A

Appendix

D. Gusak et al., Theory of Stochastic Processes, Problem Books in Mathematics, 359DOI 10.1007/978-0-387-87862-1, c© Springer Science+Business Media, LLC 2010

360 A Appendix

A Appendix 361

362 A Appendix

A Appendix 363

364 A Appendix

List of abbreviations

cadlag Right continuous having left-hand limits (p. 24)caglad Left continuous having right-hand limits (p. 24)cdf Cumulant distribution functionCLT Central limit theoremHMF Homogeneous Markov family (p. 179)HMP Homogeneous Markov process (p. 176)i.i.d. Independent identically distributedi.i.d.r.v. Independent identically distributed random variablespdf Probability density functionr.v. Random variableSDE Stochastic differential equationSLLN Strong law of large numbers

List of probability distributions

Be(p) Bernoulli, P(ξ = 1) = p,P(ξ = 0) = 1− pBi(n, p) Binomial, P(ξ = k) = Ck

n pk(1− p)n−k,k = 0, . . . ,nGeom(p) Geometric, P(ξ = k) = pk−1(1− p),k ∈ N

Pois(λ ) Poisson, P(ξ = k) = (λ k/k!)e−λ ,k ∈ Z+

U(a,b) Uniform on (a,b), P(ξ ≤ x) = 1∧ ((x−a)/(b−a))+,x ∈ R

N(a,σ2) Normal (Gaussian), P(ξ ≤ x) = (2πσ2)−1/2 ∫ x−∞ e−(y−a)2/2σ2

dy,x ∈ R

Exp(λ ) Exponential, P(ξ ≤ x) = [1− e−λx]+,x ∈ R

Γ (α,β ) Gamma, P(ξ ≤ x) = (βα/Γ (α))∫ x

0 yα−1e−βy dy,x ∈ R+

A Appendix 365

List of symbols

aξ 12 L ∗ 181aX 11 L2([a,b]) 193Aϕ 178 L2 193B(X) 177 L cl 129B(X) 3 �∞ 241C([0,T ]) 241 L∞([0,T ]) 241C([0,+∞)) 241 �p 241C2

uni(Rm) 180 Lp([0,T ]) 241

C2fin 180 l.i.m. 38

C(X,T) 2 Lip 251c0 241 〈M〉 74cap 233 〈Mc〉 75cov(ξ ,η) 11 [M] 75D([a,b],X) 24 〈M,N〉 74DX

a 253 [M,N] 74DF 242 Mc 74∂A 242 Md 74Eμ 45,108 M∗ 89F 329 M 73

F 339 Mloc 73F [−1] 4 M 2 73Ft+ 21 M 2

loc 73Ft− 21 M 2,c 74F

X ,0t 21 M 2,d 74

FXt 21 M 89

Fτ 71 Mloc 89F

X 21 M 2 74

f (n)i j 138 Mτn 73

F∗n 161 NP 21F∗0 161 N(a,B) 59Hp 89 P(s,x, t,B) 176H(X) 129 pi j 137Hk(X) 129 pi j(t) 139

HNS 254 p(n)

i j 137Hγ 251 PX

t1,...,tm 1HΛ 129 Rλ 177I[a,b]( f ) 193 RX 11,107Iab 252 RX ,Y 11It( f ) 193 Rξ 12Iξ 110 Tt f 177L(t,x) 195 U f 110

366 A Appendix

XH 61 τΓ 5Xr

n 129 τΓ 5Xs

n 129 Φ 261,282X

T 2 Φ 288(X,X) 1 Φ−1 282ZX 108 φξ 12βN(a,b) 76 φX

t1,...,tm 12ΔsMd 75 (Ω ,F,P) 1ϑN(S) 255 # 3Λ 129,244 ⇒ 242

λ 1|[0,1] 4d→ 242

λ 1|R+ 44∨α∈A Gα 21

πHΛ 129

References 367

References

1. Asmussen S (2000) Ruin Probability. World Scientist, Singapore2. Bartlett MS (1978) An Introduction to Stochastic Processes with Special Reference to

Methods and Applications. Cambridge University Press, Cambridge, UK.3. Bertoin J (1996) Levy Processes. Cambridge University Press, Cambridge, UK.4. Billingsley P (1968) Convergence of Probability Measures. Wiley Series in Probability

and Mathematical Statistics, John Wiley, New York5. Bogachev VI (1998) Gaussian Measures. Mathematical Surveys and Monographs, vol.

