Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets

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Index of Authors 721

Shiu, E.S.W., 497, 583, 624

Shortland, C.F., 165, 714

Shreve, S.E., 3, 48, 93, 116, 139, 181,

211, 232, 284, 457, 550, 709, 712

Siegel, D., 198

Siegmund, D., 155, 711

Sin, C., 87, 336, 548

Singleton, K., 333, 407

Skorokhod, A.V., 86, 592, 713

Slater, L.J., 664

Sondermann, D., 229

Song, S., 341

Sreehari, A., 597

Srivastav, R.P., 403

Stanton, R., 391

Steele, J.M., 538, 543

Stricker, Ch., 86, 538, 549, 713

Stroock, D.W., 18, 41, 48, 284, 292, 709

Suchanecki, M., 165, 256, 714

Sulem, A., 592

S¨ussmann, H.J., 51

Szatzschneider, W., 333, 365, 710

Szpirglas, J., 328

Tagliani, A., 116

Taleb, N., 165, 185, 714

Tankov, P., 592, 616, 626

Taqqu, M.S., vii, 138, 599, 713

Taylor, H., 259, 457

Tevzadze, R., 539, 709

Thalmaier, A., 57

Tian, Y.S., 372, 711, 714

Tisseyre, A., 592

Trigeorgis, L., 199

Trotter, H.F., 213

Tsirel’son, B., 49, 77, 311, 316

Vallois, P., 634, 711

Van der Hoek, J., 712

van der Weide, J.A.M., 713

van der Weide,J.A.M., 256

Van Schuppen, J. H., 69

Varadhan, S.R.S., 3, 18, 48, 284, 709

Varaiya, P., 59

Ventzel, A.D., 40

Vervaat, W., 597

Ve˘ce˘r, Y., 232, 390

Villeneuve, S., 200, 713

Viswanathan, R., 181, 714

Volkonski, V.A., 263

Volpi, A., 565

Walsh, J., 292, 312

Walter, Ch., 599

Wang, A.T., 216

Wang, H., 568, 569, 711

Warren, J., 293

Watanabe, S., 3, 11, 336, 342, 505, 556,

559, 709, 710

Weidenfeld, G., 262

Weinryb, S., 292

Werner, W., 341

Whaley, R., 582, 588

White, A., 393, 417

White,A., 394

Williams, D., 3, 9, 18, 24, 44, 211, 259,

261, 270, 275, 313, 326, 709

Williams, R.J., 3, 211, 709, 712

Wilmott, P., 165, 714

Wojakowski, R., 383

Wong, B., 336

Wong, D., 419

Wong, E., 3, 69

Wu, L., 256, 713

Wu, S., 256, 713

Xia, J., 87

Xing, H., 586, 713

Xiong, J., 105

Xu, C., 181

Xu, M., 390

Yamazato, M., 155, 711

Yan, J.A., 87

Yen, J-Y., 339

Yoeurp, Ch., 54, 525, 527

Yong, J., 65, 709

Yor, M., 3, 47, 49, 56, 60, 89, 114, 116,

119, 129, 154, 155, 159, 182, 183,

211, 226, 229, 232, 235, 243, 263,

278, 280, 292–295, 306, 312, 315,

316, 323, 325, 326, 328, 333, 339,

341, 344–347, 352, 353, 356, 375,

376, 383, 389, 390, 430, 481, 482,

522, 539, 543, 544, 597, 606, 621,

631, 643, 646, 657, 710, 711, 714

Yu, F., 425

Yuen, K.C., 501

Yuh, Y-S., 155, 711

722 Index of Authors

Zariphopoulou, T., 550

Zhang, P.G., 165, 185

Zhang, T., 46, 709

Zhang, X., 583, 587, 589, 643, 714

Zhao, Z., 278

Zhou, C., 191, 287

Zhou, X.Y., 65, 709

Zolotarev, V.M., 592, 593, 599

Index of Symbols

b(E): bounded Borel functions on

(E,E), 4

cor

,T) Corridor boost option, 185

BinDIC: down-and-in binary call, 165

BinC: binary call, 163

BM: Brownian Motion, 30

B (R): Borel sets on R,4

BS : Black and Scholes function, 572

Asian

: Asian call, 390

: double barrier call, 182

: European call, 160

∗

: European call for r = δ = 0, 160

: European bond option, 376

(x, K): European call when the

underlying is martingale, 166

(δ, α): chi-squared law, 14

: value of investment opportunity,

199

(x) is the diﬀerential w.r.t. time t

of the process depending of a

parameter x, 227

DeltaC: Delta of a call, 161

DeltaP: Delta of a put, 161

: Dirac measure at x, 282

DIB: down-and-in bond, 165

DIC

(x, K, L): DIC option price when

the underlying is martingale, 166

DOC: down-and-out call, 164

E(X)

