20 CONTENTS

74.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .249974.3 The Main Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .250274.4 A Simplification Of The Formula . . . . . . . . . . . . . . . . . . . . . .251074.5 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251174.6 The Ito Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2514

75 Some Nonlinear Operators 252575.1 An Assortment Of Nonlinear Operators . . . . . . . . . . . . . . . . . . .252575.2 Duality Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2531

76 Implicit Stochastic Equations 253776.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253776.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253776.3 The Existence Of Approximate Solutions . . . . . . . . . . . . . . . . . .254176.4 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .255476.5 Replacing Φ With σ (u) . . . . . . . . . . . . . . . . . . . . . . . . . . .257176.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .257876.7 Other Examples, Inclusions . . . . . . . . . . . . . . . . . . . . . . . . .2583

77 Stochastic Inclusions 259577.1 The General Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . .259577.2 Some Fundamental Theorems . . . . . . . . . . . . . . . . . . . . . . . .259577.3 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261177.4 Measurable Approximate Solutions . . . . . . . . . . . . . . . . . . . . .261477.5 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .262077.6 Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . .263477.7 Progressively Measurable Solutions . . . . . . . . . . . . . . . . . . . .263877.8 Adding A Quasi-bounded Operator . . . . . . . . . . . . . . . . . . . . .2642

78 A Different Approach 265578.1 Summary Of The Problem . . . . . . . . . . . . . . . . . . . . . . . . . .2655

78.1.1 General Assumptions On A . . . . . . . . . . . . . . . . . . . .265678.1.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . .2658

78.2 Measurable Solutions To Evolution Inclusions . . . . . . . . . . . . . . .266478.3 Relaxed Coercivity Condition . . . . . . . . . . . . . . . . . . . . . . . .267078.4 Progressively Measurable Solutions . . . . . . . . . . . . . . . . . . . . .2677

79 Including Stochastic Integrals 268179.1 The Case of Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . .268179.2 Replacing Φ With σ (u) . . . . . . . . . . . . . . . . . . . . . . . . . . .269679.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .270079.4 Stochastic Inclusions Without Uniqueness ?? . . . . . . . . . . . . . . . .2706

79.4.1 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .270679.4.2 A Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . .270979.4.3 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . .2723