The
Aalen counting processes 2.8
ADF regression 2.71, 2.77
ADF statistic 2.57, 2.58, 2.72, 2.77
Adjusted least squares (ALS) estimator 3.21, 3.24
Adjusted rand index (ARI) 5.21
Admissible coverings 3.120
Anticipating stochastic integral 2.34
AR processes 4.383, 4.394
ARMA processes 2.52, 4.382, 4.384, 4.388, 4.394
Asymptotic additivity (AA) 4.248
Asset price
process 2.239, 2.417
volatility 2.356
Autocovariance matrix 2.60
Autoregression 2.56, 2.61, 2.70, 2.76, 3.60, 3.68
Attributable proportion (AP) 4.110
Backward stochastic differential equations (BSDE) 4.353
BDG inequality 4.33, 4.34, 4.44, 4.47, 4.48, 4.59
Bernstein function 5.358
Bessel process 3.224–3.227, 3.230, 5.448
Bifractional Brownian motion 4.16
Bivariate VAR model 2.61
Bonferroni inequality 4.73
Borel probability measure 4.255
Brownian motion 2.31, 2.33–2.35, 2.108, 2.111, 2.186, 2.189–2.197, 2.234, 2.264, 2.328, 4.16, 4.26, 4.190, 4.193, 4.195, 4.308, 4.311, 4.316, 4.355, 5.54, 5.57, 5.58, 5.60, 5.70, 5.71, 5.76, 5.83, 5.115, 5.146, 5.147, 5.416–5.418, 5.447, 5.448, 5.485, 5.492, 5.494, 5.510
fractional 4.16, 4.17, 4.21, 4.192–4.194, 5.100, 5.101, 5.104, 5.105, 5.298, 5.416, 5.417, 5.420, 5.431
standard 2.31, 2.41, 2.42, 4.17, 4.191, 4.308, 5.120, 5.147, 5.148, 5.417
Brownian sheet 2.288–2.293
Burkholder inequality 3.9
Càdlàg progressively measurable processes 4.29
Cantor distribution 4.412
Cap style cliquet option 5.91, 5.96, 5.335
Characteristic triplet 1.50, 1.52, 1.59, 1.60
CIR process 3.2, 3.12, 5.100, 5.114, 5.115, 5.117
distribution 5.116
fractional 5.100, 5.101, 5.103, 5.105
Circulant matrix 3.183, 3.347
Circular bivariate Cauchy distribution 4.80–4.82, 4.88
Cliquet option 5.81, 5.82, 5.91, 5.92, 5.96, 5.317, 5.318, 5.325, 5.326, 5.331, 5.333
price 5.82, 5.87, 5.91–5.96, 5.318, 5.319, 5.324–5.328, 5.331–5.333, 5.335
Cointegrated sequences 3.60, 3.68–3.70, 3.72, 3.76
Cointegration 2.52, 2.62–2.64, 2.87
test 2.63
Compensated Poisson random measure 4.26
Compound Poisson process 2.2, 2.6, 4.165, 4.168, 4.173, 4.184, 4.317, 5.172, 5.176, 5.303, 5.305, 5.308, 5.315, 5.322, 5.510
Conditionally Gaussian processes 5.484, 5.485, 5.488, 5.489, 5.495
Confidence intervals (CI) 4.110
Conic section 3.20, 3.21, 3.24, 3.25, 3.27, 3.34, 3.35, 3.40, 3.44
Consistency 2.18, 2.20, 2.82, 2.132, 2.149, 2.163, 2.187, 2.299, 2.301, 2.344, 2.346, 2.352
Constant volatility 2.233, 2.365
Continuous additive functional 2.108
Continuous flow 2.189
Convex random closed set 3.326
Daley inequality 3.316
Damped stable processes 2.403
Deterministic volatility 2.356, 2.364, 2.366
Diffusion process 2.252
Dirichlet
distribution 2.4
process 2.4, 2.7
process distribution 2.4
Discounted price process 1.99, 2.236, 2.238, 2.239
Dominating distribution 3.319
Domination sequence 3.319
Doob inequality 2.207, 2.213, 3.7
Double integral 2.140
Drifted Brownian motion 5.447, 5.452, 5.454
Driftless subordinators 5.511
Economical processes 3.133
Eigenfunction representation 2.289
Entropy power (EP) 4.245
