18 CONTENTS

64.1 Real Wiener Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . .217964.2 Nowhere Differentiability of Wiener Processes . . . . . . . . . . . . . . .218464.3 Wiener Processes In Separable Banach Space . . . . . . . . . . . . . . .218564.4 Independent Increments and Martingales . . . . . . . . . . . . . . . . . .218964.5 Hilbert Space Valued Wiener Processes . . . . . . . . . . . . . . . . . . .219464.6 Wiener Processes, Another Approach . . . . . . . . . . . . . . . . . . . .2208

64.6.1 Lots Of Independent Normally Distributed Random Variables . .220864.6.2 The Wiener Processes . . . . . . . . . . . . . . . . . . . . . . .221564.6.3 Q Wiener Processes In Hilbert Space . . . . . . . . . . . . . . .221764.6.4 Levy’s Theorem In Hilbert Space . . . . . . . . . . . . . . . . .2224

65 Stochastic Integration 222765.1 Integrals Of Elementary Processes . . . . . . . . . . . . . . . . . . . . .222765.2 Different Definition Of Elementary Functions . . . . . . . . . . . . . . .223365.3 Approximating With Elementary Functions . . . . . . . . . . . . . . . . .223365.4 Some Hilbert Space Theory . . . . . . . . . . . . . . . . . . . . . . . . .223765.5 The General Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . .224265.6 The Case That Q Is Trace Class . . . . . . . . . . . . . . . . . . . . . . .224965.7 A Short Comment On Measurability . . . . . . . . . . . . . . . . . . . .225065.8 Localization For Elementary Functions . . . . . . . . . . . . . . . . . . .225165.9 Localization In General . . . . . . . . . . . . . . . . . . . . . . . . . . .225365.10 The Stochastic Integral As A Local Martingale . . . . . . . . . . . . . . .225565.11 The Quadratic Variation Of The Stochastic Integral . . . . . . . . . . . .225765.12 The Holder Continuity Of The Integral . . . . . . . . . . . . . . . . . . .226065.13 Taking Out A Linear Transformation . . . . . . . . . . . . . . . . . . . .226165.14 A Technical Integration By Parts Result . . . . . . . . . . . . . . . . . . .2263

66 The Integral∫ t

0 (Y,dM)H 2273

67 The Easy Ito Formula 228967.1 The Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .228967.2 Assumptions And A Lemma . . . . . . . . . . . . . . . . . . . . . . . .228967.3 A Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .229167.4 The Case Of Elementary Functions . . . . . . . . . . . . . . . . . . . . .229467.5 The Integrable Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .229567.6 The General Stochastically Integrable Case . . . . . . . . . . . . . . . . .229767.7 Remembering The Formula . . . . . . . . . . . . . . . . . . . . . . . . .229967.8 An Interesting Formula . . . . . . . . . . . . . . . . . . . . . . . . . . .229967.9 Some Representation Theorems . . . . . . . . . . . . . . . . . . . . . . .2300

68 A Different Kind Of Stochastic Integration 231368.1 Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . .231468.2 A Remarkable Theorem, Hermite Polynomials . . . . . . . . . . . . . . .232068.3 A Multiple Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . .232568.4 The Skorokhod Integral . . . . . . . . . . . . . . . . . . . . . . . . . . .2340

68.4.1 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . .2340