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Jacobsen M. — Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes
Jacobsen M. — Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes

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Название: Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes

Автор: Jacobsen M.


This text gives an essentially self-contained exposition of the basic theory of marked point processes, developing randomly over time, and shows how this theory may be used to treat piecewise deterministic stochastic processes in continuous time.

The focus is on point processes that generate only finitely many points in finite time intervals, resulting, in particular, in piecewise deterministic processes with "few jumps." The point processes are constructed from scratch and their distributions characterized using compensating measures and martingale structures. Piecewise deterministic processes are defined and identified with certain marked point processes, and this identification is used to construct and study a large class of piecewise deterministic Markov processes, whether time homogeneous or not.

The second part of the book shows how the theory developed earlier is used to analyze various models in statistics and applied probability with examples from survival analysis, branching processes, risk theory, finance, queueing theory and sports (soccer). Graduate students, postdocs and researchers in the above-mentioned areas will find this text an excellent resource, requiring only a solid foundation in probability theory and measure theory as well as basic knowledge of stochastic processes and martingales.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 328

Добавлена в каталог: 10.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
Предметный указатель
Absolute continuity      103
Absolute continuity, local, conditions for      105
Absolute continuity, local, definition      104
Absorbing state      28
Aggregate      120
Backward differential equation      149
Backward integral equation      149
Birth process      21
Branching diffusion      29
Branching process      231
Branching process, branching property      233
Branching process, extinction probability      235
Brownian motion      29
Cadlag      5
Canonical filtration for CP      43
Canonical filtration for RCM      43
Cauchy process      140
Censoring, right-censored survival data      221
Chapman — Kolmogorov equations, homogeneous Markov chain      27
Chapman — Kolmogorov equations, Markov process      144
Compensating measure, characterizing RCM      54
Compensating measure, definition      52
Compensating measure, general filtration      96
Compensator, basic martingale      72
Compensator, characterizing CP      54
Compensator, CP of set of marks, definition      52
Compensator, definition      51
Compensator, general filtration      96
Compensator, predictability of      52
Compensator, total, definition      52
Compound Poisson process      135
Conditional expectation      5
Conditional probability      6
Corlol      5
Counting process, canonical      12
Counting process, canonical probability      17
Counting process, definition      12
Counting process, distribution      12
Counting process, identification with SPP      12
Counting process, path space      12
Cox process      99
Cox regression model      225
Cox regression model, baseline hazard      226
Cox regression model, covariates      225
Cox regression model, partial likelihood      227
Cox regression model, partial score process      228
Cross characteristic      83
Dead process      19
Determining class      185
Differentiability w r t cadlag functions      298
Differentiability w r t cadlag functions, derivative      298
Differentiability w r t cadlag functions, partial integration      299
Doob — Meyer decomposition theorem      306
Doubly stochastic Poisson process      132
Extended generator      175
Feller — Kolmogorov equation      149
Filtered probability space      301
Filtered probability space, complete      301
Filtered probability space, completion      301
Filtered probability space, filtration      301
Filtered probability space, usual conditions      301
Filtration, canonical for CP      43
Filtration, canonical for RCM      43
Finance, arbitrage opportunity      257
Finance, attainable contingent claim      269
Finance, complete market      273
Finance, contingent claim      269
Finance, cumulated gains process      251
Finance, discounted value process      252
Finance, drift equation      258
Finance, equivalent martingale measure      258
Finance, European call option      275
Finance, Farkas’ Lemma      267
Finance, global arbitrage opportunity      257
Finance, hedging      273
Finance, pathwise bounded strategy      252
Finance, replicating strategy      269
Finance, risk-neutral measure      258
Finance, self-financing strategy      251
Finance, trading strategy      251
Finance, value process      251
Forward differential equation      149
Forward integral equation      149
Gamma-process      140
Harmonic function      211
Hazard function      34
Hazard measure, definition      34
Hazard measure, properties      35
Homogeneous Markov chain as MPP      27
Homogeneous Markov chain, compensating measure      56
Homogeneous Markov chain, intensity process      65
Homogeneous Poisson process as CP      19
Homogeneous Poisson process as SPP      19
Homogeneous Poisson process with mark space      23
Homogeneous Poisson process, basic martingale      74
Homogeneous Poisson process, compensator      55
Homogeneous Poisson process, intensity process      65
Homogeneous Poisson random measure, compensating measure      56
Homogeneous Poisson random measure, definition      24
Homogeneous Poisson random measure, intensity process      65
Innovation theorem      101
Intensity measure, definition      63
Intensity process as limit of conditional probabilities      66
Intensity process as predictable field      64
Intensity process for CP of marks in set, definition      63
Intensity process for CP, definition      63
Intensity process, conditions for existence      64
Intensity process, kappa-      63
Intensity process, positivity at jump times      67
Invariant probability      184
Invariant probability, PDMPs, generator      186
Invariant probability, PDMPs, occupation measure      195
Irrelevant mark      10
Items at risk      218
Ito’s formula for RCMs      85
Ito’s formula, PDMPs, full      180
Ito’s formula, PDMPs, special      171
Kaplan — Meier estimator      223
Key lemma      57
Levy measure      139
Levy process      127
Levy — Khinchine formula      139
Likelihood process, as martingale      104
Likelihood process, construction from local martingale      113
Likelihood process, construction from martingale      113
Likelihood process, definition      104
Likelihood process, PDMPs      204
Local martingale      308
Local martingale, reducing sequence      308
Local submartingale      308
Mark space      10
Mark space, separating points      11
Marked point process, aggregate      120
Marked point process, canonical probability      21
Marked point process, construction      22
Marked point process, definition      10
Marked point process, discrete time process as      25