Entropy power inequality (EPI) 4.238
Equivalent local martingale measure (ELMM) 2.238
Equivalent martingale measure (EMM) 2.238–2.240
Ergodic scaling transformation 1.74, 1.75, 1.78, 1.79, 1.83, 1.87
Ergodicity 1.38, 1.74, 1.79, 1.83
Erlang distribution 4.185, 4.316, 5.511
Error
distribution 3.52, 5.38, 5.51
sequences 3.48
European call option 1.96, 1.97, 1.101, 2.235, 2.241, 2.242, 2.246, 2.248, 2.356
price 2.234, 2.243, 2.355, 2.358, 2.361, 2.364, 2.366
Exponential
bound 1.169, 1.172, 1.179
Chebyshev inequality 5.133
process 5.174, 5.182, 5.185, 5.187
Extinction probability 4.2, 4.4
Faithful coverings 3.121
Family of coverings 3.120, 3.121, 3.217
Feller process 2.109–2.114, 2.125
Finite
Lévy measure 5.319, 5.361, 5.364, 5.368
symmetric Lévy measures 5.364, 5.365, 5.367
Finite mixture models (FMM) 2.344
Folded Cauchy distribution 4.80, 4.82, 4.85
Folded drifted Brownian motion 5.454
Forecast
error 2.55, 2.73
error RMSFE 2.55
estimation error 2.82
future 2.52
horizon 2.61
methods 2.51
Forecasted value 2.55
Formal information criteria 2.60
Fractional Brownian
field 1.74, 1.75
motion 1.74, 1.96, 1.102, 1.103, 1.130, 2.31, 2.33, 2.35–2.37, 2.41, 2.148–2.150, 2.219–2.222, 2.294, 2.334, 3.107, 3.108, 3.112, 3.181–3.184, 3.210, 3.303
sheet 1.74, 1.75, 1.77, 1.79, 1.91, 1.92
Fractional integral 2.220–2.223, 2.227, 2.229, 2.230
Fractional Skellam processes 4.162
Fredholm
integral equation 2.148, 2.152, 2.153
representation 2.288, 2.289, 2.292–2.294
Fuzzy adjusted Rand index (FARI) 5.9, 5.22
Fuzzy Rand index (FRI) 5.22
Gamma
process 4.162, 4.163, 4.165, 5.168, 5.178, 5.182, 5.187, 5.510
subordinators 4.162, 4.166, 5.514, 5.515, 5.517
Gaussian
process 1.140, 1.141, 1.144, 2.30, 2.32–2.35, 2.241, 2.267, 2.268, 2.282, 2.292–2.294, 2.310, 2.313, 2.314, 3.108, 3.111, 3.184, 4.194, 5.57, 5.417, 5.483–5.496
field 5.430
random matrix 3.48
stationary process 1.140, 1.183–1.185
Generalized backward stochastic differential equations (GBSDE) 4.25
Generalized inverse Gaussian (GIG) distribution 5.514
GINAR processes 5.58
Gibbs inequality 3.129
Granger causality 2.52, 2.55, 2.61, 2.62, 2.64, 2.82, 2.86, 2.89
Granger causality test (GCT) 2.55
GRETL lag length 2.85
Hellinger distance 2.5, 2.394, 2.395
Hermite processes 2.327, 2.332, 2.334, 5.431
Hitsuda representation theorem 2.293
Homogeneity hypothesis 1.203
Homogeneous
distribution 2.176
Poisson process 4.413, 4.414, 4.416
Hurst index 1.74, 1.96, 1.103, 1.105, 1.107, 1.108
Hybrid method (HM) 5.6
Hybrid stochastic volatility (HSV) 5.146
ID distribution 5.510, 5.511, 5.513, 5.515–5.518
INAR processes 5.54
Information criteria 2.61, 2.64, 2.70, 2.75, 2.76, 2.81
Information matrix 5.232, 5.233
Inhomogeneous renewal risk 2.174, 2.176
model 2.174, 2.176
Innovation process 2.276
Invariance principle 2.334
Invariant
density 2.19–2.21
probability measures 4.256, 4.266
Inverse
gamma subordinator 5.514, 5.515
Gaussian
distributions 5.514
process 5.510
subordinators 4.166, 5.510, 5.514, 5.515
stable subordinators 5.516, 5.517
