STOCHASTIC ANALYSIS

By V. JOIRA

CONTENTS PART I THEORY OF STOCHASTIC PROCESSES FUNDAMENTALS OF PROBABILITY PROBABILITY AND CONDITIONAL PROBABILITY RANDOM VARIABLES AND DISTRIBUTIONS RANDOM VECTORS INDEPENDENCE EXPECTATIONS MIXING INEQUALITIES CHARACTERISTIC FUNCTIONS GAUSSIAN RANDOM VECTORS CONDITIONAL EXPECTATIONS CONDITIONAL AND JOINT DISTRIBUTIONS CONVERGENCE OF RANDOM VARIABLES IN?NITELY DIVISIBLE DISTRIBUTIONS EXAMPLES 1 10 STABLE LAWS 1 11 MARTINGALES 1 12 EXERCISES AND ADDITIONS STOCHASTIC PROCESSES DE?NITION STOPPING TIMES CANONICAL FORM OF A PROCESS L2 PROCESSES GAUSSIAN PROCESSES KARHUNEN–LOÈVE EXPANSION MARKOV PROCESSES MARKOV DIFFUSION PROCESSES PROCESSES WITH INDEPENDENT INCREMENTS MARTINGALES THE MARTINGALE PROPERTY OF MARKOV PROCESSES THE MARTINGALE PROBLEM FOR MARKOV PROCESSES BROWNIAN MOTION AND THE WIENER PROCESS COUNTING AND POISSON PROCESSES 2 10 RANDOM MEASURES 2 10 1 POISSON RANDOM MEASURES 2 11 MARKED COUNTING PROCESSES 2 11 1 COUNTING PROCESSES 2 11 2 MARKED COUNTING PROCESSES 2 11 3 THE MARKED POISSON PROCESS 2 11 4 TIME-SPACE POISSON RANDOM MEASURES 2 12 WHITE NOISE 2 12 1 GAUSSIAN WHITE NOISE 2 12 2 POISSONIAN WHITE NOISE 2 13 LÉVY PROCESSES 2 14 EXERCISES AND ADDITIONS THE ITÔ INTEGRAL DE?NITION AND PROPERTIES STOCHASTIC INTEGRALS AS MARTINGALES ITÔ INTEGRALS OF MULTIDIMENSIONAL WIENER PROCESSES THE STOCHASTIC DIFFERENTIAL ITԒS FORMULA MARTINGALE REPRESENTATION THEOREM MULTIDIMENSIONAL STOCHASTIC DIFFERENTIALS THE ITÔ INTEGRAL WITH RESPECT TO LÉVY PROCESSES THE ITԖLÉVY STOCHASTIC DIFFERENTIAL AND THE GENERALIZED ITÔ FORMULA 3 10 FRACTIONAL BROWNIAN MOTION 3 10 1 INTEGRAL WITH RESPECT TO A FBM 3 11 EXERCISES AND ADDITIONS STOCHASTIC DIFFERENTIAL EQUATIONS EXISTENCE AND UNIQUENESS OF SOLUTIONS MARKOV PROPERTY OF SOLUTIONS GIRSANOV THEOREM KOLMOGOROV EQUATIONS MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS MULTIDIMENSIONAL DIFFUSION PROCESSES THE TIME-HOMOGENEOUS CASE APPLICATIONS OF ITԒS FORMULA FIRST HITTING TIMES EXIT PROBABILITIES ITԖLÉVY STOCHASTIC DIFFERENTIAL EQUATIONS MARKOV PROPERTY OF SOLUTIONS OF ITԖLÉVY STOCHASTIC DIFFERENTIAL EQUATIONS EXERCISES AND ADDITIONS STABILITY, STATIONARITY, ERGODICITY TIME OF EXPLOSION AND REGULARITY APPLICATION: A STOCHASTIC PREDATOR-PREY MODEL RECURRENCE AND TRANSIENCE STABILITY OF EQUILIBRIA STATIONARY DISTRIBUTIONS EXISTENCE OF A STATIONARY DISTRIBUTION—ERGODIC THEOREMS STABILITY OF INVARIANT MEASURES THE ONE-DIMENSIONAL CASE INVARIANT DISTRIBUTIONS FIRST PASSAGE TIMES ERGODIC THEOREMS PART II APPLICATIONS OF STOCHASTIC PROCESSES APPLICATIONS TO FINANCE AND INSURANCE ARBITRAGE-FREE MARKETS THE STANDARD BLACK–SCHOLES MODEL MODELS OF INTEREST RATES EXTENSIONS AND ALTERNATIVES TO BLACK–SCHOLES INSURANCE RISK APPLICATIONS TO BIOLOGY AND MEDICINE POPULATION DYNAMICS: DISCRETE-IN-SPACE–CONTINUOUS-IN- TIME MODELS INFERENCE FOR MULTIPLICATIVE INTENSITY PROCESSES POPULATION DYNAMICS: CONTINUOUS APPROXIMATION OF JUMP MODELS DETERMINISTIC APPROXIMATION: LAW OF LARGE NUMBERS DIFFUSION APPROXIMATION: CENTRAL LIMIT THEOREM POPULATION DYNAMICS: INDIVIDUAL-BASED MODELS A MATHEMATICAL DETOUR A “MODERATE” REPULSION MODEL ANT COLONIES PRICE HERDING TUMOR-DRIVEN ANGIOGENESIS THE CAPILLARY NETWORK NEUROSCIENCES EVOLUTIONARY BIOLOGY STOCHASTIC POPULATION MODELS LOGISTIC POPULATION GROWTH STOCHASTIC PREY–PREDATOR MODELS AN SIS EPIDEMIC MODEL A STOCHASTIC SIS EPIDEMIC MODEL WITH TWO CORRELATED ENVIRONMENTAL NOISES A VECTOR-BORNE EPIDEMIC SYSTEM STOCHASTICALLY PERTURBED SIR AND SEIR EPIDEMIC MODELS ENVIRONMENTAL NOISE MODELS