A
Absorbing state 110, 111, 114
Actuarial schema 334
Almost sure central limit theorem (ASCLT) 347, 350, 351, 353, 356, 357, 359
Arbitrage opportunities 487
Autocovariance functions 197
Autoregressive moving average 195, 196
B
Bernoulli random field 252–255, 257, 258
Bernstein function 420, 421, 425, 430–433, 436, 437
Beta function 64
Bifractional Brownian motion 399, 400, 404, 408, 409, 411, 413
Bilateral gamma processes 82
BINAR 228–234, 237, 238, 241, 243–245
BINAR models for loan data 245
Bivariate copulas 231, 232
Bivariate INAR 229
Bivariate MGF 381
Bivariate Poisson distribution 138
Bondora issues loans 241
Boundedness 169, 176, 180, 191
Bounds 63, 73, 74, 77
Brownian sheet 168, 170, 171, 173
BSDEs 479, 480
BSDEs for Lévy processes 482
C
Censoring 211, 213
Censoring time 211
Central limit theorem 169, 183
Chaos expansion 146, 150, 156
Characterisation 196, 205
Characteristic triplet 481, 482
CIR process 13, 14
Clayton copula 138–141, 233, 238, 243
CLS estimates 238, 240, 241, 243
CLS estimators 233–235, 238
Compound Poisson approximation 111
Compound Poisson distribution 112, 119
Compound Poisson processes 83, 286
Conditional independence 278–281
Conditional least squares (CLS) 233
Conditional maximum likelihood (CML) 237
Confidence intervals 504, 506
Confidence sets for regression coefficients 504
Consecutive subordinations 420
Copulas 228, 231–233, 236, 243–245
definition 231
dependence parameter 233, 245
functions 241
in BINAR models 228
theory 232
Cox model 212, 214, 218, 220, 223
Cumulative hazard functions 210
D
Daley inequality 334
Defaulted loan data 244
Defaulted loans 241, 243, 244
Degree distribution 42, 47
Deviation scale 168, 169
Diffuse 271, 275, 277, 278
Dirichlet boundary conditions 168, 170, 174, 183
Discrete time risk model 133–136, 142
Distributivity 271, 273, 275, 279–281
Dividend
payments 286–288, 292, 293, 302, 304, 305
strategies 286
Dominating sequence 336–339, 341
E
Euler approximation scheme 146
Euler approximations 32, 34
Explosive autoregressive models 389
Exponential generating function 422, 428, 429, 432, 434
External Independent Testing (EIT) 505
F
FGM copula 232, 236, 243–245
Finite Mixture Models (FMM) 495
Fractional
Brownian motion 1, 14–17, 24, 34, 35, 58, 60, 68, 196, 206, 377, 379, 380, 398–401, 404, 408, 410–412, 414, 416
Laplacian 397–399, 408, 409, 416
stochastic heat equation 397, 399, 416
Fractional Brownian process 58, 60, 61
Fractional CIR process 15, 16
Fractional noise 57, 58
Fractional Vasicek model 377, 379, 389
Frank copula 228, 231, 233, 239, 243–245
Fundamental solution 347, 348, 353, 358, 365, 373
G
Gamma process 85, 86
Gamma subordinator 430, 431, 440
Gaussian martingale 380
Gaussian noise 397, 398, 409, 411, 416
Gaussian process 347, 348, 351, 353, 400, 401, 403, 405, 408, 411
Gaussian property 62, 65, 67, 69, 73, 76
Gaussian random variable 456, 462, 472
Gerber–Shiu function 285–288
GINAR 235
Globally Lipschitz 168, 174, 175, 178, 183, 185
Ground process 421, 429, 431, 435
H
Haar basis 153, 159, 160
Hannan type condition 255
Hazard functions 210
Hazard rates 210, 211
Heat equation 398–400, 408, 416
Helicoidal filaments 3
Homogeneous Poisson process 43–45
Hurst index 1, 16, 24, 377, 380
Hurst parameters 400, 409
I
