This volume begins with auxiliary results in partial differential equations (Chapter 10) that are needed in the sequel. In Chapters 11 and 12 we study the behavior of the sample paths of solutions of stochastic differential equations in the same spirit as in Chapter 9. Chapter 11 deals with the question whether the paths can hit a given set with positive probability. Chapter 12 is concerned with the stability of paths about a given manifold, and (in case of two dimensions) with spiraling of paths about this manifold. Chapters 13-15 are concerned with applications to partial differential equations. In Chapter 13 we deal with the Dirichlet problem for degenerate elliptic equations. The results of Chapter 12 play here a fundamental role. In Chapter 14 we consider questions of singular perturbations. Chapter 15 is concerned with the existence of fundamental solutions for degenerate parabolic equations.Chapters 16 and 17 deal with stopping time problems, stochastic games and stochastic differential games.
Author(s): Avner Friedman
Series: Probability and Mathematical Statistics
Publisher: Academic Press
Year: 1976
Language: English
Pages: 315
Tags: Математика;Теория вероятностей и математическая статистика;Теория случайных процессов;
Friedman A. Stochastic Differential Equations and Applications. Vol. 2, ......Page 4
Copyright ......Page 5
Contents ......Page 6
Preface ix ......Page 10
General Notation xi ......Page 11
1.Schauder’s estimates for elliptic and parabolic equations 229 ......Page 13
2.Sobolev’s inequality 233 ......Page 17
3.L^p estimates for elliptic equations 236 ......Page 20
4.L^p estimates for parabolic equations 238 ......Page 22
Problems 240 ......Page 24
1.Basic definitions; a lemma 242 ......Page 26
2.A fundamental lemma 246 ......Page 30
3.The case d(x)>3 250 ......Page 34
4.The case d (x)> 2 253 ......Page 37
5.M consists of one point and d = 1 259 ......Page 43
6.The case d(x) = 0 263 ......Page 47
7.Mixed case 265 ......Page 49
Problems 267 ......Page 51
1.Criterion for stability 270 ......Page 54
2.Stable obstacles 278 ......Page 62
3.Stability of point obstacles 283 ......Page 67
4.The method of descent 286 ......Page 70
5.Spiraling of solutions about a point obstacle 290 ......Page 74
6.Spiraling of solutions about any obstacle 300 ......Page 84
7.Spiraling for linear systems 303 ......Page 87
Problems 306 ......Page 90
1.A general existence theorem 308 ......Page 92
2.Convergence of paths to boundary points 315 ......Page 99
3.Application to the Dirichlet problem 318 ......Page 102
Problems 322 ......Page 106
1.The functional IT(ф) 326 ......Page 110
2.The first Ventcel-Freidlin estimate 332 ......Page 116
3.The second Ventcel-Freidlin estimate 334 ......Page 118
4.Application to the first initial-boundary value problem 346 ......Page 130
5.Behavior of the fundamental solution as €—>0 348 ......Page 132
6.Behavior of Green’s function as c-»0 354 ......Page 138
7.The problem of exit 359 ......Page 143
8.The problem of exit (continued) 367 ......Page 151
9.Application to the Dirichlet problem 371 ......Page 155
10.The principal eigenvalue 373 ......Page 157
11.Asymptotic behavior of the principal eigenvalue 376 ......Page 160
Problems 383 ......Page 167
1.Construction of a candidate for a fundamental solution 388 ......Page 172
2.Interior estimates 396 ......Page 180
3.Boundary estimates 399 ......Page 183
4.Estimates near infinity 406 ......Page 190
5.Relation between K and a diffusion process 409 ......Page 193
6.The behavior of e(t) near S 414 ......Page 198
7.Existence of a generalized solution in the case of a two-sided obstacle 420 ......Page 204
8.Existence of a fundamental solution in the case of a strictly one-sidedobstacle 423 ......Page 207
9.Lower bounds on the fundamental solution 426 ......Page 210
10.The Cauchy problem 428 ......Page 212
Problems 432 ......Page 216
1.Statement of the problem 433 ......Page 217
2.Characterization of saddle points 436 ......Page 220
3.Elliptic variational inequalities in bounded domains 440 ......Page 224
4.Existence of saddle points in bounded domains 444 ......Page 228
5.Elliptic estimates for increasing domains 447 ......Page 231
6.Elliptic variational inequalities 457 ......Page 241
7.Existence of saddle points in unbounded domains 462 ......Page 246
8.The stopping time problem 463 ......Page 247
9.Characterization of saddle points 464 ......Page 248
10.Parabolic variational inequalities 466 ......Page 250
11.Parabolic variational inequalities (continued) 478 ......Page 262
12. Existence of a saddle point 486 ......Page 270
13.The stopping time problem 488 ......Page 272
Problems 490 ......Page 274
1.Auxiliary results 494 ......Page 278
2.N-person stochastic differential games with perfect observation 498 ......Page 282
3.Stochastic differential games with stopping time 502 ......Page 286
4.Stochastic differential games with partial observation 507 ......Page 291
Problems 518 ......Page 302
Bibliographical Remarks 520......Page 304
References 523 ......Page 307
Index 527 ......Page 311
cover ......Page 1
