Stochastic Differential Equations and Applications

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ISBN: 9780486453590 | 0486453596
Cover: Paperback
Copyright: 12/1/2006

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This text develops the theory of systems of stochastic differential equations and presents applications in probability, partial differential equations, and stochastic control problems. Originally published in two volumes, it combines a book of basic theory and selected topics with a book of applications.

Preface-Volume 1

General Notation

xvi

Stochastic Processes

The Kolmogorov construction of a stochastic process

(5)

Separable and continuous processes

(3)

Martingales and stopping times

(9)

Problems

(3)

Markov Processes

Construction of Markov processes

(5)

The Feller and the strong Markov properties

(7)

Time-homogeneous Markov processes

(6)

Problems

(5)

Brownian Motion

Existence of continuous Brownian motion

(3)

Nondifferentiability of Brownian motion

(1)

Limit theorems

(4)

Brownian motion after a stopping time

(2)

Martingales and Brownian motion

(4)

Brownian motion in n dimensions

(5)

Problems

(2)

The Stochastic Integral

Approximation of functions by step functions

(4)

Definition of the stochastic integral

(8)

The indefinite integral

(5)

Stochastic integrals with stopping time

(6)

Ito's formula

(7)

Applications of Ito's formula

(4)

Stochastic integrals and differentials in n dimensions

(9)

Problems

(5)

Stochastic Differential Equations

Existence and uniqueness

(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

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

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Stochastic Differential Equations and Applications

Summary

Table of Contents

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