Jacques J., Raimondo M. Semi-markov risk models for finance, insurance and reliability
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AUTHOR INDEX
Andersen 83, 281, 287, 288, 315,
317
Anton 71
Asmussen 84
Bacinello 375
Baker 133
Balcer 374
Barlow 93, 336, 337, 341, 348
Bartels 188
Bartholomew 399, 400
Benevento 130
Bergendhäl 304
Billingsley 141
Black 18, 171, 174, 180, 182, 183,
186, 188, 189, 191, 196, 200, 201,
202, 205, 210, 212, 213, 215, 216,
217, 218, 220, 221, 222, 232, 352
Blackwell 52, 53
Bluhm 351
Bühlmann 232
Chung 11, 58, 63, 64, 68, 104, 119
Çinlar 51, 52, 85, 216, 295
Corradi 133, 134, 143
Cox D. R. 301, 307
Cox J. C. 15, 16, 174, 179, 180, 181,
182, 209, 210, 211, 212, 215, 222,
224
Cramer 83, 281, 287, 288, 294, 295,
296, 297, 298, 299, 300, 301, 302,
304, 317
Crouhy 351, 361
D’Amico 351
De Dominicis 71, 124, 266, 374,
400, 406
De Vylder 127
Di Biase 174
Doob 38, 39
Duffie 232
Dufresne 374
Ethier 141
Evans 134
Fama 173
Feller 51, 68, 115, 121, 122
Forbes 399
Gerber 297, 301, 333
Gikhman 31, 32, 190
Haberman 248, 374
Haezendonck 127
Halmos 4, 26
Hoem 128, 129
Howard 150
Hu 351
Hull 216
Isaacson 130
Israel 351
Itô 184
Janssen 1, 38, 43, 54, 71, 74, 77, 83,
84, 101, 104, 106, 108, 114, 123,
124, 128, 133, 134, 136, 139, 143,
150, 152, 156, 174, 201, 202, 213,
215, 216, 217, 219, 221, 222, 223,
227,228, 245, 247, 248, 290, 302,
304, 309, 323, 332, 374, 399, 400,
405
Author Index
424
Jarrow 351, 362
Jousseaume 174
Kellison 151, 168
Kingman 79
Kolman 71
Khorasanee 374, 396
Kurtz 141
Lando 351
Limnios 83, 84
Loeve 11, 27, 29, 31, 32
Lotka 51
Lundberg 83, 281, 287, 288, 290,
296, 297, 298, 299, 300, 301, 302,
304, 317
Madsen 130
Manca 1, 43, 54, 71, 77, 84, 101,
114, 123, 128, 133, 134, 136, 139,
143, 150, 152, 156, 174, 201, 202,
213, 215, 216, 217, 219, 222, 223,
227, 228, 245, 247, 248, 266, 269,
374, 375, 399, 400, 405, 406
Manly 374
Markowitz 18
McClean 399
Merton 301, 352
Meyer 38
Miller 152, 242
Mohan 343, 344, 345
Myers 374
Neuts 153
Nickell 351
Norberg 232
Oprişan 84
Osaki 84
Papadopoulou 130
Parzen 58, 67, 68
Pitacco 248
Proschan 336, 337, 341, 348
Protter 37, 38, 41, 184
Pyke 102, 104, 111, 123, 318, 319
Reinhard 309, 332, 333
Riesz 139
Ross S. A. 16, 174
Royden 11
Rubinstein 15, 16, 174, 180, 182,
183, 209, 210, 211, 212, 215, 222,
224
Sahin 374
Samuelson 184
Schaufele 108, 111
Scholes 18, 171, 174, 181, 183, 184,
186, 188, 189, 191, 196, 200, 201,
202, 205, 210, 212, 213, 215, 216,
218, 220, 221, 222, 232, 352
Skiadas 38
Skodeberg 351
Skorokhod 31, 32, 141, 142, 190
Smith 51, 52
Snoussi 333
Spitzer 122
Standard & Poor’s 353, 358, 361,
362, 365
Stein 46, 53
Thorin 316
Thorisson 130
Thornton 396
Tomassetti 374, 375
Tsantas 374
Vajda 399
Vasicek 232
Vasileiou 351
Vassiliou 130
Volpe di Prignano, 151, 168, 216,
231, 233, 375, 399, 405
White 216
Wilkie 397
Wilson 396
Yntema 74, 75, 266
SUBJECT INDEX
Absence of arbitrage, 176
Absolute continuity, 5
Absolute moment of order r, 12
Absorbing (recurrent) class of states,
67
Absorbing state, 65
Absorption probability in a class
given a state, 67
Accumulated claim amount process,
284
Age distribution, 104
Algorithm for semi-Markov process,
142
Algorithm for semi-Markov reward
process, 160
Almost sure convergence, 3
American option, 171
Aperiodic Markov chain, 62
Aperiodic renewal process, 47
Aperiodic state of a Markov chain,
62
Associated counting process with
the CNHSM kernel, 85
Availability function, 346
Average return time, 88
Bernoulli random walk, 113
Binomial distribution, 15
Blackwell theorem, 52
Boole inequality, 4
Borel σ-algebra, 5
Borel set, 5
Brownian motion, 40
Càdlàg stochastic process, 36
Call option, 171
Central internal scenario, 397
Central limit theorem for functionals
of (J,X) processes, 109
Central limit theorem for the two
dimensional (J,X) processes, 110
Central limit theorem, 18, 57
Characteristic function, 20
