Peszat S., Zabczyk J. — Stochastic partial differential equations with Levy noise: An evolution equation approach :: Электронная библиотека попечительского совета мехмата МГУ

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Peszat S., Zabczyk J. — Stochastic partial differential equations with Levy noise: An evolution equation approach

Peszat S., Zabczyk J. — Stochastic partial differential equations with Levy noise: An evolution equation approach

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Название: Stochastic partial differential equations with Levy noise: An evolution equation approach

Авторы: Peszat S., Zabczyk J.

Аннотация:

Recent years have seen an explosion of interest in stochastic partial differential equations where the driving noise is discontinuous. In this comprehensive monograph, two leading experts detail the evolution equation approach to their solution. Most of the results appear here for the first time in book form, and the volume is sure to stimulate further research in this important field. The authors start with a detailed analysis of L?vy processes in infinite dimensions and their reproducing kernel Hilbert spaces; cylindrical L?vy processes are constructed in terms of Poisson random measures; stochastic integrals are introduced. Stochastic parabolic and hyperbolic equations on domains of arbitrary dimensions are studied, and applications to statistical and fluid mechanics and to finance are also investigated. Ideal for researchers and graduate students in stochastic processes and partial differential equations, this self-contained text will also interest those working on stochastic modeling in finance, statistical physics and environmental science.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2007

Количество страниц: 424

Добавлена в каталог: 24.11.2013

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Предметный указатель

$C_{0}$ -generator      75 140
$C_{0}$ -semigroup      75 140 365—367
$\alpha$ -stable families      61
$\omega$ -excessive measure      376
$\pi$ and $\lambda$ systems      20
Adapted process      21
Adjoint operator      355
Angle bracket      35 107
Arbitrage opportunity      327
Aronson estimate      18
Aronson theorem      18
Bochner integral      24
Bochner theorem      72
Bonel isomorphism theorem      4
Bonel space      5
Boundary conditions, Dirichlet      220
Boundary conditions, Neumann      221
Bounded range      301
Brace — Gatarek — Musiela (BGM) equation      347 349
Brownian sheet      72 100
Burgers equation      317
Burgers system      312
Burkhoelder — Davies — Gundy inequality      37
Cadlag, definition      26
Caratheodory theorem      23
Cauchy family, symmetric      61
Chapman — Kolmogorov equation      7
Characteristics      56 299
Compact operator      355
Composition operator      16
Compound Poisson process      9
Configuration      308
Consistency      350
Consistency conditions      22
Consistency problem      350
Continuous part      391
Contraction      6
Convolution semigroup of measures      39
Core      149
Courrege's theorem      8
Covariance      29 248
Covariance form      31
Covariance operator      30 67
Cylindrical process, impulsive      103
Cylindrical process, Wiener      97
De Acosta theorem      39
Dilation theorem      160
Dirichlet boundary conditions      220
Discounted capital      326
Discounted price      326
Dissipative mapping      180
distribution      21
Distribution, exponential      41
Distribution, finite-dimensional      22
Distribution, gamma      191
Distribution, geometric      41
Distribution, Poisson      41
Dobrushin — Lanford — Ruelle (DLR) equation      308
Doleans measure      115
Doob — Meyer decomposition      35
Doob — Meyer theorem      35
Doob, optional sampling theorem      32
Doob, regularity theorem      33
Doob, submartingale inequality      32—34
Douglas theorem      364
Dynamical system, deterministic, continuous-time      5
Dynamical system, deterministic, discrete-time      3
Dynamical system, stochastic, continuous-time      6
Dynamical system, stochastic, discrete-time      3
Dynkin $\pi-\lambda$ theorem      20
Equation, BGM      347 349
Equation, BGM, classical      349
Equation, Burgers      317
Equation, Chapman — Kolmogorov      7
Equation, DLR      308
Equation, Focker — Planck      306
Equation, heat      374
Equation, HJMM      333
Equation, stochastic delay      238
Equation, stochastic difference      4
Equation, stochastic heat      220 221 255 261 279 283 284
Equation, stochastic wave      226 228 255
Equation, transport      374
Equation, wave      375
Euler $\Gamma$ -function      190
Evolution operator      156
Excessive measure      376
Expectation      24
Expectation, conditional      25
Exponential distribution      41
Exponential mixing      288
Exponentially tempered one-sided stable coordinates      64
Factorization      190
Family, $\alpha$ -stable      61
Feller property, family      170
Fernique theorem      30
Filtered probability space      21
Filtration      21
Filtration, generated by X      22
Filtration, right-continuous      21
Filtration, satisfying usual assumption      22
Fortet — Mourier norm      288
Forward curve      333
Forward rate function      324
Fourier transform      231
Fubini stochastic theorem      119
Function, non-negative-definite      71 360
Function, symmetric      71
Gaussian measure      28—31
Gaussian random process      28—31
Gaussian random variable      28—31
Gaussian random variable, centered (zero-mean)      29
Generator of arbitrary Levy process      80
Generator of compound Poisson      78
Generator of Wiener process      79
Geometric distribution      41
Gibbs density      308
Gibbs measure      308 309
Gradient equation      307
Green function      17—19
Haar system      45
Hamiltonian      308
Heath — Jarrow — Morton — Musiela (HJMM) equation      333
Hilbert — Schmidt norm      356
Hilbert — Schmidt operator      356
Hille — Yosida theorem      365
HJM condition      329
HJM condition, classical      329
HJM postulate      327
Hoelder spaces      13—14
Independent identically distributed (i.i.d.) random variables      4
Infinite divisible family      see "Convolution semigroup of measures"
Integrated volatility      348
Invariance problem      351
Invariant measure      287
Ising model      308
Ito formulae      391—393
Ito — Nisio theorem      23
Jump intensity measure      see "Levy measure"
Jump position intensity      103
Jump size intensity      103
Kinney theorem      27
Kolmogorov existence theorem      22
Kolmogorov formula      306—307
Kolmogorov test      26
Kolmogorov — Loeve — Chentsov theorem      26
Kotelenez inequality      156
Kruglov theorem      39
Kuratowski — Ryll — Nardzewski theorem      114 115
Laplace operator      220 221 230