INAR(1) 206, 229, 233, 234
Incarceration 223
Independent
complements 271
conditioning 272
equiprobable signs 270, 273, 276, 279, 281
Independent random censoring 211
Infinite velocity 5
Infinitely divisible characteristic functions 452, 472
Infinitely divisible distributions 420
Infinitely divisible logarithmic series 419–421, 423, 425, 427
Infinitely divisible random measure 446, 447, 450
Innovations 228, 231, 232, 243, 244, 247
Insurance models 1, 109, 110, 112, 114
Inversion formula 126–128
Isometry 349–351, 354, 399, 448, 458, 473, 474, 482
Isonormal Gaussian process 351
Isotropic fBm 410
Issuing loans 228
J
Jackknife 496, 499
ACM estimators 501
estimates 496
estimation 499
estimator 496, 499, 500
technique 495, 502, 512
Joint MGF 382
K
Kolmogorov 352, 355, 359
L
Laplace principle 169, 171–173
Lévy
characteristic 446, 448
density 443, 444, 446, 448, 451
process 443, 444
Lévy measure 420, 421, 425, 426, 428–431, 433, 434, 436, 437, 439
Lévy process 419–421, 425–427, 429, 430, 433, 440, 479–483, 486
Linear growth 60, 65
Linear growth conditions 16, 152, 154
Linear regression coefficients 496
Linear span 348, 351, 353
Lipschitz continuity 60, 65, 69, 70
Liquidity 196, 206
Loans 227, 228, 241–243
defaulted 228
non-defaulted 228
Loans data 241, 245
Loans rating system 241, 242
Logarithmic series 420, 421, 430
Logarithmic utility maximization 480, 486
M
Macroevolution 41, 51
Main stochastic term 458, 461, 462
Malliavin calculus 146, 147, 152
Malliavin derivative 148, 149, 153, 154
Marginal 138, 140, 141
Marginal distribution 46, 135, 138–140, 228, 231, 232, 235, 237–239, 243–245, 247, 313, 316, 318, 320
Marginal distribution functions 228
Marine bacteria 4
Markov binomial distribution 111
Markov chain 109–111
Markov chains 334, 335, 342
Martingale convergence 271, 272, 274, 280, 281
Martingale optimality principle 480, 488, 491, 493
Martingales 390, 391
Maximum likelihood estimator (MLE) 378, 380, 389
Mild solution 58, 59, 61–63, 73, 347–351, 353, 359, 373, 398, 400, 409, 416
Minimal cost functions 169
Moderate deviations 167–169, 171, 173, 192, 193
Moment generating functions (MGF) 378
Motile bacteria 3
Multiplicative intensities 211
Mutually independent 83, 86, 92
MVC model 496
N
Natural topology 26, 29
Negative binomial 235, 237, 243–245
Negative binomial distribution 231, 235, 239, 240, 243–245, 247
Negative binomial marginal distribution 231, 244
Negative dependence 138, 139
Net profit condition 288, 291, 295, 297, 298, 302, 303
Normalised probability convolution distribution 428, 432, 434
Numerical maximization 238–240
O
Obscure structure 15, 33
Older loans 241
Orthogonalized Teugels martingales 480
Orthonormal basis 146, 147, 153
OS property 45
P
Pairwise independent 270, 275, 280
Parabolic Green’s function 58, 61, 62
Parabolic SPDEs 57, 183
Parameter estimation 378
Partial Bell polynomials 420, 422, 425, 427, 428
Partial differential equations (PDE) 345, 346
Pathwise Riemann integral 37
Pathwise stochastic integration 61
Pathwise Stratonovich integral 14, 15, 35, 37, 38
Periodic chains 336
Peripheral
vertex 313–315
weight 313, 315, 316, 318, 327
Persistent random walk 4, 5, 7–9
Perturbed fBm 400, 405–407, 414
Pointwise limit 15, 17, 22, 33
Poisson 138–141
Poisson innovations 236
