Contents

Preface vii

1. Introduction, Survey and Background Material 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 DeterministicDifferentialGames: ABriefSurvey . . . . . . . . . . . . . 5

1.3 Stochastic Differential Games: Definition and Brief Discussion . . . . . . 14

1.4 Formulation of theProblem. . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 BasicDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2. Stochastic Linear Pursuit-Evasion Game 25

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Preliminaries and an Existence Theorem . . . . . . . . . . . . . . . . . . 26

2.3 Existence of a Solution for a Stochastic Linear Pursuit-Evasion Game . . 30

2.4 The Solution of a Stochastic Linear Pursuit-Evasion Game With

Nonrandom Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3. Two Person Zero-Sum Differential Games-General Case 47

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Two Person Zero-sum Games: Martingale methods . . . . . . . . . . . . 47

3.3 TwoPersonZero-sumGames andViscositySolutions . . . . . . . . . . . 58

3.4 Stochastic differential games with multiple modes . . . . . . . . . . . . . 61

4. Formal Solutions for Some Classes of Stochastic Linear Pursuit-

Evasion Games 65

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3 Formal solution for a Stochastic Linear Pursuit-Evasion game with

perfect information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

ix

x Stochastic Differential Games

4.4 On Stochastic Pursuit-Evasion games with imperfect information . . . . . 69

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5. N-Person Noncooperative Differential Games 73

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 A stochastic Pursuit-Evasion Game . . . . . . . . . . . . . . . . . . . . 73

5.3 General solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6. Weak Convergence in Two Player Stochastic Differential Games 95

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Weak Convergence Preliminaries . . . . . . . . . . . . . . . . . . . . . . 96

6.3 Some Popular Payoff Structures . . . . . . . . . . . . . . . . . . . . . . 98

6.4 Two Person Zero-sum Stochastic Differential Game with Multiple

Modes, Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.5 Partially Observed Stochastic Differential Games . . . . . . . . . . . . . 125

6.6 Deterministic Approximations in Two-Person Differential Games . . . . 135

7. Weak Convergence in Many Player Games 147

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.2 Some Popular Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8. Some Numerical Methods 165

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

8.2 Discounted Payoff Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8.3 Ergodic Payoff case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8.4 Non-zeroSumCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

9. Applications to Finance 215

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

9.2 Stochastic Equity Investment Model with Institutional Investror

Speculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

9.3 Competitive Advertising under Uncertainty . . . . . . . . . . . . . . . . 221