Marked point process, distribution      11
Marked point process, explosion      11
Marked point process, identification with RCM      15
Marked point process, Ito’s formula      85
Marked point process, superposition      120
Markov chain, construction as step process      147
Markov chain, general state space      146
Markov chain, transition intensity      146
Markov kernel, transition probability      6
Markov modulated Poisson process      132
Markov process, definition      143
Markov process, homogeneous, definition      143
Markov process, initial distribution      145
Markov process, invariant probability      184
Markov process, piecewise deterministic      152
Markov process, stationary distribution      184
Markov process, transition operator      145
Markov process, transition probability      144
Markov property, homogeneous Markov chain      60
Markov property, PDMPs      158
Markov renewal process      28
Martingale      302
Martingale measure      79
Martingale, basic on canonical space      72
Martingale, main theorem, continuous time      304
Martingale, predictable on canonical space      69
Martingale, representation theorem      79
Minimal jump chain      28
Multi-self-similarity      161
Multiplicative functional      207
Multiplicative functional, homogeneous      207
Nelson — Aalen estimator      222
Occupation measure      187
Optional sampling theorem      307
Partial likelihood      227
Path-continuous      178
Path-differentiable      178
Piecewise continuous process      26
Piecewise deterministic Markov process      15
Piecewise deterministic Markov process, backward differential equation      183
Piecewise deterministic Markov process, backward integral equation      181
Piecewise deterministic Markov process, construction, general      157
Piecewise deterministic Markov process, construction, homogeneous general      157
Piecewise deterministic Markov process, construction, homogeneous special      155
Piecewise deterministic Markov process, construction, special      155
Piecewise deterministic Markov process, forward differential equation      183
Piecewise deterministic Markov process, full infinitesimal generator      179
Piecewise deterministic Markov process, homogeneous Poisson measure      165
Piecewise deterministic Markov process, infinitesimal generator, special      174
Piecewise deterministic Markov process, Ito’s formula      171
Piecewise deterministic Markov process, Ito’s formula, full      180
Piecewise deterministic Markov process, likelihood process      204
Piecewise deterministic Markov process, Markov property      158
Piecewise deterministic Markov process, multiplicative functional      207
Piecewise deterministic Markov process, path-continuity      178
Piecewise deterministic Markov process, path-differentiability      178
Piecewise deterministic Markov process, renewal process      162
Piecewise deterministic Markov process, solving SDE      165
Piecewise deterministic Markov process, strong Markov property      168
Piecewise deterministic process      26
Piecewise linear process      26
Poisson process, compound      135
Poisson process, curtailed      127
Poisson process, doubly stochastic      132
Poisson process, existence      128
Poisson process, general, definition      127
Poisson process, homogeneous      19
Poisson process, Markov modulated      132
Positive recurrent      192
Predictable field, definition      78
Predictable sigma-algebra      302
Predictable sigma-algebra, canonical      43
Process on canonical space      43
Process on canonical space, adapted      43
Process on canonical space, measurable      43
Process on canonical space, predictable      43
Process on canonical space, previsible      43
Process on canonical space, representation of adapted, CP      45
Process on canonical space, representation of adapted, RCM      45
Process on canonical space, representation of predictable, CP      45
Process on canonical space, representation of predictable, RCM      45
Product integral      35
Quadratic characteristic      83
Quadratic variation      84
Queueing, GI/G/1 queue      277
Queueing, Jackson network      287
Queueing, M/M/1 queue      277
Queueing, throughput rates      288
Queueing, virtual waiting time      287
Random counting measure, aggregate      120
Random counting measure, canonical probability      21
Random counting measure, definition      13
Random counting measure, deterministic compensating measure, distribution      13
Random counting measure, identification with MPP      15
Random counting measure, independence      121
Random counting measure, independent increments      123
Random counting measure, space of      13
Random counting measure, superposition      120
Regular conditional distribution      6
Renewal process, as PDMP      162
Renewal process, as SPP      20
Renewal process, backward recurrence time      163
Renewal process, compensator      55
Renewal process, forward recurrence time      164
Renewal process, intensity process      65
Renewal process, invariant probability, backward      197
Renewal process, invariant probability, forward      199
Renewal process, zero-delayed      20
Right-censoring      221
Ruin probability      236
Ruin problem      236
Ruin problem, difficult      236
Ruin problem, Lundberg equation      241
Ruin problem, simple      236
Ruin problem, undershoot      236
Semi-Markov process      28
Shift      56
Shift, translated      56
Simple point process, canonical probability      17
Simple point process, construction      18
Simple point process, definition      9
Simple point process, distribution      10
Simple point process, explosion      10
Simple point process, identification with CP      12
Simple point process, stable      19
Space-time generator      171
Stability problem      19
Stability, criterion for compensating measure      62
Stability, criterion for intensity process      68
Stability, criterion for likelihood process      113
Stability, criterion for total compensator      61
Stationary distribution      184
Step process      26
Stochastic integral, wrt compensating measure      78
Stochastic integral, wrt RCM      78
Stochastic process      301
Stochastic process, adapted      302
Stochastic process, cadlag      302
Stochastic process, continuous      302
Stochastic process, increasing      302
Stochastic process, indistinguishability      302
Stochastic process, left-continuous      302
Stochastic process, measurable      301
Stochastic process, predictable      302
Stochastic process, previsible      302
Stochastic process, right-continuous      302
Stopped process      307
Stopping time      306
Stopping time, on canonical space      69
Stopping time, sigma-algebra, canonical space      69
Stopping time, strict      306
Submartingale      302
Supermartingale      303
Superposition      120
Survivor function      19
Termination point      34
Time change      98
Time to ruin      236
Transition intensity, general Markov chain      149
Transition intensity, homogeneous Markov chain      27
Transition probability, homogeneous Markov chain      27
Transition probability, Markov process      144
Transition-closed set      187
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