Stochastic Differential Games. Theory and Applications

Cover......Page 1
Atlantis Studies in Probability and Statistics 2......Page 2
Stochastic Differential Games......Page 4
ISBN Print: 9789491216466......Page 5
Preface......Page 8
Contents......Page 10
1.1 Introduction......Page 12
1.2.1 Two-person, zero-sum differential games state and control variables......Page 16
1.2.2 Pursuit-Evasion Differential Games......Page 19
1.2.4 The Lanchester Combat Model......Page 21
1.2.5 Nonzero-sum N-person Differential Games......Page 22
1.2.6 Friedman’s approach to differential games......Page 23
1.3 Stochastic Differential Games: Definition and Brief Discussion......Page 24
1.3.1 Stochastic Linear Pursuit-Evasion Games......Page 25
1.3.2 The Definition of a Stochastic Differential Game......Page 28
1.4 Formulation of the Problem......Page 31
1.5 Basic Definitions......Page 32
Definition 1.5.5......Page 33
Definition 1.5.7.......Page 34
Definition 1.5.10......Page 35
2.1 Introduction......Page 36
2.2 Preliminaries and an Existence Theorem......Page 37
2.2.1 An Existence Theorem......Page 39
2.3.1 A General Stochastic Linear Pursuit-Evasion Game......Page 41
2.3.2 A Special Case of Equation (2.2.1)......Page 42
2.4 The Solution of a Stochastic Linear Pursuit-Evasion Game WithNonrandom Controls......Page 43
2.4.1 Preliminaries......Page 44
Theorem 2.4.1......Page 46
Corollary 2.4.1......Page 47
Lemma 2.4.1......Page 48
3.2 Two Person Zero-sum Games: Martingale methods......Page 58
Theorem 3.2.1.......Page 59
Theorem 3.2.2.......Page 60
Theorem 3.2.3.......Page 61
Theorem 3.2.4.......Page 62
Theorem 3.2.5.......Page 65
3.2.1 The Isaacs condition......Page 66
Theorem 3.2.7.......Page 67
3.3 Two Person Zero-sum Games and Viscosity Solutions......Page 68
Theorem 3.3.1.......Page 70
Theorem 3.3.4.......Page 71
3.4 Stochastic differential games with multiple modes......Page 72
Theorem 3.4.1.......Page 74
4.1 Introduction......Page 76
4.2 Preliminaries......Page 77
4.3 Formal solution for a Stochastic Linear Pursuit-Evasion game with perfectinformation......Page 78
4.4 On Stochastic Pursuit-Evasion games with imperfect information......Page 80
5.2.1 Two Person Non-Zero Sum Game......Page 84
5.2.2 Preliminaries......Page 86
Definition 5.2.2.......Page 87
5.2.3 Main Results......Page 88
5.2.4......Page 91
5.3 General solution......Page 95
5.3.1 Discounted Payoff on the Infinite Horizon......Page 97
5.3.3 Occupation Measures......Page 98
6.1 Introduction......Page 106
6.2 Weak Convergence Preliminaries......Page 107
Lemma 6.2.1......Page 108
6.3 Some Popular Payoff Structures......Page 109
6.3.1 Ergodic Payoff......Page 110
6.3.2 Problem Description......Page 111
6.3.3 Chattering Lemma......Page 113
6.3.4 Main Result......Page 114
6.3.6 Discounted Payoff......Page 121
6.4 Two Person Zero-sum Stochastic Differential Game with Multiple Modes,Weak Convergence......Page 125
6.4.1 Problem Description......Page 126
6.4.2 Weak Convergence and near optimality......Page 130
6.5 Partially Observed Stochastic Differential Games......Page 136
6.5.1 The Diffusion Model......Page 138
6.6 Deterministic Approximations in Two-Person Differential Games......Page 146
6.6.1 Preliminaries......Page 147
6.6.2 Fluid Approximation......Page 149
6.6.3 δ -Optimality......Page 152
7.2.1 Avergage Payoffs......Page 158
7.2.3 Discrete Parameter Games......Page 167
7.3 Deterministic Approximations in N-Person Differential Games......Page 168
7.3.1 Main Convergence Results......Page 170
8.1 Introduction......Page 176
8.2 Discounted Payoff Case......Page 177
8.2.1 The Markov Chain Approximation Method......Page 181
8.2.2 Continuous Time Interpolations......Page 187
8.2.3 Bounds and Approximations......Page 189
8.2.4 Approximations under the condition (A8.2.4)......Page 190
8.2.5 Finite-Valued and Piecewise Constant Approximations rε (·) in (8.2.25)......Page 193
8.2.7 Near Optimal Polici......Page 195
8.2.8 Convergence of the Numerical Solutions......Page 196
8.2.9 Stopping Time Problems and Pursuit-Evasion Games......Page 197
8.3 Ergodic Payoff case......Page 198
8.4 Non-zero-Sum Case......Page 207
8.4.1 The Model......Page 208
8.4.2 Randomized Stopping......Page 209
8.4.3 Comment on proof......Page 210
8.4.4 Approximating the Controls......Page 212
8.4.5 Equilibria and Approximations......Page 214
8.4.6 A Convenient Representation of the Values in (8.4.17)......Page 215
8.4.7 The Markov Chain Approximation Method......Page 216
8.4.9 First Approximations to the Chain......Page 221
8.4.10 Representations of the Chain With Control-Independent Driving Noise......Page 222
9.1 Introduction......Page 226
9.2 Stochastic Equity Investment Model with Institutional InvestrorSpeculation......Page 227
9.3 Competitive Advertising under Uncertainty......Page 232
9.3.1 The Model......Page 233
9.3.2 Symmetric Firms......Page 236
References......Page 244