62, American Mathematical Society, Providence, RI6. Borovkov AA (1976) Stochastic Processes in Queueing Theory. Springer-Verlag, Berlin7. Bratijchuk NS, Gusak DV (1990) Boundary problems for processes with independent

increments [in Russian]. Naukova Dumka, Kiev8. Brzezniak Z, Zastawniak T (1999) Basic Stochastic Processes. Springer-Verlag, Berlin9. Bulinski AV, Shirjaev AN (2003) Theory of Random Processes [in Russian]. Fizmatgiz,

Laboratorija Bazovych Znanij, Moscow10. Buhlmann H (1970) Mathematical Methods in Risk Theory Springer-Verlag, New-York11. Chaumont L, Yor M (2003) Exercises in Probability: A Guided Tour from Measure The-

ory to Random Processes, Via Conditioning. Cambridge University Press, Cambridge,UK

12. Chung KL (1960) Markov Chains with Stationary Transition Probabilities. Springer,Berlin

13. Chung KL, Williams RJ, (1990) Introduction to Stochastic Integration. Springer-VerlagNew York, LLC

14. Cramer H, Leadbetter MR (1967) Stationary and Related Stochastic Processes. SampleFunction Properties and Their Applications. John Wiley, New York

15. Doob JL (1990) Stochastic Processes. Wiley-Interscience, New York16. Dorogovtsev AY, Silvesrov DS, Skorokhod AV, Yadrenko MI (1997) Probability Theory:

Collection of Problems. American Mathematical society, Providence, RI17. Dudley, RM (1989) Real Analysis and Probability. Wadsworth & Brooks/Cole, Belmont,

CA18. Dynkin EB (1965) Markov processes. Vols. I, II. Grundlehren der Mathematischen Wis-

senschaften, vol. 121, 122, Springer-Verlag, Berlin19. Dynkin EB, Yushkevich AA (1969) Markov Processes-Theorems and Problems. Plenum

Press, New York20. Elliot RJ (1982) Stochastic Calculus and Applications. Applications of Mathematics 18,

Springer-Verlag, New York21. Etheridge A (2006) Financial Calculus. Cambridge University Press, Cambridge, UK22. Feller W (1970) An Introduction to Probability Theory and Its Applications (3rd ed.).

Wiley, New York23. Follmer H, Schied A (2004) Stochastic Finance: An Introduction in Discrete Time. Wal-

ter de Gruyter, Hawthorne, NY24. Gikhman II, Skorokhod AV (2004) The Theory of Stochastic Processes: Iosif I. Gikhman,

Anatoli V. Skorokhod. In 3 volumes, Classics in Mathematics Series, Springer, Berlin25. Gikhman II, Skorokhod AV (1996) Introduction to the Theory of Random Processes.

Courier Dover, Mineola26. Gikhman II, Skorokhod AV (1982) Stochastic Differential Equations and Their Applica-

tions [in Russian]. Naukova dumka, Kiev27. Gikhman II, Skorokhod AV, Yadrenko MI (1988) Probability Theory and Mathematical

Statistics [in Russian] Vyshcha Shkola, Kiev

368 References

28. Gnedenko BV (1973) Priority queueing systems [in Russian]. MSU, Moscow29. Gnedenko BV, Kovalenko IN (1989) Introduction to Queueing Theory. Birkhauser

Boston, Cambridge, MA30. Grandell J (1993) Aspects of Risk Theory. Springer-Verlag, New York31. Grenander U (1950) Stochastic Processes and Statistical Inference. Arkiv fur Matematik,

