Kloeden P/, Platen E., Schurz H. — Numerical solution of SDE through computer experiments :: Электронная библиотека попечительского совета мехмата МГУ

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Kloeden P/, Platen E., Schurz H. — Numerical solution of SDE through computer experiments

Kloeden P/, Platen E., Schurz H. — Numerical solution of SDE through computer experiments

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Название: Numerical solution of SDE through computer experiments

Авторы: Kloeden P/, Platen E., Schurz H.

Аннотация:

The book provides an easily accessible computationally oriented introduction into the numerical solution of stochastic differential equations using computer experiments. It develops in the reader an ability to apply numerical methods solving stochastic differential equations in their own fields. Furthermore, it creates an intuitive understanding of the necessary theoretical background from stochastic and numeric analysis. A downloadable softward containing programs for over 100 problems is provided at each of the following homepages:

http://www.math.uni-frankfurt.de/~numerik/kloeden/ http://www.business.uts.edu.au/finance/staff/eckhard.html http.//www.math.siu.edu/schurz/SOFTWARE/

to enable the reader to develop an intuitive understanding of the issues involved. Applications include stochastic dynamical systems, filtering, parametric estimation and finance modeling.

The book is intended for readers without specialist stochastic background who want to apply such numerical methods to stochastic differential equations that arise in their own filed.

Язык:

Рубрика: Computer science/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1994