Claim amounts process, 281
Claim number process, 281
Closed pension fund, 377
Class of states of a Markov chain,
62
Closed martingale, 38
Coherent structure of a complex
component, 340
Complement, 1
Complete market, 173
Complete probability space, 8
Completely non-homogeneous
dependent renewal process, 129
Completely non-homogeneous
Markov renewal process, 126
Completely non-homogeneous
semi-Markov chain, 125
Completely non-homogeneous
semi-Markov kernel, 125
Completely non-homogeneous
semi-Markov process, 126
Subject Index
426
Conditional density, 32
Conditional distribution of a random
variable, 32
Conditional distribution of sojourn
time, 80
Conditional expectation, 24
Conditional independence of n
variables, 32
Conditional independent sequences,
312
Conditional probability, 22, 23
Conditional probability measure, 24
Semi-Markov reward processes,
154, 157
Convergence sequence of random
variable, 8
Convolution product, 45
Correlation coefficient, 21
Counting processes associated with
a Markov renewal process, 85
Covariance, 21
Dated sum, 232
de Moivre theorem, 18
de Morgan law, 2
Decomposition theorem, 65
Defective positive (J,X) process,
117
Delayed Markov renewal process,
87
Delayed renewal process, 57
Delta Greek parameter, 193
Density function, 5
Discount factor, 151
Discrete time stochastic process, 35
Discrete time homogeneous semi-
Markov process 131
Discrete time non-homogeneous
semi-Markov process 131
Distribution function, 5
Dominated convergence theorem,
11
Drift random walk, 115
Economic scenario, 373
Efficient market, 173
Ergodic Markov chain, 62
Ergodic Markov renewal process, 91
Ergodic property of a semi-Markov
process, 97
Essential class of states, 63
Essential state of a Markov chain,
62
European option, 171
Event, 1
Event almost sure, 3
Event certain, 2
Event disjoint, 2
Event elementary , 1
Event empty, 2
Event mutually exclusive, 3
Expectation, 10
Explosive associated counting
process with the CNHSMP, 127
Extended semi-Markov chain, 78,
106
Extended semi-Markov kernel, 78
Failure rate, 336
Filtered probability space, 35
Filtration, 34
Filtration right continuity property,
34
Frequency, 3
Gambler ruin problem, 113
Gamma Greek parameter, 194
Generalized stochastic annuity, 248
Generating function, 15
Geometric brownian motion, 186
Hitting time, 63
Homogeneous Markov chain, 59
Imbedded multiple Markov chain,
80
Impulse reward (transition reward),
151
Independence, 13
Indistinguishable stochastic
processes, 36
Inessential class of states, 63
Subject Index
427
Inessential state of a Markov chain,
62
Initial distribution of a Markov
chain, 59
Integrable random variable, 9
Integral equation of Markov renewal
type, 92
Integral equation of renewal type,
50
Integral of a discrete random
variable, 9
Interarrival time, 44
Interest rate intensity, 151
Intersection, 2
Irreducible Markov chain, 62
Irreducible Markov renewal process,
91
Irrelevant component of a complex
structure, 340
J-process, 79
Key renewal theorem, 52, 94
k-out of n structure, 341
Ladder epoch, 114
Ladder height, 114
Ladder points, 115
Ladder variables, 114
Laplace Stieltjes transform, 64
Laplace transform, 55
Length-biased sampling, 55, 56
Lifetime of a defective positive
(J,X) process, 120
Lifetime of a renewal process, 47
Loading premium, 283
Log-normal distribution, 19
Maintainability function, 347
Markov chain, 59
Markov renewal equation, 92
Markov renewal matrix, 89
Markov renewal process, 82
Markov renewal time, 85
Martingale property, 37
Mean recurrence time of a state, 64
Mean time to failure, 339
Mean time to repair, 339
Mean total rewards, 153
Measurable space, 2
Mixed conditional distribution of a
random variable, 32
Modification of a stochastic process,
36
Monotone convergence theorem, 12
Monotone structure of a complex
component, 340
Multidimensional normal
distribution, 20
Multiple life models, 269