subordinators 4.174, 5.451, 5.510–5.512, 5.514, 5.515
Inverse tempered stable subordinators (ITSS) 5.513
Inverse Wishart distribution 2.7
Isonormal Gaussian process 2.33–2.36
Isotropic random
flight 5.462
set 3.343
Iterated
Brownian motion 5.448
function system (IFS) 4.254
multivariate forecasts 2.61
process 5.185
stochastic integral 2.37
Itô
formula 1.53, 1.55, 1.59, 1.153
integral 2.37
Karhunen representation 2.289, 2.292
Kato class 2.108–2.110, 2.114
Kernel
function 5.298, 5.300, 5.302
representation 2.159
symmetric 2.291
Lack of decrease (LOD) 4.224
Lack of increase (LOI) 4.223
Lack of monotonicity (LOM) 4.224
Lack of negativity (LON) 4.229
Lack of positivity (LOP) 4.228
Lack of sign (LOS) 4.229
Lag length 2.56, 2.60, 2.61, 2.71, 2.76, 2.78, 2.82
in VAR models 2.60
selection 2.60
Lamperti transformation 1.74, 1.78
LAN property 1.34–1.39
Laplace exponent 4.95, 4.163, 4.165–4.167, 4.174, 5.177, 5.178, 5.186, 5.511–5.515
Laplace transform (LT) 5.513
Large deviation principle (LDP) 3.96, 3.145, 4.5, 5.486
Least favorable
densities 3.70, 3.73
spectral densities 3.71, 3.72, 3.74–3.77
Least squares estimator (LSE) 2.298, 5.191
Lebesgue dominated convergence 3.233
Length vector distributions 3.351
LePage series 3.139, 3.140, 3.239, 3.242, 3.244
Level crossing probability 5.483, 5.495
Lévy
kernel 1.50, 1.52, 1.59–1.61
martingale 2.191
measure 1.37, 1.50, 1.51, 1.59, 1.60, 2.111, 2.125, 2.126, 2.403, 2.404, 2.411, 2.412, 4.163–4.167, 5.88–5.90, 5.169, 5.178, 5.182, 5.185, 5.186, 5.298, 5.322, 5.355, 5.357, 5.358, 5.360, 5.361, 5.364, 5.367, 5.368, 5.371, 5.377, 5.380, 5.381, 5.446, 5.511
process 2.2, 2.112, 2.126, 2.210, 2.252, 2.402, 2.403, 2.411, 4.163–4.165, 4.169, 5.87, 5.91, 5.178, 5.181, 5.300, 5.301, 5.310, 5.318–5.321, 5.324, 5.333, 5.335, 5.446, 5.510, 5.511, 5.516
process independent 5.181
type process 2.112
Lévy driven SDE 1.34, 1.37, 1.38, 1.118
Lévy fractional Brownian field 1.77, 1.78
Linear Lebesgue probability measures 1.4
Linear programming problem (LPP) 2.299
Linear regression 2.297, 2.301, 2.303
Local martingale 2.236, 2.238
Local time 1.110, 1.112
distribution 1.112
Logistic distribution 2.137, 2.145
Lognormal distribution 5.425
Loss function 3.290
Lundberg inequality 2.175, 2.176, 5.131
Lyapunov condition 1.44, 1.205, 1.206
Malliavin calculus 2.30, 2.32, 2.35, 2.166, 2.288, 2.402
Marginal probability measure 5.489
Markov
binomial distribution 5.212
process 1.33, 1.34, 1.38, 1.50, 1.53, 1.119, 2.107–2.111, 2.115, 2.165, 2.166, 2.251, 2.252, 2.265, 2.401, 2.402, 2.417, 3.96, 3.100, 3.147, 3.192, 3.224, 3.304, 4.17, 4.18
Martingale convergence theorem 4.262
Maximum likelihood
estimator 2.17, 2.18, 2.132, 2.163, 3.29, 3.30, 3.35, 3.37, 3.39, 3.52, 3.107, 3.109–3.111, 3.113, 3.270, 3.276, 3.277, 5.6, 5.230
estimator construction 2.149, 2.150
MB distribution 5.212
Mean squared error (MSE) 5.12
Meixner
distribution 5.82, 5.84–5.87, 5.90, 5.96
process 5.82, 5.85, 5.87, 5.91
Mild solution 3.137, 3.139