Количество страниц: 309

Добавлена в каталог: 14.01.2014

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Предметный указатель

$A \cap B$ , intersection of sets A and B      2
$A \cup B$ , union of sets A and B      2
$a \in A$ , a element of set A      1
$a \ll b$ , b is much larger then a      132
$a(t,X_{t})$ , drift coefficient of an SDE      73
$a^{i}(\cdot)$ , ith drift component      73
$b^{i,j}(\cdot)$ , ith component of diffusion coefficient $b^{j}(\cdot)$       73
$b^{j}(t,X_{t})$ , diffusion coefficient corresponding to $W^{j}$       73
$cov(X_{1},X_{2})$ , covariance of $X_{1}$ and $X_{2}$       24
$exp(\cdot)$ , exponential function      4
$F^{-1}$ , inverse of function F      10
$F_{X}$ , distribution function of random variable X      3
$Im(\lambda)$ , imaginary part of a complex number $\lambda$       132
$I_{(j_{1},j_{2},...,j_{l}),t}$ , multiple Ito integral      89 140
$I_{A}(\cdot)$ , $I_{A}$ , indicator function of event A      3 90
$J^{p}_{(j_{1},j_{2},...,j_{l}),t}$ , approximate multiple Stratonovich integral      8
$J_{(j_{1},j_{2},...,j_{l}),t}$ , multiple Stratonovich integral      82 140
$lim_{n\to\infty}$ , limes for $n \to \infty$       27
$ln(\cdot)$ , logarithm (to the base e)      11
$L^{0}$ , $L^{1}$ , $L^{j}$ , differential operators      78 139
$max\{\cdot,\cdot\}$ , maximum      65
$N(\mu;\sigma^{2})$ , Gaussian distributed with mean $\mu$ and variance $\sigma^{2}$       21
$n_{t}$ , largest integer for which $\tau_{n} \le t$       111
$Re(\lambda)$ , real part of a complex number $\lambda$       132
$t_{1-\alpha,n-1}$ , quantile of Student t-distribution      33
$V_{j_{1},j_{2}}$ , random variable for double Wiener integrals      182
$W_{t}^{0}=t$ , time (in compact notation)      82 140
$W_{t}^{j}$ , jth component of a Wiener process at time t      73
$X^{T}$ , transpose of matrix X      74
$Y_{n}$ , approximation Y at discretization time $\tau_n$       110
$\Delta W_{n}$ , $\Delta W_{n}^{j}$ , Wiener process increment      110 140
$\Delta Z_{n}$ , $\Delta Z_{n}^{j}$ , multiple stochastic integral      146
$\Delta$ , $\delta$ , $\Delta_{n}$ , time step size      70 110
$\emptyset$ , empty set      2
$\epsilon$ , absolute error      115
$\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}a$ , second order partial derivative of a      73
$\frac{\partial}{\partial x}a$ , partial derivative of a with respect to x      73
$\hat{\epsilon}$ , estimate for absolute error $\epsilon$       115 118
$\hat{\mu}$ , estimate for mean error $\mu$       124
$\hat{\mu}_{N}$ , sample average      17 33
$\hat{\sigma}_{N}^{2}$ , sample variance      17 33
$\infty$ , infinity      27
$\int ... dW_{t}$ , Ito integral      66
$\int ... \circdW_{t}$ , Stratonovich integral      67
$\int...dt$ , (Lebesgue, Riemann) integral      63
$\mathcal{A}$ , $\sigma$ -algebra      2 56
$\mathfrac{R}^{+}$ , positive real line      3
$\mathfrac{R}^{d}$ , d-dimensional Euclidean space      3
$\mu$ , mean error      124
$\omega$ , event, outcome      1
$\Omega$ , sample space      1
$\sigma$ -algebra      2 56 64 238
$\tau_{n}$ , discretization time      110
$\underline{a}(\cdot)$ , Stratonovich corrected drift      75 140
$\underline{L}^{0}$ , $\underline{L}^{1}$ , $\underline{L}^{j}$ , differential operators      80 139
$|\cdot|$ , Euclidean norm      15
(a,b), open interval from a to b      4
100 $\alpha$ %, level of significance      32
a,b,..., abbreviation in schemes for $a(Y_{n},b(Y_{n}),...$       140
A-stable      136
a.s., almost surely      2
Absolute error      118 170
Absolute error, criterion      115
Absolute stability region      136
Adams — Bashford method      99
Additive noise      69 142
Almost surely (a.s.)      2
Approximation, discrete time      110
Approximation, multiple integral      144
Approximation, pathwise      115
Approximation, variance reducing      207
Asymptotic stability      256
batch      33 118 125
Batch average      118 125
Bernoulli trials      31
Bifurcation      255 262
Bonhoeffer — Van der Pol equation      219
Box — Muller method      11
Brownian bridge      59
Brownian bridge, process      81
Brownian motion      50
Brusselator equation      260
Central limit theorem      30 32
Chebychev inequality      32
Commutative noise      145
Commutativity condition      145
Commutativity condition, second      148
computer time      14 171
Conditional density      22
Conditional expectation      22 56
Conditional probability      22
Confidence interval      32 117 118 125
Congruential generator      40
Congruential generator, linear      40
Convergence in distribution      27
Convergence in law      27
Convergence with probability one (w.p.1)      27
Convergence, almost sure      27
Convergence, mean-square      27
Convergence, strong      27 122
Convergence, weak      27 128
Converges weakly      128
Converges weakly with order $\beta$       128
Correlated Gaussian random variables      24
Covariance      24
d, dimension of Ito process X      139
Density function      4 30
Density, conditional      22
Density, optimal      211
Diffeomorphism      222
Difference method      91
Diffusion coefficient      72 73
Diffusion process      72
Discretization error, global      92
Discretization error, local      92
Distribution function      3
Distribution, exponential      4
Distribution, Gaussian      5 22
Distribution, Poisson      4
Distribution, uniform      4
Drift coefficient      72 73
Drift, corrected      139
Duffing — Van der Pol oscillator      222
Dynamical system      219
E(X), expectation of X      17
E(X|A), conditional expectation of X under A      22
e, Eulers number      10
Ergodic criterion      254
Ergodicity      46
Error, statistical      127
Error, systematic      127
Estimator, one-point      210
Estimator, testing parametric      229
Estimator, unbiased      214
Estimator, unbiased, variance reducing      214
Estimator, variance reducing      209
Euler approximation      110
Euler estimator, direct      211
Euler estimator, direct, unbiased      215
Euler estimator, variance reducing      211
Euler estimator, variance reducing, unbiased      215
Euler method      110
Euler method, deterministic      91
Euler method, fully implicit      196
Euler method, implicit      136 158
Euler method, simplified weak      180
Euler scheme      111 140 179
Euler scheme, implicit      136 158 194 196 242
Euler scheme, implicit, fully      196
Euler — Maruyama approximation      110 140

