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
Top Related