Minimal martingale measure (MMM) 2.238, 2.241
Minimum Hellinger distance 2.394
Molchan martingale 2.149
Moving average (MA) 2.52
Multifractional Brownian motion 4.16
Multiperiod VAR forecasts 2.61
Multiple
regression line 2.71
sclerosis (MS) 4.111, 4.117
Wiener integrals 2.36, 2.43
Multiplicative process 3.12
Multivariate forecast 2.61
Mutually independent 4.145, 4.170, 4.316–4.319, 5.130
Nonlocal porous medium equation (NPME) 5.457
Nonlogarithmic convergence rates 3.2
Nonpositive Bessel process 3.225
Numerical forecasts 2.92
Objective option price 1.96, 1.97, 1.101, 1.103, 1.105, 1.107, 1.108
Occupation time option 2.402, 2.404, 2.417
Offspring distribution 4.2, 4.3, 4.5, 4.6
Open set condition (OSC) 3.216
Optimal linear estimate 3.63, 3.66, 3.67
Ordinary least squares (OLS) 2.53, 3.24
cointegrating regression 2.63
regression 2.58
Orthogonal regression (OR) 3.24
Packing dimension 2.372, 2.373, 2.378–2.384, 2.386, 2.388
Parametrix method 2.412
Partial differential equation (PDE) 5.114, 5.318
Pareto distribution 3.171
Pathwise
integral 2.36, 2.37
volatility 2.30
Periodogram 1.181, 1.182, 1.185, 1.186
Permanent insurance policy (PIP) 4.128
Planar Lebesgue probability measures 1.4
Poisson
point process 2.3, 2.12, 2.13
process 2.2, 4.162, 4.163, 4.165–4.167, 4.170, 4.174, 4.412–4.414, 4.419, 5.167–5.169, 5.172, 5.177, 5.179, 5.182, 5.185–5.187, 5.319, 5.322, 5.510, 5.511, 5.517, 5.518
random measure (PRM) 4.26, 5.82, 5.310, 5.319
Policyholder 4.128, 4.129, 4.132–4.136, 4.138, 4.141–4.148, 4.151, 4.153, 4.156, 4.158
Posterior contraction rate 2.2, 2.6, 2.8, 2.9
Predictable processes 4.29, 4.50
Price processes 4.92
Process
exponential 5.174, 5.182, 5.185, 5.187
gamma 4.162, 4.165, 5.168, 5.178, 5.182, 5.187, 5.510
inverse 4.163, 4.173–4.182, 5.168
inverse Gaussian 5.510
stable 5.510
volatility 2.234, 2.240, 2.356
Progressively measurable
function 4.353
processes 4.377
Pseudomoments 2.96–2.98, 2.105
QLR statistic 2.58, 2.72, 2.75, 2.79, 2.80
Quantile function 4.290
Rand index (RI) 5.21
Random
convolution 4.67
distribution 3.213, 3.214, 3.35
errors 3.49, 3.288
flights 4.79, 4.82, 5.459, 5.462, 5.464, 5.465
matrix 3.48, 3.51, 3.52, 3.288, 5.250, 5.274
measure 4.29
models 5.459, 5.466
motion 5.463
Poisson measure 4.28
polygons 3.326
realization 3.49, 3.289
rectangle 3.359
regular zonotope 3.362
set 3.238, 3.325, 3.342, 3.344–3.346, 3.359
stable noises 5.429
symmetric body 3.342, 3.343
symmetric convex set 3.326, 3.327, 3.343, 3.346, 3.357–3.360, 3.362
walk 4.97, 4.98, 4.316, 5.138, 5.142, 5.462, 5.463
zonotopes 3.326, 3.327, 3.342, 3.344–3.347, 3.352, 3.357, 3.362
Randomized periodogram 2.31, 2.32, 2.37
estimator 2.30
Randomized time 5.167
Randomly stopped
processes 4.91
sums 3.168
Randomness 4.26
Regression 2.53, 2.55, 2.57, 2.58, 2.63, 2.64, 2.71, 2.72, 2.80, 2.272, 2.299, 2.300
analysis 2.51, 2.63
coefficients 2.53, 2.55
design matrix 2.297
function 2.58, 5.195, 5.202, 5.203
line 2.53, 2.71
model 2.53, 2.58, 2.68, 2.301
problem 2.269