Negative exponential distribution,
20
Non-homogeneous Markov chain,
59, 129
Non-homogeneous semi-Markov
chain, 125
Non-homogeneous semi-Markov
kernel, 125
Non-homogeneous semi-Markov
process, 126
Non-return state, 65
Non-ruin probability, 113
Normal distribution, 16
Occupancy probability, 237
Occupation time of a state in the
first n transtions, 66
Option premium, 172
Option pricing, 172
Optional sampling (Doob's)
theorem, 38
Oscillating random walk, 115
Parallel structure, 340
Partially non-homogeneous Markov
additive process, 129
Partially non-homogeneous semi-
Markov kernel, 129
Partially non-homogeneous semi-
Markov process, 129
Periodic renewal process, 47
Periodic state of a Markov chain, 62
Subject Index
428
Physical measure, 177
Poisson distribution, 16
Point-wise availability, 340
Polya process family, 128
Positive (J-X) process, 77
Preference relation, 232
Premium income process, 282
Probability continuity property, 4
Probability measure, 3
Probability space complete, 8
Probability space, 3
Pure premium, 283
Put option, 171
Quadrature formula 134
Random experiment, 1
Random variable continuous, 7
Random variable discrete, 7
Random variable extended n-
dimensional real, 6
Random variable n-dimensional
real, 6
Random variable real extended
value, 6
Random variable real, 6
Random variable, 6
Random walk, 113
Rate reward (permanence reward),
151
Real multidimensional stochastic
process, 35
Real stochastic process, 35
Recurrence time , 64
Recurrent renewal process, 47
Recurrent state, 64
Regular conditional probability, 31
Regular Markov matrix, 60
Reliability function, 335
Renewal counting process, 45
Renewal density, 53
Renewal equation,50
Renewal function, 45
Renewal process, 149
Renewal time, 44
Reward mean present value, 156
Reward process classification, 150
Reward processes, 150
Reward, 150
Rho Greek parameter, 195
Risk model, 281
Risk neutral measure, 177
Risk process, 284
Risk reserve process, 285
Ruin probability, 113, 288
Ruin problem, 114, 285
σ-algebra, 2
Sample space denumerable, 4
Sample space finite, 4
Sample space, 1
Semi-Markov chain, 78
Semi-Markov kernel, 78
Semi-Markov process, 86
Semi-Markov random walk, 106
Series structure, 340
Set indicator, 8
Skorohod topology, 141
Smoothing property of conditional
expectation, 29
Stady state availability, 340
Standard normal distribution, 17
Stationary Markov renewal process,
100
Stationary renewal process, 58
Stochastic annuity, 247
Stochastic process, 35
Stochastic process adapted to a
filtration, 36
Stopped stochastic process, 40
Stopping time σ-algebra, 37
Stopping time, 35
Strict ascending ladder point, 115
Strong law of large numbers for
(J,X) processes, 199
Strong law of large numbers for
functionals of (J,X) processes, 109
Subject Index
429
Strong law of large numbers for
functionals of positive (J,X)
processes, 108
Strong law of large numbers, 57
Sub-martingale, 37
Successive entrance time into a
state, 87
Sum of random number of random
variables, 29
Super-martingale, 37
Sure convergence, 8
Terminated defective positive (J,X)
process, 117
Theta Greek parameter, 194
Times continuous Markov process,
104
Total number of renewals, 45
Total occupation time of a state, 66
T-process, 82
Trajectory of a stochastic process,
36
Transient renewal process, 47
Transient state, 65
Transition graph, 59
Transition matrix, 59
Transition probabilities of a semi-
Markov process, 94
Transition probability of a delayed
semi-Markov process, 99
Transition probability, 59
Two-dimensional normal
distribution, 22
Unconditional distribution of
sojourn time, 80
Unconditional waiting time
distribution, 118
Uniformly integrable family, 38
Union, 1
Uni-reducible Markov chain, 69
Variance covariance matrix, 21
Variance, 12
Vega Greek parameter, 194
Volatility, 187
Wald identities, 29
Wald lemma, 56
Wiener Hopf integral equation
system, 124
X-process, 79