: Dol´eans-Dade exponential

of X, 532

esssup, 4

, τ-stopping time, 22

τ−

, τ-stopping time, 22

, τ random time, 24

: natural ﬁltration of the process

X,10

(X): last time before t at which the

process X is at the level b, 247

c,2

square integrable continuous

martingales, 29

: space of square integrable

martingales, 21

[0,T]: space of square integrable

martingales on [0,T], 21

κ: Laplace exponent of the ladder

process, 629

(KM)

,36

([0,T] × Ω; R

): square integrable

processes, 62

(M): square integrable processes, 28

(λ, F )-compound Poisson process, 483

ψ,γ

: density of e.m.m. in a

jump-diﬀusion model, 570

loc

(M) locally square integrable

martingales, 28

L(Z): stochastic logarithm, 533

: local time at x, 213

: maximum over [0,t]ofthe

process X, 135

M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial

Markets, Springer Finance, DOI 10.1007/978-1-84628-737-4,

 Springer-Verlag London Limited 2009

723

724 Index of Symbols

M(X): subset of probability measures

for which the process X is a

martingale, 58

: minimum over [0,t] of the process

X, 135

(S): set of probability measures Q

equivalent to P such that S is a

Q-local martingale, 537

: stopped martingale, 23

N(x): cumulative distribution function

of the standard Gaussian law, 135

ν-compound Poisson process, 488

PCSM : pseudo-continuous

semi-martingale, 529

: law of the Bessel process of

dimension δ,startingatr, 339

: European put, 160

∗

: European put for r = δ = 0, 160

λ↓

(x), 278

λ↑

(x), 278

P − lim: limit in probability, 515

(ν)

: law of the Bessel process of index

ν,startingatr , 339

(x, y): probability transition density,

P ∼ Q: equivalent probabilities, 7

PRP: predictable representation

property, 58

(ν)

: transition density for a BES

(ν)

343

η,z

expbar

: Exponential Down Barrier

Put, 186

k,θ

: Generalized Vasicek process, 124

: law of the r.v. X,6

Q: set of e.m.m., 570

Q << P: absolute continuity, 7

kθ,σ

: law of a CIR process, 357

:lawofBESQ

starting at x , 339

δ,t

x→y

: law of a CIR bridge, 375

(ν)

: transition density for a BESQ

(ν)

343

: discounting factor, 90

r.v.: random variable, 4

, 158

σ(X):σ-algebra generated by the r.v.

X,4

(X): ﬁrst hitting time of a by X, 135

UIC: up-and-in call, 165

UOC: up-and-out call, 164

:Wienermeasure,73

(ν)

: canonical law of a drifted BM,

(T )

x→y

law of the Brownian bridge

between x and y during the time

interval [0,T], 238

(f)

