AN INTRODUCTION TO MATHEMATICS
CONTENTS INTRODUCTION INTEGRALS RANDOM WALKS EXERCISES BROWNIAN MOTION DE?NITION OF BROWNIAN MOTION SIMPLE PROPERTIES OF BROWNIAN MOTION WIENER INTEGRAL CONDITIONAL EXPECTATION MARTINGALES SERIES EXPANSION OF WIENER INTEGRALS CONSTRUCTIONS OF BROWNIAN MOTION WIENER SPACE BOREL–CANTELLI LEMMA AND CHEBYSHEV INEQUALITY KOLMOGOROV’S EXTENSION AND CONTINUITY THEOREMS L´EVY’S INTERPOLATION METHOD STOCHASTIC INTEGRALS BACKGROUND AND MOTIVATION FILTRATIONS FOR A BROWNIAN MOTION SIMPLE EXAMPLES OF STOCHASTIC INTEGRALS DOOB SUBMARTINGALE INEQUALITY STOCHASTIC PROCESSES DE?NED BY ITˆO INTEGRALS RIEMANN SUMS AND STOCHASTIC INTEGRALS AN EXTENSION OF STOCHASTIC INTEGRALS A LARGER CLASS OF INTEGRANDS A KEY LEMMA GENERAL STOCHASTIC INTEGRALS STOPPING TIMES ASSOCIATED STOCHASTIC PROCESSES STOCHASTIC INTEGRALS FOR MARTINGALES POISSON PROCESSES PREDICTABLE STOCHASTIC PROCESSES DOOB–MEYER DECOMPOSITION THEOREM MARTINGALES AS INTEGRATORS EXTENSION FOR INTEGRANDS THE ITˆO FORMULA ITˆO’S FORMULA IN THE SIMPLEST FORM PROOF OF ITˆO’S FORMULA ITˆO’S FORMULA SLIGHTLY GENERALIZED ITˆO’S FORMULA IN THE GENERAL FORM MULTIDIMENSIONAL ITˆO’S FORMULA ITˆO’S FORMULA FOR MARTINGALES APPLICATIONS OF THE ITˆO FORMULA EVALUATION OF STOCHASTIC INTEGRALS DECOMPOSITION AND COMPENSATORS STRATONOVICH INTEGRAL L´EVY’S CHARACTERIZATION THEOREM MULTIDIMENSIONAL BROWNIAN MOTIONS TANAKA’S FORMULA AND LOCAL TIME EXPONENTIAL PROCESSES TRANSFORMATION OF PROBABILITY MEASURES GIRSANOV THEOREM MULTIPLE WIENER–ITˆO INTEGRALS A SIMPLE EXAMPLE DOUBLE WIENER–ITˆO INTEGRALS HERMITE POLYNOMIALS HOMOGENEOUS CHAOS ORTHONORMAL BASIS FOR HOMOGENEOUS CHAOS WIENER–ITˆO THEOREM REPRESENTATION OF BROWNIAN MARTINGALES STOCHASTIC DI?ERENTIAL EQUATIONS 10 1 SOME EXAMPLES 10 2 BELLMAN–GRONWALL INEQUALITY 10 3 EXISTENCE AND UNIQUENESS THEOREM 10 4 SYSTEMS OF STOCHASTIC DI?ERENTIAL EQUATIONS 10 5 MARKOV PROPERTY 10 6 SOLUTIONS OF STOCHASTIC DI?ERENTIAL EQUATIONS 10 7 SOME ESTIMATES FOR THE SOLUTIONS 10 8 DI?USION PROCESSES 10 9 SEMIGROUPS AND THE KOLMOGOROV EQUATIONS SOME APPLICATIONS AND ADDITIONAL TOPICS 11 1 LINEAR STOCHASTIC DI?ERENTIAL EQUATIONS 11 2 APPLICATION TO FINANCE 11 3 APPLICATION TO FILTERING THEORY 11 4 FEYNMAN–KAC FORMULA 11 5 APPROXIMATION OF STOCHASTIC INTEGRALS 11 6 WHITE NOISE AND ELECTRIC CIRCUITS REFERENCES