Laplace operator, discrete      302
Laplace operator, fractional      223
Laplace transform      59
Laplacian      302 see
Law      see "Distribution"
Levy measure      11 45 56
Levy process, double-sided      290
Levy semigroup      8 75—82
Levy sheet      100 103
Levy — Ciesielski construction      45
Levy — Khinchin decomposition      53
Levy — Khinchin formula      56
Levy — Ottaviani inequality      23
LIBOR rates      348
Local martingale      28
Lumer — Phillips theorem      366
m-dissipative mapping      180
Mapping, almost m-dissipative      180
Mapping, dissipative      180
Mapping, m-dissipative      180
Mapping, maximal dissipative      see "m-dissipative"
Markov chain      4
Markov property      167
Markov time      see "Stopping time"
Martingale      28
Martingale covariance      107 111
Maximal dissipative mapping      180
Mean      67
Mean-reversion property      350
Measure, $\omega$ -excessive      376
Measure, equivalent martingale      327
Measure, Gibbs      308 309
Modification      21
Musiela parametrization      332
Nagy theorem      see "Dilation theorem"
Nelson — Siegel family      350
Nelson — Siegel family, augmented      350
Nelson — Siegel family, degenerate      350
Nemytskii operator      see "Composition operator"
Neumann boundary conditions      221
Norm, Fortet — Mourier      288
Norm, graph      149 196
Norm, Hilbert — Schmidt      356
Norm, nuclear      356
Norm, operator      355
Normalizing constant      308
Nuclear space      241
Operator, adjoint      355 366
Operator, compact      355
Operator, convolution      191 248
Operator, fractional power of      368
Operator, Hilbert — Schmidt      356
Operator, Laplace      230
Operator, Laplace, Dirichlet boundary conditions      220
Operator, Laplace, Neumann boundary conditions      221
Operator, Liouville — Riemann      190 193
Operator, negative-definite      366
Operator, non-negative-definite      355
Operator, norm      355
Operator, nuclear      356
Operator, self-adjoint      355 366
Orthonormal basis      356
Parseval identity      355
Path      3
Poincare inequality      14
Poisson distribution      41
Poisson process      40—44
Poisson random measure      47 83
Poisson random measure, compensated      47 83
Poisson random measure, stationary      84
Poisson random measure, stochastic integral      85—87
Polarization      37
Polish space      4
Portfolio      325
Positive-definite mapping      160
Potential      308
Potential of local interactions      299 310
Predictable modification      22
Predictable process      21
Predictable variation process      see "Angle bracket"
Process of finite variation      28
Process of jumps      39
Process of mean zero      25
Process with independent increments      38
Process, adapted      21
Process, cadlag      26
Process, Cauchy      11
Process, compensated      48
Process, compound Poisson      9 45
Process, continuous      26
Process, continuous in probability      26
Process, E-valued      21
Process, general Markov      27
Process, Levy      38
Process, Levy, in S'( $\mathbb{R}^{d}$ )      241
Process, Levy, Ornstein — Uhlenbeck      183
Process, Levy, spatially homogeneous, stationary      247
Process, Markov      27
Process, mean-square continuous      26
Process, measurable      21
Process, predictable      21
Process, right-continuous      26
Process, square integrable      25
Process, stochastically continuous      26
Process, Wiener, in Hilbert space      50
Process, Wiener, in S'( $\mathbb{R}^{d}$ )      242
Quadratic variation      36
Radon measure      105
Radonifying norm      135
Radonifying operator      135
Random element      see "Random variable"
Random variable      21
Random variable of mean zero      24
Random variable, integrable      23
Random variable, square integrable      24
Reaction-diffusion equation      187
Renormalization      307
Reproducing kernel Hilbert space (RKHS)      91—95
Resolvent      365
Self-financing strategy      326
Semigroup of contractions      156 181
Semigroup, analytic      368
Semigroup, Feller      170
Semigroup, property      5
Semigroup, spatially homogeneous      76
Semigroup, translation-invariant      76
Semimartingale      28 36
Shift semigroup      333
Skorokhod embedding theorem      5
Sobolev spaces      13—14
Solution, fundamental      232
Solution, generalized      139
Solution, mild      140 142 183 314
Solution, weak      141 149 313
Space, real-interpolation      277
Spatial correlation      70
Spectral measure      248
Stable families of order $\beta \in (0, 1)$       60
Stable families of order $\beta \in (1, 2)$       61
State space      3
Stochastic convolution      155
Stochastic process      21
Stopping time      22
Subdifferential      179
Submartingale      31
Subordinated      59
Supermartingale      31
Supremum norm      13
Symmetric Cauchy family      61
System of Dirichlet conditions      17

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