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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 ProcessesАвтор: Jacobsen M. Аннотация: 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.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2006Количество страниц: 328Добавлена в каталог: 10.05.2008Операции: Положить на полку |
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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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