Euler — Maruyama scheme      140
Event      1
Expectation, approximating      115
Expectation, conditional      22 56
Expected value      17
Experiment      3
Explicit, strong scheme, order 1.0      151 152
Explicit, strong scheme, order 1.5      153
Explicit, strong scheme, order 2.0      154
Explicit, weak scheme      186
Explicit, weak scheme, order 2.0      186 203
Explicit, weak scheme, order 3.0      188 204
Exponential functional      251
Exponentially distributed      21 23 49
Extrapolation method      100
Extrapolation, fourth order      192 205 249
Extrapolation, order 6.0 weak      192
Extrapolation, second order      189 203
Extrapolation, third order      192 204
Functional      246
Functional integral      246
Gaussian density      5
Gaussian process      59
Gaussian process, independent increments      50
Gaussian standard distribution      5
Girsanov transformation      208
Goodness-of-fit test      35
Heun method      98
histogram      17 20 35 38
Histogram, frequency      222
I, unit matrix      158
Implicit, filter      242
Implicit, scheme      225
Implicit, weak scheme order 2.0      197 203 248
Independent      3 26 39
Independent identically distributed (i.i.d)      29
Independent increments      50
Indicator function      3 90
Inequality, Chebyshev      32
Inequality, Lyapunov      27
infty, infimum      10
Initial value problem (IVP)      91
Integral, Ito      64 66
Integral, Stratonovich      67
Intensity matrix      48
Interpolation, linear      111
Interpolation, piecewise constant      111
Invariant measure      246
Invariant probability law      253
Inverse transform method      11
Ito diffusion process      72
Ito formula      68 73 78
Ito integral      64 66
Ito integral, multiple      79 89
Ito process ergodic      253
Ito stochastic differential equation      75
Ito — Taylor expansion      78 79
Joint distribution function      26
Karhunen — Loeve expansion      57
Kolmogorov — Smirnov test      38
l!, factorial of l      77
Lagged-Fibonacci generator      16
Langevin equation      69
Law of Large Numbers      29 32 55
level of significance      32
Linear congruential generator      40
Linear congruential method      15
Logarithmic coordinates      94 126
Lyapunov exponent      134 228 256ff.
Lyapunov inequality      27
m, dimension of Wiener process W      139
Markov chain      43
Markov chain filter      239
Markov chain, continuous time      48
Markov chain, continuous time finite-state      238
Markov chain, discrete time      43
Markov chain, ergodic      47 50
Markov chain, filtering      238
Markov chain, finite-state      238
Markov chain, hidden      244
Markov chain, homogeneous      44 48
Markov property      43
Martingale      56 66 266
Maximum Likelihood Estimation      230
Mean error      124 128 202
Mean value      17
Mean-square error      28
Measure transformation method      207 251
Midpoint method      104
Milstein scheme      142 145
Milstein scheme, implicit      161
mod c, modulus c      8
Moment pth-      22
Multi-index      82
Multiple Ito integral      140
Multiple Stratonovich integral      82 140
Multiple Stratonovich integral, approximation      82
Multiplicative noise      70 223
Multistep method      98
Noise, additive      69 142
Noise, multiplicative      70 223
Noise, scalar      148
Noisy limit cycle      221 263
Nonanticipativeness      66
Normally distributed      5
Null hypothesis      33
Numerical stability      129
ODE, ordinary differential equation      XIII
Operator differential      139
Optimal density      211
Option price      266
Order of strong convergence      122
Order of weak convergence      128
Ornstein — Uhlenbeck process      69 231 25
P(.|.), conditional probability      2
P, probability measure      2
Parabolic equation      247
PDE, partial differential equation      XIV
Pearson statistic      35
Period      16
Poisson distribution      4
Polar Marsaglia method      12
Population growth model      234
Predictor-corrector method      98
Predictor-corrector method, order 1.0 weak      198
Predictor-corrector method, order 2.0 weak      199f. 203
probability      1 2
Probability measure      2
Probability one      2
Probability space      2 43
Probability vector      44 48
Probability vector stationary      46 49
Probability, conditional      2 43
Pseudo-random number      8
Pseudo-random number, Gaussian      11
Pseudo-random number, generator linear congruential      8
Radon — Nikodym derivative      230
Random number exponential      11
Random telegraphic noise      49
Random variable      3 26 43
Random variable, continuous      4
Random variable, Gaussian      5 22
Random variable, independent      26
Random variable, N-state      10
Random variable, standard Gaussian      5 21
Random variable, three-point      182
Random variable, two-point      3 10 52 180
Random walk      52 55
Realization      43
Rectangular density function      4

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