: canonical law of a drifted BM,

X ∈H: the r.v. X is H-measurable, 4

law

= Y : X and Y have the same law,

X =(f, g,h): a mixed process , 551

mart

= Y : the process X − Y is a local

martingale, 94

: stopped process, 22

Subject Index

Absolute continuity, 148

BES, 339

Brownian motion, 73

for GV process, 124

relationship, 71

Square CIR and BESQ, 358

Square OU and BESQ, 356

Airy function, 155, 663

Algebra

σ-algebra, 3

American option

in a Black and Scholes model, 191

in a CEV model, 373

in a jump-diﬀusion model, 586

in an exponential L´evy model, 641

Arc sine law, 141, 220, 233, 234

Asset

riskless, 80

risky, 80

Az´ema

martingale, 243, 531

supermartingale, 324, 419, 427

Az´ema-Emery martingales, 579

Background driving L´evy process, 597

Backward s.d.e., 61

Bessel

bridges, 373

modiﬁed - function, 150, 343, 662

process, 274, 279, 333, 335

squared process, 334

Black and Scholes

formula, 97, 101

framework, 96

function, 581, 590

model, 93

BMO martingale, 25, 76, 532

Borel

function, 4

set, 4

Boundary

exit, 273

natural, 273

non-singular, 273

Bracket

of two continuous semi-martingales,

of L´evy processes, 614

predictable, 27, 28, 519

predictable - for semi-martingales,

527

Brownian

bridge, 237, 318

meander, 242

zeros, 232

Brownian motion, 30

geometric, 279

d-dimensional, 34

complex, 47, 262, 521

correlated, 35

geometric, 45, 274, 279

reﬂected, 223, 336, 338

skew, 291

sticky, 293

with drift, 279

Call option, 97

Cameron-Martin’s formula, 69

M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial

Markets, Springer Finance, DOI 10.1007/978-1-84628-737-4,

 Springer-Verlag London Limited 2009

725

726 Subject Index

Canonical decomposition

of Brownian bridge, 239

Carrying cost, 162

Cauchy law, 595

Change

of num´eraire, 105, 162, 377, 383

of probability, 66, 478

of space, 276

of time, 260, 276, 347, 348, 357, 390,

395

of time for CIR, 357

Chapman-Kolmogorov equation, 15

Characteristic measure, 508

Class D, 20, 75

Class DL, 26

Comparison

theorem for BSDE, 63

theorem for SDE, 51

Compensated sum of jumps, 513, 521

Compensation formula, 507, 603

Compensator, 413, 428, 525

of a marked point process, 502

predictable, 266

Completion of a ﬁltration, 5

Compound Poisson process, 520, 593,

599, 605, 609, 635

Conditional

expectation, 9

variance, 10

Conditionally independent

σ-algebra, 10

random variable, 10

Conformal local martingale, 262

Conservative Markov Process, 16

Contingent claim, 81, 87

Continuous part

martingale, 515

of an increasing process, 510

of the quadratic variation, 518

Convergence of processes

in L

,12

uniformly on compacts in

probability, 12

Copula, 423

survival, 423

Counting process, 457

Cramer-Lundberg

model, 497

process, 492

Criterion

Feller, 77, 275

Kazamaki, 75

Novikov, 75

of no-explosion, 76, 77

Cumulant, 608

Cumulative distribution function, 6

Decomposition

Doob-Meyer, 20, 303, 304

multiplicative, 54, 303, 304, 414

Default probability, 409

Default risk, 188

Defaultable zero-coupon, 411

Delta, 99, 161

of a call, 99, 161

Density of hitting time

of a two sided barrier for a BM, 158

of barrier for a BM, 140

Derivative in a distribution meaning,

215

Diﬀusion

recurrent, 277

transient, 277

Discount factor, 80

Discounted loss given default, 409

Distance-to-default, 189

Dividend paying asset, 102, 130

Dol´eans-Dade exponential, 52, 463,

465, 477, 486, 532, 617

Doob-Meyer decomposition, 20, 26,

413, 421, 428, 440, 443, 445

Double exponential model, 491, 568

Dual predictable projection, 300

Dual process, 609, 630

Dupire’s formula, 227, 284

Equation

backward Kolmogorov, 282

forward Kolmogorov, 282

Equivalent martingale measure, 85

Esssup, 4

European call price

in a jump-diﬀusion model, 584

in Privault model, 581

Evaluation PDE, 93

Examples for

local martingale which is not a

martingale, 77, 336

Subject Index 727

orthogonal martingales which are

