MODELING WITH ITÔ STOCHASTIC DIFFERENTIAL EQUATIONS

By BARBARA MOORE

RANDOM VARIABLES INTRODUCTION PROBABILITY SPACE RANDOM VARIABLE, PROBABILITY DISTRIBUTION EXPECTATION MULTIPLE RANDOM VARIABLES A HILBERT SPACE OF RANDOM VARIABLES CONVERGENCE OF SEQUENCES OF RANDOM VARIABLES COMPUTER GENERATION OF RANDOM NUMBERS MONTE CARLO EXERCISES COMPUTER PROGRAMS STOCHASTIC PROCESSES DISCRETE STOCHASTIC PROCESSES CONTINUOUS STOCHASTIC PROCESSES A HILBERT SPACE OF STOCHASTIC PROCESSES COMPUTER GENERATION OF STOCHASTIC PROCESSES EXAMPLES OF STOCHASTIC PROCESSES STOCHASTIC INTEGRATION INTEGRALS OF THE FORM T A F(S, ?) DS ITˆO STOCHASTIC INTEGRALS APPROXIMATION OF STOCHASTIC INTEGRALS STOCHASTIC DI?ERENTIALS AND ITˆO’S FORMULA STRATONOVICH STOCHASTIC INTEGRALS MULTIDIMENSIONAL ITˆO’S FORMULA STOCHASTIC DI?ERENTIAL EQUATIONS EXISTENCE OF A UNIQUE SOLUTION PROPERTIES OF SOLUTIONS TO STOCHASTIC DI?ERENTIAL EQUATIONS ITˆO’S FORMULA AND EXACT SOLUTIONS APPROXIMATING STOCHASTIC DI?ERENTIAL EQUATIONS SYSTEMS OF STOCHASTIC DI?ERENTIAL EQUATIONS FORWARD KOLMOGOROV (FOKKER-PLANCK) EQUATION STABILITY PARAMETER ESTIMATION FOR STOCHASTIC DI?ERENTIAL EQUATIONS A MAXIMUM LIKELIHOOD ESTIMATION METHOD A NONPARAMETRIC ESTIMATION METHOD MODELING POPULATION BIOLOGY EXAMPLES GENERAL MODEL OF TWO INTERACTING POPULATIONS EPIDEMIC MODEL AND PREDATOR-PREY MODEL PERSISTENCE-TIME ESTIMATION A POPULATION MODEL WITH A TIME DELAY A MODEL INCLUDING ENVIRONMENTAL VARIABILITY PHYSICAL SYSTEMS MECHANICAL VIBRATION SEED DISPERSAL ION TRANSPORT NUCLEAR REACTOR KINETICS PRECIPITATION CHEMICAL REACTIONS COTTON ?BER BREAKAGE SOME STOCHASTIC FINANCE MODELS A STOCK-PRICE MODEL OPTION PRICING INTEREST RATES A GOODNESS-OF-FIT TEST FOR AN SDE MODEL ALTERNATE EQUIVALENT SDE MODELS REFERENCES