Stochastic Differential Equations And Applications

Preface-Volume 1 xv General Notation xvi Stochastic Processes The Kolmogorov construction of a stochastic process 1(5) Separable and continuous processes 6(3) Martingales and stopping times 9(9) Problems 15(3) Markov Processes Construction of Markov processes 18(5) The Feller and the strong Markov properties 23(7) Time-homogeneous Markov processes 30(6) Problems 31(5) Brownian Motion Existence of continuous Brownian motion 36(3) Nondifferentiability of Brownian motion 39(1) Limit theorems 40(4) Brownian motion after a stopping time 44(2) Martingales and Brownian motion 46(4) Brownian motion in n dimensions 50(5) Problems 53(2) The Stochastic Integral Approximation of functions by step functions 55(4) Definition of the stochastic integral 59(8) The indefinite integral 67(5) Stochastic integrals with stopping time 72(6) Ito's formula 78(7) Applications of Ito's formula 85(4) Stochastic integrals and differentials in n dimensions 89(9) Problems 93(5) Stochastic Differential Equations Existence and uniqueness 98(4) Stronger uniqueness and existence theorems 102(6) The solution of a stochastic differential system as a Markov process 108(6) Diffusion processes 114(3) Equations depending on a parameter 117(6) The Kolmogorov equation 123(5) Problems 125(3) Elliptic and Parabolic Partial Differential Equations and Their Relations to Stochastic Differential Equations Square root of a nonnegative definite matrix 128(4) The maximum principle for elliptic equations 132(2) The maximum principle for parabolic equations 134(5) The Cauchy problem and fundamental solutions for parabolic equations 139(5) Stochastic representation of solutions of partial differential equations 144(8) Problems 150(2) The Cameron--Martin--Girsanov Theorem A class of absolutely continuous probabilities 152(4) Transformation of Brownian motion 156(8) Girsanov's formula 164(8) Problems 169(3) Asymptotic Estimates for Solutions Unboundedness of solutions 172(2) Auxiliary estimates 174(6) Asymptotic estimates 180(5) Applications of the asymptotic estimates 185(3) The one-dimensional case 188(3) Counterexample 191(5) Problems 193(3) Recurrent and Transient Solutions Transient solutions 196(4) Recurrent solutions 200(3) Rate of wandering out to infinity 203(4) Obstacles 207(6) Transient solutions for degenerate diffusion 213(4) Recurrent solutions for degenerate diffusion 217(2) The one-dimensional case 219(5) Problems 222(2) Bibliographical Remarks 224(2) References 226(294) Preface-Volume 2 iii General Notation iv Auxiliary Results in Partial Differential Equations Schauder's estimates for elliptic and parabolic equations 229(4) Sobolev's inequality 233(3) Lp estimates for elliptic equations 236(2) Lp estimates for parabolic equations 238(4) Problems 240(2) Nonattainability Basic definitions; a lemma 242(4) A fundamental lemma 246(4) The case d(x) ≥ 3 250(3) The case d(x) ≥ 2 253(6) M consists of one point and d = 1 259(4) The case d(x) = 0 263(2) Mixed case 265(5) Problems 267(3) Stability and Spiraling of Solutions Criterion for stability 270(8) Stable obstacles 278(5) Stability of point obstacles 283(3) The method of descent 286(4) Spiraling of solutions about a point obstacle 290(10) Spiraling of solutions about any obstacle 300(3) Spiraling for linear systems 303(5) Problems 306(2) The Dlrichlet Problem for Degenerate Elliptic Equations A general existence theorem 308(7) Convergence of paths to boundary points 315(3) Application to the Dirichlet problem 318(8) Problems 322(4) Small Random Perturbations of Dynamical Systems The functional IT(φ) 326(6) The first Ventcel-Freidlin estimate 332(2) The second Ventcel--Freidlin estimate 334(12) Application to the first initial-boundary value problem 346(2) Behavior of the fundamental solution as ε→0 348(6) Behavior of Green's function as ε→0 354(5) The problem of exit 359(8) The problem of exit (continued) 367(4) Application to the Dirichlet problem 371(2) The principal eigenvalue 373(3) Asymptotic behavior of the principal eigenvalue 376(12) Problems 383(5) Fundamental Solutions for Degenerate Parabolic Equations Construction of a candidate for a fundamental solution 388(8) Interior estimates 396(3) Boundary estimates 399(7) Estimates near infinity 406(3) Relation between K and a diffusion process 409(5) The behavior of ξ(t) near S 414(6) Existence of a generalized solution in the case of a two-sided obstacle 420(3) Existence of a fundamental solution in the case of a strictly one-sided obstacle 423(3) Lower bounds on the fundamental solution 426(2) The Cauchy problem 428(5) Problems 432(1) Stopping Time Problems and Stochastic Games Part I. The Stationary Case Statement of the problem 433(3) Characterization of saddle points 436(4) Elliptic variational inequalities in bounded domains 440(4) Existence of saddle points in bounded domains 444(3) Elliptic estimates for increasing domains 447(10) Elliptic variational inequalities 457(5) Existence of saddle points in unbounded domains 462(1) The stopping time problem 463(1) Part II. The Nonstationary Case Characterization of saddle points 464(2) Parabolic variational inequalities 466(12) Parabolic variational inequalities (continued) 478(8) Existence of a saddle point 486(2) The stopping time problem 488(6) Problems 490(4) Stochastic Differential Games Auxiliary results 494(4) N-person stochastic differential games with perfect observation 498(4) Stochastic differential games with stopping time 502(5) Stochastic differential games with partial observation 507(13) Problems 518(2) Bibliographical Remarks 520(3) References 523(4) Index-Volume 1 527(3) Index-Volume 2 530