not independent, 38

Excursion, 232

Expectation, 6

Explosion time, 46, 275, 458

Explosive default, 444

Exponent

characteristic, 607

L´evy, 564

Laplace, 607

Extremal measures, 58

Filtered probability space, 5

Filtration, 5

larger than another, 5, 11

natural, 10

slow Brownian, 241

strictly slow Brownian, 242

strongly Brownian, 310

weakly Brownian, 310

wide slow Brownian, 242

Finite activity, 608

Finite variation process, 12

First time

entrance, 22, 135, 140

when the supremum is reached, 302

when the supremum is reached, 220

Fokker-Planck equation, 286

Formula

Bayes, 67

Black and Scholes, 160

Cameron-Martin, 69

Clark-Ocone, 57

Dupire, 227, 284

Feynman-Kac, 112, 557

Itˆo, 38, 529, 553

Itˆo-Kunita-Ventzel, 39

Itˆo-Tanaka, 226

Tanaka, 214, 224

Forward

contract, 109

measure, 109

Function

Airy, 155, 663

Bessel, 279, 662

Beta, 662

Gamma, 662

Kummer, 280, 664

Parabolic cylinder, 154, 280, 663

scale, 271

Whittaker, 281, 372, 664

Fundamental theorem of asset pricing,

Gamma, 99

law, 595

process, 635

Gaussian

process, 15, 237

Generator for a BSDE, 61

Girsanov’s Theorem, 69

Green function, 278

h-transform, 341

Harness, 481, 620

Hazard

function, 409, 410, 413

process, 426

Hitting time

for a Bessel, 346

for an OU process, 375

for Bessel processes with dimension

smaller than 2, 356

of a drifted BM, 143

process, 219

Hitting times

process, 608

Hypotheses

usual, 5

Identity

Kendall, 141, 634

Pecherskii-Rogozin, 631

Immersion, 316

Implied

volatility, 227

Increasing process

integrable, 12

Independent random variables, 6

Index of a BES, 335

Indistinguishable, 11

Inequality

Burkholder-Davis-Gundy, 28

Doob, 21

Kunita-Watanabe, 28

Inﬁnitely divisible random variable,

592

728 Subject Index

Inﬁnitesimal generator

of a Bessel process, 337

of a diﬀusion, 271

of a Markov process, 17

of a Poisson process, 464

of a squared Bessel process, 338

of an homogeneous Markov process,

of compound Poisson process, 495

Innovation process, 331

Instantaneously reﬂecting, 273

Integral

Stieltjes, 458

stochastic - w.r.t. counting process,

458

Integration by parts, 526

for continuous martingales, 38

for mixed processes, 553

for Poisson processes, 469

for processes, 519

for Stieltjes, 511

Intensity

approach for default time, 445

of a Poisson process, 462

of a random time, 413

of an inhomogeneous Poisson

process, 467

Interest rate, 80

Inverse

of a time change, 260

of an increasing process, 260

of local time, 219

Inverse Gaussian law, 596

Itˆo’s formula

for Poisson processes, 470

for L´evy processes, 614

for compound Poisson processes, 484

for continuous semi-martingales, 38

for general semi-martingales, 528

for mixed processes, 552

Itˆo-Tanaka’s formula for

semi-martingales, 531

Kendall’s identity, 141, 634

Kolmogorov

backward equation, 282

continuity criteria, 11

forward equation, 282

Kusuoka’s example, 431

L´evy

process, 599

characterization of a BM, 32

Geometric - process, 614

measure, 592, 601

process, 575

L´evy’s equivalence theorem, 224

Ladder process, 629, 633

Laplace transform of

hitting time for a drifted BM, 148

a Gaussian law, 13

hitting time for a BES, 345

hitting time for a BM, 143

hitting time for a GBM, 152

Last passage time, 232, 294

at a level b before time t, 247

before hitting a level, 297

before maturity, 298

of a transient diﬀusion, 294, 295

Law

Arc sine, 114, 234

Arcsine, 220, 233

Cauchy, 233, 595

chi-squared, 14

Gamma, 460, 592, 595, 657

Gaussian, 13

Inverse Gaussian, 148, 596

NIG, 592

Normal Inverse Gaussian, 150

of hitting time of a barrier for a

drifted BM, 148

Rayleigh, 240, 242, 245

Variance Gamma, 592

Lemma

Skorokhod, 218

Life time, 16

Local

martingale, 25

super-martingale, 26

volatility, 227

Local time, 212, 213

diﬀusion, 290

for a semi-martingale, 222, 531

Itˆo-McKean, 290

occupation time, 290

Tanaka-Meyer, 290

Localizing sequence, 25

Locally

integrable increasing process, 26