Vol. 1, no. 3:1871-2487, Springer, Netherlands32. Gross D, Shortle JF, Thompson JM, Harris CM (2008) Fundamentals of Queueing The-

ory (4th ed.). Wiley Series in Probability and Statistics, Hoboken, NJ33. Gusak DV (2007) Boundary Value Problems for Processes with Independent Increments

in the Risk Theory. Pratsi Instytutu Matematyky Natsional’noı Akademiı Nauk Ukraıny.Matematyka ta ıı Zastosuvannya 65. Instytut Matematyky NAN Ukraıny, Kyıv

34. Hida T (1980) Brownian Motion. Applications of Mathematics, 11, Springer-Verlag,New York

35. Ibragimov IA, Linnik YuV (1971) Independent and Stationary Sequences of RandomVariables. Wolters-Noordhoff Series of Monographs and Textbooks on Pure and AppliedMathematics, Wolters-Noordhoff, Groningen

36. Ibragimov IA, Rozanov YuA (1978) Gaussian Random Processes. Applications of Math.,vol. 9, Springer-Verlag, New York

37. Ibramkhalilov IS, Skorokhod AV (1980) Consistent Estimates of Parameters of RandomProcesses [in Russian]. Naukova dumka, Kyiv

38. Ikeda N, and Watanabe S (1989) Stochastic Differential Equations and Diffusion Pro-cesses, Second edition. North-Holland/Kodansya, Tokyo

39. Ito K (1961) Lectures on Stochastic Processes. Tata Institute of Fundamental Research,Bombay

40. Ito K, McKean H (1996) Diffusion Processes and Their Sample Paths. Springer-Verlag,New York

41. Jacod J, Shiryaev AN (1987) Limit Theorems for Stochastic Processes. Grundlehren derMathematischen Wissenschaften, vol. 288, Springer-Verlag, Berlin

42. Johnson NL, Kotz S (1970) Distributions in Statistics: Continuous Univariate Distribu-tions. Wiley, New York

43. Kakutani S (1944) Two-dimensional Brownian motion and harmonic functions Proc.Imp. Acad., Tokyo 20:706–714

44. Karlin S (1975) A First Course in Stochastic Processes. Second edition, Academic Press,New York

45. Karlin S (1966) Stochastic Service Systems. Nauka, Moscow46. Kijima M (2003) Stochastic Processes with Application to Finance. Second edition.

Chapman and Hall/CRC, London47. Klimov GP (1966) Stochastic Service Systems [in Russian]. Nauka, Moscow48. Kolmogorov AN (1992) Selected Works of A.N. Kolmogorov, Volume II: Probability

theory and Mathematical statistics. Kluwer, Dordrecht49. Koralov LB, Sinai YG (2007) Theory of Probability and Random Processes, Second

edition. Springer-Verlag, Berlin50. Korolyuk VS (1974) Boundary Problems for a Compound Poisson Process. Theory of

Probability and its Applications 19, 1-14, SIAM, Philadelphia51. Korolyuk VS, Portenko NI, Skorokhod AV, Turbin AF (1985) The Reference Book on

Probability Theory and Mathematical Statistics [in Russian]. Nauka, Moscow52. Krylov NV (2002) Introduction to the Theory of Random Processes. American Mathe-

matical Society Bookstore, Providence, RI53. Lamperti J (1977) Stochastic Processes. Applied Mathematical Sciences, vol. 23,

Springer-Verlag, New York

References 369

54. Lamberton D, Lapeyre B (1996) Introduction to Stochastic Calculus Applied to Finance.Chapman and Hall/CRC, London

55. Leonenko MM, Mishura YuS, Parkhomenko VM, Yadrenko MI (1995) Probabilisticand Statistical Methods in Ecomometrics and Financial Mathematics. [in Ukrainian] In-formtechnika, Kyiv

56. Levy P (1948) Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris57. Liptser RS, Shiryaev AN (2008) Statistics Of Random Processes, Vol. 1. Springer-Verlag

New York58. Liptser RS, Shiryaev AN (1989) Theory of Martingales. Mathematics and Its Applica-

tions (Soviet Series), 49, Kluwer Academic, Dordrecht59. Lifshits MA (1995) Gaussian Random Functions. Springer-Verlag, New York60. Meyer PA (1966) Probability and Potentials. Blaisdell, New York61. Øksendal B (2000) Stochastic Differential Equations, Fifth edition. Springer-Verlag,

