AN INTRODUCTION TO THE THEORY OF POINT PROCESSES
CONTENTS PART I THEORY 1 INTRODUCTION 3 1 1 OVERVIEW 3 1 2 CONDITIONAL EXPECTATIONS AND PROBABILITIES 5 2 SIMPLE AND MARKED POINT PROCESSES 9 2 1 THE DE NITION OF SPPS AND MPPS 9 2 2 COUNTING PROCESSES AND COUNTING MEASURES 11 3 CONSTRUCTION OF SPPS AND MPPS 17 3 1 CREATING SPPS 17 3 2 CREATING MPPS 21 3 3 FROM MPPS TO PDPS 25 4 COMPENSATORS AND MARTINGALES 33 4 1 HAZARD MEASURES 33 4 2 ADAPTED AND PREDICTABLE PROCESSES 41 4 3 COMPENSATORS AND COMPENSATING MEASURES 50 4 4 INTENSITY PROCESSES 63 4 5 THE BASIC MARTINGALES 68 4 6 STOCHASTIC INTEGRALS AND MARTINGALES 77 4 7 IT O S FORMULA FOR MPPS 85 4 8 COMPENSATORS AND LTRATIONS 94 5 LIKELIHOOD PROCESSES 103 5 1 THE STRUCTURE OF THE LIKELIHOOD 103 5 2 CONSTRUCTING RCMS FROM MARTINGALES 110 6 INDEPENDENCE 119 6 1 INDEPENDENT POINT PROCESSES 119 6 2 INDEPENDENT INCREMENTS L EVY PROCESSES 123 7 PDMPS 143 7 1 MARKOV PROCESSES 143 7 2 MARKOV CHAINS 146 7 3 CONSTRUCTION AND BASIC PROPERTIES OF PDMPS 152 7 4 EXAMPLES OF PDMPS 163 7 4 1 RENEWAL PROCESSES 163 7 4 2 PROCESSES DERIVED FROM HOMOGENEOUS POISSON MEASURES 165 7 4 3 A PDMP THAT SOLVES AN SDE 165 7 5 THE STRONG MARKOV PROPERTY 167 7 6 IT O S FORMULA FOR HOMOGENEOUS PDMPS 170 7 7 THE FULL IN NITESIMAL GENERATOR 177 7 8 STATIONARITY 184 7 9 LIKELIHOOD PROCESSES FOR PDMPS 203 PART II APPLICATIONS 8 SURVIVAL ANALYSIS 217 8 1 INDEPENDENT SURVIVAL TIMES RIGHT CENSORING 217 8 2 THE COX REGRESSION MODEL 225 9 BRANCHING RUIN SOCCER 231 9 1 A BRANCHING PROCESS 231 9 2 RUIN PROBABILITIES 235 9 3 THE SOCCER MODEL 243 10 A MODEL FROM FINANCE 247 10 1 THE MODEL 247 10 2 PORTFOLIOS AND SELF NANCING STRATEGIES 251 10 3 ARBITRAGE AND MARTINGALE MEASURES 257 10 4 CONTINGENT CLAIMS AND PRICING 267 11 EXAMPLES OF QUEUEING MODELS 277 11 1 THE GI G 1 QUEUE 277 11 2 NETWORK MODELS 287 PART I THEORY 1 INTRODUCTION 1 1 OVERVIEW WHILE POINT PROCESSES IN GENERAL DESCRIBE THE RANDOM SCATTERING OF POINTS IN A CER TAIN SPACE THE POINT PROCESSES CONSIDERED IN THIS BOOK ARE SPECIAL IN THE SENSE THAT THE SPECI C POINTS ARE ASSOCIATED WITH THE TIMES AT WHICH CERTAIN RANDOM EVENTS OCCUR THE RESULTING STOCHASTIC PHENOMENA DEVELOPING OVER TIME THEN FORM THE MAIN OBJECT OF STUDY JUST RECORDING THE TIME OCCURRENCES OF THE EVENTS RESULTS IN A SIMPLE POINT PROCESS SPP A MARKED POINT PROCESS MPP IS OBTAINED IF AT EACH TIME OCCURRENCE THE TYPE OR MARK OF THE EVENT IS ALSO REGISTERED APART FROM DISCUSSING THE THEORY OF SUCH PROCESSES ONE OF THE MAIN POINTS OF THE BOOK IS TO SHOW HOW CERTAIN ORDINARY STOCHASTIC PROCESSES MAY BE VIEWED AS MPPS E G BY CONSIDERING THE TIME POINTS AT WHICH A GIVEN STOCHASTIC PROCESS JUMPS WITH THE TYPE OF EVENT OR MARKS ASSOCI ATED WITH EACH OCCURRENCE SOME CHARACTERISTIC OF THE JUMP THE RESULT IS A PIECEWISE DETERMINISTIC PROCESS PDP FORMALLY DE NED AS A PROCESS ADAPTED TO THE LTRATION GENERATED BY THE MPP OF TIMEPOINTS OF JUMPS AND THEIR CHARACTERISTICS IN THE RST CHAPTERS CHAPTERS 2 AND 3 THE DE NITION AND CONSTRUCTION OF SIMPLE AND MARKED POINT PROCESSES ARE GIVEN TOGETHER WITH THE EQUIVALENT DESCRIPTION OF RESPEC TIVELY AN SPP AS A COUNTING PROCESS CP AND AN MPP AS A RANDOM COUNTING MEASURE RCM IT IS PART OF THE DE NITION THAT THE PROCESSES SHOULD BE STABLE I E ONLY NITELY MANY EVENTS ARE ALLOWED IN NITE INTERVALS OF TIME BUT POINT PROCESSES WITH EXPLOSION WHERE IN NITELY MANY TIME POINTS ACCUMULATE IN NITE TIME ARISE NATURALLY AND THEY ARE THEREFORE DISCUSSED AS WELL ALBEIT SOMEWHAT SUMMARILY AT VARIOUS PLACES IN THE TEXT THE LONG CHAPTER 4 CONTAINS THE BASIC THEORY OF CPS AND RCMS INCLUDING THE CHARACTERIZATION OF THEIR DISTRIBUTIONS THROUGH COMPENSATORS COMPENSATING MEASURES AND MARTINGALE PROPERTIES IN CHAPTER 5 LIKELIHOOD PROCESSES PROCESSES OF RADON NIKODYM DERIVATIVES ARE TREATED CHAPTER 6 TREATS STOCHASTIC DEPENDENCE BETWEEN NITELY MANY RCMS AS WELL AS RCMS WITH INDEPENDENT INCREMENTS ONE OF THE MOST IMPORTANT CHAPTERS CHAPTER 7 DEVELOPS THE THEORY OF PIECEWISE DETERMINISTIC MARKOV PROCESSES PDMPS USING THE THEORY OF RCMS FROM THE EARLIER CHAPTERS IN PARTICULAR IT INVESTIGATES THE DESCRIPTION OF GENERAL PDPS AS MPPS USING