Regressors 2.55, 2.59, 2.63, 2.64, 2.68, 2.71, 2.347
vector 2.344
Regular best asymptotically normal (RBAN) estimators 3.31
Renewal
process 5.517, 5.518
risk 2.173–2.177
sequence 5.517
Representation
integral 2.164
kernel 2.159
Reproducing kernel 2.290, 2.292, 2.294
Hilbert space (RKHS) 2.290, 2.292, 5.485
Response
function 4.95, 4.98
process 4.94
Restricted Oppenheim expansion (ROE) 4.275
Risk model 2.173–2.176, 2.422, 2.424, 2.426, 2.427
Rosenthal inequality 5.261, 5.271, 5.272
Ruin probability 1.168, 1.169, 1.172, 1.174, 1.175, 2.175, 2.422, 2.429, 4.315–4.318, 4.328, 4.335, 4.341, 4.345, 4.347, 4.348, 5.131, 5.132, 5.136, 5.137, 5.139, 5.141
Scaling transformation 1.74, 1.75, 1.78–1.81, 1.83
Semimartingale
process 2.236
quadratic variance 2.30
Series expansion 2.288, 2.294
Shannon differential entropy (SDE) 4.234
Shannon entropy (SE) 4.233
Significance level 2.57, 2.71, 2.72, 2.75, 2.77, 2.79, 2.80, 2.82, 2.83, 2.86
Skellam
distribution 4.164
process 4.162–4.164, 4.166, 4.170, 4.171, 4.177, 4.181, 4.182
Skewed offspring distributions 4.410
Skorokhod
approach 3.270
conditions 3.271
integral 2.34, 2.36, 2.37
selection theorem 3.306, 3.307
Sociological data analysis 1.196, 1.203
Solution
pathwise uniqueness 3.15, 3.274, 3.304
Spectral
densities 1.182–1.186, 2.268, 2.270, 2.272, 2.273, 2.276, 2.284, 3.60, 3.62, 3.66, 3.68, 3.70–3.77
function 3.61
Spurious regression 2.57
Stable
convergence 5.299, 5.300, 5.302–5.304, 5.307
process 3.134, 3.137, 5.510
subordinator 4.103, 4.162, 4.166, 4.167, 4.170, 5.100, 5.167, 5.172, 5.187, 5.446, 5.447, 5.511–5.513, 5.516
subordinators distributions 5.513
State dependent
characteristic triplet 1.50, 1.52
parameter 1.49, 1.50
State process upward 4.354
Stationarity 2.56, 2.60, 2.75, 2.225, 2.276, 2.277
Stationary
ARMA processes 4.382
distribution 2.55, 4.254, 4.256, 4.266, 4.267, 5.100
field 1.78, 1.79
Gaussian process 1.139, 1.140, 1.143, 1.145, 1.147, 2.267, 3.250, 5.72
Gaussian series 1.185
increment stochastic sequences 3.61
independent increments 4.174
probability 4.146, 4.153
processes 2.321, 2.335
sequences 3.60, 3.69
sequences linear functionals 3.60
version 1.38, 1.44, 1.45
Stiffness matrix 5.529
Stochastic
boundedness 1.15, 1.18, 1.30
differential equation (SDE) 3.2, 3.3, 3.15, 3.223, 3.270–3.273, 3.304, 5.85, 5.114, 5.319, 5.522
distributions 3.250
heat equation 1.129, 1.130
integral 2.37, 2.219, 2.220
measure (SM) 5.430
representation formulae 2.402
trend 2.52, 2.56, 2.57, 2.60, 2.63, 2.72, 2.77, 2.80, 2.81
volatility 2.234, 2.235, 2.355, 2.356, 5.146
volatility models 2.361, 2.366
Stochastically independent 5.39
Strike price 2.265, 2.362, 2.417
Strong asymptotic arbitrage (SAA) 5.416, 5.418
Subexponential distributions 3.81, 3.167, 4.67
Subfractional Brownian motion 4.15–4.17, 4.23
Subjective income 1.204
Submartingale 2.191
Subordinated
Lévy process 4.168, 5.168, 5.180
Poisson process 5.170, 5.176
Subordinator
independent 4.164
stable 4.103, 4.162, 4.166, 4.167, 4.170, 5.100, 5.167, 5.172, 5.187, 5.446, 5.447, 5.511–5.513, 5.516