Berlin62. Pliska SR (1997) Introduction to Mathematical Finance. Discrete Time Models. Black-

well, Oxford63. Port S, Stone C (1978) Brownian Motion and Classical Potential Theory. Academic

Press, New York64. Protter P (1990) Stochastic Integration and Differential Equations. A New Approach

Springer-Verlag, Berlin65. Prokhorov AV, Ushakov VG, Ushakov NG (1986) Problems in Probability Theory. [in

Russian] Nauka, Moscow66. Revuz D, Yor M (1999) Continuous martingales and Brownian Motion. Third edition.

Springer-Verlag, Berlin67. Robbins H, Sigmund D, Chow Y (1971) Great Expectations: The Theory of Optimal

Stopping. Houghton Mifflin, Boston68. Rolski T, Schmidli H, Schmidt V, Teugels J (1998) Stochastic Processes for Insurance

and Finance. John Wiley and Sons, Chichester69. Rozanov YuA (1977) Probability Theory: A Concise Course. Dover, New York70. Rozanov YuA (1982) Markov Random Fields. Springer-Verlag, New York71. Rozanov YuA (1995) Probability Theory, Random Processes and Mathematical Statis-

tics. Kluwer Academic, Boston72. Rozanov YuA (1967) Stationary Random Processes. Holden-Day, Inc., San Francisco73. Sato K, Ito K (editor), Barndorff-Nielsen OE (editor) (2004) Stochastic Processes.

Springer, New York74. Sevast’yanov BA (1968) Branching Processes. Mathematical Notes, Volume 4, Number

2 / August, Springer Science+Business Media, New York75. Sevastyanov BA, Zubkov AM, Chistyakov VP (1988) Collected Problems in Probability

Theory. Nauka, Moscow76. Skorokhod AV (1982) Studies in the Theory of Random Processes. Dover, New York77. Skorokhod AV (1980) Elements of the Probability Theory and Random Processes [in

Russian]. Vyshcha Shkola Publ., Kyiv78. Skorohod AV (1991) Random Processes with Independent Increments. Mathematics and

Its Applications, Soviet Series, 47 Kluwer Academic, Dordrecht79. Skorohod AV (1996) Lectures on the Theory of Stochastic Processes. VSP, Utrecht80. Spitzer F (2001) Principles of Random Walk. Springer-Verlag New York81. Shiryaev AN (1969) Sequential Statistical Analysis. Translations of Mathematical Mono-

graphs 38, American Mathematical Society, Providence, RI82. Shiryaev AN (1995) Probability. Vol 95. Graduate Texts in Mathematics, Springer-Verlag

New York

370 References

83. Shiryaev AN (2004) Problems in Probability Theory. MCCME, Moscow84. Shiryaev AN (1999) Essentials of Stochastic Finance, in 2 vol. World Scientific, River

Edge, NJ85. Steele JM (2001) Stochastic Calculus and Financial Applications. Springer-Verlag, New

York86. Striker C, Yor M (1978) Calcul stochastique dependant d’un parametre. Z. Wahrsch.

Verw. Gebiete, 45: no. 2: 109–133.87. Stroock DW, Varadhan SRS (1979) Multidimensional Diffusion Processes Springer-

Verlag, New York88. Vakhania NN, Tarieladze VI, Chobanjan SA (1987) Probability Distributions on Banach

Spaces. Mathematics and Its Applications (Soviet Series), 14. D. Reidel, Dordrecht89. Ventsel’ ES and Ovcharov LA (1988). Probability Theory and Its Engineering Applica-

tions [in Russian]. Nauka, Moscow90. Wentzell AD (1981) A Course in the Theory of Stochastic Processes. McGraw-Hill, New

York91. Yamada T, Watanabe S (1971) On the uniquenes of solutions of stochastic differential

equations J. Math. Kyoto Univ., 11: 155–16792. Zolotarev VM (1997) Modern Theory of Summation of Random Variables. VSP, Utrecht

