP <P,

33 J-e: class of all square-integrable

martingales, 9 'Hfoe: space of all locally square­

integrable martingales, 10 M: class of all uniformly integrable

martingales, 9 N: random counting measure, 30 0: optional a-field, 5 P: predictable a-field, 6 S: collection of simple predictable

processes, 15 V: set of all real-valued processes A

that are rcll, adapted with Ao = 0

and whose each path has finite variation, 11

V+: set of all real-valued processes A that are rcll, adapted with Ao = 0 and whose each path is non-decreasing, 11

(M, N): predictable quadratic covaria­tion of two locally square-integrable martingales, 12

«: absolute continuity, 11, 109 lim = lim sup, 117

- 2 II X Ilit: norm defined on 'H , 24

II Z 111-2= )lE[Z2], 24 p(lP', Q): Kakutani-Hellinger distance

between the probabilities lP' and Q, _ }08

'H : space of all special semimartingales - 2

with finite 'H norm, 24 lim = lim inf, 117 b£: space of all adapted processes with

bounded lerl paths, 24 bP: set of bounded processes that are

predictable, 24 h( a; lP', Q): Hellinger process of order a

between lP' and Q, 109 s('H, XIlP'H; B, C, v): (martingale

problem), 82 wN(a, B): modulus of continuity, 65 [xl: integer part of x, 10

Class C 2 : twice continuously­differentiable, 22

Class (D), 10 Convergence ucp: convergence uni­

formly on compacts in probability, 16

MPP: Marked Point Process, 29 MZ: Meyer-Zheng, 74

UT: Uniform Tightness, 101