Surplus process 1.169, 1.174, 1.175
Survival probability 4.135
Symmetric
convex set 3.326, 3.327, 3.331, 3.333, 3.336, 3.338, 3.340–3.343, 3.357, 3.362
distribution 2.134
kernel 2.291
Lévy measure 5.355, 5.360, 5.364, 5.367, 5.368, 5.378, 5.379
Lévy process 5.298
matrix 3.293, 3.299
stable process 2.108, 5.451
Synergy index (SI) 4.110
Tail probability 5.141
Tapered data 1.181, 1.182
Target distribution 4.308
Tempered
Hermite process 2.327, 2.328, 2.331, 2.332, 2.335
stable processes 2.403
stable subordinators (TSS) 5.446, 5.510, 5.513
subordinators 5.446
Temporary insurance policy (TIP) 4.128
Term insurance policy (TIP) 4.132
Total least squares (TLS)
estimator 3.48–3.50, 3.52, 3.288–3.290, 3.292, 3.301, 5.248, 5.252
problem 3.48, 3.49, 3.288, 3.289
Transition
density 2.108, 2.114, 2.125, 2.126
matrix 4.146
probability 4.247, 5.212
probability density 1.34, 1.35, 1.38, 1.118, 1.119, 2.108–2.114, 2.118, 2.126, 2.166, 2.167, 2.252, 2.403, 2.412, 2.413, 3.224
probability function 4.130, 4.131, 4.138
Translated process 3.344
Transportation distance 1.50–1.52, 1.59, 1.61, 1.62
Trimmed regions 1.152, 1.156, 1.157, 1.163
Truncated pseudomoments 2.96, 2.97
Ultimate ruin probability 2.422–2.426, 2.438, 5.130
Unbiased criterion 1.5, 1.10
Uncentered packing 2.374–2.377, 2.379
dimension 2.373, 2.374, 2.379
Unconditional probability mass 4.145
Uncorrelated
processes 2.240
Wiener processes 2.234
Undiscounted process 2.238
Univariate
distributions 4.287
forecasts 2.61
time series 2.52
Unobserved nonrandom vector 5.247
Unstable solution 3.192
VAR
coefficients 2.61
model 2.59–2.61, 2.68, 2.80, 2.82, 2.84, 2.88, 2.89
model for exports 2.81
Vector
autoregression 2.52, 2.59
autoregressive process 5.58
Vitali coverings 3.120
Volatility
function 2.235, 2.357
process 2.234, 2.240, 2.356, 3.2, 4.354
risk market price 2.240
stochastic 2.234, 2.235, 2.355, 2.356
Volterra
kernel 2.291, 2.293
representation 2.291
Weak convergence 2.269, 2.327, 2.328, 2.334, 2.335, 2.345, 2.349, 3.2, 3.10, 3.12, 3.14, 3.192, 3.195, 3.204, 3.224
Weak large deviation principle (WLDP) 5.487
Weak solution 3.156, 3.197, 3.271, 3.272, 3.277, 3.306
Weekly separable family 1.8, 1.10
Weight function (WF) 4.234, 5.354, 5.355
Weighted differential entropy (WDE) 4.234
Weighted entropy power inequality (WEPI) 4.239
Weighted entropy power (WEP) 4.245
Weighted entropy (WE) 4.234
Weighted Fisher information inequality (WFII) 4.244
Weighted Fisher information matrix (WFIM) 4.237
Weighted information (WI) 4.247
Wiener process 1.52, 1.96, 1.104, 1.110, 1.154, 2.17, 2.19, 2.149–2.151, 2.203, 2.204, 2.215, 2.223, 2.235, 2.239, 2.240, 2.356, 2.357, 3.2, 3.3, 3.15, 3.98, 3.100, 3.146, 3.150, 3.182, 3.192, 3.196, 3.197, 3.199, 3.224, 3.225, 3.232, 3.233, 3.271–3.277, 3.281, 4.204, 4.210, 4.213, 4.216, 5.100, 5.204
Zeta distribution 3.170, 3.171, 4.71
Zonotope 3.326–3.328, 3.331–3.333, 3.339, 3.347, 3.357, 3.360, 3.362
approximation 3.326, 3.327, 3.362
rotational 3.343