Index

Symbolsσ–algebra

cylinder, 2generated by Markov moment, 71predictable, 72

“0 and 1” rule, 52

BBayes method, 280boundary functional, 329Brownian bridge, 60

Ccall (put) option

American, 305claim causing ruin, 329classic risk process, 328Coding by pile of books method, 146contingent claim

American, 305attainable, 305European, 304

continuity set, 242convergence

of measuresweak, 242

of random elementsby distribution, 242weak, 242

correlation operator of measure, 272coupling, 246

optimal, 247covariance, 11criterion

for the regularity, 130Neyman–Pearson, 273recurrence, 138

critical region, 271cumulant, 45cumulant function, 327

Ddecision rule

nonrandomized, 271randomized, 271

decompositionDoob’s

for discrete-time stochastic processes,87

Doob–Meyerfor supermartingales, 72for the general supermartingales,

73Krickeberg, 87Kunita–Watanabe, 90Riesz, 87Wald, 129

densityof measure, 272posterior, 280prior, 280spectral, 107

diffusion process, 180distribution

finite-dimensional, 1Gaussian, 59marginal, 251

371

372 Index

of Markov chaininvariant, 138stationary, 138

Eequation

Black–Sholes, 316Fokker–Planck, 181Langevin, 218Lundberg, 333Ornstein–Uhlenbeck, 218

ergodic transformation, 110errors of Type I and II, 271estimator

Bayes, 280consistent

for singular family of measures, 281strictly for regular family of measures,

280of parameter, 279

excessive majorant, 229

Ffair price, 305family of measures

tight, 243weakly compact, 243

filtration, 21complete, 21continuous, 21left-hand continuous, 21natural, 21right-hand continuous, 21

financial market, 303Black–Scholes/(B,S) model, 316complete, 305dividend yield, 317Greeks, 316

flow of σ -algebras, 21formula

Black–Sholes, 316Dynkin, 202Feynman–Kac, 219Ito, 194

multidimensional, 194Levy–Khinchin, 44Pollaczek–Khinchin

classic, 337generalized, 333

Tanaka, 195

fractional Brownian motion, 61function

bounded growth, 77characteristic, 12

m-dimensional, 12common, 12

covariance, 11, 107excessive, 229, 230generalized inverse, 4Holder, 23, 251Lipschitz, 251lower semicontinuous, 230mean, 11mutual covariance, 11nonnegatively defined, 11, 12payoff, 229

continuous, 230premium, 229, 230renewal, 161spectral, 107structural, 108superharmonic, 230

Ggenerator, 178Gronwall–Bellman lemma,

219

Iinequality

Burkholder, 78Burkholder–Davis,

77, 78Doob’s, 75, 77

integral, 76Khinchin, 78Marcinkievich–Zygmund, 78

infinitesimal operator, 178

KKakutani alternative, 258Kolmogorov equation

backward, 181forward, 181

Kolmogorov system of equationsfirst (backward), 139second (forward), 140

Kolmogorov–Chapman equations,137, 176

Index 373

Llocal time, 195loss function

all-or-nothing, 280quadratic, 280

Mmain factorization identity, 331Markov

homogeneous family, 179Markov chain, 137

continuous-time, 138homogeneous, 137regular, 139

Markov moment, 71predictable, 71

Markov process, 175homogeneous, 176weakly measurable, 177

Markov transition function, 176martingale, 71

inverse, 85Levy, 79local, 73

martingale measure, 304martingale transformation, 80martingales orthogonal, 90matrix

covariance, 12joint, 60

maximal deficit during a period, 330mean value of measure, 272mean vector, 12measure

absolutely continuous, 272Gaussian, 59

on Hilbert space, 273intensity, 45Levy, 45locally absolutely continuous, 80locally finite, 44random

point, 45Poisson point, 45

spectral, 107stochastic

orthogonal, 108structural, 108Wiener, 242

measuresequivalent, 272singular, 272

modelbinomial (Cox–Ross–Rubinstein), 307Ehrenfest P. and T., 146Laplace, 145

modificationcontinuous, 22measurable, 22

Nnumber of crossings of a band, 76

Ooutpayments, 328overjump functional, 329

Pperiod of a state, 137Polya scheme, 81portfolio, 315

self-financing, 315price of game, 229principle

invariance, 244of the fitting sets, 10reflection, 260

processadapted, 71almost lower semicontinuous, 328almost upper semicontinuous, 328Bessel, 88birth-and-death, 147claim surplus, 329differentiable

in Lp sense, 33in probability, 33with probability one, 33

discounted capital, 304Galton—Watson, 85geometrical Brownian motion, 316integrable, 71