Poisson marginals 139
Poisson processes 83, 88, 93
Polynomial chaos expansions 159, 160
Power law 311–313, 316, 318, 329
Power variations 379, 380, 398, 416
Prabhakar function 43, 50
Predictable
compensator 490
function 481, 491
process 486–488, 490, 492
Predictable increasing process 481
Predictable representation property 479
Predictable square variation 481
Predictable variation 215, 216, 218
Preferential attachment 41, 42, 312, 314
Preferential attachment random graph 42
Price process 487, 490, 491
Probability generating function 44
Product copula 232
Propagator system 150, 151, 161
Proportional hazards (PH) 209, 210
Q
QL estimators 206
QL methods 206
R
Random
field 252–255, 258, 262, 443, 444, 447, 449, 450, 455, 457, 461, 464, 476
graphs 42
intensities 47
measure 443, 446, 454
motion 7
times 41
variable 44–47, 49, 50, 52, 251, 252, 254, 255, 260, 262
walk 5, 7, 8
walker 6
Random processes 17, 34, 35
Regression coefficients 496, 501, 504, 507
Regression data 209, 211
Regression model 252, 257
Regularly growing 444–446, 455, 456, 462, 463, 472, 476
Remainder term 458, 464
Renewal
proceses 334
sequences 333–335, 341
times 335, 336, 339
trials 337, 338
Reorientation frequency 4, 10
Residential treatment 223
Residential treatment for drug abuse 223
Reversed triangular inequality 263
Riemann integrable 22, 29
Riesz kernel 398, 408, 409
Risk models 285–287
Risky assets 88, 95
Ruin probability 134, 138, 140–142, 285–287, 291, 295, 299, 304, 305
S
Signed compound Poisson measure 114
Signed Stirling numbers 426, 429
Skorokhod definition 61
Sparse truncation 159, 161
SPDEs 168, 173, 174, 191
Stability estimates 334, 342
Stable subordinator 43, 46, 47
Standard CIR model 14
Standard Laplacian 398, 408, 416
Stationary probability distribution 7, 8
Stationary processes 195, 197, 198, 206
Stirling numbers 420, 423, 424, 435, 436
Stochastic Burgers equation 168, 169, 174, 183
Stochastic heat equation 346, 347, 350, 358, 359, 398, 399, 408, 409
Stochastic heat equation fractional 397, 399, 416
Stochastic integral 352, 353, 481, 487, 490
Stochastic partial differential equation (SPDE) 346, 347, 358, 372
Stochastic premiums 286, 287
Subgeometrical Markov chains 334
Subordinated Lévy process 433
Subordinated process 420, 421, 429–431, 433, 437, 440
Subordination 420, 429, 431, 434, 440
Subordinators 86, 88, 90, 94
Surplus process 287, 288, 292, 296, 302
Survival probability 135
T
Telegraph equation 4–6
Telegraph process 4, 5
Teugels martingales 480, 483
TfPp 43, 44, 46, 47
Thinning operator 229, 230
Tightness 169, 174–177, 179
Transition probability 110–112, 420, 421, 425–431, 433, 435
Trigonometric basis 147, 153, 159
Tumbling rate 4, 9, 10
U
Ultimate ruin probability 1, 134
Unconditional probability distribution 7
V
Variational solution 58
Velocity 4–9
Velocity random motion 5
Vertex 312–315, 318
Vertex peripheral 313–315
W
Wasserstein 352, 355, 359
Wasserstein distance 398, 400, 402, 411, 416
Weak convergence 169, 171, 172, 191
Weak topology 172, 174, 181
Wealth process 488–491
Wiener 349, 350
integral 348, 373, 399, 409, 410
Wiener chaos 146, 147, 153, 351, 352
Wiener chaos expansion 146–149, 151
Wiener process 83, 346
Y
Yule model 1, 41, 42, 46, 47
Yule process 42, 45, 46
Z
Zeta distribution 141
