Stochastic Differential Equations and Applications

Friedman A. Stochastic Differential Equations and Applications. Vol. 1 ......Page 4
Copyright ......Page 5
Contents v ......Page 6
Preface ix ......Page 9
General Notation xi  1......Page 10
Contents of Volume 2 xiii ......Page 11
1.The Kolmogorov construction of a stochastic process 1 ......Page 12
2.Separable and continuous processes 6 ......Page 17
3.Martingales and stopping times 9 ......Page 20
Problems 15 ......Page 26
1.Construction of Markov processes 18 ......Page 29
2.The Feller and the strong Markov properties 23 ......Page 34
3.Time-homogeneous Markov processes 30 ......Page 41
Problems 31 ......Page 42
1.Existence of continuous Brownian motion 36 ......Page 47
2.Nondifferentiability of Brownian motion 39 ......Page 50
3.Limit theorems 40 ......Page 51
4.Brownian motion after a stopping time 44 ......Page 55
5.Martingales and Brownian motion 46 ......Page 57
6.Brownian motion in n dimensions 50 ......Page 61
Problems 53 ......Page 64
1.Approximation of functions by step functions 55 ......Page 66
2.Definition of the stochastic integral 59 ......Page 70
3.The indefinite integral 67 ......Page 78
4.Stochastic integrals with stopping time 72 ......Page 83
5.Ito’s formula 78 ......Page 89
6.Applications of Ito’s formula 85 ......Page 96
7.Stochastic integrals and differentials in n dimensions 89 ......Page 100
Problems 93......Page 104
1.Existence and uniqueness 98 ......Page 109
2.Stronger uniqueness and existence theorems 102 ......Page 113
3.The solution of a stochastic differential system as a Markov process 108 ......Page 119
4.Diffusion processes 114 ......Page 125
5.Equations depending on a parameter 117 ......Page 128
6.The Kolmogorov equation 123 ......Page 134
Problems 125 ......Page 136
1.Square root of a nonnegative definite matrix 128 ......Page 139
2.The maximum principle for elliptic equations 132 ......Page 143
3.The maximum principle for parabolic equations 134 ......Page 145
4.The Cauchy problem and fundamental solutions for parabolic equations 139 ......Page 150
5. Stochastic representation of solutions of partial differential equations 144 ......Page 155
Problems 150 ......Page 161
1.A class of absolutely continuous probabilities 152 ......Page 163
2.Transformation of Brownian motion 156 ......Page 167
3.Girsanov’s formula 164 ......Page 175
Problems 169 ......Page 180
1.Unboundedness of solutions 172 ......Page 183
2.Auxiliary estimates 174 ......Page 185
3.Asymptotic estimates 180 ......Page 191
4.Applications of the asymptotic estimates 185 ......Page 196
5.The one-dimensional case 188 ......Page 199
6.Counterexample 191 ......Page 202
Problems 193 ......Page 204
1.Transient solutions 196 ......Page 207
2. Recurrent solutions 200 ......Page 211
3.Rate of wandering out to infinity 203 ......Page 214
4.Obstacles 207 ......Page 218
5.Transient solutions for degenerate diffusion 213 ......Page 224
6.Recurrent solutions for degenerate diffusion 217 ......Page 228
7.The one-dimensional case 219 ......Page 230
Problems 222 ......Page 233
Bibliographical Remarks 224 ......Page 235
References 226 ......Page 237
Index I ......Page 240
cover ......Page 1