Subject Index 729

square integrable martingale, 26

Locally equivalent probabilities, 66

Malliavin’s derivative, 57

Market

complete, 88

Markov

process, 15, 464, 600

strong - property, 17

Martingale

Asmussen-Kelly-Whitt, 620

Az´ema, 243, 522

exponential, 532

local, 25, 517

orthogonality, 28

purely discontinuous, 521

square integrable, 21, 513

Wald, 616

Maximum

of a L´evy process, 628

of a BM over [0,t], 136, 218

Mean reverting property, 120

Measurable

map, 3

space, 3

Measure

L´evy , 592

random, 458, 490, 501, 502, 505

Mellin transform, 403

Merton’s model, 491, 568, 584

Minimal entropy measure, 549

Mixed process, 551

Model

Black and Scholes, 93

CEV, 365

CGMY, 641

Cox-Ingersoll-Ross, 333

Dritschel and Protter, 579

Garman-Kohlhagen, 130

Hull and White, 120

Privault, 579

Vasicek, 120

Modiﬁcation, 11

Multiplicative decomposition, 54, 303,

304, 414

Natural scale, 271

Negligible sets, 5

Num´eraire, 105, 162

Occupation time

for a Brownian motion, 113

for a drifted BM, 114

formula, 212, 223

Options

American - for exponential

L´evy model, 641

American - in a Black and Scholes

model, 191

American - in a CEV model, 373

Asian, 381

barrier, 164

barrier forward-start, 184

boost, 184

call - in a CEV model, 372

cumulative, 116

down-and-in binary call, 165

down-and-in bond, 165

down-and-out call, 164

exchange, 130

knock-in, 164

knock-out, 164

Knock-out BOOST, 230

out-of-the-money, 160

Parisian, 247

power, 97

quanto, 132

real, 198

regular, 165

regular binary, 167

regular up-and-in put, 167

up-and-in call, 165

up-and-out call, 164

Orthogonality, 28, 513, 521

Overshoot, 564, 631

Parabolic cylinder functions, 663

Parity put-call, 84, 226

Partial information, 418, 437

Pitman’s Theorem, 307

Poisson

bridge, 480

measures, 505

Poisson process

doubly stochastic, 476

inhomogeneous - with stochastic

intensity, 476

Polar set, 18

Polarization identity, 28, 519

730 Subject Index

Portfolio, 81

hedging, 88

num´eraire, 93

self-ﬁnancing, 82

Position

long, 83

short, 83

Predictable

σ-algebra, 23

bracket, 27, 515

compensator, 266

function, 43, 513

process, 513

stopping time, 32, 513

Predictable representation theorem, 55

for a pair BM and PP, 558

for compound Poisson processes, 496

for default risk, 416, 432

for marked point processes, 504

for Poisson process, 473

Principal value, 335

Probabilities

absolutely continuous, 7

equivalent, 8

Probability measure, 5

Probability space, 5

Probability transition density

for BM, 31

Process

BESQ

, 335

BES

, 335

adapted, 10

Cauchy, 261, 635

CEV, 116, 280, 365

CIR, 357, 361

compensated, 462

compound Poisson, 483, 608

counting, 457

Cramer-Lundberg, 492

elementary predictable, 36

ﬁnite variation, 12

Gamma, 634, 635

Gaussian, 15

generalized Vasicek, 120

homogeneous, 16

increasing, 10, 510

inhomogeneous Poisson, 467

inverse Gaussian, 625

L´evy, 599

L´evy-Itˆo, 613

marked point, 502

multidimensional OU, 128

Normal Inverse Gaussian, 636

optional, 264, 513

Ornstein-Uhlenbeck, 120, 153, 280,

374, 605

point, 506

Poisson, 459

Poisson point, 506

predictable, 23, 264, 513

progressively measurable, 11

SABR, 366

square of a GV, 127

square OU, 128

square-root, 357

squared radial Ornstein-Uhlenbeck,

356

stopped, 23

tempered stable, 640

Variance-Gamma, 639

Vasicek, 120, 275, 280

with independent increments, 12

with stationary increments, 12

Projection

dual predictable, 265

optional, 264

predictable, 264

Pure jump process, 510

Put option, 97

Quadratic covariation

of a square integrable martingale,

515

of local martingales, 519, 520

of semi-martingales, 525

predictable - of continuous

semi-martingales, 29

Quadratic variation

of a continuous local martingale, 27

of a local martingale, 517

Quantiles, 119

Radon-Nikod´ym density, 7, 66

Random time, 22, 24

honest, 327

Random variable, 4

Range of prices, 415, 572

Rayleigh law, 240, 245