Lp sense, 34in probability, 34with probability one, 34

Levy, 44lower continuous, 328nonbusy, 160

374 Index

of the fractional effect, 37Ornstein–Uhlenbeck, 61, 183Poisson, 44

compound, 49, 328with intensity measure κ, 44with parameter λ , 44

predictable, 72discrete-time, 72

progressively measurable, 25registration, 44renewal

delayed, 161pure, 161

semicontinuous, 328stepwise, 327stochastic, 1uniformly integrable, 72upper continuous, 328Wiener, 44

two-sided, 61with discrete time, 1with independent increments, 43

homogeneous, 43

Qquadratic characteristic

joint, 74quantile transformation, 4

Rrandom element, 1

generated by a random process, 242distribution, 242

generated by a random sequence, 242distribution, 242

random field, 1Poisson, 45

random function, 1centered, 11compensated, 11continuous

a.s., 33in mean, 33in mean square, 33in probability, 22, 33in the Lp sense, 33with probability one, 33

measurable, 22separable, 22

stochastically continuous, 33random functions

stochastically equivalent, 21in a wide sense, 21

random walk, 138realization, 1red period, 330renewal

epoch, 161equation, 161theorem, 162

representationspectral, 109

reserve process, 328resolvent operator, 177risk process, 328risk zone, 330ruin probability with finite horizon,

329ruin time, 329

Ssafety security loading, 329second factorization identity, 332security of ruin, 329sequence

ergodic, 110random, 1regular, 129singular, 129

setcontinuation, 230cylinder, 2of separability, 22stopping, 230supporting, 230total, 118

shift operator, 110Snell envelope, 81space

functional, 241Skorohod, 24

Spitzer–Rogozin identity, 331square variation, 74square characteristic, 74square variation

joint, 74State

regular, 139

Index 375

stateaccessible, 137communicable, 137essential, 137inessential, 137recurrent, 138transient, 138

stationarityin wide sense, 107strictly, 109

stationary sequenceinterpolation, 129, 130prediction, 129, 130

stochastic basis, 71stochastic differential, 194stochastic differential equation, 215

strong solution, 215weak solution, 216

stochastic integralIto, 193discrete, 80over orthogonal measure, 108

stoppingoptimal, 230

stopping time, 71optimal, 229

strategyoptimal, 229

strong Markov family, 179submartingale, 71superharmonic majorant, 230

least, 230supermartingale, 71surplus prior to ruin, 329

Ttelegraph signal, 183theorem

Birkhoff–Khinchin, 110Bochner, 12Bochner–Khinchin, 107Donsker, 244Doob’s

on convergence of submartingale, 76on number of crossings, 76optional sampling, 73

ergodic, 138

Fubini for stochastic integrals, 195functional limit, 244Hajek–Feldman, 273Herglotz, 107Hille –Yosida, 178Kolmogorov

on finite-dimensional distributions, 2on continuous modification, 23on regularity, 130

Levy, 204on normal correlation, 60Poincare on returns, 119Prokhorov, 243Ulam, 241

total maximal deficit, 330trading strategy, 303

arbitrage possibility, 304self-financing, 303

trajectory, 1transform

Fourier–Stieltjes, 340Laplace, 164Laplace–Karson, 336Laplace–Stieltjes, 163

transition function, 176substochastic, 185

transition intensity, 139transition probabilities matrix, 137

Uultimate ruin probability, 329uniform integrability

of stochastic process, 72of totality of random variables, 72

Vvector

Gaussian, 59virtual waiting time, 160

Wwaiting process, 3Wald identity

first, 85fundamental, 85generalized, 85second, 85

white noise, 109